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+arXiv:2301.04627v1  [quant-ph]  11 Jan 2023
+An Improved Approximation for Sparse Fermionic Hamiltonians
+Daniel Hothem∗,
+Ojas Parekh† and Kevin Thompson‡
+Abstract
+We give a classical 1/(qk+1)-approximation for the maximum eigenvalue of k-sparse fermionic
+Hamiltonians with q-local terms as well as a 1/(4k + 1)-approximation when the Hamiltonian
+has both 2-local and 4-local terms.
+We consider approximations for extremal eigenvalues of a k-sparse fermionic Hamiltonian:
+H =
+�
+Γ
+HΓcΓ.
+(1)
+Here H is a fermionic Hamiltonian with real coefficients HΓ, where ignoring phase factors, each
+term cΓ is a product of q Majorana operators (i.e. H is q-local with q even) and each Majorana
+operator appears in at most k non-zero terms (i.e. H is k-sparse). We let m = �
+Γ |HΓ|.
+Herasymenko, Stroeks, Helsen, and Terhal [2] show that λmax(H) ≥ m/Q, where λmax(H) is the
+largest eigenvalue of H and Q = q(q−1)(k−1)2+q(k−1)+2. We demonstrate that this is true with
+Q = qk + 1 and also that Q = 4k + 1 is attainable for k-sparse H with a mix of 2-local and 4-local
+terms. All of these results are obtained by efficient classical algorithms producing descriptions of
+Gaussian states. We refer the reader to [2] for further background, motivation, and applications to
+the SYK model.
+Results of the above flavor were obtained for traceless k-sparse qubit Hamiltonians with constant
+locality by Harrow and Montanaro [1], who show that λmax(H) ≥ Ω(m/k) using product states,
+where m is defined analogously as above. They also give an improved bound with respect to the
+operator norm instead of the maximum eigenvalue: ∥H∥ ≥ Ω(m/
+√
+k). In the fermionic case, we
+give a 2-local example with λmax(H) = ∥H∥ = Θ(m/k), showing that such an improvement is not
+possible.
+We specify our algorithm for the case when H has 2-local and 4-local terms and point out how it
+generalizes when terms are q-local. Concretely, we are given n fermionic modes and corresponding
+traceless Majorana operators {ci}2n
+i=1 satisfying the canonical anticommutation rules {ci, cj} = 2δij.
+We assume that in Equation (1), each Γ corresponds to a subset of [2n]: Γ = {j1, j2, ..., jq} ⊆ [2n]
+with q ∈ {2, 4} and j1 < j2 < ... < jq. The local terms are defined as cΓ = icj1cj2 if q = 2 and
+cΓ = cj1cj2cj3cj4 if q = 4. We let E = {Γ | HΓ ̸= 0}. As noted above, we assume H is k-sparse, i.e.
+for all i ∈ [2n], |{Γ ∈ E | i ∈ Γ}| ≤ k.
+Theorem 1. There is a classical polynomial time algorithm that given as input the weights {HΓ},
+returns a description of a quantum state ρ achieving energy
+Tr(Hρ) ≥
+1
+4k + 1
+�
+Γ
+|HΓ| ≥
+1
+4k + 1λmax(H).
+∗Sandia National Laboratories, email: [email protected]
+†Sandia National Laboratories, email: [email protected]
+‡Sandia National Laboratories, email: [email protected]
+1
+
+Proof. Define a graph G = (V, E) with vertices corresponding to the nonzero terms in the Hamilto-
+nian, i.e. V = E. The graph G may contain vertices corresponding to 2-local or 4-local terms. We
+include an edge (vΓ, vΓ′) ∈ E if and only if one of the following conditions is met:
+(i) cΓ and cΓ′ share one or more Marjorana operators, i.e. Γ ∩ Γ′ ̸= ∅, or
+(ii) Γ and Γ′ are disjoint and Γ ∪ Γ′ ∈ E.
+If there are m nonzero terms in the Hamiltonian then the graph G has m vertices, and the degree
+of a vertex in the graph is at most 4k. We can see the latter as follows. Fix some vertex vΓ. By
+construction,
+deg(vΓ) = |{(Γ, Γ′) ∈ E × E | Γ and Γ′ satisfy (i) or (ii)}|.
+(2)
+We consider two cases:
+- Γ is 4-local. Consider an edge (vΓ, vΓ′). As H contains no 6-local or 8-local terms, Γ ∩ Γ′ ̸= ∅.
+As H is k sparse, there are at most 4k Γ′ for which this can occur.
+- Γ is 2-local. Let a equal the number of 4-local Hamiltonian terms overlapping with Γ, and let
+b equal the number of 2-local terms overlapping with Γ. We claim that the degree of vΓ is at
+most 2a + b.
+There are b 2-local Γ′ satisfying (i) with Γ. Each 2-local Γ′ satisfying (ii) results in a unique
+4-local Γ ∪ Γ′ ∈ E overlapping with Γ, hence there at most a such Γ′. Finally, no 4-local Γ′
+may satisfy (ii), and there are a 4-local Γ′ satisfying (i).
+Since Γ overlaps with at most 2k Γ′, we have a + b ≤ 2k so that 2a + b ≤ 4k.
+By Brooks Theorem we can in polynomial time find a coloring of the vertices of G with at
+most 4k + 1 colors. This means we can partition the vertices into at most 4k + 1 independent sets,
+{S1, ..., St}, with one of these sets having at least a 1/(4k + 1) fraction of the sum of the absolute
+values of the weights:
+�
+Γ
+|HΓ| =
+�
+Si
+�
+Γ∈Si
+|HΓ| ≤ (4k + 1) max
+i
+�
+Γ∈Si
+|HΓ|.
+(3)
+It follows from Equation (3) that
+max
+i
+�
+Γ∈Si
+|HΓ| ≥
+1
+(4k + 1)
+�
+Γ
+|HΓ|.
+Define Sj = arg maxj
+�
+Γ∈Sj |HΓ|, and consider the following state:
+ρ = 1
+2n
+�
+Γ∈Sj
+(I + sign(HΓ)cΓ).
+(4)
+We claim that ρ is a valid quantum state and obtains objective �
+Γ∈Sj |HΓ|. The state ρ is
+proportional to a projector on a stabilizer state with stabilizer generators given by cΓ for Γ ∈ Sj:
+Observe that [cΓ, cΓ′] = 0 for all Γ, Γ′ ∈ Sj since Sj is an independent set. Hence, ρ is the product
+of commuting projectors and must be positive semidefinite.
+We expand the product in Equation (4) as a sum and consider products of two or more terms,
+σ = �
+p cΓp for Γp ∈ Sj. If any of the Γp are 4-local or p ≥ 3, σ cannot be proportional to a term of
+2
+
+H since the Γ ∈ Sj are disjoint, and no cancellation in products of Majorona operators can occur.
+The remaining case is a product of two 2-local operators. For any such Γ, Γ′ ∈ Sj, by (ii) and
+because Sj is an independent set, the product cΓcΓ′ cannot be proportional to cΓ′′ for any Γ′′ ∈ E.
+Hence we have
+Tr(Iρ) = 1,
+Tr(cΓρ) = sign(HΓ)
+∀Γ ∈ Sj, and
+Tr(cΓρ) = 0
+∀Γ ∈ E \ Sj.
+This yields the desired claim that ρ is a normalized state for which
+Tr(Hρ) =
+�
+Γ
+HΓTr(cΓρ) =
+�
+Γ∈Sj
+HΓTr(cΓρ) =
+�
+Γ∈Sj
+|HΓ| ≥
+1
+4k + 1
+�
+Γ
+|HΓ|.
+Gaussian states.
+The ρ constructed in Theorem 1 is, in fact, a mixture of Gaussian states. This
+is proven in the following lemma. This implies the existence of a Gaussian state with at least the
+same objective as ρ.
+Lemma 2. The state ρ defined in Equation (4) is a mixture of Gaussian states.
+Proof. For each Γ ∈ Sj let MΓ be the perfect matching of the operators in Γ induced by the lexico-
+graphic ordering of Γ, and let M be a perfect matching of the Majorana operators in {c1, ...c2n}\{ci |
+∃Γ ∈ Sj with i ∈ Γ} induced by the lexicographic ordering. Define the following Gaussian state:
+ρ′(z) = 1
+2n
+�
+Γ∈Sj
+�
+gh∈MΓ
+(I + zgh icgch)
+�
+rs∈M
+(I + zrs icrcs),
+(5)
+where all zgh, zrs ∈ {±1}.
+Consider the state ρ′′ = Ez[ρ′(z)] where for each Γ the set {zgh}gh∈MΓ is uniformly random
+distributed over {±1}|MΓ| subject to the constraint:
+sign
+
+
+
+ �
+gh∈MΓ
+zgh icgch
+
+ cΓ
+
+ = sign(HΓ)
+∀Γ ∈ Sj,
+(6)
+where sign(±I) is defined as ±1. In other words, {zgh}gh∈MΓ is chosen as the uniform distribution
+over strings in {±1}|MΓ| which satisfy Equation (6). We will assume further that {zgh}gh∈MΓ is
+independent of all other {zgh}gh∈MΓ′ and that each zrs for rs ∈ M is uniform and independent of
+all other random variables.
+We claim that ρ = ρ′′. Begin by using independence to push the expectation past the first and
+third products in Equation (5):
+ρ′′ = 1
+2n
+�
+Γ∈Sj
+�
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+�� �
+rs∈M
+�
+Ez
+�
+(I + zrs icrcs)
+��
+,
+(7)
+We first focus on the final product. Observe that:
+�
+rs∈M
+�
+Ez
+�
+(I + zrs icrcs)
+��
+= I
+(8)
+3
+
+This follows from the independence of the {zrs | rs ∈ M} and because Ez[zrs] = 0 for all rs ∈ M.
+Hence:
+ρ′′ = 1
+2n
+�
+Γ∈Sj
+�
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+��
+.
+(9)
+For fixed Γ ∈ Sj, we claim that:
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+�
+= I + sign(HΓ)cΓ.
+(10)
+Lemma 2 follows immediately from Equation (10). For any strict subset Γ′ ⊊ Γ, define
+MΓ′∩Γ := {gh ∈ MΓ : g ∈ Γ′, h ∈ Γ′}.
+We may then expand the left-hand side of Equation (10) as:
+Ez
+�
+�
+gh∈MΓ
+(I + zgh icgch)
+�
+= I +
+�
+Γ′⊊Γ
+Ez
+�
+�
+gh∈MΓ′∩Γ
+zgh icgch
+�
++ Ez
+�
+�
+gh∈MΓ
+zgh icgch
+�
+(11)
+= I + sign(HΓ)cΓ
+(12)
+The final expectation in Equation (11) evaluates to sign(HΓ)cΓ due to constraint 6. The sum of
+expectations in Equation (11) disappears as the marginal distribution of the z when restricted to
+a matching on a strict subset Γ′ ⊊ Γ of size |MΓ′∩Γ| = p is totally uniform over {±1}p. Therefore
+Ez[zgh] = 0 for any such matching.
+Although ρ′(z) in Lemma 2 is a Gaussian state for any z, the state ρ′′ is a mixture of Gaussian
+states by definition. However, we may derandomize the choice of z to obtain a Gaussian state.
+We only require pairwise independence of the elements of z, hence using standard derandomization
+approaches, we can obtain a Gaussian state ρ′(z) in polynomial time such that Tr(Hρ′(z)) ≥
+Tr(Hρ′′).
+Extension to strictly q-local Hamiltonians.
+A simple modification of the proof of Theorem 1
+produces a 1/(qk + 1)-approximation to k-sparse Hamiltonians where each term is q-local. In this
+case we only need to include edges in G between vΓ and vΓ′ precisely when condition (i) holds, since
+(ii) is vacuous. Consequently we may omit the second case below Equation (2) and simply bound
+the degree as qk. We then effectively replace “4” with q in the remaining proof.
+Extension to Hamiltonians with terms of different sizes.
+An additional modification of the
+proof of Theorem 1 produces a 1/O(qk2)-approximation to a k-sparse Hamiltonian with terms of
+different sizes, where q = maxΓ∈E(|Γ|). In this case we need an appropriate generalization of (ii).
+Let us start by defining G using only the condition (i); the maximum possible degree in G is qk.
+The purpose of (ii) in the proof is to ensure that for Γ, Γ′ in the independent set Sj, cΓcΓ′ cannot
+be proportional to cΓ′′ for any Γ′′ ∈ E. Note that if this happens, then Γ′′ must contain both Γ and
+Γ′. Thus it would suffice for our independent set Sj in G to satisfy the additional property that
+no vΓ, vΓ′ ∈ Sj could have a common neighbor vΓ′′ ∈ V with Γ, Γ′ ⊂ Γ′′. We could satisfy this by
+adding an edge in G between all pairs vΓ and vΓ′ with such a common neighbor. By k-sparsity, the
+vertex vΓ has at most k neighbors vΓ′′ in G with Γ ⊂ Γ′′. Since any such vΓ′′ has degree at most
+qk, the degree of vΓ increases by at most k(qk − 1), and maximum degree in the resulting graph G′
+is O(qk2). Applying Brook’s Theorem in G′ produces the desired approximation.
+4
+
+Optimality.
+For k-sparse H where all terms are q-local, since ∥H∥ ≥ λmax(H), our results show
+that
+∥H∥ ≥ λmax(H) ≥
+m
+qk + 1,
+where we recall m = �
+Γ |HΓ|. We give an explicit family of fermionic 2-local n-sparse Hamiltonians
+{Hn}∞
+n=1 demonstrating this bound is asymptotically tight (i.e., cannot be improved for all q and
+k, up to constant factors).
+Each Hn is expressed as a sum of monomials in 2n Majorana operators {c1, c2, ..., c2n} satisfying
+the usual canonical anti-commutation relations. For each n, partition [2n] evenly into A = {1, ..., n}
+and B = {n + 1, ..., 2n}. Then:
+Hn :=
+�
+a∈A,b∈B
+icacb = i
+��
+a∈A
+ca
+� ��
+b∈B
+cb
+�
+.
+The eigenvalues of Hn are easy to determine, define R ∈ O(2n) as some orthogonal matrix
+satisfying:
+Ra,1 = 1/√n
+∀a ∈ A and Rb,2 = 1/√n
+∀b ∈ B.
+Note that this is well defined since the first two columns are orthonormal. We can then define a
+new set of Majorana operators (also satisfying the canonical anti-commutation relations) by:
+˜ci =
+2n
+�
+i=1
+Rj,icj.
+In particular, we have
+˜c1 =
+1
+√n
+�
+a∈A
+ca and ˜c2 =
+1
+√n
+�
+b∈B
+cb,
+so
+H = ni ˜c1 ˜c2.
+Since i ˜c1 ˜c2 is Hermitian and satisfies (i ˜c1 ˜c2)2 = I, it has eigenvalues in {±1}. Thus the eigenvalues
+of Hn are {±n}. Note that Hn is n-sparse, m = n2, and ∥Hn∥ = λmax(Hn) so that
+∥Hn∥ = λmax(Hn) = n = Θ
+�
+n2
+2n + 1
+�
+= Θ
+�
+m
+qk + 1
+�
+.
+Acknowledgements
+We thank Yaroslav Herasymenko for an insightful contribution to Lemma 2.
+This article has been authored by an employee of National Technology & Engineering Solutions
+of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE).
+The employee owns all right, title and interest in and to the article and is solely responsible for
+its contents. The United States Government retains and the publisher, by accepting the article
+for publication, acknowledges that the United States Government retains a non-exclusive, paid-up,
+irrevocable, world-wide license to publish or reproduce the published form of this article or allow
+others to do so, for United States Government purposes.
+The DOE will provide public access
+to these results of federally sponsored research in accordance with the DOE Public Access Plan
+https://www.energy.gov/downloads/doe-public-access-plan.
+5
+
+This material is based upon work supported by the U.S. Department of Energy, Office of Science,
+Office of Advanced Scientific Computing Research, National Quantum Information Science Research
+Centers, Exploratory Research for Extreme Scale Science program. Support is also acknowledged
+from the Accelerated Research in Quantum Computing program under the same office.
+References
+[1] Aram W. Harrow and Ashley Montanaro. Extremal eigenvalues of local Hamiltonians. Quantum,
+1:6, April 2017. doi:10.22331/q-2017-04-25-6.
+[2] Yaroslav Herasymenko, Maarten Stroeks, Jonas Helsen, and Barbara Terhal. Optimizing sparse
+fermionic hamiltonians. arXiv preprint arXiv:2211.16518, 2022.
+6
+

2NE3T4oBgHgl3EQfnwo5/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf,len=161
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='04627v1  [quant-ph]  11 Jan 2023  An Improved Approximation for Sparse Fermionic Hamiltonians  Daniel Hothem∗,  Ojas Parekh† and Kevin Thompson‡  Abstract  We give a classical 1/(qk+1)-approximation for the maximum eigenvalue of k-sparse fermionic  Hamiltonians with q-local terms as well as a 1/(4k + 1)-approximation when the Hamiltonian  has both 2-local and 4-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We consider approximations for extremal eigenvalues of a k-sparse fermionic Hamiltonian:  H =  �  Γ  HΓcΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  (1)  Here H is a fermionic Hamiltonian with real coefficients HΓ, where ignoring phase factors, each  term cΓ is a product of q Majorana operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' H is q-local with q even) and each Majorana  operator appears in at most k non-zero terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' H is k-sparse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We let m = �  Γ |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Herasymenko, Stroeks, Helsen, and Terhal [2] show that λmax(H) ≥ m/Q, where λmax(H) is the  largest eigenvalue of H and Q = q(q−1)(k−1)2+q(k−1)+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We demonstrate that this is true with  Q = qk + 1 and also that Q = 4k + 1 is attainable for k-sparse H with a mix of 2-local and 4-local  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' All of these results are obtained by efficient classical algorithms producing descriptions of  Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We refer the reader to [2] for further background, motivation, and applications to  the SYK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Results of the above flavor were obtained for traceless k-sparse qubit Hamiltonians with constant  locality by Harrow and Montanaro [1], who show that λmax(H) ≥ Ω(m/k) using product states,  where m is defined analogously as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' They also give an improved bound with respect to the  operator norm instead of the maximum eigenvalue: ∥H∥ ≥ Ω(m/  √  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In the fermionic case, we  give a 2-local example with λmax(H) = ∥H∥ = Θ(m/k), showing that such an improvement is not  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We specify our algorithm for the case when H has 2-local and 4-local terms and point out how it  generalizes when terms are q-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Concretely, we are given n fermionic modes and corresponding  traceless Majorana operators {ci}2n  i=1 satisfying the canonical anticommutation rules {ci, cj} = 2δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We assume that in Equation (1), each Γ corresponds to a subset of [2n]: Γ = {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', jq} ⊆ [2n]  with q ∈ {2, 4} and j1 < j2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' < jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The local terms are defined as cΓ = icj1cj2 if q = 2 and  cΓ = cj1cj2cj3cj4 if q = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We let E = {Γ | HΓ ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' As noted above, we assume H is k-sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  for all i ∈ [2n], |{Γ ∈ E | i ∈ Γ}| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' There is a classical polynomial time algorithm that given as input the weights {HΓ},  returns a description of a quantum state ρ achieving energy  Tr(Hρ) ≥  1  4k + 1  �  Γ  |HΓ| ≥  1  4k + 1λmax(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  ∗Sandia National Laboratories, email: dhothem@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov  †Sandia National Laboratories, email: odparek@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov  ‡Sandia National Laboratories, email: kevthom@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov  1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Define a graph G = (V, E) with vertices corresponding to the nonzero terms in the Hamilto-  nian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' V = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The graph G may contain vertices corresponding to 2-local or 4-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We  include an edge (vΓ, vΓ′) ∈ E if and only if one of the following conditions is met:  (i) cΓ and cΓ′ share one or more Marjorana operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Γ ∩ Γ′ ̸= ∅, or  (ii) Γ and Γ′ are disjoint and Γ ∪ Γ′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  If there are m nonzero terms in the Hamiltonian then the graph G has m vertices, and the degree  of a vertex in the graph is at most 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We can see the latter as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Fix some vertex vΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' By  construction,  deg(vΓ) = |{(Γ, Γ′) ∈ E × E | Γ and Γ′ satisfy (i) or (ii)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  (2)  We consider two cases:  Γ is 4-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Consider an edge (vΓ, vΓ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' As H contains no 6-local or 8-local terms, Γ ∩ Γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  As H is k sparse, there are at most 4k Γ′ for which this can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Γ is 2-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Let a equal the number of 4-local Hamiltonian terms overlapping with Γ, and let  b equal the number of 2-local terms overlapping with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We claim that the degree of vΓ is at  most 2a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  There are b 2-local Γ′ satisfying (i) with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Each 2-local Γ′ satisfying (ii) results in a unique  4-local Γ ∪ Γ′ ∈ E overlapping with Γ, hence there at most a such Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Finally, no 4-local Γ′  may satisfy (ii), and there are a 4-local Γ′ satisfying (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Since Γ overlaps with at most 2k Γ′, we have a + b ≤ 2k so that 2a + b ≤ 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  By Brooks Theorem we can in polynomial time find a coloring of the vertices of G with at  most 4k + 1 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This means we can partition the vertices into at most 4k + 1 independent sets,  {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', St}, with one of these sets having at least a 1/(4k + 1) fraction of the sum of the absolute  values of the weights:  �  Γ  |HΓ| =  �  Si  �  Γ∈Si  |HΓ| ≤ (4k + 1) max  i  �  Γ∈Si  |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  (3)  It follows from Equation (3) that  max  i  �  Γ∈Si  |HΓ| ≥  1  (4k + 1)  �  Γ  |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Define Sj = arg maxj  �  Γ∈Sj |HΓ|, and consider the following state:  ρ = 1  2n  �  Γ∈Sj  (I + sign(HΓ)cΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  (4)  We claim that ρ is a valid quantum state and obtains objective �  Γ∈Sj |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The state ρ is  proportional to a projector on a stabilizer state with stabilizer generators given by cΓ for Γ ∈ Sj:  Observe that [cΓ, cΓ′] = 0 for all Γ, Γ′ ∈ Sj since Sj is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Hence, ρ is the product  of commuting projectors and must be positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We expand the product in Equation (4) as a sum and consider products of two or more terms,  σ = �  p cΓp for Γp ∈ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' If any of the Γp are 4-local or p ≥ 3, σ cannot be proportional to a term of  2  H since the Γ ∈ Sj are disjoint, and no cancellation in products of Majorona operators can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  The remaining case is a product of two 2-local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For any such Γ, Γ′ ∈ Sj, by (ii) and  because Sj is an independent set, the product cΓcΓ′ cannot be proportional to cΓ′′ for any Γ′′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Hence we have  Tr(Iρ) = 1,  Tr(cΓρ) = sign(HΓ)  ∀Γ ∈ Sj, and  Tr(cΓρ) = 0  ∀Γ ∈ E \\ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  This yields the desired claim that ρ is a normalized state for which  Tr(Hρ) =  �  Γ  HΓTr(cΓρ) =  �  Γ∈Sj  HΓTr(cΓρ) =  �  Γ∈Sj  |HΓ| ≥  1  4k + 1  �  Γ  |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  The ρ constructed in Theorem 1 is, in fact, a mixture of Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This  is proven in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' This implies the existence of a Gaussian state with at least the  same objective as ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The state ρ defined in Equation (4) is a mixture of Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For each Γ ∈ Sj let MΓ be the perfect matching of the operators in Γ induced by the lexico-  graphic ordering of Γ, and let M be a perfect matching of the Majorana operators in {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='c2n}\\{ci |  ∃Γ ∈ Sj with i ∈ Γ} induced by the lexicographic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Define the following Gaussian state:  ρ′(z) = 1  2n  �  Γ∈Sj  �  gh∈MΓ  (I + zgh icgch)  �  rs∈M  (I + zrs icrcs),  (5)  where all zgh, zrs ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Consider the state ρ′′ = Ez[ρ′(z)] where for each Γ the set {zgh}gh∈MΓ is uniformly random  distributed over {±1}|MΓ| subject to the constraint:  sign  \uf8ee  \uf8f0  \uf8eb  \uf8ed �  gh∈MΓ  zgh icgch  \uf8f6  \uf8f8 cΓ  \uf8f9  \uf8fb = sign(HΓ)  ∀Γ ∈ Sj,  (6)  where sign(±I) is defined as ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In other words, {zgh}gh∈MΓ is chosen as the uniform distribution  over strings in {±1}|MΓ| which satisfy Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We will assume further that {zgh}gh∈MΓ is  independent of all other {zgh}gh∈MΓ′ and that each zrs for rs ∈ M is uniform and independent of  all other random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We claim that ρ = ρ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Begin by using independence to push the expectation past the first and  third products in Equation (5):  ρ′′ = 1  2n  �  Γ∈Sj  �  Ez  �  �  gh∈MΓ  (I + zgh icgch)  �� �  rs∈M  �  Ez  �  (I + zrs icrcs)  ��  ,  (7)  We first focus on the final product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Observe that:  �  rs∈M  �  Ez  �  (I + zrs icrcs)  ��  = I  (8)  3  This follows from the independence of the {zrs | rs ∈ M} and because Ez[zrs] = 0 for all rs ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Hence:  ρ′′ = 1  2n  �  Γ∈Sj  �  Ez  �  �  gh∈MΓ  (I + zgh icgch)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  (9)  For fixed Γ ∈ Sj, we claim that:  Ez  �  �  gh∈MΓ  (I + zgh icgch)  �  = I + sign(HΓ)cΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  (10)  Lemma 2 follows immediately from Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For any strict subset Γ′ ⊊ Γ, define  MΓ′∩Γ := {gh ∈ MΓ : g ∈ Γ′, h ∈ Γ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We may then expand the left-hand side of Equation (10) as:  Ez  �  �  gh∈MΓ  (I + zgh icgch)  �  = I +  �  Γ′⊊Γ  Ez  �  �  gh∈MΓ′∩Γ  zgh icgch  �  + Ez  �  �  gh∈MΓ  zgh icgch  �  (11)  = I + sign(HΓ)cΓ  (12)  The final expectation in Equation (11) evaluates to sign(HΓ)cΓ due to constraint 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The sum of  expectations in Equation (11) disappears as the marginal distribution of the z when restricted to  a matching on a strict subset Γ′ ⊊ Γ of size |MΓ′∩Γ| = p is totally uniform over {±1}p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Therefore  Ez[zgh] = 0 for any such matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Although ρ′(z) in Lemma 2 is a Gaussian state for any z, the state ρ′′ is a mixture of Gaussian  states by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' However, we may derandomize the choice of z to obtain a Gaussian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  We only require pairwise independence of the elements of z, hence using standard derandomization  approaches, we can obtain a Gaussian state ρ′(z) in polynomial time such that Tr(Hρ′(z)) ≥  Tr(Hρ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Extension to strictly q-local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  A simple modification of the proof of Theorem 1  produces a 1/(qk + 1)-approximation to k-sparse Hamiltonians where each term is q-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In this  case we only need to include edges in G between vΓ and vΓ′ precisely when condition (i) holds, since  (ii) is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Consequently we may omit the second case below Equation (2) and simply bound  the degree as qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We then effectively replace “4” with q in the remaining proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Extension to Hamiltonians with terms of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  An additional modification of the  proof of Theorem 1 produces a 1/O(qk2)-approximation to a k-sparse Hamiltonian with terms of  different sizes, where q = maxΓ∈E(|Γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' In this case we need an appropriate generalization of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Let us start by defining G using only the condition (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' the maximum possible degree in G is qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  The purpose of (ii) in the proof is to ensure that for Γ, Γ′ in the independent set Sj, cΓcΓ′ cannot  be proportional to cΓ′′ for any Γ′′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Note that if this happens, then Γ′′ must contain both Γ and  Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Thus it would suffice for our independent set Sj in G to satisfy the additional property that  no vΓ, vΓ′ ∈ Sj could have a common neighbor vΓ′′ ∈ V with Γ, Γ′ ⊂ Γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We could satisfy this by  adding an edge in G between all pairs vΓ and vΓ′ with such a common neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' By k-sparsity, the  vertex vΓ has at most k neighbors vΓ′′ in G with Γ ⊂ Γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Since any such vΓ′′ has degree at most  qk, the degree of vΓ increases by at most k(qk − 1), and maximum degree in the resulting graph G′  is O(qk2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Applying Brook’s Theorem in G′ produces the desired approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  4  Optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  For k-sparse H where all terms are q-local, since ∥H∥ ≥ λmax(H), our results show  that  ∥H∥ ≥ λmax(H) ≥  m  qk + 1,  where we recall m = �  Γ |HΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We give an explicit family of fermionic 2-local n-sparse Hamiltonians  {Hn}∞  n=1 demonstrating this bound is asymptotically tight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', cannot be improved for all q and  k, up to constant factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Each Hn is expressed as a sum of monomials in 2n Majorana operators {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', c2n} satisfying  the usual canonical anti-commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' For each n, partition [2n] evenly into A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', n}  and B = {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=', 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Then:  Hn :=  �  a∈A,b∈B  icacb = i  ��  a∈A  ca  � ��  b∈B  cb  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  The eigenvalues of Hn are easy to determine, define R ∈ O(2n) as some orthogonal matrix  satisfying:  Ra,1 = 1/√n  ∀a ∈ A and Rb,2 = 1/√n  ∀b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Note that this is well defined since the first two columns are orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' We can then define a  new set of Majorana operators (also satisfying the canonical anti-commutation relations) by:  ˜ci =  2n  �  i=1  Rj,icj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  In particular, we have  ˜c1 =  1  √n  �  a∈A  ca and ˜c2 =  1  √n  �  b∈B  cb,  so  H = ni ˜c1 ˜c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Since i ˜c1 ˜c2 is Hermitian and satisfies (i ˜c1 ˜c2)2 = I, it has eigenvalues in {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Thus the eigenvalues  of Hn are {±n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Note that Hn is n-sparse, m = n2, and ∥Hn∥ = λmax(Hn) so that  ∥Hn∥ = λmax(Hn) = n = Θ  �  n2  2n + 1  �  = Θ  �  m  qk + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  Acknowledgements  We thank Yaroslav Herasymenko for an insightful contribution to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  This article has been authored by an employee of National Technology & Engineering Solutions  of Sandia, LLC under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' DE-NA0003525 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Department of Energy (DOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  The employee owns all right, title and interest in and to the article and is solely responsible for  its contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' The United States Government retains and the publisher, by accepting the article  for publication, acknowledges that the United States Government retains a non-exclusive, paid-up,  irrevocable, world-wide license to publish or reproduce the published form of this article or allow  others to do so, for United States Government purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  The DOE will provide public access  to these results of federally sponsored research in accordance with the DOE Public Access Plan  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='gov/downloads/doe-public-access-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  5  This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Department of Energy, Office of Science,  Office of Advanced Scientific Computing Research, National Quantum Information Science Research  Centers, Exploratory Research for Extreme Scale Science program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Support is also acknowledged  from the Accelerated Research in Quantum Computing program under the same office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  References  [1] Aram W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Harrow and Ashley Montanaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Extremal eigenvalues of local Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Quantum,  1:6, April 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='22331/q-2017-04-25-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  [2] Yaroslav Herasymenko, Maarten Stroeks, Jonas Helsen, and Barbara Terhal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' Optimizing sparse  fermionic hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='16518, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE3T4oBgHgl3EQfnwo5/content/2301.04627v1.pdf'}

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+arXiv:2301.02398v1  [gr-qc]  6 Jan 2023
+Glitch subtraction from gravitational wave data
+using adaptive spline fitting
+Soumya D. Mohanty, Mohammad A. T. Chowdhury
+Department of Physics and Astronomy, University of Texas Rio Grande Valley, One
+West University Blvd., Brownsville, Texas 78520, USA
+E-mail: [email protected]
+E-mail: [email protected]
+Abstract.
+Transient signals of instrumental and environmental origins (“glitches”)
+in gravitational wave data elevate the false alarm rate of searches for astrophysical
+signals and reduce their sensitivity. Glitches that directly overlap astrophysical signals
+hinder their detection and worsen parameter estimation errors.
+As the fraction of
+data occupied by detectable astrophysical signals will be higher in next generation
+detectors, such problematic overlaps could become more frequent. These adverse effects
+of glitches can be mitigated by estimating and subtracting them out from the data,
+but their unpredictable waveforms and large morphological diversity pose a challenge.
+Subtraction of glitches using data from auxiliary sensors as predictors works but not
+for the majority of cases. Thus, there is a need for nonparametric glitch mitigation
+methods that do not require auxiliary data, work for a large variety of glitches, and have
+minimal effect on astrophysical signals in the case of overlaps. In order to cope with
+the high rate of glitches, it is also desirable that such methods be computationally fast.
+We show that adaptive spline fitting, in which the placement of free knots is optimized
+to estimate both smooth and non-smooth curves in noisy data, offers a promising
+approach to satisfying these requirements for broadband short-duration glitches, the
+type that appear quite frequently. The method is demonstrated on glitches drawn
+from three distinct classes in the Gravity Spy database as well as on the glitch that
+overlapped the double neutron star signal GW170817. The impact of glitch subtraction
+on the GW170817 signal, or those like it injected into the data, is seen to be negligible.
+1. Introduction
+In a fairly short time since the first direct detection of a gravitational wave (GW) signal
+(GW150914) in 2015 [1] by the twin LIGO [2] detectors, GW astronomy has emerged as
+an information-rich field that will revolutionize our understanding of compact objects
+such as black holes and neutron stars. By now, the network of LIGO and Virgo [3]
+detectors has reported 90 confirmed detections of GW signals from compact binary
+coalescences (CBCs) across the first observing run (O1) [4] to the third (O3) [5]. The
+majority of these are binary black hole (BBH) mergers but the haul also includes a
+double neutron star (DNS) system (GW170817) [6].
+
+Adaptive spline glitch removal
+2
+The rate of detectable GW signals will grow as more detectors, namely KAGRA [7]
+and LIGO-India [8], join the network and increase its distance reach for GW sources.
+Design studies are already underway for the successors to the current generation of GW
+detectors [9, 10, 11] with the goal of achieving an order of magnitude improvement in
+sensitivity across the current operational frequency band. In addition, next-generation
+detectors will seek to expand the operational range to lower frequencies (≈ 1 Hz),
+thereby increasing the duration of in-band GW signals across the board: for example, a
+DNS signal starting at ≈ 10 Hz will last for days compared to the ≈ 1 min for GW170817.
+Thus, future detectors will not only see a higher rate but also longer signals, raising the
+prospect [12] that there will be no data segment free of detectable GW signals.
+The false alarm rate, hence the sensitivity, of searches for CBCs as well as generic
+short duration GW signals, or bursts, is dominated [13] by transient non-GW signals
+of instrumental or environmental origins, commonly called glitches.
+This is because
+glitches that populate the same frequency band as CBC or burst signals and happen to
+be transient in duration can falsely trigger the respective search pipelines. A glitch has
+a particularly adverse effect if it overlaps with a GW signal, as happened in the case of
+GW170817 [6], and causes the glitch rejection step of a search pipeline to also discard
+the signal. Even a non-overlapping glitch can severely degrade parameter estimation
+if it is close enough to a GW signal [14]. In the third observing run of the LIGO and
+Virgo detectors, ≈ 20% of detected GW signals overlapped with glitches [15] due to the
+high rate of the latter. For future detectors, the frequency of chance overlaps will be
+enhanced by the higher rate of detectable GW signals as well as, for CBC signals, their
+longer durations.
+Glitches have dissimilar and unpredictable waveforms but many of the observed ones
+tend to fall into distinct morphological classes. This has motivated the investigation
+of automated glitch classification using machine learning where a range of different
+methods have been proposed, such as Support Vector Machine [16], t-Sne [17], random
+forests [16], S-means [18], and Deep Convolutional Neural Networks [19]. The Gravity
+Spy [20] project uses a citizen science approach to engage the lay public in labeling
+glitches by visual inspection of their constant Q-transform [21, 22] images. This has
+created a high quality training dataset for machine learning methods. By now, there
+exist more than 20 named glitch classes in the Gravity Spy database, collected over
+multiple observing runs of the LIGO detectors [17].
+Several different approaches have been developed to mitigate the adverse effects
+of glitches on GW searches.
+GW search pipelines typically compute secondary
+functionals, called vetoes, of the data besides the primary detection statistic that help
+in distinguishing genuine GW signals from glitches. A well-known example is the Chi-
+square [23] veto used in CBC search pipelines. For LIGO-Virgo data, a set of Data
+Quality flags have been developed that use information from a large number of auxiliary
+sensors to quantify the safety of analyzing a given segment of GW strain data [24]. For
+glitches that overlap a GW signal, the gating [25] method excises the rectangular time-
+frequency block, or just the time interval, containing an identified glitch from the data.
+
+Adaptive spline glitch removal
+3
+Cross-channel regression using data from auxiliary sensors
+[26, 27, 28, 29] has been
+used to reduce excess broadband noise and a few types of glitches [30].
+A relatively recent approach is that of estimating the waveform of a glitch
+from the data time series itself and subtracting it out.
+Glitch subtraction was of
+critical importance in the case of GW170817 and has been shown to be an important
+requirement in reducing bias in the estimation of GW signal parameters [31].
+The
+GW170817 glitch subtraction was carried out using the multi-detector BayesWave
+pipeline [32, 33], which has also been used for other types of glitches [15]. Another
+method, Glitschen [34], follows the approach of constructing parametrized waveform
+models for identified glitch classes using principal component analysis of training sets.
+A strong motivation for developing glitch estimation and subtraction methods is that
+one could, in principle, preprocess the data to clean out every sufficiently loud glitch of
+a known type and make glitch rejection in all downstream GW searches safer.
+In this paper, we present a method for the estimation and subtraction of broadband,
+short-duration glitches that have appeared frequently in the observation runs of the
+LIGO detectors.
+The method is computationally cheap, works with single-detector
+data, does not require a training set of pre-identified glitches, and is not predicated on
+auxiliary sensor data. The core component of the method is SHAPES (Swarm Heuristics
+based Adaptive and Penalized Estimation of Splines), an adaptive spline curve fitting
+algorithm introduced in [35]‡. SHAPES uses splines with free placement of knots to fit
+both smooth and non-smooth curves in noisy data. In particular, point discontinuities
+in the curve or its derivatives (up to some order) can be accommodated in the fit by
+allowing knots to merge. The ability to handle both sharp and slow changes in a curve
+is a built-in form of multiresolution analysis in SHAPES and a critical requirement for
+effective estimation of broadband glitches. We examine the performance of our glitch
+subtraction method on the GW170817 glitch in LIGO-Livingston data and instances of
+glitches from three morphologically distinct classes, namely, Blip, Koi Fish, and Tomte,
+in the Gravity Spy database. In each of the latter three cases, we inject a DNS signal
+overlapping with the glitch to mimic the case of GW170817. We find that the impact
+of glitch subtraction on the signals, real or injected, is negligible.
+The rest of the paper is organized as follows. Sec. 2 reviews SHAPES with the
+goal of providing a self-contained description of the algorithm that is pertinent to this
+paper. Further details, such as the motivation and justification for certain features of
+the algorithm, can be found in [35]. Sec. 3 describes the dataset used in this paper and
+the details of how SHAPES is used for glitch subtraction. Sec. 4 presents the results.
+Our conclusions and discussion of future work are presented in Sec. 5.
+‡ The SHAPES code is available from the Github repository mohanty-sd/SHAPES.git.
+
+Adaptive spline glitch removal
+4
+2. Adaptive spline fitting: the SHAPES algorithm
+SHAPES is derived under the following models for the noisy data, y, and the signal
+s(θ).
+y = s(θ) + ǫ ,
+(1)
+where y, s, and ǫ are row vectors with N elements, yi = y(ti) and si(θ) = s(ti; θ),
+i = 0, 1, . . . , N −1, are samples taken at ti = i/fs with fs being the sampling frequency,
+and θ denotes the set of signal parameters that need to be estimated from the data.
+The noise samples, ǫi, are drawn independently from the zero mean and unit variance
+normal (Gaussian) probability density function N(0, 1). This assumption, namely, that
+of a white Gaussian noise process does not entail a loss of generalization since GW data
+can always be whitened using the estimated noise power spectral density (PSD).
+The signal s(t; θ) is assumed to be a spline of polynomial order k and, as such, can
+be represented by a linear combination of B-spline functions [36],
+s (t; θ = {α, τ}) =
+P −k−1
+�
+j=0
+αjBj,k(t; τ) ,
+(2)
+where α = (α0, α1, . . . , αP −k−1), and τ = (τ0, τ1, . . . , τP −1), τi+1 ≥ τi, is a sequence of P
+knots that marks the end points of the contiguous intervals containing the polynomial
+pieces of the spline. Note that knots are allowed to be equal, leading to knots with
+multiplicity higher than one. Repeating knots create discontinuity in either the value
+of a B-spline function or its derivatives (up to order k − 2). This allows the s(t; θ) in
+Eq. 2 to model signals with point discontinuities in value or derivatives. In the rest of
+the paper, we will set k = 4, making s(t; θ) a cubic spline.
+The best fit spline parameters, �α and �τ, are the ones that minimize a penalized
+least-squares function,
+Lλ(α, τ) = L(α, τ) + λR(α) ,
+(3)
+L(α, τ) =
+N−1
+�
+i=0
+(yi − si(α, τ))2 ,
+(4)
+where the penalty term,
+R(α) =
+P −k−1
+�
+j=0
+α2
+j ,
+(5)
+is found to be useful in the suppression of spurious clustering of the knots.
+These
+clusters are observed when the method tries to minimize Lλ(α, τ) by fitting out outlier
+data points arising from the noise alone. The strength of the penalty is controlled by
+the gain factor λ, with higher values of λ leading to smoother estimates.
+The optimization of Lλ(α, τ) over the non-linear parameters τ has been a long-
+standing computational barrier [37, 38, 39, 40] for adaptive spline fitting. At the same
+time, the benefits of optimizing the placement of knots have also been demonstrated
+extensively [38, 41]. It was shown in [42], and independently in [43], that Particle Swarm
+Optimization (PSO) [44, 45], a widely used nature-inspired metaheuristic for global
+
+Adaptive spline glitch removal
+5
+optimization, has good performance on the free knot placement problem. Moreover,
+being a continuous optimization method, PSO can explore all arrangements of knots,
+including the ones where knots are sufficiently close to be merged into a single knot of
+higher multiplicity. This allows the fitting of functions that have a mix of smooth and
+non-smooth parts.
+There are many variations [46] among the algorithms that fall under the umbrella
+of the PSO metaheuristic but they all share the following common features. (i) The
+function to be optimized, called the fitness function, is sampled at multiple locations,
+called particles, that move iteratively to explore the domain, called the search space, over
+which the the global optimum of the fitness is to be found. The set of particles is called
+a swarm. (ii) The location of each particle is updated following a dynamical rule that
+incorporates randomness. The rule typically uses the best location found by a particle
+in its history, called its personal best, and the best location found by the particles in
+its neighborhood, called its local best. Here, the fitness value at a location defines how
+good it is: for a minimization problem, the lower the fitness, the better the location.
+(iii) Each particle explores the search space independently but is constantly attracted
+towards the personal and local bests. This leads to a form of communication between
+the particles that speeds up convergence to a promising region, followed by refinement
+of the solution until the iterations are terminated.
+The best location among all the particles at termination is the final solution found
+by the swarm for the global optimum. While there is no guarantee that the final solution
+is the true global optimum, the probability of successful convergence can be boosted
+exponentially by running multiple independent runs of PSO and picking the one with
+the best final solution. Most of the parameters involved in the PSO algorithm, such
+as the number of particles or the weights attached to the attractive forces, have very
+robust values across a wide variety of benchmark optimization problems [47] and rarely
+need to be changed. In our experience, there are typically only two quantities that
+need tuning: the number of iterations, Niter, to termination and the hyper-parameter
+Nruns, the number of independent PSO runs. In this paper, we fix Niter = 2000 and
+Nruns = 8 throughout. The number of particles is always set to 40 and the settings for
+the remaining parameters, as well as the definition of the neighborhood used for the
+local best, are described in [35].
+The description above was for the case where the number of knots, P, is fixed. The
+complete SHAPES algorithm incorporates model selection using the Akaike Information
+Criterion (AIC) [48], where the optimum number of knots minimizes,
+AIC = 4P + Lλ(�α, �τ) .
+(6)
+While, given sufficient computing resources, model selection could be performed over all
+values of P until the minimum value of AIC is found, practical considerations dictate
+that the set of knot numbers used be a finite and small one. In this paper, for example,
+we use knot numbers in the set starting at 5 and ending at 60 in increments of 5. It is
+important to note that this restriction of knot numbers is not a fundamental limitation
+
+Adaptive spline glitch removal
+6
+but a technical one meant to manage the computational burden of model selection.
+Thus, the only significant free parameter that needs to be set by the user in the current
+version of SHAPES is λ.
+Since SHAPES assumes that the noise in the data is white, GW strain data must
+be whitened prior to glitch estimation and subtraction. The data conditioning steps
+involved are as follows (in sequential order).
+(a) Suppression of the seismic noise
+below 10 Hz, (b) robust estimation of the power spectral density (PSD) noise floor,
+(c) whitening of the noise floor using the estimated PSD [49], and (d) automated
+identification of high-power narrowband noise features (“lines”) and their suppression
+using notch filters. These steps are common to all GW search pipelines, so they do not
+need to be elaborated further here.
+3. Demonstration data
+The glitches considered in this paper for demonstrating the performance of SHAPES are
+listed in Table 1. The corresponding GW strain data files can be located and downloaded
+from the Gravitational Wave Open Science Center (GWOSC) [50] using the information
+provided in this table. We have used the standard 4096 sec long GWOSC data files
+sampled at 4 kHz.
+The GW170817 glitch presents a particularly interesting example of the deleterious
+effect of glitches on GW searches. The GW signal appeared in both LIGO-Hanford (H1)
+and LIGO-Livingston (L1) with a combined network signal to noise ratio (SNR) of 32.4.
+Such a strong signal would have been detected easily in coincidence across L1 and H1
+by the GW search pipelines in operation at the time. However, a coincident detection
+was prevented by a large overlapping glitch in L1 causing the release of only an unusual
+single-detector GW detection alert to the astronomical community. About 11 hours
+elapsed between the initial alert and the release of the skymap localizing GW170817, a
+process that included the subtraction of the glitch using BayesWave.
+In addition to the GW170817 glitch, we have taken three representative glitches
+from the Blip, Koi Fish, and Tomte, classes in the Gravity Spy [20] database [51].
+These glitches were selected by taking the loudest 5 events, in terms of their signal-to-
+noise ratio (SNR) as given in the Gravity Spy database, for each class and then picking
+the first one in this list for which the corresponding GWOSC file had 100% science data
+that was also reasonably stationary. As can be seen from Table 1, this results in the
+selected glitches spanning a wide range in SNR.
+After conditioning the data, we use the start time of a glitch, recorded in Table 1,
+to locate the glitch. Starting from the peak of the glitch, the data time series is scanned
+visually in both directions to identify a segment, containing the glitch, that tapers off
+at both its boundaries to the general noise level of the conditioned data.
+To mimic the case of GW170817 and to study the effect of glitch subtraction on
+an overlapping GW signal, we injected a whitened restricted-2PN circularized binary
+inspiral signal with equal 1.4 M⊙ components in the conditioned data. The SNR (in
+
+Adaptive spline glitch removal
+7
+Glitch Name
+GPS start (sec)
+SNR
+Detector
+run
+GW170817 glitch
+1187008880
+–
+L1
+O2
+Blip
+1182397347
+109.1
+H1
+O2
+Koi Fish
+1169847108
+608.1
+H1
+O2
+Tomte
+1173086299
+19.6
+H1
+O2
+Table 1. Glitches considered in this paper along with their GPS start times, SNRs,
+the detectors in which they appeared, and the observation runs. For the Blip, Koi Fish,
+and Tomte glitches, the start times are taken from the Gravity Spy database. To the
+best of our knowledge, there is no SNR available in the literature for the GW170817
+glitch.
+white noise with unit variance) of the injected signal is set at 37.3, which is an ad
+hoc factor of
+√
+2 higher than the observed SNR of 26.4 of GW170817 in L1 [6]. The
+enhancement in SNR allows clearer visibility of the signal in time-frequency images
+while also posing a stronger challenge to SHAPES in terms how well it ignores the GW
+signal when estimating a glitch. The segment containing the glitch, taken from the
+conditioned data with the injected signal, is passed to SHAPES for estimation of the
+glitch waveform followed by its subtraction.
+4. Results
+In common with other papers on glitch estimation and subtraction, we present our
+results in the form of constant Q-transform (CQT) time-frequency images and time
+series plots. These are obtained by taking projections of the data on a set of windowed
+sinusoids. The width of the window decreases with an increase in the carrier frequency,
+fc, such that Q = fc/∆f, where ∆f is the −3 dB bandwidth of the Fourier transform
+of the window, remains constant. We use the CQT code provided in the librosa [52]
+Python package for audio processing. For each glitch, we show CQTs of the conditioned
+data with injected signal and the residual after subtraction of the glitch estimate.
+Fig. 1 shows the data segments that were processed using SHAPES and the
+corresponding estimated glitch waveforms. Except for GW170817, each segment was
+processed as a whole to obtain the glitch estimate. In the case of GW170817, SHAPES
+was applied independently to three separate but contiguous time intervals to estimate
+the complete glitch. This was necessitated by the presence of extended wings, preceding
+and trailing the core broadband (and rapidly varying) part in the middle, that dominate
+the conditioned data for ≈ 0.5 sec on each side. Applying SHAPES to the complete
+segment would have required using a very large number of knots (> 60), making it
+unnecessarily expensive computationally given that splitting the segment achieves a
+good solution.
+As mentioned in Sec. 2, the penalty gain λ controls the smoothness of the estimate
+and is a user-specified parameter of the SHAPES algorithm. Typically, when a glitch
+is loud and has a complex shape, λ = 0.01 allows SHAPES to provide a better fit. For
+
+Adaptive spline glitch removal
+8
+932.67
+932.68
+932.69
+932.7
+932.71
+-60
+-40
+-20
+0
+20
+40
+60
+80
+Data
+Estimate
+839.1
+839.12
+839.14
+839.16
+Time (sec)
+-100
+-50
+0
+50
+100
+150
+Whitened Strain
+Data
+Estimate
+93.04
+93.06
+93.08
+93.1
+93.12
+93.14
+Time (sec)
+-4
+-2
+0
+2
+4
+Data
+Estimate
+370
+370.5
+371
+-100
+-50
+0
+50
+100
+Whitened Strain
+Data
+Estimate
+Figure 1. The conditioned strain data and the glitch waveform estimated by SHAPES
+for each of the glitches considered in this paper. Top row: GW170817 (left) and Blip
+(right). Bottom row: Koi Fish (left) and Tomte (right). The X-axis in each plot shows
+the time (sec) since the start of the open data file containing the glitch as provided
+by GWOSC. For GW170817, the dashed vertical lines demarcate the three adjacent
+segments that were analyzed separately.
+low SNR and simple glitch waveforms, or if the data is just plain white noise, λ = 0.1
+does an adequate job. In general, estimates from SHAPES are not sensitive to small
+variations of λ around these values because the model selection is able to compensate
+for a lower value of λ by selecting a higher knot number and vice versa. Without much
+fine tuning, we found that the values of λ listed in Table 2 work well for the glitches
+studied in this paper. We have also listed in this table the number of knots for the best
+fit models selected by the AIC.
+Fig. 2 to Fig. 5 show the CQTs of the conditioned data and residuals after glitch
+subtraction for the glitches in the sequence GW170817, Blip, Koi Fish, and Tomte,
+respectively. In all cases, we see that the subtraction of the glitch does not affect the
+overlapping GW signal (real or injected) in any significant way. Some overfitting to the
+data, seen as very small CQT values, is visible in the residual for the GW170817 glitch at
+frequencies below ≈ 32 Hz but this band has no overlap with the signal. The overfitted
+parts are the two wings of the GW170817 glitch mentioned earlier. The CQTs of the
+residuals for the Blip and Tomte glitches show near perfect removal of the glitch. (For
+
+Adaptive spline glitch removal
+9
+Glitch Name
+Penalty gain (λ)
+Number of knots
+GW170817 glitch
+0.1, 0.01, 0.1
+60,40,50
+Blip
+0.01
+15
+Koi Fish
+0.01
+30
+Tomte
+0.1
+15
+Table 2. The penalty gain λ used for the glitches and the number of knots in the best
+fit model. For the GW170817 glitch, there are three segments with the middle one
+containing the principal glitch and adjacent ones containing the wings. The penalty
+gains and best fit model are listed for all three segments in sequential order from left
+to right.
+Tomte, the coalescence time of the GW signal was kept further away from the glitch
+in order to create an overlap between the signal track and the glitch.) The residual for
+Koi Fish shows effective removal of the glitch with the exception of a transient and low
+frequency narrowband component. This leftover component does not overlap with the
+signal.
+The principal computational cost in SHAPES is the global optimization of the
+fitness function in Eq. 3. The time taken by the current MATLAB [53] code for a single
+PSO run on a segment with ≈ 300 samples and knot numbers P ∈ [10, 60] (in steps of
+5) is < 10 min on an Intel Xeon E5 processor (clock rate 3 GHz). The runtime increases
+with the number of knots used, mainly due to an increase in the number of B-spline
+functions that need to be computed. With a code currently under construction in the
+C language, and implementation of further hardware acceleration (e.g., using Graphics
+Processing Units), the runtime is expected to decrease substantially. We also note that
+the segments containing glitches can be processed in parallel since SHAPES is a purely
+time-domain method. Hence, the computational cost will scale slower than linearly with
+the number of glitches when analyzing data containing multiple glitches.
+5. Discussion and Conclusions
+We have presented a new approach to glitch subtraction using an adaptive spline fitting
+method called SHAPES. The method was demonstrated on the GW170817 glitch as
+well as other representative short duration and broadband glitches. In a single detector
+and in the absence of strong prior information about the signal, it is not possible to
+distinguish a GW signal from a glitch in the part where they overlap.
+Hence, it is
+expected that the signal power will be removed in that part along with the glitch when
+the latter is estimated and subtracted out. Nonetheless, as far as the DNS signal used
+in this paper is concerned, we observe very little impact on the signal across a wide
+range of glitch SNRs. While this conclusion will be quantified in future studies using a
+much larger number of glitches, it is clear that SHAPES is effective at addressing glitch
+subtraction.
+SHAPES is not well adapted to fitting highly oscillatory waveforms since these
+
+Adaptive spline glitch removal
+10
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+12
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+12
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+Figure 2. Subtraction of the GW170817 Glitch. The top and bottom panels show
+the CQT of the data and residual, respectively. The glitch is the vertical feature at
+≈ 10.5 sec. In order to show both the glitch and the signal in the same image, a
+threshold has been applied to the CQT as indicated by the maximum value in the
+colorbar of the top panel.
+are are not represented well by splines without using an inordinate number of knots.
+Therefore, the direct use of SHAPES for glitches in the Gravity Spy database such as
+whistlers or wandering lines is not viable. However, chirp signals such as these could be
+estimated using the method proposed in [54, 55], where adaptive splines figure indirectly
+
+Adaptive spline glitch removal
+11
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+Figure 3. Subtraction of the Blip Glitch. The top and bottom panels show the CQT
+of the data and residual, respectively. The glitch is the vertical feature at ≈ 6 sec. In
+order to show both the glitch and the signal in the same image, a threshold has been
+applied to the CQT as indicated by the maximum value in the colorbar of the top
+panel.
+in a non-linear signal model. This is an interesting direction that will be pursued in
+future work.
+Other current limitations of SHAPES, which are technical in nature, are that the
+penalty gain parameter λ as well as the segment length to be processed must be specified
+
+Adaptive spline glitch removal
+12
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+1.5
+3
+4.5
+6
+7.5
+9
+10.5
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+Figure 4.
+Subtraction of the Koi Fish glitch.
+The top and bottom panels show
+the CQT of the data and residual, respectively. The glitch is the vertical feature at
+≈ 9.0 sec.
+In order to show both the glitch and the signal in the same image, a
+threshold has been applied to the CQT as indicated by the maximum value in the
+colorbar of the top panel.
+by the user. The choice of the latter, along with the nature of the data, influences the
+number of knots used in the fit and led to the necessity of breaking up the data for the
+GW170817 glitch into three ad hoc parts. Work is in progress to address both of these
+limitations.
+
+Adaptive spline glitch removal
+13
+0
+2
+4
+6
+8
+10
+12
+14
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+0
+5
+10
+15
+Time (sec)
+16
+32
+64
+128
+256
+512
+Frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 5
+Figure 5. Subtraction of the Tomte Glitch. The top and bottom panels show the CQT
+of the data and residual, respectively. The glitch is the vertical feature at ≈ 8.0 sec.
+In order to show both the glitch and the signal in the same image, a threshold has
+been applied to the CQT as indicated by the maximum value in the colorbar of the
+top panel.
+Our results show that SHAPES is a promising addition to the toolbox of glitch
+subtraction methods that will become increasingly important as GW detectors become
+more sensitive. SHAPES is computationally inexpensive, taking on the order of a few
+minutes for each glitch, and will be made much faster by planned code improvements.
+
+Adaptive spline glitch removal
+14
+This could allow, in principle, the subtraction of a large number of broadband glitches
+of known types as part of data conditioning and provide significantly cleaner data for
+any type of GW search.
+Acknowledgments
+S.D.M is supported by U.S. National Science Foundation (NSF) grant PHY-2207935 and
+partially supported by the U.S. Department of Defense grant W911NF2110169. MATC
+acknowledges support from the Presidential Graduate Research Award at the University
+of Texas Rio Grande Valley. We acknowledge the Texas Advanced Computing Center
+(TACC) at the University of Texas at Austin (www.tacc.utexas.edu) for providing high
+performance computing resources.
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+

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+arXiv:2301.04645v1  [math.CA]  11 Jan 2023
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP FOR
+SETS OF DIMENSION BETWEEN 2 AND 3
+TERENCE L. J. HARRIS
+Abstract. It is shown that vertical projections in the Heisenberg group al-
+most surely do not decrease Hausdorff dimension for Borel sets of dimension
+between 2 and 3.
+The proof uses the method of point-plate incidences in-
+troduced by F¨assler and Orponen, and uses a similar approach to a recent
+theorem of Zahl.
+1. Introduction
+Let H be the Heisenberg group, identified as a set with C × R, and equipped
+with the group law
+(z, t) ∗ (ζ, τ) =
+�
+z + ζ, t + τ + 1
+2ω(z, ζ)
+�
+,
+where
+ω(x + iy, u + iv) := xv − uy.
+For each θ ∈ [0, π), let
+Vθ =
+��
+λeiθ, 0
+�
+: λ ∈ R
+�
+,
+and let V⊥
+θ be the Euclidean orthogonal complement of Vθ. Each (z, t) ∈ H can be
+uniquely decomposed as a product
+(z, t) = PV⊥
+θ (z, t) ∗ PVθ(z, t),
+of an element of V⊥
+θ on the left, with an element of Vθ on the right. This defines
+the vertical projection maps PV⊥
+θ . Let dH be the Kor´anyi metric on H, given by
+dH((z, t), (ζ, τ)) =
+��(ζ, τ)−1 ∗ (z, t)
+��
+H ,
+where
+∥(z, t)∥H =
+�
+|z|4 + 16t2�1/4 .
+The Kor´anyi metric is bi-Lipschitz equivalent to the more natural Carnot-Carath´eodory
+metric on H, and thus induces the same Hausdorff dimension. Let dim refer to the
+Hausdorff dimension of a set in H with respect to the Kor´anyi metric. This work
+gives a proof of the following theorem.
+Theorem 1. Let A be an analytic subset of H. Then
+dim PV⊥
+θ (A) ≥ min{dim A, 3}
+for a.e. θ ∈ [0, π).
+2020 Mathematics Subject Classification. 28A78; 28A80.
+Key words and phrases. Heisenberg group, Hausdorff dimension, vertical projections.
+1
+
+2
+TERENCE L. J. HARRIS
+This was first conjectured by Balogh, Durand-Caragena, F¨assler, Mattila and
+Tyson [1, Conjecture 1.5], who proved the conjecture in the range dim A ≤ 1.
+Recently, this conjecture was proved for dim A ∈ [0, 2]∪{3} by F¨assler and Orponen
+(and thus also for dim A > 3, though Conjecture 1.5 in [1] also predicts positive area
+in this range). They introduced a method of point-plate incidences, and proved
+(1) in the case dim A = 3 by using a square function estimate for the cone of
+Guth, Wang and Zhang [4] to control the average L2 norm of pushforwards of 3-
+dimensional measures. The point-line duality principle they used is due to Liu [5].
+Theorem 1 resolves the conjecture in the remaining range dim A ∈ (2, 3). The proof
+of Theorem 1 uses the incidence approach of F¨assler and Orponen, but rather than
+using the square function estimate for the cone, it uses a broad-narrow approach
+to Kakeya-type inequalities for tubes arranged in fractal families of planks. This is
+based on recent work of Zahl [6], which used a broad-narrow approach to Kakeya-
+type inequalities for fractal families of tubes.
+Acknowledgements
+I thank Shaoming Guo for some discussion in the earlier stages of working on
+this problem, when I visited UW Madison in October 2022. I also thank UW for
+their hospitality.
+2. Preliminaries
+For each θ ∈ [0, π] let H2 be the 2-dimensional Lebesgue measure on V⊥
+θ . A line
+ℓ in H is called horizontal if it is a left translate of a horizontal subgroup Vθ for
+some θ ∈ [0, π); meaning that there exists p ∈ H such that ℓ = p ∗ Vθ. For each
+horizontal line ℓ ⊆ H, let H1 be the Lebesgue measure on ℓ with respect to the
+Euclidean metric. Given a non-negative Borel function f and a horizontal line ℓ,
+define
+Xf(ℓ) =
+�
+ℓ
+f dH1.
+Given a measure µ on H, let
+cα(µ) =
+sup
+x∈H,r>0
+µ (BH(x, r))
+rα
+.
+More generally, given δ > 0, define
+cα,δ(µ) =
+sup
+x∈H,r>δ
+µ (BH(x, r))
+rα
+.
+Definition 1. Define ℓ∗ : H → P(R3) by
+ℓ∗(x, y, t) = (0, x, t − xy/2) + Ly,
+where
+Ly =
+�
+λ(1, −y, y2/2) : λ ∈ R
+�
+.
+Define ℓ : R3 → P(H) by
+ℓ(a, b, c) = {(as + b, s, (bs)/2 + c) : s ∈ R} .
+The following lemma is the point-line duality principle.
+Lemma 1 ([3, Lemma 4.11]). Let p∗ ∈ H and let p ∈ R3. Then
+p ∈ ℓ∗(p∗)
+if and only if
+p∗ ∈ ℓ(p).
+
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP
+3
+Lemma 2. Given a non-negative continuous function f supported in the unit ball
+of H, let µf be the measure whose Radon-Nikodym derivative with respect to the
+Lebesgue measure on H is equal to f. Then for any q ∈ (1, ∞) and ε > 0,
+� π−ε
+ε
+���PV⊥
+θ #µf
+���
+q
+Lq(H2) dθ ∼q,ε
+�
+Uǫ
+|Xf(ℓ(p))|q dH3(p),
+where Uǫ is the set of p ∈ R3 such that ℓ(p) is a horizontal line with corresponding
+angle in [ε, π − ε].
+Proof. This is outlined in [3, Eq. 4.15] in the case q = 2. One version of the proof
+uses the coarea formula, and the formula for L2 norms in terms of the distribution
+function, which naturally extends to the case q ∈ (1, ∞).
+□
+Lemma 3. Let f be a non-negative continuous function supported in the unit ball
+of H, let µf be the measure whose Radon-Nikodym derivative with respect to the
+Lebesgue measure on H is equal to f. Then for any q ∈ (1, ∞) and any p ∈ BH(0, 1),
+� π
+0
+���PV⊥
+θ # (Lp#µf)
+���
+q
+Lq(H2) dθ
+� π
+0
+���PV⊥
+θ #µf
+���
+q
+Lq(H2) dθ ∼q,
+where Lp(z, t) = p ∗ (z, t).
+Proof. The Radon-Nikodym derivative of Lp#µf is f ◦ L−1
+p , since left translation
+has Jacobian equal to 1. Hence
+� π
+0
+���PV⊥
+θ # (Lp#µf)
+���
+q
+Lq(H2) dθ ∼
+� ����
+�
+ℓ
+(f ◦ L−1
+p ) dH1
+����
+q
+dh(ℓ),
+where h is the natural left-invariant measure on the set of horizontal lines; see [3,
+Eq. 4.14]. If ℓ is such that the integrand is nonzero, then ℓ = (z, t) ∗ Vθ for some
+θ ∈ [0, π) and (z, t) ∈ V⊥
+θ with dH((z, t), 0) ≲ 1 (since f is supported in the unit
+ball), which implies that the Euclidean measure H1 is equivalent to the Heisenberg
+Hausdorff measure H1
+H on ℓ. Similarly, these two measures are equivalent on p ∗ ℓ
+(which is also a horizontal line). Since H1
+H is left-invariant, and h is left-invariant,
+it follows that
+� ����
+�
+ℓ
+(f ◦ L−1
+p ) dH1
+����
+q
+dh(ℓ) ∼
+� ����
+�
+ℓ
+f dH1
+����
+q
+dh(ℓ) ∼
+� π
+0
+���PV⊥
+θ # (µf)
+���
+q
+Lq(H2) dθ.
+□
+3. Main results
+For the statement of the following theorem, let dθ refer to the Lebesgue measure
+on [0, π). Let
+�
+∗ refer to the lower integral (this is only used to avoid measurability
+issues in the statement).
+Theorem 2. Let t ∈ [0, 3], and let µ be a Borel measure supported in the unit
+Heisenberg ball, with ct(µ) ≤ 1. Let ǫ > 0. Let δ > 0, and suppose that for each θ,
+Dθ is a disjoint collection of at most δ
+√ǫ−tµ(H) Heisenberg δ-balls in V⊥
+θ . Then
+�
+∗
+�
+PV⊥
+θ #µ
+� � �
+D∈Dθ
+D
+�
+dθ ≲ǫ δǫµ(H).
+Before proving Theorem 2, it will be shown that it implies Theorem 1.
+
+4
+TERENCE L. J. HARRIS
+Proof of Theorem 1. Measurability issues will be ignored since they can be easily
+adjusted for. By scaling it may be assumed that A is contained in the unit ball.
+Let ǫ > 0, and (using Frostman’s lemma) let µ be a Borel probability measure on
+A with cα(µ) < ∞, where α = dim A − ǫ. Let E ⊆ [0, π) be a compact set such
+that dim PV⊥
+θ (supp µ) < α − ǫ for all θ ∈ E. Let ε > 0, and for each θ ∈ E,
+let Dθ = {BH(pj(θ), rj(θ))}j be a finitely overlapping cover of PV⊥
+θ (supp µ) by
+Heisenberg balls of dyadic radii at most ε, such that
+(1)
+�
+j
+rj(θ)α−ǫ < cα(µ)−1.
+For each k, let Dθ,k be the subcollection of balls in Dθ with dyadic radii equal to
+2−k. Then for each θ ∈ E,
+1 ≤
+�
+k
+�
+PV⊥
+θ #µ
+�
+
+
+�
+D∈Dθ,k
+D
+
+ .
+Integrating over E gives
+H1(E) ≤
+�
+k
+�
+E
+�
+PV⊥
+θ #µ
+�
+
+
+�
+D∈Dθ,k
+D
+
+ dθ.
+By (1), each set Dθ,k satisfies
+|Dθ,k| ≲ 2k(α−ǫ)cα(µ)−1.
+By applying Theorem 2 and summing the geometric series, this yields
+H1(E) ≤
+�
+k
+�
+E
+�
+PV⊥
+θ #µ
+�
+
+
+�
+D∈Dθ,k
+D
+
+ dθ ≲ cα(µ)
+�
+k≥| log2 ε|
+2−kǫ2 ≲ cα(µ)εǫ2,
+Letting ε → 0 gives H1(E) = 0. By inner regularity of the Lebesgue measure on
+[0, π), it follows that
+dim PV⊥
+θ (A) ≥ dim PV⊥
+θ (supp µ) ≥ α − ǫ ≥ dim A − 2ǫ.
+for a.e. θ ∈ [0, π). Since the outer parts of this inequality hold for any ǫ > 0, this
+implies the theorem.
+□
+It remains to prove Theorem 2.
+Proof of Theorem 2. Let φ be a fixed non-negative bump function supported in the
+unit Euclidean ball of H around the origin, such that
+�
+φ = 1 and such that φ ≳ 1
+on BE(0, 1/10). For each λ > 0 let φλ = λ−3φ(x/λ). Define ν by the Euclidean
+convolution ν = µ ∗ φδ2, and let
+q = 3 + t
+1 + t ∈ [3/2, 3].
+By H¨older’s inequality with respect to the Lebesgue measure on each plane V⊥
+θ ,
+and the fact that the vertical projections are uniformly 1
+2-H¨older continuous when
+considered as maps from (H, dE) to (H, dH),
+�
+∗
+�
+PV⊥
+θ #µ
+� � �
+D∈Dθ
+D
+�
+dθ ≲ µ(H)1− 1
+q δ(3+√ǫ−t)(1− 1
+q)
+�� π
+0
+���PV⊥
+θ #ν
+���
+q
+Lq(H2)
+�1/q
+.
+
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP
+5
+Therefore, it suffices to prove that
+(2)
+� π
+0
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ≲ǫ ν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1),
+for any δ > 0, whenever ν = µ ∗ φδ2 for some finite Borel measure µ supported in
+the unit Heisenberg ball. This will be shown via induction on δ. Thus, let δ > 0
+be given and assume that (2) holds for all �δ < δ1−ǫ2, for any finite Borel measure
+µ supported in the unit Heisenberg ball.
+It will now be shown that (2) holds for δ. By scaling, it may be assumed that ν
+is a probability measure. By rotational symmetry, it may be assumed that
+� π
+0
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ≲
+� 3π/4
+π/4
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ.
+By pigeonholing, it may be assumed that
+ν =
+1
+ω3|B|δ6
+�
+B∈B
+χB,
+where B is a disjoint collection of Euclidean δ2-balls in BH(0, c), for a small constant
+c (to be chosen in a moment) and ω3 = 4π
+3 is the volume of the unit Euclidean ball
+in R3. Following the argument in [3, Proof of Theorem 5.2]:
+� 3π/4
+π/4
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ∼
+�
+L∠
+|Xν(ℓ)|q dm(ℓ)
+∼
+1
+|B|qδ6q
+�
+L∠
+�����
+�
+B∈B
+H1(ℓ ∩ B)
+�����
+q
+dm(ℓ)
+≲
+1
+|B|qδ4q
+�
+{p:ℓ(p)∈L∠}
+|{B ∈ B : B ∩ ℓ(p) ̸= ∅}|q dH3(p)
+(3)
+∼
+1
+|B|q δ4q
+�����
+�
+B∈B
+χℓ∗(B)
+�����
+q
+Lq(B(0,1))
+,
+where the integration restricts to B(0, 1) if c is small enough. Let ρ = δǫ2. Let {τ}
+be a covering of Γ by boxes of dimensions ρ × ρ2 × 1 in the standard way, where
+Γ = {(ξ, |ξ|) ∈ R3 : ξ ∈ B2(0, 1)}.
+Call x ∈ B(0, 1) “narrow” if there is a 2-dimensional subspace V of R3 (depending
+on x), such that at least half of the tubes ℓ∗(B) passing through x have direction
+vectors in a ρ2-neighbourhood of V . If x is narrow, then because of the curvature
+of Γ,
+�
+B∈B
+χℓ∗(B)(x) ≲
+��
+τ
+|{B ∈ B : dir(ℓ∗(B)) ∈ τ and x ∈ ℓ∗(B)}|q
+�1/q
+.
+
+6
+TERENCE L. J. HARRIS
+If x is not narrow then it is called “broad”. If x is a broad point, then
+�
+B∈B
+χℓ∗(B)(x) ≲ ρ−100×
+� �
+B1∈B
+�
+B2∈B
+�
+B3∈B
+χℓ∗(B1)χℓ∗(B2)χℓ∗(B3)
+��uℓ∗(B1) ∧ uℓ∗(B2) ∧ uℓ∗(B3)
+��
+�1/3
+.
+Clearly
+(4)
+�����
+�
+B∈B
+χℓ∗(B)
+�����
+Lq(B(0,1))
+≲
+�����χbroad
+�
+B∈B
+χℓ∗(B)
+�����
+Lq(B(0,1))
++
+�����χnarrow
+�
+B∈B
+χℓ∗(B)
+�����
+Lq(B(0,1))
+.
+If the broad part dominates in (4), then using q ≥ 3/2 and applying the trilinear
+Kakeya theorem (see e.g. [2, Theorem 1]) gives
+�����χbroad
+�
+B∈B
+χℓ∗(B)
+�����
+q
+Lq(R3)
+≲ ρ−100
+� 
+
+�
+B1,B2,B3∈B
+χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))|
+
+
+q/3
+≲ ρ−100 |B|q− 3
+2
+� 
+
+�
+B1,B2,B3∈B
+χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))|
+
+
+1/2
+,
+≲ ρ−100 |B|q δ6−ǫ
+= ρ−100 |B|q δ4q−(q−1)(3−t)−ǫ,
+by the definition of q. That ν is a probability measure implies that ct,δ(ν) ≳ 1, so
+this proves (2) when the broad part dominates.
+If the narrow part dominates in (4), then
+�����
+�
+B∈B
+χℓ∗(B)
+�����
+q
+Lq(R3)
+≲
+�
+τ
+������
+�
+B∈B:dir(ℓ∗(B))∈τ
+χℓ∗(B)
+������
+q
+q
+.
+For each cap τ, let Tτ be a finitely overlapping cover of physical space by ∼ ρ×ρ2×1
+planks dual to τ. If δ is smaller than some absolute constant, then each δ2-tube
+ℓ∗(B) with dir(ℓ∗(B)) ∈ τ is contained in at least 1 plank from Tτ, and intersects
+≲ 1 planks from Tτ. We associate each ℓ∗(B) with exactly one plank T ∈ Tτ such
+that dir(ℓ∗(B)) ∈ τ and ℓ∗(B) ⊆ T , and abbreviate this by writing ℓ∗(B) ≤ T .
+Then
+�
+τ
+������
+�
+B∈B:dir(ℓ∗(B))∈τ
+χℓ∗(B)
+������
+q
+q
+≲
+�
+τ
+�
+T ∈Tτ
+������
+�
+B∈B:ℓ∗(B)≤T
+χℓ∗(B)
+������
+q
+q
+.
+
+VERTICAL PROJECTIONS IN THE HEISENBERG GROUP
+7
+The point-line duality argument at Eq. (3) is reversible provided the Euclidean
+balls are enlarged by a factor of 2. This gives
+� 3π/4
+π/4
+���PV⊥
+θ #ν
+���
+q
+Lq(H2) dθ ≲
+�
+τ
+�
+T ∈Tτ
+� 3π/4
+π/4
+���PV⊥
+θ #νT
+���
+q
+Lq(H2) dθ,
+where
+νT =
+1
+|B|6δ6
+�
+B∈B:ℓ∗(B)≤T
+χ2B.
+Each measure νT is essentially the restriction of ν to a Heisenberg ball of radius
+ρ. By left translation (using Lemma 3), followed by a Heisenberg dilation (which
+commutes with vertical projections), and then by applying the induction hypothesis,
+�
+τ
+�
+T ∈Tτ
+� 3π/4
+π/4
+���PV⊥
+θ #νT
+���
+q
+Lq(H2) dθ
+≲
+�
+τ
+�
+T ∈Tτ
+ρ(t−3)(q−1)ct,δ(ν)q−1ν
+
+
+�
+B∈B:ℓ∗(B)≤T
+B
+
+ (δ/ρ)−ǫ−(3−t)(q−1)
+≲ ρǫν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1).
+The power of ρ is positive, so the induction closes provided δ is sufficiently small
+(depending only on ǫ). For the rescaled measures, the Euclidean δ2-balls are sent
+to δ2/ρ × δ2/ρ × δ2/ρ2 ellipsoids, which are essentially unions of (δ/ρ)2-balls. This
+means that the rescaled measures can be convolved with φδ/ρ without significantly
+affecting their properties (at least on Heisenberg balls of radius ≥ δ/ρ).
+□
+References
+[1] Balogh, Z. M., Durand-Cartagena, E, F¨assler, K., Mattila, P., Tyson, J. T.: The effect of
+projections on dimension in the Heisenberg group. Rev. Mat. Iberoam. 29, 381–432 (2013)
+[2] Carbery, A., Valdimarsson, S. I.: The endpoint multilinear Kakeya theorem via the Borsuk-
+Ulam theorem. J. Funct. Anal. 264, 1643–1663 (2013)
+[3] F¨assler, K., Orponen, T.: Vertical projections in the Heisenberg group via cinematic functions
+and point-plate incidences. arXiv:2210.00458v2 (2022)
+[4] Guth. L., Wang. H., Zhang, R.: A sharp square function estimate for the cone in R3. Ann. of
+Math. (2) 192, 551–581 (2020)
+[5] Liu,
+J.:
+On
+the
+dimension
+of
+Kakeya
+sets
+in
+the
+first
+Heisenberg
+group.
+Proc. Amer. Math. Soc. 150, 3445–3455 (2022)
+[6] Zahl, J.: Unions of lines in Rn. To appear in Mathematika. arXiv:2208.02913v1 (2022)
+Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
+Email address: [email protected]
+

6tE3T4oBgHgl3EQfpwpy/content/tmp_files/load_file.txt

@@ -0,0 +1,185 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf,len=184
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='04645v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='CA]  11 Jan 2023  VERTICAL PROJECTIONS IN THE HEISENBERG GROUP FOR  SETS OF DIMENSION BETWEEN 2 AND 3  TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' It is shown that vertical projections in the Heisenberg group al-  most surely do not decrease Hausdorff dimension for Borel sets of dimension  between 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  The proof uses the method of point-plate incidences in-  troduced by F¨assler and Orponen, and uses a similar approach to a recent  theorem of Zahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Introduction  Let H be the Heisenberg group, identified as a set with C × R, and equipped  with the group law  (z, t) ∗ (ζ, τ) =  �  z + ζ, t + τ + 1  2ω(z, ζ)  �  ,  where  ω(x + iy, u + iv) := xv − uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  For each θ ∈ [0, π), let  Vθ =  ��  λeiθ, 0  �  : λ ∈ R  �  ,  and let V⊥  θ be the Euclidean orthogonal complement of Vθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Each (z, t) ∈ H can be  uniquely decomposed as a product  (z, t) = PV⊥  θ (z, t) ∗ PVθ(z, t),  of an element of V⊥  θ on the left, with an element of Vθ on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This defines  the vertical projection maps PV⊥  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let dH be the Kor´anyi metric on H, given by  dH((z, t), (ζ, τ)) =  ��(ζ, τ)−1 ∗ (z, t)  ��  H ,  where  ∥(z, t)∥H =  �  |z|4 + 16t2�1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  The Kor´anyi metric is bi-Lipschitz equivalent to the more natural Carnot-Carath´eodory  metric on H, and thus induces the same Hausdorff dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let dim refer to the  Hausdorff dimension of a set in H with respect to the Kor´anyi metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This work  gives a proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let A be an analytic subset of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then  dim PV⊥  θ (A) ≥ min{dim A, 3}  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' θ ∈ [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 28A78;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 28A80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Heisenberg group, Hausdorff dimension, vertical projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  1  2  TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS  This was first conjectured by Balogh, Durand-Caragena, F¨assler, Mattila and  Tyson [1, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='5], who proved the conjecture in the range dim A ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Recently, this conjecture was proved for dim A ∈ [0, 2]∪{3} by F¨assler and Orponen  (and thus also for dim A > 3, though Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='5 in [1] also predicts positive area  in this range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' They introduced a method of point-plate incidences, and proved  (1) in the case dim A = 3 by using a square function estimate for the cone of  Guth, Wang and Zhang [4] to control the average L2 norm of pushforwards of 3-  dimensional measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The point-line duality principle they used is due to Liu [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Theorem 1 resolves the conjecture in the remaining range dim A ∈ (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The proof  of Theorem 1 uses the incidence approach of F¨assler and Orponen, but rather than  using the square function estimate for the cone, it uses a broad-narrow approach  to Kakeya-type inequalities for tubes arranged in fractal families of planks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This is  based on recent work of Zahl [6], which used a broad-narrow approach to Kakeya-  type inequalities for fractal families of tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Acknowledgements  I thank Shaoming Guo for some discussion in the earlier stages of working on  this problem, when I visited UW Madison in October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' I also thank UW for  their hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Preliminaries  For each θ ∈ [0, π] let H2 be the 2-dimensional Lebesgue measure on V⊥  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' A line  ℓ in H is called horizontal if it is a left translate of a horizontal subgroup Vθ for  some θ ∈ [0, π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' meaning that there exists p ∈ H such that ℓ = p ∗ Vθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each  horizontal line ℓ ⊆ H, let H1 be the Lebesgue measure on ℓ with respect to the  Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Given a non-negative Borel function f and a horizontal line ℓ,  define  Xf(ℓ) =  �  ℓ  f dH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Given a measure µ on H, let  cα(µ) =  sup  x∈H,r>0  µ (BH(x, r))  rα  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  More generally, given δ > 0, define  cα,δ(µ) =  sup  x∈H,r>δ  µ (BH(x, r))  rα  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Define ℓ∗ : H → P(R3) by  ℓ∗(x, y, t) = (0, x, t − xy/2) + Ly,  where  Ly =  �  λ(1, −y, y2/2) : λ ∈ R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Define ℓ : R3 → P(H) by  ℓ(a, b, c) = {(as + b, s, (bs)/2 + c) : s ∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  The following lemma is the point-line duality principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Lemma 1 ([3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let p∗ ∈ H and let p ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then  p ∈ ℓ∗(p∗)  if and only if  p∗ ∈ ℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  VERTICAL PROJECTIONS IN THE HEISENBERG GROUP  3  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Given a non-negative continuous function f supported in the unit ball  of H, let µf be the measure whose Radon-Nikodym derivative with respect to the  Lebesgue measure on H is equal to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then for any q ∈ (1, ∞) and ε > 0,  � π−ε  ε  ���PV⊥  θ #µf  ���  q  Lq(H2) dθ ∼q,ε  �  Uǫ  |Xf(ℓ(p))|q dH3(p),  where Uǫ is the set of p ∈ R3 such that ℓ(p) is a horizontal line with corresponding  angle in [ε, π − ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This is outlined in [3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='15] in the case q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' One version of the proof  uses the coarea formula, and the formula for L2 norms in terms of the distribution  function, which naturally extends to the case q ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let f be a non-negative continuous function supported in the unit ball  of H, let µf be the measure whose Radon-Nikodym derivative with respect to the  Lebesgue measure on H is equal to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then for any q ∈ (1, ∞) and any p ∈ BH(0, 1),  � π  0  ���PV⊥  θ # (Lp#µf)  ���  q  Lq(H2) dθ  � π  0  ���PV⊥  θ #µf  ���  q  Lq(H2) dθ ∼q,  where Lp(z, t) = p ∗ (z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' The Radon-Nikodym derivative of Lp#µf is f ◦ L−1  p , since left translation  has Jacobian equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Hence  � π  0  ���PV⊥  θ # (Lp#µf)  ���  q  Lq(H2) dθ ∼  � ����  �  ℓ  (f ◦ L−1  p ) dH1  ����  q  dh(ℓ),  where h is the natural left-invariant measure on the set of horizontal lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' see [3,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If ℓ is such that the integrand is nonzero, then ℓ = (z, t) ∗ Vθ for some  θ ∈ [0, π) and (z, t) ∈ V⊥  θ with dH((z, t), 0) ≲ 1 (since f is supported in the unit  ball), which implies that the Euclidean measure H1 is equivalent to the Heisenberg  Hausdorff measure H1  H on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Similarly, these two measures are equivalent on p ∗ ℓ  (which is also a horizontal line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Since H1  H is left-invariant, and h is left-invariant,  it follows that  � ����  �  ℓ  (f ◦ L−1  p ) dH1  ����  q  dh(ℓ) ∼  � ����  �  ℓ  f dH1  ����  q  dh(ℓ) ∼  � π  0  ���PV⊥  θ # (µf)  ���  q  Lq(H2) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Main results  For the statement of the following theorem, let dθ refer to the Lebesgue measure  on [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let  �  ∗ refer to the lower integral (this is only used to avoid measurability  issues in the statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let t ∈ [0, 3], and let µ be a Borel measure supported in the unit  Heisenberg ball, with ct(µ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let δ > 0, and suppose that for each θ,  Dθ is a disjoint collection of at most δ  √ǫ−tµ(H) Heisenberg δ-balls in V⊥  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then  �  ∗  �  PV⊥  θ #µ  � � �  D∈Dθ  D  �  dθ ≲ǫ δǫµ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Before proving Theorem 2, it will be shown that it implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  4  TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Measurability issues will be ignored since they can be easily  adjusted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By scaling it may be assumed that A is contained in the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Let ǫ > 0, and (using Frostman’s lemma) let µ be a Borel probability measure on  A with cα(µ) < ∞, where α = dim A − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let E ⊆ [0, π) be a compact set such  that dim PV⊥  θ (supp µ) < α − ǫ for all θ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ε > 0, and for each θ ∈ E,  let Dθ = {BH(pj(θ), rj(θ))}j be a finitely overlapping cover of PV⊥  θ (supp µ) by  Heisenberg balls of dyadic radii at most ε, such that  (1)  �  j  rj(θ)α−ǫ < cα(µ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  For each k, let Dθ,k be the subcollection of balls in Dθ with dyadic radii equal to  2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Then for each θ �� E,  1 ≤  �  k  �  PV⊥  θ #µ  �  \uf8eb  \uf8ed  �  D∈Dθ,k  D  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Integrating over E gives  H1(E) ≤  �  k  �  E  �  PV⊥  θ #µ  �  \uf8eb  \uf8ed  �  D∈Dθ,k  D  \uf8f6  \uf8f8 dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  By (1), each set Dθ,k satisfies  |Dθ,k| ≲ 2k(α−ǫ)cα(µ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  By applying Theorem 2 and summing the geometric series, this yields  H1(E) ≤  �  k  �  E  �  PV⊥  θ #µ  �  \uf8eb  \uf8ed  �  D∈Dθ,k  D  \uf8f6  \uf8f8 dθ ≲ cα(µ)  �  k≥| log2 ε|  2−kǫ2 ≲ cα(µ)εǫ2,  Letting ε → 0 gives H1(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By inner regularity of the Lebesgue measure on  [0, π), it follows that  dim PV⊥  θ (A) ≥ dim PV⊥  θ (supp µ) ≥ α − ǫ ≥ dim A − 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' θ ∈ [0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Since the outer parts of this inequality hold for any ǫ > 0, this  implies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  □  It remains to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let φ be a fixed non-negative bump function supported in the  unit Euclidean ball of H around the origin, such that  �  φ = 1 and such that φ ≳ 1  on BE(0, 1/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For each λ > 0 let φλ = λ−3φ(x/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Define ν by the Euclidean  convolution ν = µ ∗ φδ2, and let  q = 3 + t  1 + t ∈ [3/2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  By H¨older’s inequality with respect to the Lebesgue measure on each plane V⊥  θ ,  and the fact that the vertical projections are uniformly 1  2-H¨older continuous when  considered as maps from (H, dE) to (H, dH),  �  ∗  �  PV⊥  θ #µ  � � �  D∈Dθ  D  �  dθ ≲ µ(H)1− 1  q δ(3+√ǫ−t)(1− 1  q)  �� π  0  ���PV⊥  θ #ν  ���  q  Lq(H2)  �1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  VERTICAL PROJECTIONS IN THE HEISENBERG GROUP  5  Therefore, it suffices to prove that  (2)  � π  0  ���PV⊥  θ #ν  ���  q  Lq(H2) dθ ≲ǫ ν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1),  for any δ > 0, whenever ν = µ ∗ φδ2 for some finite Borel measure µ supported in  the unit Heisenberg ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This will be shown via induction on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Thus, let δ > 0  be given and assume that (2) holds for all �δ < δ1−ǫ2, for any finite Borel measure  µ supported in the unit Heisenberg ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  It will now be shown that (2) holds for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By scaling, it may be assumed that ν  is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By rotational symmetry, it may be assumed that  � π  0  ���PV⊥  θ #ν  ���  q  Lq(H2) dθ ≲  � 3π/4  π/4  ���PV⊥  θ #ν  ���  q  Lq(H2) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  By pigeonholing, it may be assumed that  ν =  1  ω3|B|δ6  �  B∈B  χB,  where B is a disjoint collection of Euclidean δ2-balls in BH(0, c), for a small constant  c (to be chosen in a moment) and ω3 = 4π  3 is the volume of the unit Euclidean ball  in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Following the argument in [3, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='2]:  � 3π/4  π/4  ���PV⊥  θ #ν  ���  q  Lq(H2) dθ ∼  �  L∠  |Xν(ℓ)|q dm(ℓ)  ∼  1  |B|qδ6q  �  L∠  �����  �  B∈B  H1(ℓ ∩ B)  �����  q  dm(ℓ)  ≲  1  |B|qδ4q  �  {p:ℓ(p)∈L∠}  |{B ∈ B : B ∩ ℓ(p) ̸= ∅}|q dH3(p)  (3)  ∼  1  |B|q δ4q  �����  �  B∈B  χℓ∗(B)  �����  q  Lq(B(0,1))  ,  where the integration restricts to B(0, 1) if c is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let ρ = δǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Let {τ}  be a covering of Γ by boxes of dimensions ρ × ρ2 × 1 in the standard way, where  Γ = {(ξ, |ξ|) ∈ R3 : ξ ∈ B2(0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Call x ∈ B(0, 1) “narrow” if there is a 2-dimensional subspace V of R3 (depending  on x), such that at least half of the tubes ℓ∗(B) passing through x have direction  vectors in a ρ2-neighbourhood of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If x is narrow, then because of the curvature  of Γ,  �  B∈B  χℓ∗(B)(x) ≲  ��  τ  |{B ∈ B : dir(ℓ∗(B)) ∈ τ and x ∈ ℓ∗(B)}|q  �1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  6  TERENCE L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' HARRIS  If x is not narrow then it is called “broad”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If x is a broad point, then  �  B∈B  χℓ∗(B)(x) ≲ ρ−100×  � �  B1∈B  �  B2∈B  �  B3∈B  χℓ∗(B1)χℓ∗(B2)χℓ∗(B3)  ��uℓ∗(B1) ∧ uℓ∗(B2) ∧ uℓ∗(B3)  ��  �1/3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Clearly  (4)  �����  �  B∈B  χℓ∗(B)  �����  Lq(B(0,1))  ≲  �����χbroad  �  B∈B  χℓ∗(B)  �����  Lq(B(0,1))  +  �����χnarrow  �  B∈B  χℓ∗(B)  �����  Lq(B(0,1))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  If the broad part dominates in (4), then using q ≥ 3/2 and applying the trilinear  Kakeya theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' [2, Theorem 1]) gives  �����χbroad  �  B∈B  χℓ∗(B)  �����  q  Lq(R3)  ≲ ρ−100  � \uf8eb  \uf8ed  �  B1,B2,B3∈B  χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))|  \uf8f6  \uf8f8  q/3  ≲ ρ−100 |B|q− 3  2  � \uf8eb  \uf8ed  �  B1,B2,B3∈B  χℓ∗(B1)χℓ∗(B2)χℓ∗(B3) |u(ℓ∗(B1)) ∧ u(ℓ∗(B2)) ∧ u(ℓ∗(B3))|  \uf8f6  \uf8f8  1/2  ,  ≲ ρ−100 |B|q δ6−ǫ  = ρ−100 |B|q δ4q−(q−1)(3−t)−ǫ,  by the definition of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' That ν is a probability measure implies that ct,δ(ν) ≳ 1, so  this proves (2) when the broad part dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  If the narrow part dominates in (4), then  �����  �  B∈B  χℓ∗(B)  �����  q  Lq(R3)  ≲  �  τ  ������  �  B∈B:dir(ℓ∗(B))∈τ  χℓ∗(B)  ������  q  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  For each cap τ, let Tτ be a finitely overlapping cover of physical space by ∼ ρ×ρ2×1  planks dual to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' If δ is smaller than some absolute constant, then each δ2-tube  ℓ∗(B) with dir(ℓ∗(B)) ∈ τ is contained in at least 1 plank from Tτ, and intersects  ≲ 1 planks from Tτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' We associate each ℓ∗(B) with exactly one plank T ∈ Tτ such  that dir(ℓ∗(B)) ∈ τ and ℓ∗(B) ⊆ T , and abbreviate this by writing ℓ∗(B) ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Then  �  τ  ������  �  B∈B:dir(ℓ∗(B))∈τ  χℓ∗(B)  ������  q  q  ≲  �  τ  �  T ∈Tτ  ������  �  B∈B:ℓ∗(B)≤T  χℓ∗(B)  ������  q  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  VERTICAL PROJECTIONS IN THE HEISENBERG GROUP  7  The point-line duality argument at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' (3) is reversible provided the Euclidean  balls are enlarged by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This gives  � 3π/4  π/4  ���PV⊥  θ #ν  ���  q  Lq(H2) dθ ≲  �  τ  �  T ∈Tτ  � 3π/4  π/4  ���PV⊥  θ #νT  ���  q  Lq(H2) dθ,  where  νT =  1  |B|6δ6  �  B∈B:ℓ∗(B)≤T  χ2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Each measure νT is essentially the restriction of ν to a Heisenberg ball of radius  ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' By left translation (using Lemma 3), followed by a Heisenberg dilation (which  commutes with vertical projections), and then by applying the induction hypothesis,  �  τ  �  T ∈Tτ  � 3π/4  π/4  ���PV⊥  θ #νT  ���  q  Lq(H2) dθ  ≲  �  τ  �  T ∈Tτ  ρ(t−3)(q−1)ct,δ(ν)q−1ν  \uf8eb  \uf8ed  �  B∈B:ℓ∗(B)≤T  B  \uf8f6  \uf8f8 (δ/ρ)−ǫ−(3−t)(q−1)  ≲ ρǫν(H)ct,δ(ν)q−1δ−ǫ−(3−t)(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  The power of ρ is positive, so the induction closes provided δ is sufficiently small  (depending only on ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' For the rescaled measures, the Euclidean δ2-balls are sent  to δ2/ρ × δ2/ρ × δ2/ρ2 ellipsoids, which are essentially unions of (δ/ρ)2-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' This  means that the rescaled measures can be convolved with φδ/ρ without significantly  affecting their properties (at least on Heisenberg balls of radius ≥ δ/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  □  References  [1] Balogh, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=', Durand-Cartagena, E, F¨assler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=', Mattila, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=': The effect of  projections on dimension in the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
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+page_content=', Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=': A sharp square function estimate for the cone in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' of  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' (2) 192, 551–581 (2020)  [5] Liu,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=':  On  the  dimension  of  Kakeya  sets  in  the  first  Heisenberg  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' 150, 3445–3455 (2022)  [6] Zahl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=': Unions of lines in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' To appear in Mathematika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='02913v1 (2022)  Department of Mathematics, Cornell University, Ithaca, NY 14853, USA  Email address: tlh236@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE3T4oBgHgl3EQfpwpy/content/2301.04645v1.pdf'}

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+Wave function-based emulation for nucleon-nucleon scattering in momentum space
+A. J. Garcia
+,1, ∗ C. Drischler
+,2, 3, † R. J. Furnstahl
+,1, ‡ J. A. Melendez
+,1, § and Xilin Zhang
+3, ¶
+1Department of Physics, The Ohio State University, Columbus, OH 43210, USA
+2Department of Physics and Astronomy and Institute of Nuclear and Particle Physics, Ohio University, Athens, OH 45701, USA
+3Facility for Rare Isotope Beams, Michigan State University, MI 48824, USA
+(Dated: January 13, 2023)
+Emulators for low-energy nuclear physics can provide fast & accurate predictions of bound-state
+and scattering observables for applications that require repeated calculations with different param-
+eters, such as Bayesian uncertainty quantification. In this paper, we extend a scattering emulator
+based on the Kohn variational principle (KVP) to momentum space (including coupled channels)
+with arbitrary boundary conditions, which enable the mitigation of spurious singularities known as
+Kohn anomalies. We test it on a modern chiral nucleon-nucleon (NN) interaction, including emu-
+lation of the coupled channels. We provide comparisons between a Lippmann-Schwinger equation
+emulator and our KVP momentum-space emulator for a representative set of neutron-proton (np)
+scattering observables, and also introduce a quasi-spline-based approach for the KVP-based emula-
+tor. Our findings show that while there are some trade-offs between accuracy and speed, all three
+emulators perform well. Self-contained Jupyter notebooks that generate the results and figures in
+this paper are publicly available.
+I.
+INTRODUCTION
+Nucleon-nucleon (NN) scattering has long been used
+to fix parameters of microscopic Hamiltonians designed
+for ab initio few- and many-body calculations. But the
+uncertainty in most existing nuclear models has been un-
+derestimated because they have lacked two key ingredi-
+ents: a rigorous accounting of Hamiltonian uncertainty
+and a complete estimate of parameter uncertainty.
+In the case of chiral effective field theory (χEFT) [1–
+4], Hamiltonian uncertainty manifests as a truncation er-
+ror, which has been statistically modeled in Refs. [5–8].
+A holistic parameter estimation study would then both
+account for truncation errors in the likelihood, and esti-
+mate and propagate all plausible values of the low-energy
+constants (LECs) rather than finding a single parame-
+ter value maximizing the likelihood. Bayesian statisti-
+cal methods are particularly suitable for these tasks [9–
+13], but are computationally demanding, especially when
+generalizing to include few-body forces.
+Emulators—
+surrogate models that allow for fast & accurate (but ap-
+proximate) model predictions—have the potential to al-
+leviate some of these demands [14]. In this paper, we
+extend our recent explorations of emulators for NN scat-
+tering [15–17] to momentum-space wave functions and
+coupled channels, and test against a representative set of
+neutron-proton (np) scattering observables.
+The demand for emulators has led nuclear physics to
+the general field of parametric model order reduction
+(PMOR), where the goal is to extract the relevant in-
+formation from a model while reducing the computa-
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+¶ [email protected]
+tional cost significantly.
+An efficient offline-online de-
+composition is crucial to construct an efficient emula-
+tor.
+In the offline stage, the emulator is trained with
+high-fidelity calculations1 for selected sets of parameters,
+also known as snapshots, while making predictions for
+any other set of parameters are performed in the online
+stage. The end result is a reduced-order model (ROM)
+that serves as an emulator. For general overviews of the
+literature on PMOR techniques and their applications,
+we refer the reader to Refs. [19, 20]. A pedagogical intro-
+duction to projection-based emulators for both scatter-
+ing and bound-state calculations, including interactive,
+open-source Python code, can be found in Refs. [21, 22].
+A particular snapshot-based ROM known as the re-
+duced basis method (RBM)2 has emerged as an efficient
+emulator for the prediction of both bound state and scat-
+tering observables [15, 23, 24]. The foundation of the first
+emulators for scattering is the Kohn variational princi-
+ple (KVP) [e.g., for the K matrix], whose snapshots are
+based on scattering solutions to the Schr¨odinger equa-
+tion [25, 26].
+It has been demonstrated for a variety
+of real and optical potentials that such emulators can be
+trained for two- and three-body3 scattering in coordinate
+space, then evaluated in the form of matrix inversions
+with low-dimensional matrices [15, 16, 27].
+Subsequently, an emulator of the Lippmann-Schwinger
+(LS) equation using the Newton variational principle
+(NVP) [28] was introduced in Ref. [17]. In contrast to the
+KVP emulator, the variational trial basis is composed of
+1 Following the terminology of Ref. [18], we will refer to the cal-
+culational machinery that generates high-fidelity solutions (e.g.,
+LS equation solver) as a simulator.
+2 The RBM has been rediscovered in the low-energy nuclear theory
+community as eigenvector continuation (EC). See Ref. [19] for
+more details.
+3 In Ref. [27], the offline training stage involves calculations in both
+momentum and coordinate space.
+arXiv:2301.05093v1  [nucl-th]  12 Jan 2023
+
+ID2
+TABLE I. Notation used in this work.
+Notation
+Description
+θ
+vector of parameters; θi are the parameters
+for the ith snapshot
+s, s′
+indices for the exit and entrance channels of
+the scattering process, e.g., 3S1 and 3D1
+t, t′
+indices for available channels (summation
+convention implied)
+ψs
+i
+wave function in the channel s used for train-
+ing and associated with the ith snapshot with
+θi [high-fidelity solution of Eq. (1)]
+�ψs
+snapshot-based trial wave function in the
+channel s (3) applied to the KVP func-
+tional (2)
+Lss′
+E
+a generic scattering matrix at energy E
+Lss′[ �ψ]
+a functional whose stationary point is an ap-
+proximation of the generic L-matrix; i.e.,
+L[ �ψ + δ �ψ] = Lss′
+E + O(δL2)
+βi
+to-be-determined coefficient of the ith snap-
+shot in the trial wave function with �
+i βi = 1
+∆�U ss′
+ij (θ)
+nb × nb kernel matrix defined in Eq. (5)
+scattering matrices (e.g., K matrices) rather than scat-
+tering wave functions. Both approaches were shown to
+quickly and accurately predict the np phase shifts from a
+chiral Hamiltonian across a range of parameter values. In
+this paper, we compare a momentum-space KVP-based
+emulator, including emulation of coupled channels and
+allowing for arbitrary boundary conditions, to the NVP
+emulator for a representative set of np observables. For a
+comparison of the KVP and NVP emulators in a Galerkin
+framework and a survey on other emulators see Ref. [21].
+The paper is organized as follows. In Sec. II, we review
+the underlying formalism of the KVP emulators and its
+extension to momentum space and coupled channels. We
+then show results for the momentum-space KVP emula-
+tor and compare them to the K matrix (NVP) emulator
+in Sec. III. We demonstrate that spurious singularities
+known as Kohn (or Schwartz) anomalies [29, 30] are mit-
+igated using methods from Ref. [16]. Section IV has a
+summary and outlook and additional details of the im-
+plementation are given in several appendices. The self-
+contained set of codes that generate all results and figures
+shown in this paper is publicly available [31].
+II.
+FORMALISM
+Our goal is to emulate the partial-wave Schr¨odinger
+equation for NN scattering at the center-of-mass energy
+E > 0
+�H(θ) |ψs⟩ ≡
+� �T + �V (θ)
+�
+|ψs⟩ = E |ψs⟩ ,
+(1)
+where the vector θ is composed of parameters used by
+the theoretical model to match results with experimen-
+tal observations (e.g., the LECs of χEFT). Building our
+snapshot-based MOR emulator begins by writing Eq. (1)
+in integral form.
+Here we choose the general (con-
+strained4) KVP, which is based on the functional [16, 32]
+Lss′[ �ψ] = �Lss′
+E − 2µk0
+det u ⟨ �ψs| �H(θ) − E| �ψs′⟩ ,
+(2)
+where �ψ is a trial scattering wave function, �Lss′
+E
+is a
+generic trial scattering matrix, u is a non-singular ma-
+trix [16, 32] used to parameterize the asymptotic bound-
+ary condition associated with �Lss′
+E (see Appendix A), and
+k0 = √2µE is the on-shell energy with µ being the re-
+duced mass.5 More details can be found in Ref. [16] and
+Appendix A. Table I summarizes the notation we use in
+this work. Note that we adopt the convention that the
+wave functions in a bra symbol ⟨·| in bra-ket notation are
+not complex conjugated [e.g., ⟨ �ψs| in Eq. (2)] [15, 16, 34].
+In Eq. (2), the superscripts s and s′ index the coupled
+channels (e.g., 3S1 and 3D1); for the uncoupled case this
+reduces to a single equation with s′ = s. Each combi-
+nation of (s′, s) will have their own, distinct emulator in
+our formulation. As an example, for a coupled-channel
+np interaction in Eq. (2), the (s′, s) pair could be one
+of 3S1–3S1, 3S1–3D1, 3D1–3S1, or 3D1–3D1, and for an
+uncoupled channel s′ = s could be 1S0.
+We use the
+np spin-triplet coupled channels as an exemplary case,
+but the general emulation procedure applies to general
+channel coupling (including spin-singlet spin-triplet np
+coupling [35]).
+The functional (2) yields Lss′[ �ψ] = Lss′
+E
+when �ψ is the
+exact wave function, and provides a stationary approxi-
+mation otherwise: Lss′[ �ψ + δ �ψ] = Lss′
+E + O(δL2). Rather
+than finding a wave function |ψ⟩ that satisfies Eq. (1),
+our task has now changed to finding a wave function that
+makes Eq. (2) stationary for a given choice of E.
+The key to creating an efficient PMOR emulator from
+Eq. (2) is to use a snapshot trial wave function,
+| �ψs⟩ ≡
+nb
+�
+i=1
+βi |ψs
+i ⟩ ,
+(3)
+where nb is the number of parameter vectors {θi}nb
+i=1 in
+the training set and {|ψs
+i ⟩}nb
+i=1 the associated high-fidelity
+solutions to Eq. (1), obtained by solving the LS equation
+directly (see also Sec. III). These solutions are determined
+once in the offline stage. The to-be-determined basis co-
+efficients ⃗β will not be the same for all the channels, re-
+sulting in independent emulators for each (s′, s) pair (see
+Appendix B for more details).
+For the np spin-triplet
+4 For a description of constrained and unconstrained emulators see
+Ref. [21]
+5 Throughout this paper we use boldface symbols to indicate vec-
+tors in parameter-space, arrows to indicate vectors in snapshot-
+space, natural units in which ℏ = c = 1, and follow the conven-
+tions for scattering matrices in Refs. [26, 33].
+
+3
+coupled channels, this will result in three distinct varia-
+tional principles being enforced: one for each of angular
+momentum s′ = s = j ± 1 and one for the off-diagonal
+component.
+The other off-diagonal component can be
+inferred through the unitarity of the S matrix.6
+Upon inserting the snapshot trial wave function (3)
+into the functional (2), the functional takes the form [15]
+Lss′[⃗β ] = βiLss′
+E,i − 1
+2βi∆�U ss′
+ij βj,
+(4)
+with the symmetric matrix
+∆�U ss′
+ij (θ) ≡ 2µk0
+det u
+�
+⟨ψs
+i | �H(θ) − E|ψs′
+j ⟩ + (i ↔ j)
+�
+= 2µk0
+det u
+�
+⟨ψs
+i |�V (θ) − �Vj|ψs′
+j ⟩ + (i ↔ j)
+�
+,
+(5)
+where, as in Eq. (2), s′ and s correspond to the entrance
+and exit channels. Equation (4) is a stationary approx-
+imation to the generic L-matrix at one energy, hence
+we build independent emulators for each value of an en-
+ergy grid. Equation (5) is obtained [15] by adding and
+subtracting �Vi ≡ �V (θi) and �Vj ≡ �V (θj) and applying
+Eq. (1). In this form, the constant terms in the poten-
+tials, such as a long-range Coulomb interaction (assuming
+the fine-structure constant is not varied), will cancel, and
+the matrix elements will only involve short-range physics.
+Emulating the scattering wave function [via Eq. (3)],
+and hence Lss′
+E ≈ Lss′[ �ψ] [via Eq. (4)], has now been re-
+duced to choosing an appropriate training set {θi} and
+then determining the values of βi that make Eq. (4) sta-
+tionary under the constraint that �
+i βi = 1. The latter
+is a consequence of maintaining a consistent asymptotic
+normalization for the scattering wave functions in Eq. (3)
+as required by the constrained KVP [15, 21]. A numeri-
+cally robust solution can be found by introducing a La-
+grange multiplier λ, and solving the matrix equation [16]
+�
+∆�U ss′ ⃗1
+⃗1 ⊺
+0
+� �⃗β⋆
+λ⋆
+�
+=
+�⃗Lss′
+E
+1
+�
+,
+(6)
+where ⃗1 is an nb × 1 vector of ones, ⃗Lss′
+E
+are the basis
+states used in the offline stage, and ⃗β⋆ is a vector of co-
+efficients of the trial wave function associated with the
+KVP’s stationary approximation. Since Eq. (6) is a lin-
+ear system, it will be a highly computationally efficient
+emulator for scattering systems if the number nb of basis
+functions is much smaller than the size of the high-fidelity
+wave function ψ.
+Thus far we have not specified whether the matrix el-
+ements ∆�U ss′
+ij
+are to be calculated in coordinate space
+or momentum space. The only difference between these
+6 For (complex-valued) optical potentials with two coupled chan-
+nels, one has four (instead of three) distinct variational principles
+because the S matrix is not unitary.
+implementations is the way we obtain the basis functions
+ψi used to construct the trial ansatz in Eq. (3), and thus
+the manner in which ∆�U ss′ is evaluated. To formulate a
+momentum-space wave function approach to MOR emu-
+lators for scattering, we initially solve for the K matrix
+and relate ψ to K before using Eq. (5). The scattering
+wave function in momentum space takes the form [36]
+ψst(k; k0) = 1
+k2 δ(k − k0)δst + 2
+π PKst(k, k0)/k0
+k2 − k2
+0
+,
+(7)
+which vanishes as k → ∞, but is singular at k = k0 =
+√2µE (the superscripts used for the K matrix in Eq. (7)
+are opposite Ref. [36]). Here, Kst is the reactance ma-
+trix (or just the K matrix), k0 the on-shell energy, P the
+Cauchy principal value, and the labeling st indicates the
+partial-wave or reaction channels.
+One can also write
+Eq. (5) in the momentum-space representation by insert-
+ing complete sets of states,7 resulting in
+∆�U ss′
+ij (θ) =
+¨ ∞
+0
+dk dp k2p2�
+ψts
+i (k)V tt′
+θ,j(k, p)ψt′s′
+j
+(p)
++ (i ↔ j)
+�
+,
+(8)
+with
+V tt′
+θ,j(k, p) ≡ 2µk0
+det u
+�
+V tt′(k, p; θ) − V tt′
+j
+(k, p)
+�
+,
+(9)
+where t and t′ are summed over the available channels
+and the dependence of ψ on k0 is left implicit. Moving
+forward, we will drop the channel superscripts on ∆�U.
+This is the general form of the momentum-space ∆ ˜U
+matrix. Note the ordering of the channel indices (t, s)
+in the left-hand wave function in Eq. (8), which follows
+from ψts(k) ≡ ⟨kt|ψs⟩ and the convention that ⟨ψ| = |ψ⟩⊺
+(without a complex conjugate), so that ψts(k) = ⟨ψs|kt⟩.
+Thus, if ψ has outgoing (ψ(+)), incoming (ψ(−)), or
+standing wave (ψ(0)) boundary conditions, then the same
+version of ψ(x) is used for both ψ(k) and ψ(p) in Eq. (8).
+No modification of Eq. (8) is needed in the case of optical
+potentials, where again the left-hand wave function is not
+conjugated relative to the right-hand wave function. For
+more details on how to build the general KVP emulator
+we refer the reader to Appendix C. Different boundary
+conditions will be used below to mitigate Kohn anomalies
+(see Sec. III B).
+The efficient evaluation of ∆�U across a range of θ val-
+ues is critical to the applicability of the emulator. If the
+Hamiltonian operators have an affine (i.e., factorizable)
+parameter dependence, denoted as
+�H(θ) =
+�
+n
+hn(θ) �Hn,
+(10)
+7 For example, for np scattering as in Sec. III, the complete set
+of states are relative-momentum partial-wave states with orbital
+angular momentum and spin coupled to total J and MJ.
+
+4
+then matrix elements of the Hn operators in a given basis
+only need to be calculated once in the offline stage rather
+than for every parameter set θi. Chiral NN interactions
+have the form of Eq. (10) and, when varying only the
+contact LECs, can even be cast into the form8
+�V (θ) = �V 0 + θ · �V 1,
+(11)
+so that Eq. (5) can then be written as
+∆�U(θ) = ∆�U 0 + θ · ∆ �U 1.
+(12)
+The matrices �V 0 and ∆�U 0 and vectors of matrices �V 1
+and ∆ �U 1, can now be pre-calculated during the emu-
+lator’s offline stage, allowing for considerable speed-up
+factors in the online stage where the value of ∆�U(θ) at
+any new parameter value is efficiently constructed.
+III.
+RESULTS
+In this section, we apply the KVP momentum-space
+emulator to calculate np scattering observables. We use
+the Reinert et al. semilocal momentum-space (SMS) reg-
+ularized chiral potential at N4LO+ with the momentum
+cutoff Λ = 450 MeV [37], which is a state-of-the-art chiral
+NN interaction. The parameters θ are composed of the
+NN contact LECs contributing to this potential.
+A.
+Emulator overview
+The snapshots used in the offline stage are the scatter-
+ing solutions given by Eq. (7). The K matrices used to
+calculate the second term in Eq. (7) are obtained from
+numerically solving the LS equation. The LS equation is
+reduced to a set of linear equations by approximating the
+integral as a sum over N quadrature points obtained from
+a Gauss–Legendre rule with corresponding weights (see
+Refs. [36, 38]). If the potential was calculated merely on
+the quadrature points, without appending the on-shell
+values, interpolation must be performed to obtain the
+(half-)on-shell potential so that one can (1) account for
+the singularity of the Green’s function when solving the
+LS equation [38], and (2) integrate the delta distribution
+in Eq (7) (explained in next paragraph). To generate the
+figures in this paper, we use a compound Gauss-Legendre
+quadrature mesh of N = 80 momentum points. For the
+observables, we use a lab energy range of 0 to 350 MeV
+with 350 points. For the partial waves plots, we use a fine
+8 Note that hn(θ) would include higher-order polynomials when
+also emulating the pion-nucleon coupling c2 (at N3LO) and axial
+coupling constant gA (already at LO). Nevertheless, the Hamil-
+tonian remains affine and thus the emulators discussed here are
+directly applicable.
+energy mesh of 3500 points over the same energy range
+previously mentioned.
+When performing the KVP emulation, we calculate
+Eq. (5) two different ways.
+The first is by inserting
+Eq. (7) into Eq. (5) and analytically integrating the delta
+distribution, which corresponds to appending the exact
+on-shell value of the potential. The remaining integrals
+are then solved numerically (see Appendix C). We re-
+fer to this method as the Standard method. The second
+is based on the global Gl¨ockle spline interpolation [39],
+which belongs to the family of quasi-spline methods that
+perform the mapping
+�
+k
+f(k)Sk(k0) ≈ f(k0),
+(13)
+for smooth functions f(k) sampled on a grid k that en-
+compasses k0 using the cubic spline polynomials Sk(k0)
+constructed in Ref. [39].
+This allows us to calculate
+Sk(k0) once in the offline stage and save the result for
+the online stage since it has no dependence on f(k) it-
+self. Using this method, we interpolate the solutions to
+the integrals that appear in Eq. (5) (i.e., k0 does not need
+to be appended to the mesh as opposed to the Standard
+method), thus decreasing the computational cost needed
+in the offline stage significantly at the expense of accu-
+racy. We compare the KVP emulator results using the
+Gl¨ockle and Standard method and compare those results
+to the NVP emulator described in Ref. [17].
+To reduce numerical errors in both the simulator and
+emulator, we compute snapshots {Ki} of the LS equation
+using non-interpolated potentials for partial waves that
+have a LEC-dependence and interpolated potentials for
+LEC-independent partial waves. When referring to in-
+terpolated potentials, we mean calculating the potential
+using only the momentum mesh and then using an in-
+terpolation method (such as the bivariate Gl¨ockle spline
+method) to interpolate the potential to k0.
+By non-
+interpolated, we mean that each k0 is appended to the
+momentum mesh and the potential evaluated at these
+points, which improves the accuracy of our potentials
+compared to interpolating the potential to k0. We chose
+to use non-interpolated potentials for the LEC-dependent
+partial waves since these are the only ones used to cal-
+culate Eq. (5) in the offline phase.
+The same LEC-
+independent partial waves are employed by the simulator
+and emulator. All potentials used for the emulators and
+simulator are pre-calculated for efficiency.
+The simulator used in this paper numerically solves
+the LS equation for each partial wave.
+The accuracy
+of our simulator was tested by comparing the simulator
+results to the analytical solution of a Gaussian separable
+potential, producing relative errors of ≈ 10−7 or better.
+Additionally, the simulator’s speed was roughly 4x slower
+when we doubled the mesh size from N = 80 to N = 160
+quadrature points.
+The accuracy of emulated observables depends on the
+size of the basis (see Sec. III C); here we use a basis size
+nb = 2na, where na is the number of LECs associated
+
+5
+with a given partial wave channel. The training points
+θi are randomly sampled within an interval of [−5, 5]
+using a Latin-hypercube for each partial wave, with the
+fitted coupling constants and appropriate units given in
+Ref. [37].
+The matrix ∆�U is increasingly ill-conditioned as the
+basis size nb increases. One can reduce numerical noise
+by (1) adding a regularization parameter (“nugget”) to
+the diagonal elements of the near-singular matrix [15], or
+(2) using a solver that performs some type of regulariza-
+tion. For the KVP emulator results in the figures, we use
+NumPy’s least-squares solver linalg.lstsq() [40] with
+a cut-off ratio for small singular values of 10−10 [16]. For
+the NVP emulator, we add a nugget of 10−10 to the di-
+agonal and use NumPy’s linalg.solve().
+The general KVP functional may not always provide
+a (unique) stationary approximation, giving rise to spu-
+rious singularities known as Kohn (or Schwartz) anoma-
+lies [29, 30]. The energies at which those anomalies occur
+depends on the training parameters θ used in the offline
+stage and the evaluation set used in the online stage.
+Reference [16] proposed detecting and mitigating these
+numerical instabilities by considering an array of KVPs
+with different boundary conditions (i.e., scattering ma-
+trices) within a partial wave and using the emulator so-
+lutions to obtain an estimated S matrix by a weighted
+sum of averages [see also Refs. [32, 41]].
+For our KVP emulator, the mitigation process involves
+first calculating Eq. (5) using the K matrix boundary
+condition. Once we have calculated ∆�U, the terms in
+Eq. (4) are rescaled to match the boundary conditions
+we want to emulate (here, L = K, K−1, and T). The
+anomalies are then detected by applying a consistency
+check to the (independent) emulated solutions of the dif-
+ferent boundary conditions. The emulator solutions that
+do not pass this check are discarded while those that
+pass are averaged to obtain an anomaly-free scattering
+matrix (here, the S matrix). All KVP emulator results
+in this paper are shown with anomaly mitigation unless
+otherwise stated. So far, such a mitigation protocol has
+not been implemented for the NVP emulator. However,
+one approach would be to use multiple emulators based
+on different variational principles [21] instead of multiple
+boundary conditions. See Appendix A for our implemen-
+tation and Ref. [16] for more information on emulation
+with arbitrary boundary conditions and ways to mitigate
+Kohn anomalies.
+B.
+Emulation of phase shifts
+We first apply the emulators to the uncoupled 1S0
+channel using Eq. (5) to calculate ∆�U (see Appendix C
+for explicit expressions).
+At N4LO+, this channel de-
+pends on na = 3 non-redundant LECs [37], and thus
+we choose our basis to be composed of nb = 6 training
+points. Figure 1 shows the phase shifts calculated using
+our simulator (black line) and the KVP emulator Stan-
+FIG. 1. Simulated (black solid line) and KVP emulated Stan-
+dard method (orange dots) 1S0 phase shifts for the N4LO+
+SMS potential with Λ = 450 MeV (top panel). The bottom
+panel shows the relative errors between the simulated and em-
+ulated phase shifts for the Gl¨ockle method (red dashed line),
+Standard method (blue solid line), and NVP emulator (green
+dotted line), respectively. The spike at Elab ≈ 270 MeV is
+due to the phase shift crossing zero.
+dard method prediction (orange dots) as a function of
+the laboratory energy in the top panel. The phase shifts
+associated with the training points are depicted by the
+light gray lines. In addition, the bottom panel shows the
+relative errors
+Rel. Error = 2
+����
+Simulator − Emulator
+Simulator + Emulator
+����
+(14)
+between the simulated and emulated phase shifts for
+the Gl¨ockle method (red dashed line), Standard method
+(blue solid line), and NVP emulator (blue dotted line).
+We find that our KVP emulator accurately reproduces
+the high-fidelity phase shifts over a large energy range
+for both methods, but the Standard method is much
+more accurate than the Gl¨ockle method.
+On average,
+the relative error for the Gl¨ockle method is on the order
+of ≈ 10−6 −10−5, while the Standard method has a rela-
+tive error on the order of ≈ 10−12 for the same basis size.
+The NVP emulator’s relative error is similar to the KVP
+Standard method, with an error of ≈ 10−13.
+We now turn to the coupled 3S1–3D1 channel. This
+channel depends on na = 6 non-redundant LECs [37]
+at N4LO+, which means that our basis will be com-
+posed of nb = 12 training points. Figure 2 shows the
+on-shell K matrix for the simulator calculation (black
+lines) and KVP emulator prediction (orange dots) as
+a function of the laboratory energy for each different
+partial-wave component. The errors are similar to the
+
+Basis
+- Simulator
+oooEmulator
+200
+[deg]
+100
+0
+-100
+100
+ Glockle
+Standard
+NVP
+4
+10
+Error
+0
+100
+200
+300
+Eiab
+[MeV]6
+FIG. 2. As in Fig. 1, but for the on-shell K matrix in the coupled 3S1–3D1 as a function of the laboratory energy. From left
+to right: pure D–wave, pure S–wave, and mixed S–D-wave component.
+1S0 channel, with the Standard method being much more
+accurate than the Gl¨ockle method, and the NVP emula-
+tor’s relative error being slightly better than the Stan-
+dard method. In all cases, we see a spike in the relative
+error at Elab ≈ 20 MeV where the K matrix is singular.
+The small spikes seen in the Standard method error are
+not Kohn anomalies, but can be attributed to a numerical
+instability of the principal value integral in the LS equa-
+tion. These spikes are mesh-dependent and appear when
+a k0 value is close to a momentum mesh point, thus caus-
+ing the denominator of the Green’s function to approach
+zero faster than the numerator. A way to decrease the
+relative error produced by these spikes is to not allow the
+k0 values to be close to momentum mesh points by mov-
+ing energies that are close to any momentum mesh point
+until the relative distance is greater than some threshold
+value; e.g., ε ≳ 10−2 MeV (see Appendix D for details).
+The oscillations that appear in the Gl¨ockle method’s rel-
+ative errors plots are potential-dependent, and increase
+in number, but decrease in separation, when increasing
+the mesh size.
+Overall, the emulators accurately predict the partial
+waves for the uncoupled 1S0 and coupled 3S1–3D1 chan-
+nels.
+When comparing the Gl¨ockle method emulation
+with the Standard method, we see that the relative error
+for the Standard method is much less than the Gl¨ockle
+method. For both partial waves shown, the NVP emula-
+tor is the one that most accurately reproduces its high-
+fidelity solution. Results for the other channels are sim-
+ilar to the ones presented here, with the only difference
+being that the relative error decreases as na gets smaller.
+This can be further explored with the Jupyter notebooks
+provided [31].
+C.
+Emulation of scattering observables
+Next, we examine the performance of the emulator for
+nuclear observables. As a demonstration, we use the SMS
+regularized chiral potential at N4LO+ for np scatter-
+ing with partial waves having total momentum quantum
+numbers j ⩽ jmax = 20. Overall, there are a total of 25
+parameters in θ that are being sampled using a Latin-
+hypercube design.
+As previously mentioned, the basis
+size is chosen as nb = 2na, where na is the number of
+LECs associated with the specific partial-wave, for a to-
+tal of 50 training points. Since these parameters are only
+present in the channels j ⩽ 4, the emulator only needs to
+be trained over these channels. The remaining channels
+do not change as the parameters are varied, therefore,
+they do not undergo a training process and need to be
+calculated only once by solving the LS equation directly.
+The emulation of observables is carried out by com-
+bining multiple emulators across different partial-wave
+channels. The total np cross section can be calculated
+using
+σtot(k0) =
+π
+2k2
+0
+jmax
+�
+j=0
+(2j + 1) Re{Tr[Sj(k0) − 14]},
+(15)
+where Sj = 14 − 2i(1 − iKj)−1Kj is the S matrix, Kj
+is the predicted on-shell K matrix, and Tr[·] denotes the
+trace. Both Sj and Kj are 4 × 4 matrices that contain
+both the triplet-triplet and the singlet-triplet channels.
+Figure 3 shows the simulator and emulator prediction
+for the total np cross section, which are calculated us-
+ing the fit values for the LECs determined in Ref. [37].
+The inset in Fig. 3 depicts the mean relative errors for
+all three emulators when randomly sampling 500 differ-
+ent combinations of np LECs (chosen within the same
+range as the training points), using these to calculate the
+emulated and simulated total cross section, and compar-
+ing the results.
+On average, the relative errors for all
+three emulators are similar to those for the partial-wave
+calculations discussed in Sec. III B. Although the mean
+relative errors for the Standard method and NVP emula-
+tors are very similar, the NVP emulator seems to be the
+one that most accurately reproduces its simulator.
+
+Basis
+1
+K
+K+
+Simulator
+(oy)M
+·o Emulator
+IOOOI
+Glockle
+2
+10°
+Standard
+NVP
+10
+Rel.
+10-16
+0
+100
+200
+300
+0
+100
+200
+300
+0
+100
+200
+300
+Eiab [MeV]
+Eiab[MeV]
+Eiab [MeV]7
+FIG. 3.
+Simulated (black solid line) and emulated (orange
+dots) np cross section with jmax = 20 for the N4LO+ SMS
+potential with Λ = 450 MeV as a function of the laboratory
+energy.
+The inset shows the relative mean errors between
+the emulator and the simulator using the Gl¨ockle, Standard
+method, and NVP emulator for 500 different sets of np LECs
+obtained from Latin-hypercube sampling. See the main text
+for details.
+As mentioned in Sec. III A and following Ref. [16],
+the Kohn anomalies found in the calculation were mit-
+igated by emulating with different boundary conditions
+and building the estimated S matrix. Figure 6 in Ap-
+pendix D shows a total cross section emulation result
+with one boundary condition, hence no anomaly mitiga-
+tion. From the figure, we see that anomalies contribute
+to the Standard method mean relative error at higher en-
+ergies with a magnitude of approximately 10−3. These
+spikes are reduced to approximately 10−9 with mitiga-
+tion. The Gl¨ockle method result exhibits anomaly con-
+tributions of order 10−3 at lower energies, which get re-
+duced to approximately 10−5–10−7 with mitigation. For
+additional information, see the discussion in Appendix D.
+Although the NVP emulator is subject to anomalies, they
+are not evident in the figures shown in this section, even
+though no mitigation strategy was applied. An example
+of noticeable anomaly contributions as large as 10−3 in
+the NVP emulation are seen in Fig. 7 in Appendix D.
+The remaining spikes in Fig. 3 (e.g., at Elab
+≈
+140 MeV) can be traced back to singularities in the on-
+shell K matrix for the 3S1–3D1 channel at those energies
+and are only seen for a few (specific) LEC values out
+of the 500 sampled (see also Fig. 2). The mesh-induced
+spikes seen in the Standard method relative error were
+also reduced in magnitude by preventing the on-shell k0
+value from being too close to a momentum mesh value
+(see Fig. 8 for result comparisons).
+We now turn our attention to spin-dependent observ-
+ables for non-identical particles. A detailed description
+of NN observables and their different conventions can be
+found in Refs. [35, 42–46]. In general, one can write the
+spin observables in terms of Saclay parameters, which are
+complex functions of the center-of-mass energy and an-
+gle θ. Here we only consider the differential cross section
+and analyzing power:
+dσ
+dΩ = 1
+2
+�
+|a|2 + |b|2 + |c|2 + |d|2 + |e|2 + |f|2�
+,
+(16)
+dσ
+dΩAy = Re(a∗ e + b∗ f),
+(17)
+where dσ/dΩ is the unpolarized differential cross sec-
+tion and Ay the analyzing power (also known as Pb).
+For identical particles, one has f = 0. More informa-
+tion on the description of the spin observables can be
+found in Refs. [44, 45]; see also Appendix D, which con-
+tains our emulation results for more spin observables.
+The Saclay parameters can be obtained from the spin-
+scattering M = M(θ, φ) matrix written in singlet-triplet
+space,
+M =
+�
+�
+�
+�
+M11
+M10e−iφ M1−1e−2iφ MST e−iφ
+M01eiφ
+M00
+M0−1e−iφ
+0
+M−11e2iφ M−10eiφ
+M−1−1
+MST eiφ
+MST eiφ
+0
+−MST e−iφ
+MSS
+�
+�
+�
+� ,
+(18)
+where the subscripts SS and ST represent the singlet-
+singlet and singlet-triplet channel,
+respectively [43].
+Equation (18) can be calculated using spherical harmon-
+ics and Clebsch-Gordan coefficients, and can be related
+to the Saclay parameters from the expressions:
+a = 1
+2(M11 + M00 − M1−1),
+(19)
+b = 1
+2(M11 + Mss − M1−1),
+(20)
+c = 1
+2(M11 − Mss − M1−1),
+(21)
+d = −
+1
+√
+2 sin θ(M01 + M01),
+(22)
+e = i
+2(M10 − M01),
+(23)
+f = −i
+√
+2MST .
+(24)
+The emulation process is performed similarly to the
+one for the total cross section, where multiple trained em-
+ulators are combined across different partial-wave chan-
+nels. Figures 4 and 5 show the simulator and emulator
+prediction for the differential cross section and analyzing
+power at three different energies calculated using the fit
+values for the LECs determined in Ref. [37]. The relative
+mean errors shown are obtained by randomly sampling
+500 different combinations of np LECs (the same LECs
+used for the sampled relative error calculation in Fig. 3)
+and comparing them against their respective simulator
+
+Standard
+Glockle
+NVP
+Error
+10-
+Mean Rel.
+10-7
+mb
+11
+10
+Otot
+-15
+10-
+0
+100
+200
+300
+Eiab[MeV]
+10
+01010101010101010
+Simulator
+Emulator
+000
+101
+0
+100
+200
+300
+Eiab[MeV]8
+FIG. 4. Simulated (solid lines) and emulated (dots) unpolar-
+ized differential cross section for the N4LO+ SMS potential
+with Λ = 450 MeV as a function of the center-of-mass angle
+at the three energies 60, 160, and 320 MeV (top panel). The
+bottom panel shows the mean relative errors between the em-
+ulators and their respective simulators for 500 different sets
+of np LECs obtained from Latin-hypercube sampling. The
+colors for the relative mean errors correspond to the energies
+in the top panel. The gray arrows point from the label asso-
+ciated with the emulator to its error. See the main text for
+details.
+calculation. On average, the spin observables emulator
+has a relative mean error on the order of ≈ 10−5 when
+employing the Gl¨ockle method and ≈ 10−14–10−11 when
+using the Standard method and NVP emulators, which
+are similar to the total cross section results. The results
+are similar to those obtained over the entire energy grid
+and for other observables (see Appendix. D).
+Table II details the angle-averaged relative errors be-
+tween the simulator and KVP emulators (based-10 log-
+arithm) for different spin observables with varying ba-
+sis size at a variety of energies. As can be seen, when
+training the emulator with basis size nb = na both the
+Standard and Gl¨ockle method emulators have large rela-
+tive errors of roughly 10−1 when compared to the high-
+fidelity model calculation. However, if we increase the
+basis size by doubling the parameters used per partial-
+wave, nb = 2na, the relative mean errors are signifi-
+cantly smaller, roughly 10−12–10−9 and 10−6–10−3, re-
+spectively.
+According to Ref. [47], the relative errors
+given by nb = 2na are below experimental uncertainties.
+When increasing the basis size to nb = 4na, the mean er-
+rors have mostly saturated and the improvement in accu-
+racy is insignificant compared to the basis size nb = 2na.
+Note that although only four energies are shown, these
+results are similar over the entire energy grid.
+FIG. 5. As in Fig. 4, but for the analyzing power Ay (also
+known as Pb). See the main text for details.
+The speed-up between the emulators and the simu-
+lator is highly implementation dependent (e.g., to-be-
+considered factors are the desired accuracy, idiosyncrasies
+of the solver, programming language, level of paralleliza-
+tion, hardware, etc.).
+The emulator speed-up will de-
+pend on the size of the quadrature mesh used by the
+simulator to obtain the high-fidelity solution. For repro-
+ducing the total cross section using the NVP emulator,
+Ref. [17] states an emulator speed-up factor of > 300x
+faster than the simulator in CPU time. When doubling
+the quadrature mesh size this factor becomes > 1000x.
+When comparing the KVP and NVP emulator speeds
+using one boundary condition (no anomaly checking) for
+the 1S0 uncoupled partial wave, the KVP emulator is
+slightly slower due to the Lagrange multiplier in Eq. (6)
+and numerical operations needed to solve Eq.(4). Mitiga-
+tion of Kohn anomalies (by emulating multiple boundary
+conditions) will further contribute to slowing down the
+KVP emulator, or any other emulator.
+IV.
+SUMMARY AND OUTLOOK
+We showed that the coordinate space KVP emulator
+for NN scattering [15, 16] can be extended to momen-
+tum space and coupled channels, and demonstrated its
+efficiency in accurately reproducing phase shifts and np
+observables using a modern chiral interaction at N4LO+.
+In addition, we provided two methods to implement the
+emulator, with the Gl¨ockle spline interpolation method
+having a faster offline stage, but less accurate online stage
+than the Standard method. By emulating (independent)
+scattering solutions associated with different asymptotic
+
+60 MeV
+160 MeV
+320 MeV
+15.0
+do/d2[mb/sr]
+10.0
+5.0
+0.0
+Glockle/NVP/Standard
+Error
+Mean Rel.
+10
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+0.5
+0.2
+9
+0.0
+-0.2
+Glockle/NVP/Standard
+Error
+10
+Tean Rel.
+10
+10
+0
+50
+100
+150
+Ocm [deg]9
+TABLE II. Comparison of the angle-averaged relative errors (base-10 logarithm) between high-fidelity model and emulator
+for various angular observables with different basis size for 500 sets of np LECs using the N4LO+ SMS potential [37] with
+momentum cutoff Λ = 450 MeV (rounded to two significant figures). These results are similar over the entire energy mesh.
+Here, “Std.” refers to the Standard method emulator. See the main text for details.
+dσ/dΩ
+D
+Ay
+Ayy
+A
+Basis size
+E [MeV]
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+Std.
+Gl¨ockle
+nb = na
+5
+−1.2
+−1.2
+−0.93
+−0.93
+−0.46
+−0.46
+−0.72
+−0.72
+−0.78
+−0.78
+100
+−0.73
+−0.73
+−0.47
+−0.47
+−0.12
+−0.12
+−0.20
+−0.20
+−0.28
+−0.28
+200
+−0.54
+−0.64
+−0.30
+−0.30
+−0.028
+−0.028
+−0.035
+−0.035
+−0.12
+−0.12
+300
+−0.49
+−0.49
+−0.24
+−0.24
+−0.066
+−0.066
+−0.037
+−0.037
+−0.043
+−0.043
+nb = 2na
+5
+−10
+���7.0
+−8.8
+−6.1
+−8.8
+−5.6
+−8.5
+−5.8
+−8.3
+−5.9
+100
+−12
+−6.3
+−11
+−5.1
+−10
+−4.9
+−10
+−4.9
+−11
+−5.3
+200
+−10
+−4.0
+−8.8
+−3.2
+−7.8
+−2.7
+−8.4
+−2.9
+−8.0
+−3.0
+300
+−12
+−4.9
+−11
+−4.0
+−11
+−3.9
+−9.9
+−3.8
+−11
+−3.9
+nb = 4na
+5
+−10
+−7.3
+−8.8
+−6.4
+−8.8
+−6.1
+−8.5
+−6.4
+−8.3
+−6.1
+100
+−13
+−6.5
+−12
+−5.3
+−11
+−5.1
+−11
+−5.0
+−11
+−5.4
+200
+−10
+−4.4
+−9.3
+−3.6
+−8.5
+−3.0
+−8.8
+−3.3
+−8.8
+−3.3
+300
+−12
+−5.1
+−11
+−4.0
+−10
+−4.1
+−10
+−3.8
+−11
+−4.0
+boundary conditions in each partial wave and weighting
+the results (e.g., for the S-matrix), spurious singularities
+known as Kohn anomalies were successfully mitigated for
+the KVP-based emulators [16].
+We also constructed an NVP-based emulator and as-
+sessed how well the three emulators reproduced their re-
+spective high-fidelity solution for the 1S0 and 3S1–3D1
+partial-waves, total and differential cross sections, and
+analyzing powers. While all emulators produced errors
+well below experimental errors [47], the KVP Standard
+method and NVP emulators most closely reproduced the
+simulator, while the KVP Gl¨ockle spline interpolation
+emulator was overall the least accurate. The KVP emula-
+tor was found to have a slower online stage than the NVP
+emulator because it has to evaluate a higher-dimensional
+matrix and perform overall more numerical operations.
+We stress, however, that the emulators’ speed-ups are
+highly implementation dependent and should be further
+investigated. Extensions of the NVP-based emulator for
+anomaly mitigation with minimal computational cost,
+similar to the KVP-based emulators, should also be in-
+vestigated [17].
+An alternative procedure for mitigat-
+ing anomalies would be constructing the estimated S
+matrix using solutions from emulator based on differ-
+ent variational principles, as opposed to emulating mul-
+tiple boundary conditions. Reference [21] provides fur-
+ther perspectives regarding different emulators (KVP-
+and NVP-based included) and efficient offline-online de-
+compositions.
+Although we considered here only χEFT NN potentials
+for np scattering, the constructed emulators are gener-
+ally applicable to two-body scattering, including pp scat-
+tering and nuclear reactions with complex-valued opti-
+cal potentials.
+To help implement these fast & accu-
+rate scattering emulators in Bayesian parameter estima-
+tions, we provide self-contained set of codes that gener-
+ate all results and figures shown in this paper [31]. Fur-
+thermore, we have written a pedagogical introduction to
+projection-based emulators [21] with interactive, open-
+source Python code [22] to facilitate implementations of
+fast & accurate emulators even further. However, taking
+full advantage of emulators for UQ in nuclear scattering
+and reaction calculations will require generalizations to
+higher-body scattering and non-affine potentials. Recent
+advances in this direction are already promising [27].
+ACKNOWLEDGMENTS
+We thank Evgeny Epelbaum for sharing a code that
+generates the SMS chiral potentials, Kyle Wendt for shar-
+ing a code that generates the spin obbservables, and
+Filomena Nunes for fruitful discussions. This work was
+supported in part by the National Science Foundation
+Award Nos.
+PHY-1913069 and PHY-2209442 and the
+NSF CSSI program under award number OAC-2004601
+(BAND Collaboration [48]), and the NUCLEI SciDAC
+Collaboration under U.S. Department of Energy MSU
+subcontract RC107839-OSU.
+This material is based
+upon work supported by the U.S. Department of Energy,
+Office of Science, Office of Nuclear Physics, under the
+FRIB Theory Alliance award DE-SC0013617.
+Appendix A: Mitigating Kohn anomalies
+We follow the method developed in Ref. [16] to detect
+and mitigate Kohn anomalies (see also Ref. [32]). The
+
+10
+estimated S matrix is calculated from the emulator so-
+lutions by using a weighted sum of averages. Letting L1
+and L2 be two independent KVP functional solutions,
+this weighted sum is computed by first calculating the
+relative residuals
+γrel(L1, L2) = max
+������
+S(L1)
+S(L2) − 1
+�����,
+�����
+S(L2)
+S(L1) − 1
+�����
+�
+,
+(A1)
+for all emulated KVP solutions without repetitions to
+avoid the trivial case where L1 = L2. Using a consistency
+check, γrel < ϵrel, with ϵrel = 10−1, we select the set
+of pairs P = {(L1, L2)} that satisfies this check. If at
+least one consistency check passes, the S matrix is now
+estimated by the weighted sum of averages
+[S](mixed)
+KVP
+=
+�
+(L1,L2)∈P
+ω(L1, L2)S(L1) + S(L2)
+2
+,
+(A2)
+ω(L1, L2) =
+γrel(L1, L2)−1
+�
+(L′
+1,L′
+2)∈P γrel(L′
+1, L′
+2)−1 .
+(A3)
+If no consistency check passes, one could change the ba-
+sis size to shift the position of the Kohn anomalies in the
+parameter space. However, we found that using Eq. (A2)
+was sufficient to mitigate Kohn anomalies in our appli-
+cations.
+We first calculate Eq. (5) using Eq. (7), then rescale
+Eq. (5) using the relations from Appendix B of Ref. [16],
+∆�U (u′) = C
+′−1(Li) C
+′−1(Lj) det u
+det u′ ∆�U (u),
+(A4)
+C′(L) = det u
+det u′
+u′
+11 − u′
+10K(L)
+u11 − u10K(L).
+(A5)
+Here, u and u′ are nonsingular matrices parameterizing
+the scattering boundary conditions; the K, K−1, and T
+scattering matrices, respectively, are given by
+uK =
+�
+1 0
+0 1
+�
+,
+uK−1 =
+�
+0 1
+1 0
+�
+,
+uT =
+�
+1 0
+i 1
+�
+.
+(A6)
+The u matrix parameterizes the initial boundary condi-
+tion associated with L, while the u′ parameterizes the
+final boundary condition associated with L′.
+The snapshots used in the emulator’s offline stage are
+transformed using the M¨obius transform [16]
+L′(L) = −u′
+01 + u′
+00K(L)
+u′
+11 − u′
+10K(L) .
+(A7)
+Once we obtain an emulator solution, we transform that
+solution back into its K matrix form using
+K(L) = u01 + u11L
+u00 + u10L.
+(A8)
+For the estimated S calculation, the KVP solution
+pairs (L1, L2) being evaluated are the K matrix solu-
+tions obtained from the different boundary conditions
+used [e.g., γrel(K(K), K(K−1)), γrel(K(K), K(T)), and
+γrel(K(K−1), K(T))]. See Ref. [16] for more details.
+Appendix B: Formalism details
+Here we provide clarifying remarks about how Eq. (4)
+arises in the coupled case.
+In particular, we focus on
+two questions about the specific manner in which the
+coefficients ⃗β enter into Eq. (4).
+Why can Lss′ be emulated separately for each ss′ pair
+rather than with one global set of coefficients for the cou-
+pled block?
+For uncoupled channels, each partial wave is inde-
+pendent of one another, thus they can be emulated
+individually using trial wave functions and coefficients
+that are specific to the channel under consideration.
+Without loss of generality, let us consider two uncoupled
+channels labeled as s = 0 and s = 1, and let ⃗β(0)
+and ⃗β(1) denote the independent sets of coefficients
+found by making each channel’s KVP stationary.
+To
+move toward the coupled regime, imagine adiabatically
+turning on the coupling between these two originally
+uncoupled channels.
+The coefficients for each channel
+should remain nearly fixed to their previously uncoupled
+values, but the coupling will introduce a new set of
+coefficients ⃗β(01) ̸= ⃗β(0) ̸= ⃗β(1) that must be determined.
+Hence, each independent channel in the coupled case
+will have its own set of coefficients. Attempting to force
+a global set of coefficients for a coupled system would be
+inconsistent with the treatment in the uncoupled case
+and also degrade accuracy in general. A more technical
+answer follows from the (Petrov-)Galerkin procedure
+described below.
+Should not each of |ψs′⟩ and ⟨ψs| have its own basis
+expansion with their own independent coefficients?
+No, there is only one set of coefficients that en-
+ter quadratically in Eq. (4).
+A way of understanding
+how the coefficients enter in Eq. (4) follows from the
+(Petrov-)Galerkin orthogonalization procedure (see also
+Ref. [21]). Rather than starting with a variational prin-
+ciple, the (Petrov-)Galerkin approach starts with the
+Schr¨odinger equation. Like the variational approach, it
+expands |ψs′⟩ as a linear combination of known functions,
+but determines the basis coefficients by enforcing orthog-
+onality against a set of test functions. For the diagonal
+channels, the test functions are chosen to have the same
+exit channel as the trial functions (standard Galerkin ap-
+proach). On the other hand, the test functions for the
+off-diagonal channels are chosen to have a different exit
+channel (s) than the trial functions (s′) (Petrov-Galerkin
+approach). The resulting set of linear equations is equiv-
+alent to those that follow from making the KVP station-
+ary for each combination of (s′, s) independently. Thus
+by following the (Petrov-)Galerkin procedure we can de-
+termine how the coefficients are to enter in Eq. (4).
+This discussion will follow closely that of Ref. [21],
+however using coupled-channel notation and more gen-
+eral boundary conditions consistent with the general
+
+11
+KVP. Starting from (the strong form of) the Schr¨odinger
+equation
+�H(θ) |ψs′⟩ = E |ψs′⟩ ,
+(B1)
+we can derive its weak form after multiplying by a test
+function ⟨ψs|
+⟨ψs| �H(θ) − E|ψs′⟩ = 0.
+(B2)
+This can be considered a Petrov-Galerkin approach be-
+cause s ̸= s′ in general. The boundary conditions can be
+made explicit via the relationship
+0 = ⟨ψs| �H(θ) − E|ψs′⟩
+= ⟨ψs| �H†(θ) − E|ψs′⟩ −
+�
+t
+W(rψts, rψts′; r)
+2µ
+�����
+∞
+r=0
+,
+(B3)
+where �H† denotes the operator acting to the left (via
+integration by parts) and where we have used ψts(r) =
+⟨rt|ψs⟩ = ⟨ψs|rt⟩ and defined the Wronskian
+W(φ, ψ; r) ≡ φ(r)ψ′(r) − φ′(r)ψ(r).
+(B4)
+The wave function rψ vanishes at the origin, so that only
+the limit as r → �� contributes. By adding Eqs. (B3)
+and (B2), we have
+⟨ψs| �H(θ) − E|ψs′⟩ + ⟨ψs| �H†(θ) − E|ψs′⟩
+=
+�
+t
+W(rψts, rψts′; r)
+2µ
+�����
+∞
+r=0
+.
+(B5)
+This is the weak form for general |ψs′⟩ and ⟨ψs|. We can
+arrive at the discrete form by inserting basis states |ψs
+i ⟩
+that satisfy the asymptotic boundary conditions
+ψst(r) −−−→
+r→∞ δst ¯φ(0)
+s (r) + Lst ¯φ(1)
+s (r) ,
+(B6)
+where
+�
+¯φ(0)
+ℓ (r)
+¯φ(1)
+ℓ (r)
+�
+∝
+�
+u00 u01
+u10 u11
+� �
+jℓ(qr)
+ηℓ(qr)
+�
+.
+(B7)
+With this substitution, we have, for i ∈ [1, nb],
+∆�U ss′
+ij βj = Lss′
+i
+�
+j
+βj − Ls′s
+j βj,
+(B8)
+where the expression for ∆�U ss′
+ij
+is given by Eq. (5). We
+must now implement the constraint �
+j βj = 1, which is
+performed here by a Lagrange multiplier λ mimicking a
+variational approach (see Ref. [19] for details):
+λ + ∆�U ss′
+ij βj = Lss′
+i
+�
+j
+βj − Ls′s
+j βj.
+(B9)
+The sum multiplying Lss′
+i
+can be evaluated using the
+constraint �
+j βj = 1, and we can make the redefinition
+λ′ ≡ λ + �
+j βjLs′s
+j
+without impacting the solution be-
+cause this term does not depend on i. Thus, we have
+λ′ − ⃗L(E) + ∆�U ⃗β⋆ = 0,
+(B10)
+which is exactly Eq. (6) found by making the KVP sta-
+tionary. This simplification can be understood by not-
+ing that if {⃗β⋆, λ⋆} satisfy Eq. (B9), then we know that
+{⃗β⋆, λ′
+⋆} is the unique solution to Eq. (B10). Therefore,
+we can solve Eq. (B10) to obtain ⃗β⋆ rather than Eq. (B9).
+In conclusion, using the Petrov-Galerkin projection of the
+homogeneous Schr¨odinger equation with trial and test
+bases of |ψs′
+i ⟩ and ⟨ψs
+i |, respectively, we were able to ob-
+tain the same coefficients as the KVP in Eq. (6), which
+yield the same on-shell Lss′ matrix when used in Eq. (4).
+Appendix C: KVP emulator construction details
+For single channel scattering over a k × p momentum
+grid using the K matrix (det u = 1), Eq. (8) becomes
+∆�Uij(θ) =
+∞
+¨
+0
+dk dp k2p2�
+ψi(k)Vθ,j(k, p)ψj(p) + (i ↔ j)
+�
+,
+(C1)
+with Vθ,j(k, p) defined as in Eq (9). We drop the super-
+scripts for the uncoupled case since s′ = s. Note that ψi
+is not complex conjugated. For the Gl¨ockle method, one
+would simply substitute Eq. (7) into Eq. (C1) and inter-
+polate the solutions to the integrals with the cubic spline
+polynomials Sk(k0). For the Standard method, the Dirac
+delta distribution is analytically integrated; thus we ob-
+tain the following expression for ∆�Uij
+∆�Uij(θ) = Vθ,j(k0, k0) + 2
+π (I1
+ij + I2
+ij) + 4
+π2 I3
+ij + (i ↔ j),
+(C2)
+with I1
+ij, I2
+ij, and I3
+ij defined as
+I1
+ij = P
+∞
+ˆ
+0
+dk k2
+k0
+Ki(k0, k)
+k2 − k2
+0
+Vθ,j(k, k0),
+(C3)
+I2
+ij = P
+∞
+ˆ
+0
+dp p2
+k0
+Vθ,j(k0, p)Kj(p, k0)
+p2 − k2
+0
+,
+(C4)
+I3
+ij = P
+∞
+¨
+0
+dk dp k2p2
+k2
+0
+Ki(k0, k)
+k2 − k2
+0
+Vθ,j(k, p)Kj(p, k0)
+p2 − k2
+0
+.
+(C5)
+
+12
+If V has an affine dependence on the parameters θ,
+applying Eq. (11) and Eq. (12) produces
+∆�U 0
+ij =
+∞
+¨
+0
+dk dp k2p2�
+ψi(k)V 0
+j (k, p)ψj(p) + (i ↔ j)
+�
+,
+(C6)
+∆ �U 1
+ij =
+∞
+¨
+0
+dk dp k2p2�
+ψi(k)V 1(k, p)ψj(p) + (i ↔ j)
+�
+,
+(C7)
+with
+V 0
+j (k, p) ≡ 2µk0
+�
+V 0(k, p) − Vj(k, p)
+�
+.
+(C8)
+For coupled-channel interactions (s′ ̸= s), the details
+of the emulation are more complex. In this case, we apply
+Eq. (4) to each individual channel in a partial-wave, but
+the real difference lies in how Eq. (5) is calculated. The
+usual way of solving for the phase shifts and mixing angle
+for the coupled channels involves building a 2 × 2 block
+matrix for the potential,
+V =
+�
+V 00 V 01
+V 10 V 11
+�
+.
+(C9)
+The same process can be applied to the emulator calcu-
+lation when calculating Eq. (5),
+∆�U =
+�
+∆�U 00 ∆�U 01
+∆�U 10 ∆�U 11
+�
+.
+(C10)
+Each of the four blocks in ∆�U has a separate functional,
+although there are contributions from the different wave
+functions and potentials (e.g., for the 3S1–3D1 partial
+wave ∆�U 00 depends on the 3S1–3S1, 3S1–3D1, and 3D1–
+3D1 wave functions and potentials).
+Additionally, Eq. (7) tells us that we can consider the
+momentum-space wave function for the individual chan-
+nels ψst. Using Eq. (8) with Eq. (9), the functionals for
+the individual channels in a coupled-channel calculation
+(using the 3S1–3D1 as an example) will be
+∆�U ss′
+ij
+=
+¨ ∞
+0
+dk dp k2p2�
+∆uss′
+ij + (i ↔ j)
+�
+,
+(C11)
+with
+∆u00
+ij = ψ00
+i (V 00
+θ,jψ00
+j + V 01
+θ,jψ10
+j )
++ ψ10
+i (V 10
+θ,jψ00
+j + V 11
+θ,jψ10
+j ),
+(C12)
+∆u01
+ij = ψ00
+i (V 00
+θ,jψ01
+j + V 01
+θ,jψ11
+j )
++ ψ10
+i (V 10
+θ,jψ01
+j + V 11
+θ,jψ11
+j ),
+(C13)
+∆u10
+ij = ψ01
+i (V 00
+θ,jψ00
+j + V 01
+θ,jψ10
+j )
++ ψ11
+i (V 10
+θ,jψ00
+j + V 11
+θ,jψ10
+j ),
+(C14)
+∆u11
+ij = ψ01
+i (V 00
+θ,jψ01
+j + V 01
+θ,jψ11
+j )
++ ψ11
+i (V 10
+θ,jψ01
+j + V 11
+θ,jψ11
+j ),
+(C15)
+where we have suppressed the arguments for compact-
+ness.
+Note that the weights βi in Eq. (4) are differ-
+ent for each channel (i.e., ∆�U 00, ∆�U 11, and ∆�U 01 =
+∆�U 10), and are determined independently of one an-
+other. Once Eqs. (C12) through (C15) are calculated,
+the steps for the uncoupled channel calculation are ap-
+plied to each ∆�U ss′
+ij
+to obtain the emulator prediction,
+in particular Eqs. (C2) through (C5), and the separation
+of ∆�U ss′(θ) into parameter-dependent and parameter-
+independent pieces as described by Eq. (12).
+Appendix D: Additional results
+FIG. 6.
+As in Fig. 3, but only emulating with the K ma-
+trix.
+The mesh-induced spikes have been removed for this
+calculation.
+Figure 6 shows the relative mean error for the total
+cross section using only the K matrix boundary condi-
+tion. Comparing to Fig. 3, where we apply the weighted
+sum (mixed) S approach, we see that for one bound-
+ary condition the relative mean error has Kohn anoma-
+lies (see Elab ≈ 270 MeV and ≈ 315 MeV for the stan-
+dard method and Elab ≈ 40 MeV and ≈ 130 MeV for
+the Gl¨ockle method) and a more spread-out error. From
+Fig. 8 and comparing to Fig. 3 and 6, we conclude that
+the mixed S approach is indeed successful in mitigating
+the Kohn anomalies.
+Figure 7 shows the relative mean error for the to-
+tal cross section with momentum cutoff 550 MeV. The
+weighted sum (mixed) S approach is used for the KVP
+emulator results. Here, the anomalies found in the NVP
+emulation are noticeable.
+Figure 8 shows the relative errors for the KVP emula-
+tors in the 1S0 channel. The figure on the left shows the
+
+Standard
+Glockle
+NVP
+Error
+Mean Rel.
+10-7
+mb
+11
+10
+Otot
+0
+100
+200
+300
+Eiab[MeV]
+10
+101001010101010101
+Simulator
+Emulator
+000
+101
+100
+0
+200
+300
+Eiab [MeV]13
+FIG. 7.
+As in Fig. 3, but for cutoff Λ = 550 MeV.
+relative error when emulating with the K−1 boundary
+condition and the one on the right shows the weighted
+sum (mixed) S errors. In the figure on the left we can
+see a spike around Elab ≈ 65 MeV, which disappears
+when using the weighted sum S approach.
+This is a
+clear example of the weighted sum S approach helping to
+mitigate these anomalies. Additionally, there are other
+smaller mesh-induced spikes (i.e., not anomalies) present
+throughout the energy grid in the figure on the left that
+are not in the figure on the right. These were mitigated
+by not allowing the k0 values to be close to any mo-
+mentum mesh points. See Sec. III for a more detailed
+description.
+Figures 9 through 12 show emulator results for the
+following spin observables:
+dσ
+dΩD = 1
+2
+�
+|a|2 + |b|2 − |c|2 − |d|2 + |e|2 + |f|2�
+, (D1)
+dσ
+dΩA = − Re(a∗ b − e∗ f) sin(α + θ
+2)
++ Re(c∗ d) sin(α − θ
+2)
+− Im(b∗ e + a∗ f) cos(α + θ
+2),
+(D2)
+dσ
+dΩAxx = Re(a∗ d) cos(θ) + Re(b∗ c) − Im(d∗ e) sin(θ),
+(D3)
+dσ
+dΩAyy = 1
+2
+�
+|a|2 + |b|2 − |c|2 − |d|2 + |e|2 + |f|2�
+, (D4)
+where D is the depolarization parameter, A is the spin-
+flip amplitude, Axx and Ayy are the spin-correlation am-
+plitudes, and α a relativistic spin rotating angle that van-
+ishes in the non-relativistic case [8]. For identical parti-
+cles, f = 0. The results and conclusions are similar to
+those described in Sec. III C.
+Figure 13 shows emulator results for the total cross
+section for the N4LO+ SMS potential with momentum
+cutoff 550 MeV. The results and conclusions are similar
+to the ones described in the text for the 450 MeV mo-
+mentum cutoff (see Sec. III C).
+Figures 14 and 15 shows emulator results for the dif-
+ferential cross section and analyzing power Ay for the
+N4LO+ SMS potential with momentum cutoff 550 MeV.
+The results and conclusions are similar to the ones de-
+scribed in the text for the 450 MeV momentum cutoff
+(see Sec. III C). These results and conclusions also ex-
+tend down to momentum cutoff 400 MeV. The spin ob-
+servables at 500 MeV show larger errors on order of 10−7
+for the NVP emulator at particular energies, which may
+come from Kohn anomalies at one or more of the sam-
+pled parameter sets (see Fig. 7); nevertheless, the errors
+are still well below experimental uncertainties [47].
+
+Standard
+Glockle
+NVP
+010101010101
+Error
+Mean Rel.
+10-
+mb
+1
+10
+Otot
+0
+100
+200
+300
+Eiab[MeV]
+10
+Simulator
+Emulator
+000
+101
+0
+100
+200
+300
+Eiab [MeV]14
+FIG. 8. Relative error of the 1S0 channel for a basis size of nb = 2na + 1 for the N4LO+ SMS potential with Λ = 450 MeV
+as a function of the laboratory energy.
+The left panel shows the relative error for an emulator using the K−1 boundary
+condition. There is a Kohn anomaly at Elab ≈ 65 MeV for both the Standard and Gl¨ockle emulators and mesh-induced spikes
+present throughout the energy grid. The right panel shows the relative error for the mixed S-matrix approach presented by
+Reference [16] with care taken to avoid the k0 values that correspond with a mesh point as described in Sec. III B. When
+comparing both graphs, the Kohn anomaly is no longer present and the mesh-induced spikes are much smaller in the right
+panel.
+
+Glockle
+Standard
+Mixed S
+emu.
+R
+Error
+10°
+Rel
+10-9
+10-12
+LA
+10-15
+0
+50
+200 0
+50
+100
+200
+100
+150
+150
+Eiab [MeV]
+Eiab [MeV]15
+FIG. 9. As in Fig. 4, but for the depolarization D.
+FIG. 10. As in Fig. 4, but for the spin-flip amplitude A.
+FIG. 11. As in Fig. 4, but for the spin-correlation amplitude
+Axx.
+FIG. 12. As in Fig. 4, but for the spin-correlation amplitude
+Ayy.
+
+60 MeV
+160 MeV
+320 MeV
+1.0
+0.5
+0.0
+-0.5
+Glockle/NVP/Standard
+Error
+Tean Rel.
+10
+0
+50
+100
+150
+cm [deg]60 MeV
+160 MeV
+320 MeV
+1.0
+0.5
+0.0
+-0.5
+Giockle/NVP/Standard
+Error
+Tean Rel.
+10
+10-
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+0.5
+0.0
+-0.5
+-1.0
+Glockle/NVP/Standard
+Error
+10
+Tean Rel.
+10
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+1.0
+0.5
+0.0
+-0.5
+-1.0
+Glockle/NVP/Standard
+Error
+10
+Rel.
+10
+Mean 
+11
+10°
+0
+50
+100
+150
+Ocm [deg]16
+FIG. 13.
+As in Fig. 3, but for cutoff Λ = 550 MeV.
+FIG. 14. As in Fig. 3, but for cutoff Λ = 550 MeV.
+FIG. 15. As in Fig. 3, but for cutoff Λ = 550 MeV.
+
+Standard
+Glockle
+NVP
+ Error
+Mean Rel.
+10-
+mb
+Otot
+0
+100
+200
+300
+Eiab [MeV]
+10
+0101001010101010
+Simulator
+Emulator
+000
+101
+100
+0
+200
+300
+Eiab[MeV]60 MeV
+160 MeV
+320 MeV
+15.0
+do/d2[mb/sr]
+10.0
+5.0
+0.0
+Glockle/NVP/Standard
+Error
+Mean Rel.
+10
+10-15
+0
+50
+100
+150
+Ocm [deg]60 MeV
+160 MeV
+320 MeV
+0.5
+0.2
+9
+0.0
+-0.2
+Glockle/NVP/Standard
+Error
+3
+10°
+Tean Rel.
+10
+10-
+10-15
+0
+50
+100
+150
+Ocm [deg]17
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@@ -0,0 +1,1626 @@
+Anisotropic Electron Heating in an Electron Cyclotron Resonance
+Thruster with Magnetic Nozzle
+J. Porto,1, 2 P.Q. Elias,1 and A. Ciardi2
+1)Physics - Instrumentation and Space Department, ONERA/DPHY, Université Paris Saclay
+F-91123 Palaiseau – France.
+2)Sorbonne Université, Observatoire de Paris, PSL Research University, LERMA, CNRS UMR 8112
+75005 Paris – France.
+(*Electronic mail: [email protected])
+(*Electronic mail: [email protected])
+(Dated: 30 January 2023)
+In a grid-less Electron Cyclotron Resonance (ECR) plasma thruster with a diverging magnetic nozzle, the magnitude
+of the ambipolar field accelerating the positive ions depends of the perpendicular energy gained by the electrons. This
+work investigates the heating of the electrons by electromagnetic waves, taking their bouncing motion into account in
+a confining well formed by the magnetic mirror force and the electrostatic potential of the thruster. An electromagnetic
+Particle-In-Cell (PIC) code is used to simulate the plasma in a magnetic field tube. The code’s Maxwell solver is based
+on a semi-Lagrangian scheme known as the Constrained Interpolation Profile (CIP) which enables larger time steps.
+The results show that anisotropic plasma heating takes place exclusively inside the coaxial chamber, along a Doppler-
+broadened zone. It is also shown that a trapped population of electrons with a larger perpendicular energy exists in the
+plume.
+I.
+INTRODUCTION
+Electric thrusters play a fundamental role in the field of
+space propulsion. Their main advantage lies in an efficient
+use of the propellant mass, and therefore a reduced consump-
+tion of propellant. Hall Effect Thrusters or Gridded Ion En-
+gines are examples of the most well-known and flight-proven
+technologies in the current propulsion market nowadays. Both
+technologies eject an ion beam which is subsequently neutral-
+ized to prevent the spacecraft from charging. Several compo-
+nents of these technologies, such as the acceleration grid or
+the neutralizer, are subject to erosion and wear and for this
+reason, meeting the challenging lifetime targets requires care-
+ful optimization and demanding testing1. The complexity of
+some of the components has driven the investigation of al-
+ternative concepts of propulsion devices that require neither
+a grid nor a neutralizer. The Electron Cyclotron Resonance
+(ECR) plasma thruster2,3 is one of these concepts and it is the
+subject of the present study.
+The ECR plasma thruster consists of a semi-open chamber
+where a quasi-neutral plasma is heated by electron cyclotron
+resonant microwaves at 2.45GHz, and accelerated by a mag-
+netic nozzle. This concept was first proposed in the 1960s in
+the works of Miller et al. 4 and Nagatomo 5, then further de-
+veloped by Sercel 6. These studies used a prototype with a
+wave-guide structure to couple the microwaves to the plasma.
+Their results showed that it was possible to achieve specific
+impulses and thrust values high enough to be of interest for
+space propulsion applications6. Nonetheless, the inefficiency,
+size and weight of the micro-wave sources and electromag-
+nets at that time led to a stagnation of the research on ECR
+thrusters for several years. Interest for this technology arose
+again recently with experimental works7,8. In particular, the
+use of coaxial microwave coupling structures and compact
+rare-earth permanent magnets were instrumental in designing
+compact sources (a schematic of the design is shown in Fig.
+1a).
+More experimental and theoretical efforts has since been
+made in order to get a deeper understanding of the phys-
+ical phenomena governing the plasma heating and acceler-
+ation in the thruster. Experimental characterizations of the
+plasma properties have been carried out using different mea-
+surement techniques such as Langmuir and Faraday probes,
+Laser Induced Fluorescence diagnostics, diamagnetic loops
+and thrust balances2,9–12. Unfortunately, most of the exper-
+imental studies so far have been limited to survey the plasma
+outside the thruster coaxial chamber.
+Recently, a resonant
+probe was developed to measure an electron density of about
+1×1011 cm−3 at the source exit plane, close to the coaxial
+chamber13. In the source, it is likely that the plasma density is
+higher (∼ 1×1012 cm−3) with electron temperatures of a few
+tens of eV.
+From a theoretical point of view, as a first step, global
+models describing the energy balance in the plasma source
+were proposed as a means to obtain the key parameters of
+the thruster14,15.
+While this approach yielded good agree-
+ment with measured electron temperature at high mass flow
+rate or high pressure, they failed at the lower mass-flow rate
+where the thruster achieves its best performance. Indeed, the
+assumptions of uniform electron temperature and isotropic
+Maxwellian electron distribution are too crude approxima-
+tions when collisionality decreases and the electron mean free
+path becomes much larger than the source length: in that range
+non-local effects become prevalent, as electrons undergo a
+bouncing motion along the magnetic field line. Those elec-
+trons which cross the ECR surface can gain energy depending
+on their phase in the gyromotion16, which leads to a strong
+anisotropy of the distribution function. An attempt to account
+for this stochastic heating in the plasma was made by consid-
+ering the electron heating as a random walk in phase space17.
+arXiv:2301.11411v1  [physics.plasm-ph]  26 Jan 2023
+
+2
+While this model provided a qualitative agreement with the
+measured ion energies, it could not account for the plasma
+feedback on the waves (assumed constant) and the collisions
+along the bouncing motion. Recently Sánchez-Villar et al. 18
+performed 2D axisymmetric simulations of the thruster with
+a hybrid model consisting of particle-in-cell (PIC) ions and a
+fluid model for the mass-less electrons. One of the main find-
+ings of this study was the identification of different regions in
+the source where the waves are either propagating or evanes-
+cent, with most of the power absorption taking place close to
+the inner conductor, near the ECR surface. By acting as a
+sink for the plasma, the inner conductor induces a decrease
+of the plasma density in its vicinity, enabling the propaga-
+tion of electromagnetic waves downstream of the ECR sur-
+face. These features lead to the formation of a hot electron
+beam close to the inner conductor, with a colder plasma in the
+bulk of the source. While these 2D results provided important
+insights on the operation of the thruster, some assumptions
+of the fluid model limit the validity of the results obtained
+from these simulations. The most important one being the
+assumption of isotropic electron temperature which excludes
+anisotropic heating in the directions parallel and perpendicu-
+lar to the magnetostatic field.
+This latter point is still an open question for this technology.
+Indeed, ECR heating is expected to lead to anisotropic heating
+of the electron translation modes. This difference affects the
+power losses near the source walls and the potential drop in
+the magnetic nozzle19. However, most of the electron temper-
+ature measurements performed in the thruster plume did not
+differentiate perpendicular and parallel electron temperature
+(with respect to the local magnetic field direction). A way to
+measure the electron temperature anisotropy is, for example,
+incoherent Thomson scattering20, but this type of measure-
+ment is not presently available in the ECR source. At any
+rate, this heating is intimately linked to the absorption of the
+electromagnetic waves in the coaxial source.
+Another issue is the non-local transport due to the bounc-
+ing magnetized electrons in the nozzle (the electron mean free
+path is greater than the source radius). In particular, the pro-
+duction and the heating of the electrons are not necessarily at
+the same location.
+While gaining a better understanding of these issues should
+firstly rely on experimental measurements, the challenges as-
+sociated with such an investigation are a strong incentive to
+use numerical models, even if simplified, to investigate the
+main physical processes at play in ECR thrusters.
+In par-
+ticular, such a model should be able to account for the self-
+consistent wave absorption and the anistropic heating, as well
+as the non-local effects and bouncing motion of the parti-
+cles. Electromagnetic kinetic models, such as Particle-In-Cell
+(PIC) or Vlasov methods, are natural candidates for this task.
+There are currently a few works using kinetic simulations of
+propulsion devices exploiting the ECR phenomenon, however
+the majority of these developments are concerned with grid-
+ded ion thrusters with ECR heating21–24, where the plasma
+acceleration is achieved by a grid-imposed electric field and
+not the plasma expansion in the magnetic nozzle, as in our
+design. Takao et al. 23 successfully modelled a gridded ion
+thruster where the ions are produced in an ECR source at 4.2
+GHz. The authors used a Particle-In-Cell (PIC) code consid-
+ering the microwave electric field as a temporal modulation
+of its initial amplitude obtained by simulating the microwave
+propagation without plasma. Therefore, in this approach the
+plasma feedback on the wave was considered negligible.
+The main purpose of the our study is to perform full-PIC
+electromagnetic simulations of the plasma in the thruster, tak-
+ing into account the plasma feedback on the wave propaga-
+tion, and to investigate the heating and confinement of the
+electrons.
+For this purpose it is necessary to simulate the
+microwave propagation and its interaction with the charged
+particles in the source and the nozzle region. However, due
+to the complexity and computational cost of simulating a full
+3D configuration (which should include the nozzle region),
+we restrict our investigation to the simplified case of an iso-
+lated magnetic flux tube. This approximation effectively re-
+stricts the phase space to 4 dimensions (1 dimension in space,
+3 dimensions in velocity space), and a 1D3V electromagnetic
+Particle-In-Cell can be used to model the ECR thruster.
+We show that the electron heating takes place over a
+broad region in the thruster source and leads to a signifi-
+cant anisotropy (ratio Te⊥/Te∥ ∈ [2.5,7.5]). The perpendicular
+electron temperature reaches a first maximum in the source
+and, surprisingly, has a second maximum in the downstream
+region. The explanation for these features lies in the confine-
+ment of electrons in the potential well formed by the com-
+bination of the diverging magnetic field and the electrostatic
+potential.
+II.
+NUMERICAL MODEL
+A.
+The Quasi-One-Dimensional Approach
+In the coaxial ECR thruster, an axially magnetized cylindri-
+cal permanent magnet creates a diverging static magnetic field
+BMS in the source and in the plume region12,14. This shape
+for the magnetostatic field was chosen to ensure a magnetic
+confinement at the close end of the coaxial chamber (called
+backplate in Fig. 1) while allowing the electrons to get accel-
+erated in the plume thanks to the divergence of the magnetic
+field lines.
+In fact, the ECR uses a diverging magnetic field whose
+magnitude decreases from approximately 100 mT at the back
+of the source to around 5 mT 10 cm downstream of the thruster
+exit plane. Under these conditions, assuming an electron tem-
+perature around Te ≃ 10eV, the Larmor radius of the elec-
+trons is between rL ≃ 0.07mm − 1.4mm. Thus electrons are
+strongly magnetized in the source and in the near-field plume
+region, while ions remain mostly unmagnetized. As a conse-
+quence, before the onset of plasma detachment, electrons and
+ions are bound to the magnetic field tube. Several mechanisms
+may account for the plasma detachment25 : collisions, stretch-
+ing of the magnetic field lines, electron demagnetization and
+plume instabilities. While it is out of the scope of this work to
+study the dominant mechanisms, several recent works have in-
+vestigated some of these effect in 2D PIC simulations26–28.In
+
+3
+(a)
+(b)
+FIG. 1: ECR thruster: (a) Schematic view of the coaxial
+source. The magnetic field lines are shown in red. The
+dashed surface corresponds to the flux tube (b) Schematic
+view flux tube used for the quasi-1D model. The exit plane of
+the coaxial source of length LS is represented by the dashed
+line. The end of the computational domain is reached at
+x = L. The axial magnetic profile and tube cross section
+along the axis are shown in red and blue, respectively.
+this work, we decided to rely on experimental evidence to de-
+fine the section of the nozzle where the plasma remains bound
+to the field lines. Recently, Little and Choueiri 29 have mapped
+the plasma potential in a magnetic nozzle to show that a good
+criterion for detachment is χp = rL/L∇B ≃ 0.1, where rL is the
+electron Larmor radius and L∇B = (∇B/B)−1 is the character-
+istic length scale of the magnetic field gradient. In the region
+of the nozzle where χp = rL/L∇B < 0.1, the plasma remains
+attached to the magnetic field. In our case, we considered a
+magnetic field with L∇B ≃ 5 − 10cm. Under this condition,
+we have χp < 0.1 up to L = 10cm downstream of the nozzle,
+and it is a reasonable assumption to consider that electrons do
+not detach from the magnetic field tube over this distance.
+As a consequence of this assumption, we decided to con-
+sider the creation and formation of the plasma enclosed in a
+magnetic field tube of length L = 10cm. More precisely, a
+portion of the thruster chamber and plume was represented
+by a quasi-1D model of a magnetic field tube with a varying
+cross-sectional area, as seen in Fig. 1b. There are several ex-
+amples of the use of quasi-1D models in the space propulsion
+field. Niewood and Martinez-Sanchez 30 used it to model a
+Magnetoplasmadynamic thruster, while De Giorgi and Fonta-
+narosa 31 studied a Vaporizing Liquid Microthruster with this
+approach. Recently, Saini and Ganesh 32 also used this ap-
+proach to model plasma expansion in a Radio-Frequency
+thruster. The moderate computational cost of a 1D3V model
+of the thruster facilitates the analysis of the plasma behavior in
+both the coaxial chamber and in the magnetic nozzle, and im-
+portantly, taking into account the nozzle is critical to resolve
+the bouncing motion of the electrons.
+The quasi-1D model assumes that the electrons and the ions
+are confined within a diverging magnetic flux tube, whose area
+is related to the axial magnetic field intensity through the con-
+servation of the magnetic flux:
+A(x)Bx(x) = A0B0
+(1)
+The model further assumes that the electromagnetic fields and
+all the plasma properties are constant across the section of the
+flux tube. For the ECR thruster under consideration12,14, the
+shape of the magnetic field lines close to the antenna is well
+approximated by an exponential function. For the sake of sim-
+plicity we approximated the magnetic field as:
+Bx(x) = B0 exp
+�
+− x
+LB
+�
+(2)
+In addition, we considered cylindrical symmetry for the static
+magnetic field around the field tube centerline.
+These assumptions mean that the particles guiding centers
+remain on the centerline. Since the plasma is assumed uni-
+form in the cross section, this approach does not allow the
+formation of a diamagnetic current and E ×B drifts.
+From now on, the term parallel and the subscript ∥ will
+refer to the direction parallel to the magnetostatic field lines.
+Similarly, perpendicular and the subscript ⊥ refer to the direc-
+tion perpendicular to the magnetostatic field lines. The source
+region, which corresponds to the coaxial cavity in Fig. 1b,
+was defined by 0 ≤ x ≤ LS, where LS is the coaxial source
+length. The plume region, which corresponds to the plasma
+expansion in vacuum, was defined by x ≥ LS.
+B.
+Particle-In-Cell Code overview
+The simulations were carried out with the Particle-In-Cell
+(PIC) code Rhei, which was developed to simulate low pres-
+sure cold plasmas and is adapted to parallel architectures. It
+
+4
+can be run with either a pure MPI or a hybrid MPI/OpenMp
+parallelization. The code integrates a Monte-Carlo Collision
+(MCC) module to simulate the collisions between the charged
+particles and a prescribed neutral background. At each it-
+eration, once the electrostatic and the electromagnetic fields
+were computed, the position of each macro particle labeled
+“p” was updated using dxp/dt = vp, and the velocity using
+Eq. 3. Each macro-particle represents W physical particles.
+The value of W used in the simulation is given in table I.
+ms
+dvp
+dt = qs
+�
+EESp +EEMp +vp ×
+�
+BMSp +BEMp
+��
+(3)
+In Eq. 3, qs is the charge of the particle, ms the mass, xp
+the position, and vp the velocity. Regarding the fields, they
+were computed at the location of the particle p using linear
+interpolation function, where EESp is electrostatic field from
+the charge distribution, BMSp is magnetostatic field from the
+permanent magnets and EEMp and BEMp are electromagnetic
+fields produced by the microwave source and by the plasma
+itself.
+The equations of motion were integrated using the leap-frog
+method and the Boris scheme to get the v × B rotation from
+the Lorentz force33. Details of the integration in the context
+of the quasi-1D model are provided in appendix A. Particle
+quantities were projected on a uniform grid using linear shape
+functions.
+The Rhei code development follows a test-driven approach
+to ensure the robustness and the maintainability of the code
+over time. Additionally, several test cases were run as a val-
+idation of the code. The first elementary test was the simu-
+lation of a magnetic bottle. The simulation domain, with a
+converging-diverging parabolic magnetic field, was uniformly
+loaded with a Maxwellian electron population. At the end of
+the simulation the electron distribution in velocity space v∥,v⊥
+was plotted to verify that the loss cone angle is coherent with
+the expected theoretical value arcsin
+��
+B0/BMax
+�
+. The sec-
+ond elementary test concerned the electromagnetic modes in
+a one-dimensional magnetized plasma. The simulation do-
+main was initialized with a uniform Maxwellian distribution
+of electrons and cold ions. The random fluctuations excited
+the modes of the plasma. The resulting dispersion curves were
+obtained by computing the discrete 2D Fourier transform of
+the electric fields during the simulation. This was compared
+to the expected theoretical description of the extraordinary and
+the ordinary wave. Finally, the third test case was the classi-
+cal capacitively coupled discharge in Helium, which verified
+in particular the collision module34.
+1.
+Collisions
+The Monte-Carlo Collision module used the Null Collision
+technique35 to speed-up the computation of the collisions by
+removing the velocity dependency of the total collision cross-
+section. Assuming Np collision processes defined by their re-
+spective cross sections σi(v),i = 1..Np, a null collision cross-
+section is defined as σ0(v) such that:
+σ0(v) = max
+v≥0
+� Np
+∑
+i=1
+σi(v)
+�
+−
+Np
+∑
+i=1
+σi(v)
+(4)
+A first test over all the particles of species s found the
+fraction of particles which undergo a collision with the back-
+ground. In that case the total cross section σT = ∑i=0 Npσi(v)
+(including the null collision process) did not depend on the
+velocity (thus avoiding a costly interpolation to get the cross
+section for all the particles). Then a second test among those
+selected particles computed all the collision cross sections
+for their given relative velocity and determined which cross
+section to use (including the null collision). When this test
+pointed to the null-collision cross section, then the particle
+did not experience an actual collision and was left unchanged.
+When the test pointed to another cross section, the the parti-
+cles experienced a collision.
+For the collisions of electrons with Xenon neutrals, we con-
+sidered a simplified set of three processes: elastic, ionization,
+and excitation. Excitation processes were lumped into a single
+process. Electron impact ionization and excitation were taken
+from the Morgan (Kinema Research & Software) database,
+while the total elastic scattering is from Ref. 36. For all elec-
+tronic processes, we assumed an isotropic scattering of the rel-
+ative velocity vector between the electron and the target dur-
+ing the collision. For the ionization collisions, the kinetic en-
+ergy of the projectile electron was equally split (after subtract-
+ing the threshold energy) between the secondaries. For the
+collisions of Xenon ions with Xenon neutrals, we considered
+isotropic scattering and backscattering34. Ion cross sections
+comes from Ref. 37. All electronic and ionic processes con-
+served momentum and total energy (kinetic plus internal). In
+order to start with a simplified description simulating weakly
+ionized plasmas, in which the collisions with the neutral par-
+ticles are the dominant process, Coulomb collisions were not
+considered in the code. Indeed, the electron-ion collision fre-
+quency νei, for Maxwellian electrons, is given by:
+νei = ωp
+Λei
+lnΛei
+(5)
+Here, ωp is the plasma frequency and lnΛ is the Coulomb log-
+arithm. For the typical simulation conditions in the thruster
+source, as it will be shown below, the maximum plasma den-
+sity was ne ∼ 1×1011 cm−3, the electron temperature was
+Te ∼ 10eV and the electron-neutral elastic collision frequency
+was νen ∼ 1×107 s−1.
+This gave lnΛ ∼ 12 − 15, ωp ∼
+1.8×1010 rads−1. Consequently, the maximum electron-ion
+collision frequency was νei ∼ 1×105 s−1, much less than the
+the electron-neutral collision frequency νen.
+The neutral gas in the thruster is injected at the backplate
+(see Fig. 1a) and expands in the source resulting in a decreas-
+ing density. Since there is no measurement of the neutral gas
+density profile in the thruster, and to avoid a costly particle
+simulation of the neutral particles, we modelled this expansion
+heuristically by assuming that the neutral background density
+
+5
+followed an exponential profile:
+nn(x) = nn0 exp
+�
+− x
+Ln
+�
+(6)
+where nn0 is the maximum density of neutrals found at the
+close end of the source, and Ln is the neutrals density char-
+acteristic length. The assumption of a time-independent neu-
+tral gas density profile means that the simulation did not con-
+serve the total mass, momentum and energy. In addition, it
+means that the neutral gas depletion due to ionizing collisions
+was not considered. However, both of those limitations are
+acceptable in the frame of this work which does not seek to
+compute the total thrust and energy balance but is concerned
+with the particle heating and trapping. To estimate the neutral
+depletion we note that the ion removal is driven by their ve-
+locity (at most 10kms−1), while the neutral removal is driven
+by their thermal speed, ∼ 200ms−1. From mass balance, the
+neutral inflow is balanced by the ion flux and the neutral out-
+flow. Using the characteristic speeds and the typical parame-
+ters for the gas density ng ∼ 1×1014 cm−3, and plasma den-
+sity nmax
+e
+∼ 3×1011 cm−3, gives a neutral depletion of at most
+10%, indicating that the assumption of a static background
+remained consistent with the assumed density profile.
+2.
+Fields solvers
+The Rhei code solved the Poisson equation to compute the
+electrostatic space potential Φ and the electric field (EES =
+Exx) using Eq.
+7 where the charge density is ρs.
+The
+solver implements a second order finite difference discretiza-
+tion and the resulting linear system is inverted using an itera-
+tive method (GMRES)38.
+∇2Φ(x,t) = −ρs(x,t)
+ε
+(7)
+In addition, an electromagnetic solver computed the fields
+produced by the microwave source and by the plasma itself:
+EEM = Eyy + Ezz and BEM = Byy + Bzz.
+This solver was
+based on the Constrained Interpolation Profile (CIP) method
+explained in detail in Ref. 39. This method considers not
+only the electromagnetic fields but also their spatial deriva-
+tives, therefore suppressing instabilities and providing lower
+numerical dispersion even when using coarse grids and large
+time steps40. The use of this method is a novel solution for a
+PIC code since most of the electromagnetic solvers are based
+on conventional approaches like the finite-difference time-
+domain method (FDTD). It was shown that it provides higher
+accuracy than the latter under the condition of identical cell
+size41.
+The CIP method is a semi-Lagrangian scheme that
+circumvents the Courant-Friedrichs-Lewy (CFL) stability
+condition42,43, i.e., (u∆t/∆x) < 1 where u is the magnitude
+of the velocity, ∆t is the time-step, and ∆x the length inter-
+val. This feature allows computations with CFL values ≥ 1.0,
+as can be seen in Ref.
+44 and 45 where the authors per-
+formed simulations using a CFL value of 2.6 in a Cartesian
+coordinate system. The gain in computational time, that is
+afforded by using high CFL values, is a key factor that en-
+ables the self-consistent kinetic simulations presented here to
+reach steady-state. In this paper, CFL values close to 3 were
+used for the simulations. As a check, simulations were also
+run with CFL=0.6 and compared to the results obtained with
+larger time steps. The results were identical to the one at larger
+time-steps, within small variations due to the noise inherent to
+the statistical nature of the PIC simulations.
+Finally, the CIP scheme does not necessarily maintain the
+divergence-free condition for the dynamic field BEM. How-
+ever, BEM is smaller than the magnetostatic field (which has
+divergence equal to zero by construction, see appendix A) by
+several orders of magnitude, over the whole computational do-
+main. Therefore, the resulting error on the total divergence
+was considered to be negligible.
+3.
+Boundary conditions
+As it was shown in Fig. 1b when describing the model, the
+domain goes from x = 0 at the left side which corresponds
+to the backplate and the microwave input, to the right-end at
+x = L, as discussed in II A.
+Electrostatic: At the right end of the computational do-
+main x = L, we imposed a Dirichlet boundary condition, with
+Φ(L) = 0, to simulate the presence of a grounded vacuum
+chamber wall. The dielectric backplate, at x = 0, is in con-
+tact with the plasma and therefore its surface voltage ΦBP is
+changed by the collection of charged particles. This can be
+modeled as a capacitor. Hence, the evolution of ΦBP is given
+by ∆ΦBP = ∆Q/(C∆t), where ∆Q is the charge deposited at
+the backplate at each time step, and C is an equivalent ca-
+pacitance under the assumption that the backplate is in con-
+tact with a grounded conductor.
+This capacitance is com-
+puted by considering that the backplate is a plane capacitor,
+its value is of a few picoFarads. Changing its magnitude mod-
+ifies the charging rate of the backplate and thus the transient
+phase of the computation. However, its does not affect the
+steady-state voltage of the backplate. This approach guaran-
+tees that at steady-state, the ion flux equates the electron flux
+on the backplate. In principle the steady-state value of the
+backplate potential is also affected by other processes such as
+secondary electron emission or charge migration. However,
+for this study, these processes were neglected.
+Electromagnetic: In the coaxial ECR thruster, the mi-
+crowaves are injected as Transverse Electro-Magnetic (TEM)
+mode. For this 1D simulation, the TEM mode can be seen
+as a linearly polarized wave, where the radial component of
+the electric field is along the transverse y axis, the azimuthal
+magnetic field defines the z axis and the wavevector direction
+is along the longitudinal x axis. Therefore, the microwaves
+were injected at the backplate as a propagating wave with a
+linear polarization along the y-axis. The incident wave was
+parametrized by its power per unit area Pin and its frequency
+fEM = ω/2π. The electric fields from the injected linearly
+polarized wave were computed as Ey = √µcPin sin(ωt) and
+Ez = 0.
+
+6
+The injected microwave input power per unit area Pin could
+be fixed, or it could be adapted to keep a roughly constant pre-
+defined number of particles Ntarget during the transient phase.
+This feature was intended to speed up the simulations by re-
+producing a faster plasma response to a given variation in the
+simulation’s parameters. The value of Pin can be regulated
+with an attenuation factor α ≤ 1 varying with the number of
+particles in the domain: α = exp(−Nparticles/Ntarget). A run
+performed without this regulation confirmed that it did not
+have an effect on the final steady state but only on the duration
+of the transient phase.
+Particles: We imposed a loss condition at both ends of the
+domain, for both ions and electrons. Particles crossing these
+boundaries are suppressed from the simulation. As a simpli-
+fying assumption, secondary emission processes on the back-
+plate were not considered in this first approach.
+4.
+Cross field diffusion loss model
+Electron cross-field diffusion is an important mechanism to
+model to get a more accurate representation of the discharge
+loss mechanisms. Previous works using PIC codes for elec-
+tric thrusters took it into account as wall losses by artificially
+increasing the collision rate or by using a profile of the cross
+field diffusion based on empirical evidence46,47. The electron
+balance equation is:
+∂ne(r,t)
+∂t
++∇⊥ ·neu⊥ +∇∥ ·neu∥ = kionne(r,t)
+(8)
+Where u⊥ and u∥ are the electron macroscopic velocity per-
+pendicular and parallel to the local magnetic field, respec-
+tively, and kion is the ionization rate.
+For our 1D3V simulations, the transport along the magnetic
+field is taken into account by the kinetic model. However, the
+perpendicular transport cannot be modeled with a 1D model.
+Therefore we simulated the particle losses into the coaxial
+chamber walls using a phenomenological, Monte Carlo loss
+model, as shown in Fig. 1b. The probability of an electron im-
+pacting the walls of the coaxial chamber was calculated from
+the diffusion equation of electrons across the magnetic field
+based on the assumption that their number density profile in
+the radial direction was independent of time and axial posi-
+tion. In a cylindrical coordinate system it can be expressed
+as the product ne(x,r,t) = ne0(x,t)g(r). The balance equation
+(for a constant diffusion coefficient D) integrated over the ra-
+dius of the flux tube rmax was then given by:
+∂ne0(x,t)
+∂t
++ ∂ne0(x,t)ux(x,t)
+∂x
+= −νLne0(x,t)+kionne0(x,t)
+(9)
+With the loss frequency given by:
+νL = −rmax
+g′(rmax)
+S
+D
+(10)
+Where rmax is the radius of the flux tube, and the weighted
+cross section S is given by:
+S =
+� rmax
+0
+rg(r)dr
+and
+(11)
+A first choice for the diffusion coefficient D would be a co-
+efficient based on classical diffusion obtained from theories on
+standard electron-neutral collisions. It can be seen in Eq. 12
+where τ = 1/ν is the collision period with the neutral back-
+ground. However, the electron mobility tends to be higher
+than the value predicted by this classical diffusion approach48.
+The cause of this discrepancy is an active area of research in
+the electric propulsion field49,50. As a consequence, we de-
+cided to use the Bohm coefficient, which is a phenomenologi-
+cal coefficient accounting for the anomalous cross-field diffu-
+sion.
+DBohm = 1
+16
+kBTe
+eB
+or
+Dclassical =
+ωcτ
+1+(ωcτ)2
+kBTe
+eB
+(12)
+The probability for a given particle to be lost between t and
+t + ∆t is given by pL = νL∆t. In our quasi-1D model, the
+flux conservation relates the magnetostatic field to the cross-
+sectional area of the magnetic field tube as shown in Eq. 1.
+In this work we assumed g(r) = J0(k0r/rmax), with k0 the first
+zero of the Bessel function. Then Eq. 10 has the form νL ∝
+k2
+0/r2
+maxD. Since D ∝ B−1 and in the model BS = Bπr2
+max is a
+constant, the loss probability does not depend on the position
+along the flux tube and is given by:
+pL(x) = 2
+3
+π
+16k2
+0
+�1
+2m⟨v(x)2⟩
+�
+dt
+eA0B0
+(13)
+Where ∆t is the time step, m⟨v(x)2⟩/2 is the electron’s mean
+kinetic energy, and A0 = A(0) is the cross-section of the mag-
+netic field tube at x = 0. The losses are computed at each
+time-step. For all electrons in the source (such as x ≤ LS, LS
+being the length of the coaxial chamber as shown in Fig. 1b),
+the probability pL is computed using equation 13. A random
+number x is drawn from a uniform distribution. If x ≤ pL, the
+electron and a neighboring ion are removed from the simula-
+tion.
+C.
+Simulation Setup
+The electron dynamics and the electromagnetic solver were
+updated every iteration. For these one-dimensional calcula-
+tions the real Xenon mass for the ions was used and to speed
+up the calculations a subcycling was used so the ion’s position
+and velocity that were updated every 10 time steps as given by
+∆tions in Table I. The collisions were also computed every 10
+time steps as given by ∆tcoll. The charged particle’s population
+was seeded using a uniform density distribution (N ∼ 103).
+The electron’s initial energy along each of the x-y-z axis was
+set to Te = 20eV, while ions were assumed cold Ti = 0.03eV.
+These values were intended to reproduce a non-equilibrium
+plasma at low density. The choice of the initial electron tem-
+perature Te = 20eV is somewhat arbitrary. Checks run with
+several energy values between 10 eV and 30 eV showed no
+impact of the initial electron energy on the final characteristic
+of the steady state. To sustain the plasma at the beginning,
+a plasma source located at 2 mm from the backplate injected
+electrons at 3×105 ms−1 and ions at 3×102 ms−1 during the
+
+7
+first 150 ns of the simulation. These velocities were specified
+along each of the x-y-z axis. Here the idea was to sustain the
+initial plasma long enough for the ionization to pick up and
+the plasma density to grow. The conditions for the simulation
+presented below are shown in table I. With this choice of mag-
+netic field profile, the resonance condition fEM = eB/2πm
+was met at x = 6.7mm.
+TABLE I: Simulation parameters for the electromagnetic full
+PIC simulations using the quasi-one-dimensional model.
+Parameter
+Description
+Value
+∆t
+Time step
+1.6 ps
+∆x
+Mesh spacing
+167 µm
+C
+CFL condition
+2.87
+fEM
+Microwave frequency
+2.45 GHz
+LS
+Coaxial chamber length
+20 mm
+xECR
+ECR surface position
+6.7 mm
+W
+Weight for the charged particles
+2×105
+LD
+Computational domain length
+100 mm
+nn0
+Maximum number density of neutrals
+8×1019m−3
+Ln
+Neutral density characteristic length
+1.0 cm
+AL
+Cross-sectional area for the loss module
+1cm2
+∆tions
+Time step to push the ions
+10∆t
+∆tcoll
+Time step for collisions
+10∆t
+The simulation was run until it reached a steady state, usu-
+ally after around 30µs which represents between 5 to 8 ion
+transit times. The definition of this steady state was done by
+following up the variation of the total number of particles in
+the domain, its mean kinetic energy, and the particle flux at
+the backplate and the plume since an equal number of ions
+and electrons must be impacting both surfaces, as shown in
+Fig. 2. At the end of the simulation, when the steady state
+was reached, the plasma properties were obtained by calcu-
+lating the time average for each parameter over several time
+steps. Overall, the wall time of the simulation was 44 hours,
+with 12 OpenMP threads.
+III.
+RESULTS
+Figure 3 shows the steady-state plasma potential distribu-
+tion over the whole computational domain. Except for slight
+random fluctuations on the instantaneous potential, no large
+scale fluctuations were observed. Time-averaging improved
+the signal to noise ratio but did not blur the shape of the pro-
+file. The backplate reached a positive steady state potential
+of around 70V. The peak of the plasma potential was 105V,
+and it was reached at around 3 mm, interestingly not at the
+ECR surface (indicated with a vertical dashed line). Indeed,
+the shapes of the plasma density and potential are driven by
+the ionization rate. For this simulation, the ionization rate
+was monotonically decreasing, because the background den-
+sity decrease was faster than the ionization rate increase due
+to the plasma heating. As a consequence, the maximum ion-
+ization was upstream of the ECR surface. This peak in the
+plasma potential formed a barrier. As a result, ions collected
+FIG. 2: Time evolution of simulation quantities. Top frame :
+total number of macro-particles (ions and electrons) in the
+simulation. Middle frame : Mean kinetic energy of the
+electrons. Bottom frame : particle fluxes at the boundaries
+(backplate and outlet) and particle source and sink terms in
+the whole computational domain. The volume loss gives the
+average number of particle lost per timestep due to the Bohm
+loss model. For the steady-state analysis, the particles
+quantities are sampled after t = 30µs
+on the backplate were necessarily created in a region where
+x ≤ 3mm, while ions collected downstream were created in
+a region where x ≥ 3mm and accelerated into the nozzle
+by the ambipolar electric field. Sheaths were formed at the
+backplate and the vacuum chamber wall. The plasma sheath
+width on the backplate was ∼ 0.1mm. At the downstream
+end x = L of the domain, as shown by Fig. 4, the plasma
+sheath began at around 90 mm. This size was consistent with
+a Debye length λD ∼ 1 − 5mm for a plasma density around
+1×108 cm−3. The electron and ion peak number density was
+1.12×1011 cm−3 at x = 1.5mm. Recall that the ECR condi-
+tion is met at x = 6.7mm.
+In Fig. 5 we plotted the electron’s mean kinetic energy in
+both the axial (e∥) and the perpendicular (e⊥) direction as a
+function of the axial position on the domain. First, we ob-
+served that the mean parallel kinetic energy remained nearly
+constant, around 4−5eV, over the whole simulation domain.
+
+e losses
+backplate
+ionization rate
+outlet
+Electrons
+Ions8
+FIG. 3: Plasma potential. The vertical dashed line indicates
+the ECR surface location. The horizontal dashed line shows
+the backplate potential ΦBP. The two colored zones delineate
+the regions where the plasma potential is above (∆Φ >> 0 or
+below (∆Φ < 0) the backplate potential.
+FIG. 4: Electron (solid) and ion (dashed) number densities.
+The dashed line indicates the ECR surface location.
+The perpendicular energy was higher than the parallel com-
+ponent, which underlined the anisotropic heating of the elec-
+trons in this thruster. More precisely, the mean perpendicular
+kinetic energy e⊥ reached a first peak at around x = 9mm and
+then decreased before reaching a global maximum of 25 eV at
+x = 45mm. After this point, e⊥ decreased until the end of the
+simulation domain. Over the whole simulation domain, the
+anisotropy ratio Te,⊥/Te,∥ was found to vary between 2.5 and
+7.5. Given that the ECR heating increases the perpendicular
+energy of the electrons, it was expected to see an anisotropic
+behavior depending on the direction parallel or perpendicular
+to the magnetic field lines. However, the second broad en-
+FIG. 5: Electron’s mean kinetic energies: e∥ longitudinal
+(blue line) and e⊥ perpendicular (red line) directions. The
+location of the ECR surface is shown by the dashed line.
+ergy peak in the downstream part of the magnetic nozzle was
+puzzling. To get a better understanding of these feature, it
+was necessary to evaluate the energy deposition by the elec-
+tromagnetic field.
+A.
+Electromagnetic Energy deposition in the source
+To understand how the field energy was transferred to the
+particles, we considered the energy balance equation, includ-
+ing the Poynting flux (its derivation is provided in appendix
+B).
+∂εEM +ε
+∂t
++∇·(Q+Π) = Scoll
+(14)
+In this equation, ε and εEM stands for the electron kinetic en-
+ergy density and the electromagnetic energy density, respec-
+tively. Q and Π are the kinetic energy flux and electromag-
+netic energy flux; Scoll, whose expression is given in Eq. B4,
+is the volume power loss term due to the collisions and the
+diffusion. This latter term account for the energy lost by elas-
+tic and inelastic collisions with the neutral background and by
+the particles removed by the loss model. Since we were inter-
+ested in the steady state regime, and considering that the field
+quantities depend on x only, this was further simplified to:
+1
+A
+∂
+∂xA(Qe +Qi +Π) = Se,coll +Si,coll
+(15)
+where we separated the time-averaged total energy flux into a
+kinetic contribution due to the electrons Qe , the ions Qi and
+the electromagnetic contribution Π. The kinetic energy flux of
+the electron was further separated into a flux of parallel energy
+Qe,∥ and perpendicular energy Qe,⊥ (see appendix B).
+To quantify the magnitude and direction of the energy ex-
+changes between the electromagnetic field and the particles,
+the different terms of Eq. 15 were evaluated. To do so, the
+particles and field quantities were sampled in the steady state
+phase (after t = 30µs, see fig. 2). First, particles were sorted
+in 120 spatial bins equally spaced along the axial direction. In
+
+△Φ<0
+△Φ>0ell
+el9
+each bin, the moments of particle distribution provided the to-
+tal energy flux Qe and Qi, as detailed in appendix B. Second,
+the cross product of the electric and magnetic field provided
+the axial component of the Poynting vector. This vector was
+time-averaged over a period corresponding to an integer num-
+ber of wave periods.
+FIG. 6: Energy source terms of eq. 15 along the axial
+direction. The location of the ECR surface is shown by the
+dashed line.
+FIG. 7: Perpendicular and parallel energy source terms for
+the electrons along the axial direction. The location of the
+ECR surface is shown by the vertical dashed line.
+The results, plotted in Fig.
+6, showed first that the en-
+ergy source terms are negligible in the plume region (x ≥
+20mm) compared to the source region (x < 20mm). In the
+plume region, the magnitude of the source terms is below
+1×104 Wm−3. For that magnitude, the signal is dominated
+by the statistical noise of the PIC simulation.
+This noise
+drowns the finer features of the source terms, especially for
+the electrons which are more prone to statistical noise. Never-
+theless, this underlines that most of the energy exchanges take
+place in the source region and cannot explain the secondary
+peak for mean perpendicular kinetic energy e⊥ observed in
+the plume region (Fig. 5).
+Second, the collision source term is negative over the whole
+source region. Its magnitude is maximum near the backplate,
+where the neutral and plasma densities are higher, and de-
+creases along the source axis. This behaviour is not unex-
+pected, since this term lumps together the contribution of in-
+elastic collisions and the diffusion model: these two processes
+are loss mechanisms for the plasma. Given that the plasma
+density and the neutral gas density decrease as we move away
+from the backplate, the collision frequency drops and the mag-
+nitude of the energy loss decreases.
+Third, the sign and magnitude of the source terms reveal
+different phenomena. For x ≤ 3mm, the terms linked to the
+Poynting vector are positive, while the source term due to the
+electron energy flux is negative. This indicates an energy con-
+version from the electron kinetic energy to the field energy. In
+parallel, the ion source term is positive, which points to a gain
+of energy in the sheath. For the 3mm < x ≤ 10mm range,
+the Poynting source term shows a negative peak, while the
+source term due to the electron energy flux is positive, with
+a peak centered on the ECR surface location. In that case,
+there is a transfer of energy from the field to the electrons.
+This latter feature corresponds to the ECR heating of the per-
+pendicular energy mode of the electrons. In fact, this appears
+when considering the parallel and perpendicular contributions
+to the source term in Fig. 7. The perpendicular source term
+dominates over the parallel source term, with a peak centered
+on the ECR surface. The axial extent of this peak shows that
+the perpendicular mode of the electrons is heated in a zone of
+about ∆xECR ≈ 6mm, i.e., from x = 3mm to x = 9mm. Con-
+sidering that the ECR condition is only met on a specific sur-
+face, the presence of an extended region may seem surprising.
+However, as will be discussed later, most of the electrons in
+the magnetic field tube are confined and undergo a bouncing
+motion in the potential well formed by the electrostatic field
+and the magnetic mirror force. These bouncing electrons can
+cross the resonance surface with a significant parallel velocity,
+thus one may expect a shift of the resonance condition due to
+the Doppler effect. The width ∆xECR can be compared with
+the expected value for a Doppler broadened resonance ∆xD in
+Eq. 1651.
+∆xD =
+�
+�
+�
+�
+2πv∥
+ωc
+BMS
+���
+∂BMS
+∂x
+���
+(16)
+Using
+the
+electrons’
+mean
+axial
+velocity
+v∥
+is
+1.1×106 ms−1 we obtain ∆xD = 4.7mm.
+However, as
+it will be shown later in Fig. 8, there is a high dispersion
+for the values of v∥. Therefore, we can expect a much larger
+Doppler broadening for the fastest electrons. The electrons’
+axial velocity can reach values up to 3.0×106 ms−1 around
+the ECR zone. With this velocity, we can compute a max-
+imum Doppler broadening of 7.8mm, which means that
+4.8mm < ∆xD < 7.8mm. The ECR heating zone obtained in
+the simulations is consistent with the one expected analyti-
+cally, indicating that the Doppler effect is a good candidate to
+explain the width of the heating zone observed in Fig. 6.
+
+1
+AQ
+Ao x
+1
+AQi
+Ao x
+1 
+AII
+Ao
+X
+S
+coll0
+AQe.ll
+Ao x
+1 0
+AQe,1
+Ao x
+A(Qe. lI +Qe,↓)10
+Because of Doppler-broadening, ECR heating of the elec-
+trons can even occur when the ECR surface is outside the
+plasma source. Indeed, the fact that the plasma in the thruster
+can be sustained even with an ECR zone outside the coax-
+ial chamber was demonstrated experimentally by Vialis 52,
+where the location of the resonance surface was placed at x =
+−0.17mm and x = −0.77mm. Doppler broadening is a pos-
+sible explanation to this finding and this hypothesis was tested
+in simulations using the same parameters described in Table
+I but with an input microwave frequency of fEM = 2.9GHz.
+It was possible to sustain a discharge for this frequency even
+though the ECR condition was meet at x = −1.78mm (i.e.,
+upstream of the plasma source).
+In summary, the analysis of the power deposition shows that
+the energy transfer occurs mainly in the source region. The
+electrons absorb the wave energy over a Doppler-broadened
+volume. This energy goes preferentially to the perpendicu-
+lar energy mode and leads to an anisotropy ratio Te,⊥/Te,∥ ∈
+[2.5,7.5]. This energy deposition from the field to the perpen-
+dicular energy mode can explain the first peak in perpendicu-
+lar energy seen in Fig. 5. However, the source terms are neg-
+ligible in the plume region and thus cannot explain the broad
+peak observed in this region.
+B.
+Electron confinement in the magnetic nozzle
+To determine the factors driving the evolution of the mean
+electron energy, we considered the electron distribution in the
+nozzle region. In Fig. 8 we plotted the normalized electron
+distribution in the velocity space v∥,v⊥ plane. The distribution
+was plotted at different locations in the computational domain.
+The results in Fig. 8 show that as we move downstream into
+the nozzle, we see an increased electron population with high
+energies in the perpendicular direction. To understand this
+phenomenon, we must first get a more detailed description of
+the electron confinement, i.e., how they get trapped and under
+which conditions they can leave the thruster.
+Three loss pathways are identified for the electrons pro-
+duced in the source.
+1. Cross field losses : electrons can diffuse across the mag-
+netic field, due to anomalous transport, collision, etc.
+This is modeled by the phenomenological cross-field
+diffusion model presented in section II B 4.
+2. Losses at the downstream of the nozzle electrons which
+have a kinetic energy sufficient to overcome the confin-
+ing potential well are lost, alongside ions accelerated by
+this same potential drop.
+3. Electrons than can overcome both the repelling poten-
+tial of the plasma sheath at the close-end of the source
+and the mirror-force are collected on the dielectric plate
+and will contribute to its surface charge.
+While the first loss mechanism does not depend on the ki-
+netic energy of a single electron but rather on the mean kinetic
+energy at a given location, the two other mechanisms depend
+on the electron kinetic energy. An electron moving along the
+magnetic field will see both an electrostatic potential Φ (Fig.
+3) and the magnetostatic field B. Depending on its initial ki-
+netic and potential energies, its trajectory might have turning
+points (where v∥ = 0) within the domain, or out of the domain.
+In the first case, this electron is confined in the potential well.
+In the latter case, the electron is lost, either downstream (loss
+pathway 2) or at the backplate (loss pathway 3). Now the lim-
+iting case between confined / unconfined electrons is when the
+turning points are located at the boundaries. This will define
+the necessary conditions for the electrons confinement. Let
+us note ΦBP the potential of the backplate. Neglecting the
+plasma-wave interaction, and noting µ the magnetic moment,
+the equation for the total energy of an electron is:
+Etotal = 1
+2mv2
+∥ + µB−eΦ
+(17)
+From Eq. 17 we can say that the electron is oscillating in
+an effective potential given by Uef f = µB − eΦ, where µB
+represents the magnetic confinement as the electron moves
+towards the backplate while −eΦ is the electrostatic confine-
+ment given by the plasma potential. It can be seen in Fig. 9 for
+different arbitrary values of the magnetic moment taken from
+the results of the simulation. Its concave shape explains the
+confinement of the electrons in the ECR thruster inside this
+potential well.
+Let us now consider an electron moving from an arbitrary
+initial point to a turning point at a position x0 along the longi-
+tudinal direction, such that v∥(x0) = 0. The energy conserva-
+tion between any initial location and the turning point gives:
+v2
+∥ +v2
+⊥
+�
+1− Bx0
+B
+�
+= −2e
+m ∆Φ
+(18)
+Where ∆Φ = Φx0 − Φ. Now let us consider the loss path-
+ways 2 and 3 identified above.
+For an electron lost to the downstream boundary (pathway
+2), the confinement condition is obtained by setting x0 = L. In
+that case, given the divergence of the magnetic field, we have
+0 < 1−B(L)/B < 1 and ∆Φ < 0. Thus equation 18 describes
+an ellipse in the v∥ − v⊥ plane. Electrons in the ellipse have
+there turning points x0 ≤ L and remain confined by the elec-
+trostatic well. Electrons out of the ellipse can overcome this
+electrostatic confinement and are lost downstream.
+For an electron lost to the backplate (pathway 3), the con-
+finement condition is obtained by setting x0 = 0. Because the
+magnetic field is monotonically decreasing, 1 − B(0)/B < 0.
+In addition, Φ(0) = ΦBP. We define the loss cone angle as:
+sin(θ) =
+�
+B
+B(0)
+(19)
+Equation 18 can be recast as:
+v2
+⊥ = tan2(θ)
+�
+v2
+∥ + 2e
+m ∆Φ
+�
+(20)
+Depending on the sign of ∆Φ, three cases are possible:
+
+11
+(a)
+(b)
+(c)
+FIG. 8: Electrons distribution in the velocity space v∥,v⊥
+plane at different locations on the simulation domain. The
+number of electrons has been normalized by the total number
+of electrons for each case independently. For each case, we
+also plotted what we call the confinement boundaries
+described by an analytical model (Eq. 19 and 20). The dotted
+line is the magnetic confinement, the dashed line the
+electrostatic potential at the backplate, and the solid line is
+the electrostatic confinement on the plume. (a) x = 5 mm, (b)
+20 mm, (c) x = 80 mm.
+FIG. 9: Schematic view of the effective potential profile for
+arbitrary values of the magnetic moment.
+• ∆Φ = ΦBP − Φ = 0: The confinement of the electron
+at the backplate is given exclusively for the topology of
+the magnetostatic field according to the loss cone angle
+θ. Those electrons with values for v∥,v⊥ such as they
+are located in the loss cone and will therefore be lost at
+the backplate (Fig. 10a).
+• ∆Φ = ΦBP −Φ < 0: The backplate potential repels the
+negative charges and thus confines the electron. Conse-
+quently, a fraction of the electron in the loss cone will
+be reflected back and stay confined (Fig. 10b).
+• ∆Φ = ΦBP −Φ > 0: The backplate potential attracts the
+electrons. Thus, even electrons out of the loss cone are
+collected. Therefore, the confined electrons are those
+that meet two conditions: they are not on the loss cone
+for the magnetic field, and they are energetic enough
+in the perpendicular direction to avoid being lost at
+the backplate thanks to the electrostatic acceleration to-
+wards it (Fig. 10c).
+Fig. 3, displays the sign of ∆Φ in the nozzle. The combina-
+tion of the loss conditions at the backplate (pathway 3) and
+downstream (pathway 2) delimits a confinement volume in
+phase space where electrons are confined, as shown in Fig.
+10. Pitch-angle scattering, either caused by collisions with the
+neutral background or by the electromagnetic field, enables
+electrons to cross the confinement volume boundaries. De-
+pending on which boundary is crossed, electrons are lost at the
+backplate or at the downstream side of the nozzle. Indeed, it is
+important to recall that the electron deconfinement is mainly
+driven by these two phenomena in the source region. Since
+the neutral background density decreases exponentially, most
+of the collisions occur in the source. In addition, as shown
+above, wave absorption happens over a few millimeters in the
+source. As a consequence, the electron deconfinement rate is
+driven by the wave interaction and the collisions:
+• Interaction with the electromagnetic wave is akin to a
+scattering of the electron momentum53. After several
+passages through the ECR heating zone, the electron
+may gain enough energy to overcome the electrostatic
+barrier and escape into the plume.
+
+12
+(a)
+(b)
+(c)
+FIG. 10: Confinement boundaries in velocity space on the
+v∥,v⊥ plane where the orange colored zones describe the
+electrons being trapped in the ECR thruster. Solid line for the
+plume electrostatic confinement, and dashed (electrostatic)
+plus dotted (magnetic) lines for the backplate confinement.
+Where: (a) ∆Φ = 0, (b) ∆Φ < 0, and (c) ∆Φ > 0.
+• If the electron undergoes an elastic collision, it will ran-
+domly scatter its velocity vector. If the electron scat-
+tered momentum falls in the loss region defined by Eq.
+20 and shown in Fig. 10, the particle is lost at the back-
+plate.
+If we now go back to the results in Fig. 8 for the elec-
+tron distribution in the velocity space v∥,v⊥ plane, we notice
+that as we move downstream into the plume, the confinement
+boundaries change. There is a transition from a confinement
+boundary as the one in Fig. 10b (∆Φ < 0) to the one in Fig.
+10c (∆Φ > 0). This transition is a consequence of the fact
+that, as shown in Fig. 3 the plasma potential is greater than the
+backplate potential in the source Φ > ΦBP and lesser than ΦBP
+downstream. Thus, inside the coaxial chamber, the plasma po-
+tential is such that ∆Φ = ΦBP − Φ < 0 (Fig. 10b), and in the
+plume section it is such that ∆Φ = ΦBP − Φ > 0 (Fig. 10c).
+As a consequence, as we move downstream into the magnetic
+nozzle, the mean perpendicular kinetic energy can increase.
+However, this is not given by an additional heating phase but
+as a result of confining only a highly energetic electron pop-
+ulation in the perpendicular direction. Those electrons with a
+low perpendicular kinetic energy (i.e., below the dashed line)
+are lost at the backplate as previously described. It can be
+seen as a filtering process where only the hot electrons are
+confined, and this is what we see when plotting the electron
+perpendicular kinetic energy in Fig. 5. Further downstream
+of the magnetic nozzle, the magnitude of the potential well to
+the end of the nozzle decreases, while the magnitude of the at-
+tracting potential drop to the backplate increases. This results
+in a narrower distribution for the confined population and fi-
+nally a decrease in the mean perpendicular kinetic energy.
+IV.
+CONCLUSIONS
+We have performed electromagnetic full-PIC simulations
+of the ECR thruster using a 1D3V model that allowed us
+to shed light onto some of its working principles. The re-
+sults confirmed the expected anisotropic behavior for the elec-
+trons’ energies in the direction perpendicular and parallel to
+the magnetic field lines and a peak for the mean perpendic-
+ular energy near the resonance zone. The microwave energy
+injected at the backplate of the thruster propagates through
+the coaxial chamber while being absorbed by the electrons
+increasing their kinetic energy perpendicular to the magnetic
+field lines. The absorption takes place exclusively inside the
+coaxial chamber on a zone of 6 mm around the resonance
+condition.
+This zone is coherent with the predicted value
+from Doppler broadening. The width of this heating zone
+may explain why the thruster works even with a configura-
+tion in which the resonance condition is met outside the coax-
+ial chamber52. From a practical point of view, this feature
+improves the reliability of the thruster, since it means that the
+thruster can still operate even when the magnetostatic field de-
+creases, for example due to excessive heating of the magnets.
+The results also show, unexpectedly, that the electrons’
+mean perpendicular energy has a second peak in the plume
+due to the confinement of highly energetic electrons.
+The
+confinement is determined by the backplate’s potential, the
+magnetostatic field, and the potential drop on the plume. As
+a consequence, there is a population of trapped electrons
+with significant perpendicular kinetic energy in the down-
+stream region of the magnetic nozzle. The existence of dou-
+bly trapped electron population has been investigated using
+a kinetic model with a paraxial approximation similar to this
+work54. While this work was assuming the shape of the initial
+distribution function, it has also been seen in the case of an
+anistropic distribution that the perpendicular electron temper-
+
+= arcsin(V B,)
+0-2e△Φ
+V=
+m2e△Φ
+*
+Vi = tan(0) 
+m13
+ature could increase in the diverging part of the nozzle55. We
+can speculate that these high temperature electrons trapped in
+the plume could be important to drive some instabilities ob-
+served in the plume that are thought to enhance cross-field
+transport of electrons and thus play a role in the detachment.
+In particular, Lower Hybrid drift instabilities have been re-
+cently observed in diverging magnetic nozzle. In these ex-
+periments, diamagnetic drift vD = ∇pe⊥ ×B/enB2 was iden-
+tified for the primary energy source for the instability56. The
+trapped electrons could enhance the radial gradient in perpen-
+dicular pressure and thus enhance the diamagnetic drift.
+Appendix A: Electron motion integration in the Quasi-1D
+model
+On one hand, the model assumes that the axial static mag-
+netic field in the flux tube depends on x only
+dBx
+dx = α(x)
+(A1)
+In the quasi-1D model, the field remains constant across the
+section of the tube. On the other hand, in the ECR thruster
+the electromagnetic part of the magnetic field (the part due
+to the propagation of the electromagnetic power injected in
+the source) remains negligible compared to the static part.
+Indeed, assuming a locally plane wave, the Poynting vector
+S = E × B/µ0. Since E ≃ cB and S ≃ 1Wcm−2, the order
+of magnitude for the electromagnetic component of the mag-
+netic field is B ≃ 1×10−2 mT. The static part of the ��eld is
+between 10 mT and 10 mT, much greater than the electromag-
+netic component. Therefore, using the divergence equation
+for the static magnetic field and neglecting the electromag-
+netic component, it is possible to obtain the radial component
+of the magnetic field :
+Br(x,r) = −α(x)r
+2
+(A2)
+Assuming this value for the radial part of the magnetic field
+ensures that the divergence condition is automatically en-
+forced for the static part of the field. The electron guiding
+center is on the flux tube centerline. The Larmor radius of the
+electrons is given by:
+rL(x) = V⊥(x)
+ωc(x)
+(A3)
+Where V⊥ =
+�
+v2y +v2z, ωc(x) = eBx(x)/me. The gyromotion
+of the electron is shown in Fig. 11. Thus, knowing the parti-
+cle velocities it is possible to obtain the phase angle θ in its
+gyromotion, as given in Eq. A5.
+cos(θ) =
+y
+rL(x) = vz
+V⊥
+(A4)
+sin(θ) =
+z
+rL(x) = − vy
+V⊥
+(A5)
+Knowing the phase angle, sine and cosine, the By an Bz com-
+ponents can be deduced from eqs. A2 and A5.
+FIG. 11: Gyromotion in the plane normal to the axial static
+field Bx.
+Appendix B: Energy equation
+For the energy equation for the particles we consider the
+second order moment of the Vlasov equation. Multiplying the
+Vlasov equation for the electrons by
+� mev2/2, we obtain:
+∂
+∂t
+� mev2
+2
+fed3v+ ∂
+∂x ·
+� mev2
+2
+v fed3v
+(B1)
++qe
+� v2
+2 (E+v×B)· ∂ fe
+∂v d3v =
+� mev2
+2
+�∂ fe
+∂t
+�
+col
+d3v
+Where the right hand side lumps the contribution of colli-
+sions and the loss model detailed in section II B 1 and II B 4,
+respectively. Eq. B1 can be rewritten as:
+∂εe
+∂t +∇·Qe =−je ·E+ScollQe =
+� mev2
+2
+v fed3v (B2)
+εe =
+� mev2
+2
+fed3vr
+(B3)
+Se,coll =
+� mev2
+2
+�
+∂ fe
+∂t
+�
+coll d3v
+(B4)
+The same procedure can be applied to the ions:
+∂εi
+∂t +∇·Qi =−ji ·E+ScollQi =
+� Miv2
+2
+vfid3v (B5)
+εi =
+� Miv2
+2
+fid3v
+(B6)
+Si,coll =
+� Miv2
+2
+�
+∂ fi
+∂t
+�
+coll d3v
+(B7)
+Considering the electron population, the total heat flux can be
+written as:
+Qe = qe +Pe ·ue +neue(eK +EK)
+(B8)
+
+14
+Where the density is given by ne =
+� fed3v and the macro-
+scopic velocity by neue =
+� fevd3v. The random part of the
+velocity is c = v−ue The different terms are then:
+qe =
+� me
+2 c2cd3c
+(B9)
+Pe =
+�
+meccd3c
+(B10)
+EK = 1
+2meu2
+e
+(B11)
+eK =
+� me
+2 c2d3c
+(B12)
+Considering the one-dimensional approximation, we are con-
+sidering only the axial (parallel) part of the total heat flux.
+Thus, after averaging over a period of the incoming wave :
+< ... >= 1
+T
+� ...dt, we obtain:
+∂
+∂xA(x)
+�
+qe∥
+�
++A
+�
+Pe∥u∥ +Pe⊥u⊥
+�
++A(x)
+�
+ue∥ +neue;∥(eK +EK)
+�
+= −A(x)⟨je ·E⟩+A(x)
+�
+Se,coll
+�
+(B13)
+A similar expression can be written for the ions. if we make
+use of the Poynting theorem, we can relate the time averaged
+joule term to the divergence of the pointing flux Π = E×B
+µ0 :
+A(x)⟨(je +ji)·E⟩+ ∂
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+(B14)
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+

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+Multiplexed random-access optical memory in warm cesium vapor
+Leon Meßner,1, 2, a) Elizabeth Robertson,2, 3 Luisa Esguerra,2, 3 Kathy L¨udge,4 and Janik Wolters2, 3
+1)Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany.
+2)Deutsches Zentrum f¨ur Luft- und Raumfahrt e.V. (DLR), Institute of Optical Sensor Systems, Rutherfordstr. 2, 12489 Berlin,
+Germany.
+3)Technische Universit¨at Berlin, Institut f¨ur Optik und Atomare Physik, Str. des 17 Juni 135, 10623 Berlin,
+Germany
+4)Technische Universit¨at Ilmenau, Institut f¨ur Physik, Weimarer Straße 25, 98693 Ilmenau,
+Germany
+(Dated: 13 January 2023)
+The ability to store large amounts of photonic quantum states is regarded as substantial for future optical quantum
+computation and communication technologies. However, research for multiplexed quantum memories has been focused
+on systems that show good performance only after an elaborate preparation of the storage media. This makes it generally
+more difficult to apply outside a laboratory environment. In this work, we demonstrate a multiplexed random-access
+memory to store up to four optical pulses using electromagnetically induced transparency in warm cesium vapor. Using
+a Λ-System on the hyperfine transitions of the Cs D1 line, we achieve a mean internal storage efficiency of 36% and a
+1/e lifetime of 3.2 µs. In combination with future improvements, this work facilitates the implementation of multiplexed
+memories in future quantum communication and computation infrastructures.
+I.
+INTRODUCTION
+Quantum memories are considered to be a main component for the realization of many future second generation quantum
+technologies. Their potential use ranges from synchronizing inputs into various types of quantum systems1 to re-configurable
+optical reservoir computing2. They enable on-demand operation of otherwise probabilistic single-photon sources and quantum
+gates3, which will significantly enhance their rate of operation4. Moreover, they have been identified as an essential device
+required to realize a quantum repeater5, a key technology needed for long-distance quantum communication. When specifically
+considering the implementation of a global quantum communication network, satellite based quantum communication has been
+hallmarked as a most promising system if enhanced with a multiplexed quantum memory. It has been shown that a significant
+increase in communication rate is already achievable with around 1000 randomly accessible storage modes6,7. Consequently,
+the realization of suitable multiplexed quantum memories will be an important milestone in extending quantum communication
+over long distances.
+Quantum memories have been demonstrated using a variety of storage protocols in a number of single emitter and ensemble-
+based matter systems. These include solid state systems, ultra-cold atoms and warm atomic vapors1. Although the routine
+formation of subnanokelvin Bose-Einstein condensates in earth’s orbit has been demonstrated8 and is pursued in future projects9,
+reducing the technological requirements for space-borne quantum memories is a key step. This makes memories based on warm
+atomic vapors favored for applications, as they require no vacuum, laser cooling or strong magnetic fields.
+The memory used in this experiment utilizes the effect of electromagnetically induced transparency (EIT) on the Λ-system
+composed by the 62S1/2F=3, F=4 and the 62P1/2F=3 atomic hyperfine levels of an ensemble of cesium atoms to map optical
+excitations to a long-lived spin-wave, i.e. a coherence of the two hyperfine ground states of the atomic ensemble10,11. Due to its
+coherent ensemble origin, this spin-wave shows comparatively low dephasing and loss. Subsequent retrieval of the spin-wave
+excitation into the input optical mode can then be performed at some chosen time that is smaller than the spin-wave lifetime.
+Light storage for up to 1 s12 and single-photon operation13,14 have been demonstrated in separate experiments in warm atomic
+vapors.
+For a quantum memory to be most useful within a quantum communication system, the memory must be scaleable with
+the possibility to access individual storage modes in a way that is not significantly limited by the used technology. Various
+forms of quantum memory multiplexing have been studied in the past, including time bin15, orbital angular momentum16, and
+spatial17,18 multiplexing. Among these approaches, Ref.18 shows a clear foreseeable path to achieving the required number of
+1000 randomly accessible modes. However, the technological overhead of a cold atom setup complicates operating these outside
+of a laboratory. In this work we demonstrate a memory that combines the advantages of using warm vapor with a path towards
+scaleable multi-rail operation.
+a)Electronic mail: [email protected]
+arXiv:2301.04885v1  [quant-ph]  12 Jan 2023
+
+2
+FIG. 1. Memory scheme. a) Scheme of the Cs D1 energy levels’ hyperfine structure that forms the Λ-system used. Signal and control fields
+are red detuned from resonance by ∆ and have angular frequencies ωs and ωc respectively. The transition F = 4 → F ′ = 4 is resonantly
+driven by the pump field with angular frequency ωp. b) Time traces from an experiment for writing a signal pulse onto an atomic spin-wave at
+time t0 = 0 and retrieving it at t0 + 0.4 µs. The bottom panel shows the operation performed on the memory (W: write, R: read).
+II.
+EXPERIMENT
+The multi-rail memory presented here uses an EIT memory scheme19 on the hyperfine ground state transitions of the Cs
+D1 line, as shown in Fig. 1(a). The control and signal lasers (14-pin butterfly external cavity diode laser (ECDL) modules by
+Sacher Lasertechnik) are set on the F=4 → F ′=3 and the F=3 → F ′=3 transitions respectively. To stabilize the frequency
+difference between signal and control laser to the Cs ground state hyperfine splitting, light from both lasers is superimposed
+on a fast photodiode (Electro Optics Technology, ET-3500FEXR) and their beat frequency is offset locked20 using a RedPitaya
+FPGA-board running the Linien21 locking software. We generate Gaussian signal and control pulses with a full width at half
+maximum (FWHM) of 25 ns and 43.75 ns respectively using an arbitrary function generator (AFG, Tek AFG31152). These
+pulses are modulated onto cw laser beams with electro-optic amplitude modulators (EOMs, Jenoptik AM905). The experiment
+is designed such that the signal and control pulses have linear and orthogonal polarization to each other to reduce the control
+light leaking into the detection path.
+The atomic storage medium is confined to a cylindrical, 25x75 mm anti-reflection coated cesium vapor cell filled with 5 torr
+N2 of buffer gas. It is kept at 60°C and shielded from ambient magnetic fields by a double-layered mu-metal housing. Spatial
+addressing of different rails within the cell is performed by one acousto-optic deflector (AOD) in front and one behind the vapor
+cell. The AODs (AA MT200-B100A0,5-800) have an aperture of 0.5x2 mm2 and a measured deflection of 0.2 mrad/MHz.
+Each of them is driven by the frequency sum of an arbitrary function generator (AFG, Tek AFG31152) and a local oscillator
+(Mini-Circuits, ZOS-300+) resulting in 200±50 MHz of carrier frequency. We refer to the position of memory rails by the
+AOD driving frequency used to deflect the beam to that position; the distance between rails is thus expressed as the difference
+in driving frequency. Changing the AOD driving frequency by 8 MHz changes the lateral position of the deflected beam by
+270 µm, equaling one signal beam radius.
+While the signal pulses enter the memory unmodified after their generation, the control laser pulses are amplified by a self-
+made tapered amplifier (TA, see22) and then spectrally filtered by a dielectric bandpass interference filter (IF) with a 1 nm
+FWHM. This results in 200 mW of coupled cw power. To increase the control laser’s on-off ratio, the TA diode’s driving current
+is only switched on for 120 ns, centered on the optical pulse, using a 4 A dc-coupled input follower driver with 2 ns rise/fall
+time. This reduces the unwanted interaction between control laser and atoms in times when no control pulse is generated.
+The beam paths of the cross-polarized signal and control lasers are overlapped at a polarizing beam splitter (PBS) on one side
+of the memory and then propagate collinearly through the cesium cell, and both AODs. At the position of the cell, control and
+signal beam have a 1/e2-level radius of 350 µm and 270 µm respectively. This yields an atomic transit time of ∆t = 3.7 µs for
+one signal beam radius when using a 2D diffusion model of ∆x =
+√
+4D∆t for the diffusion length and an assumed diffusion
+constant of D0 = 0.24 cm2s−123 at T0 = 0 K and P0 = 760 torr. After traversing the second AOD, the signal and control
+beams are split by a second PBS and are then individually coupled to fibers and detected by either a Si photodiode (Thorlabs
+DET10A2) or a Si avalanche photodiode (Menlo Systems APD210).
+Optical pumping of the Cs atoms into the 62S1/2F=3 state is performed by a third ECDL laser locked to the 62S1/2F=4 →
+
+a)
+b)
+5.
+SIGNAL
+62P1/ 2
+F'= 4
+F'= 3
+[arb.]
+CONTROL
+Intensity
+m
+m
+dm
+DETECTOR
+F= 4
+62S1/ 2
+9.2 GHz
+F= 3
+PUMP
+W
+R
+R
+- 0.5
+0.0
+0.5
+1.0
+Time [μs]3
+FIG. 2. Sketch of the experiment with A) laser sources, spectroscopy and optical pulse shaping, B) TA based pulse amplification, C) multi-rail
+storage system and D) CCD image of the four used rails, with the camera at the place of the Cs cell. HWP: half-wave plate, QWP: quarter-wave
+plate, DET: detector, (P)BS: (polarizing) beam splitter, L1;L2/L3: aspheric/cylindrical lens, AOD: acousto-optic deflector, EOM: electro-optic
+modulator, AFG: arbitrary function generator, IF: interference filter, OL: offset lock.
+62P1/2F=4 transition by saturated absorption spectroscopy. The pump light power is controlled via transmission through an
+electrically pulsed semiconductor optical amplifier (SOA) and illuminates each memory rail with 20 mW of optical power for
+900 ns prior to the memory experiment sequence.
+Figure 1(b) shows a typical time trace for a single-rail storage experiment and a sketch of the experimental setup can be seen
+in Fig. 2. Several features of storage within an EIT medium can be observed in the time trace. At t = 0 µs a signal pulse enters
+the atomic medium and is partly mapped to an atomic spin wave by the control laser field. The portion of that signal pulse that
+is transmitted through the atomic vapor is detected by the photodiode as leakage. After 0.4 µs the control laser field is switched
+on again and retrieves the spin-wave excitation back into the signal beams optical mode. A third pulse of the control laser at
+t = 0.8 µs serves to determine if all the excitation has been retrieved and also allows to estimate the signal noise induced by
+the control laser field. Not having a significant detection event during this last pulse, we conclude that nearly all the spin-wave
+excitation is mapped backed to optical and signal to noise ratio is not a limiting factor for this experiment with aforementioned
+laser pulses.
+III.
+RESULTS
+Prior to performing multi-rail storage, we first identify optimal operating conditions for the multi-rail memory by assessing
+the influence of rail separation on the interactions between two memory rails, and subsequently minimizing the cross-talk. For
+comparison of the single and multi-rail operation, we measure the 1/e lifetime and memory efficiency per rail.
+To assess the influence of rail separation on their interaction, multiple storage experiments at different rail separations are
+conducted. For effective operation of a memory, we require that operations on a given memory rail do not affect its neighbors.
+To determine the minimal separation that shows no cross-talk, we write into a rail fixed at 190 MHz, read from a neighboring rail,
+and then read from the 190 MHz rail again. This write/read/read sequence is depicted in the inset of Fig. 3. The rail separation
+is varied from 0 to 25 MHz at steps of 1 MHz, and the retrieval peak intensities after the first and second read are measured. The
+results are depicted in Fig. 3. Below 5 to 8 MHz of separation no excitation is left for the second retrieval pulse and both read
+pulses address the same ensemble of atoms. At a separation of 20 MHz the influence of the read operation on the neighboring
+rail is no longer visible.
+The AOD device used has a 100 MHz bandwidth and a 25% reduced diffraction efficiency at the edges of the frequency range;
+consequently we chose to limit this experiment to four memory rails spaced by 20 MHz. A CCD image of the four rails, taken
+at the position of the Cs cell, is shown in Fig. 2(D). A straightforward method to increase the number of rails, is to use AODs
+with a higher number of resolvable spots.
+The 1/e storage lifetime per rail is determined by performing storage experiments with increasing time delay between the
+memory write and read operation, for each rail. The delay was varied between 0.4 µs and 11.2 µs in steps of 400 ns, and for each
+
+A)
+duwnd
+EOM
+C)
+ signal
+HWP
+AOD
+L2
+AOD
+HWP
+PBS
+HwP
+L1
+QWP
+DET
+LASER
+to AWG
+BS
+PBS
+PBS
+to OL
+control
+HWP
+PBS
+LASER
+EOM
+D)
+B)
+BP
+HWP
+1nm HWP
+PBS
+20 MHz
+『』  TA-diode L2L3
+opt, isolator
+PD
+625 um4
+Peak 1
+Peak 2
+0
+5
+10
+15
+20
+25
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Rail separation [MHz]
+Intensity [arb.]
+0.0
+0.5
+1.0
+Time [µs]
+W
+R
+R
+FIG. 3. Cross-talk estimation. Intensities detected in the first (Peak 1, orange rail) and second (Peak 2, green rail) read peak depending on the
+rail separation for the experiment sequence depicted in the inset. For a difference of 20 MHz in AOD driving frequency (rail separation), the
+influence between neighboring rails vanishes. The inset shows the used experiment sequence consisting of a write on the green rail at t = 0, a
+read on a neighboring orange rail at t = 0.4 µs and finally a read on the green rail at t = 0.8 µs.
+delay, we measure the retrieved peak intensities, averaged over 500 repetitions. Uncertainties are given by the standard deviation
+of the intensities. The intensities were fitted with an exponential function to extract the 1/e lifetime. Measured retrieval peak
+intensities and fit function are displayed in Fig. 4.
+The resulting lifetime values per rail are shown in Table I together with the achieved internal memory efficiencies ηmem at
+t = 0.
+Since the measured lifetimes are consistent with the estimate using the simple diffusion model presented earlier, it is reasonable
+to assume that diffusion is the most important lifetime-limiting process for the beam diameters chosen in this work.
+Independent investigation on a single rail setup also showed that spin polarization lifetimes at least on the order of several
+hundred microseconds are possible with larger beam diameters.
+The memory efficiencies are calculated by extrapolating the pulse energy of a retrieved pulse after t = 0 µs of storage from a
+retrieved pulse after t = 0.4 µs of storage using the memory lifetime. This is then divided by the energy of a normalization pulse
+to yield the efficiency. To obtain the normalization pulse, we set the signal laser frequency 2 GHz below the F=3 → F ′=3
+transition frequency, block the control beam and record the transmitted signal pulse. Under these conditions, we assume the
+pulses not to be absorbed by the atoms.
+Using the insights and results from the measurements on lifetime, efficiency and rail separation, we now explore the possibility
+of random-access operation in the memory setup. For this purpose an experimental sequence was designed that highlights
+important criteria for use as a random-access quantum memory. Figure 5 illustrates this experiment.
+The bottom panel depicts the operation performed on each specific rail and the top panel shows the intensity detected by
+the APD over a time span of about 5 µs. The experimental sequence contains 12 operations, either read (r) or write (w). We
+define three features which are necessary for use as a memory: a) that reading or writing to a rail should not affect its neighbors
+(interaction-free), b) rails which have not been written to should not return a retrieved pulse (empty state) and c) a read should
+leave the memory empty; a subsequent read pulse should yield no excitation (full retrieval).
+Rail (MHz)
+170
+190
+210
+230 Mean
+Lifetime (µs)
+4.3(5) 5.4(7) 3.3(3) 2.6(3) 3.2(2)
+Efficiency (%)
+32
+35
+39
+36
+36
+TABLE I. Measured 1/e-lifetime and retrieval efficiency for each rail and weighted mean.
+
+5
+FIG. 4. Measured retrieval amplitudes for storage times between 0.4 and 11.4 µs together with an exponential fit to the values. The inset shows
+the per rail 1/e storage lifetime deduced from the fit.
+0.00
+0.05
+0.10
+0.15
+0.20
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Intensity [arb.]
+Operation #
+0
+1
+2
+3
+4
+230MHz
+210MHz
+190MHz
+170MHz
+W
+W
+R
+R
+W
+R
+W
+R
+R
+R
+W
+R
+Time [µs]
+FIG. 5. Storage experiment in the random-access memory using four rails with detected intensity in the signal path in the top and performed
+operations (read/write) in the bottom panel. A total of 12 operations are performed over a span of 5 µs.
+
+0.20
+1/e Lifetime [μus]
+0.15
+4
+Intensity [arb.]
+3
+王
+0.10
+2
+170
+190
+210
+230
+Rail [MHz]
+0.05
+0.00F
+0
+2
+4
+6
+8
+10
+12
+Time [μs]6
+Interaction-free operation is ensured by choosing an adequate rail separation and then confirmed by looking at the storage
+performance of a specific rail while there are operations performed on the neighboring rails. The rails at 230 MHz (blue) and
+190 MHz (green) can be used to show that operations are interaction free. Between the write (t = 1.6 µs) and read (t = 3.6 µs)
+operation on the 190 MHz rail, four operations are performed on the neighboring rails and there is no visible impact on the read
+peak shape or height. On the 230 MHz rail a pulse is written at t = 0 and then retrieved during the last operation on the memory
+at t = 4.4 µs. Taking into account the 2.6 µs lifetime of this rail, the high remaining intensity of the retrieval peak clearly shows
+that there is no significant detrimental influence from multi-rail operation.
+Reading an empty rail should not result in a significant amount of intensity. We verify this by reading the 210 MHz and
+170 MHz rail in a state that should not have excitation. In the 210 MHz rail a pulse is written to the memory at t = 0.4 µs
+and then this rail is immediately read twice. The second read operation at t = 0.8 µs yields negligible intensity compared to
+the first read operation at 0.6 µs. Additionally the same rail is read again at 3.2 µs to observe the amount of noise, which is
+found to be comparable to the read at 1.2 µs. The first operation on the 190 MHz rail at t = 2 µs is a read of a rail that has
+not been used before. This allows us to determine how well the memory was initialized by the pumping that is performed prior
+to the experimental sequence. Observing a larger intensity peak would point to insufficient polarization of the medium. As the
+observed peak is similar to the other reads of an empty rail mentioned above, we conclude that pumping is sufficient and reading
+an empty rail, regardless of its history, does not lead to the detection of a significant peak. In combination with the measurements
+on lifetime and rail interaction it follows that this setup allows random-access storage and retrieval of optical pulses for times
+comparable to the mean rail lifetime of 3.2 µs.
+IV.
+CONCLUSION
+We have presented a multiplexed optical random-access memory, realised within a single vapor cell at a temperature of 60°C.
+Using an EIT based storage scheme in a Λ-system on the cesium hyperfine transitions, we achieved a mean storage lifetime
+and internal efficiency of 3.2(2) µs and 36% respectively in multi-rail operation. According to the chosen rail separation of
+20 MHz, we performed random-access storage and retrieval in four parallel rails without observing reciprocal in��uence between
+the different rails.
+The time between successive operations was chosen to be 400 ns for the sake of simplifying experiment control. This time
+could be reduced considerably with the lower bound determined by the AODs switching time of 48 ns. Increasing the storage
+lifetime and number of addressable rails is possible by increasing the beam diameters and using AODs that have a higher
+time-bandwidth product respectively. This step will be important for applications in quantum communication and repeater
+networks. Reaching beyond the threshold number of 1000 individually addressable modes is possible by using 2-axis AODs and
+multiplexing into a two dimensional grid of parallel storage modes.
+FUNDING
+This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number
+445183921. E.R. acknowledges funding through the Helmholtz Einstein International Berlin Research School in Data Science
+(HEIBRiDS).
+DISCLOSURES
+The authors declare no conflicts of interest.
+DATA AVAILABILITY
+The data presented in this paper is available from the authors upon reasonable request.
+REFERENCES
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+

B9E4T4oBgHgl3EQfFQzj/content/tmp_files/load_file.txt

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+page_content='Multiplexed random-access optical memory in warm cesium vapor  Leon Meßner,1, 2, a) Elizabeth Robertson,2, 3 Luisa Esguerra,2, 3 Kathy L¨udge,4 and Janik Wolters2, 3  1)Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 15, 12489 Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  2)Deutsches Zentrum f¨ur Luft- und Raumfahrt e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' (DLR), Institute of Optical Sensor Systems, Rutherfordstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2, 12489 Berlin,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  3)Technische Universit¨at Berlin, Institut f¨ur Optik und Atomare Physik, Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' des 17 Juni 135, 10623 Berlin,  Germany  4)Technische Universit¨at Ilmenau, Institut f¨ur Physik, Weimarer Straße 25, 98693 Ilmenau,  Germany  (Dated: 13 January 2023)  The ability to store large amounts of photonic quantum states is regarded as substantial for future optical quantum  computation and communication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' However, research for multiplexed quantum memories has been focused  on systems that show good performance only after an elaborate preparation of the storage media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This makes it generally  more difficult to apply outside a laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In this work, we demonstrate a multiplexed random-access  memory to store up to four optical pulses using electromagnetically induced transparency in warm cesium vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Using  a Λ-System on the hyperfine transitions of the Cs D1 line, we achieve a mean internal storage efficiency of 36% and a  1/e lifetime of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In combination with future improvements, this work facilitates the implementation of multiplexed  memories in future quantum communication and computation infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  INTRODUCTION  Quantum memories are considered to be a main component for the realization of many future second generation quantum  technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Their potential use ranges from synchronizing inputs into various types of quantum systems1 to re-configurable  optical reservoir computing2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' They enable on-demand operation of otherwise probabilistic single-photon sources and quantum  gates3, which will significantly enhance their rate of operation4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Moreover, they have been identified as an essential device  required to realize a quantum repeater5, a key technology needed for long-distance quantum communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' When specifically  considering the implementation of a global quantum communication network, satellite based quantum communication has been  hallmarked as a most promising system if enhanced with a multiplexed quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' It has been shown that a significant  increase in communication rate is already achievable with around 1000 randomly accessible storage modes6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Consequently,  the realization of suitable multiplexed quantum memories will be an important milestone in extending quantum communication  over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Quantum memories have been demonstrated using a variety of storage protocols in a number of single emitter and ensemble-  based matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' These include solid state systems, ultra-cold atoms and warm atomic vapors1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Although the routine  formation of subnanokelvin Bose-Einstein condensates in earth’s orbit has been demonstrated8 and is pursued in future projects9,  reducing the technological requirements for space-borne quantum memories is a key step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This makes memories based on warm  atomic vapors favored for applications, as they require no vacuum, laser cooling or strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The memory used in this experiment utilizes the effect of electromagnetically induced transparency (EIT) on the Λ-system  composed by the 62S1/2F=3, F=4 and the 62P1/2F=3 atomic hyperfine levels of an ensemble of cesium atoms to map optical  excitations to a long-lived spin-wave, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a coherence of the two hyperfine ground states of the atomic ensemble10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Due to its  coherent ensemble origin, this spin-wave shows comparatively low dephasing and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Subsequent retrieval of the spin-wave  excitation into the input optical mode can then be performed at some chosen time that is smaller than the spin-wave lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Light storage for up to 1 s12 and single-photon operation13,14 have been demonstrated in separate experiments in warm atomic  vapors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  For a quantum memory to be most useful within a quantum communication system, the memory must be scaleable with  the possibility to access individual storage modes in a way that is not significantly limited by the used technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Various  forms of quantum memory multiplexing have been studied in the past, including time bin15, orbital angular momentum16, and  spatial17,18 multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Among these approaches, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='18 shows a clear foreseeable path to achieving the required number of  1000 randomly accessible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' However, the technological overhead of a cold atom setup complicates operating these outside  of a laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In this work we demonstrate a memory that combines the advantages of using warm vapor with a path towards  scaleable multi-rail operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  a)Electronic mail: messner@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='de  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='04885v1  [quant-ph]  12 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Memory scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a) Scheme of the Cs D1 energy levels’ hyperfine structure that forms the Λ-system used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Signal and control fields  are red detuned from resonance by ∆ and have angular frequencies ωs and ωc respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The transition F = 4 → F ′ = 4 is resonantly  driven by the pump field with angular frequency ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' b) Time traces from an experiment for writing a signal pulse onto an atomic spin-wave at  time t0 = 0 and retrieving it at t0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The bottom panel shows the operation performed on the memory (W: write, R: read).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  EXPERIMENT  The multi-rail memory presented here uses an EIT memory scheme19 on the hyperfine ground state transitions of the Cs  D1 line, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The control and signal lasers (14-pin butterfly external cavity diode laser (ECDL) modules by  Sacher Lasertechnik) are set on the F=4 → F ′=3 and the F=3 → F ′=3 transitions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To stabilize the frequency  difference between signal and control laser to the Cs ground state hyperfine splitting, light from both lasers is superimposed  on a fast photodiode (Electro Optics Technology, ET-3500FEXR) and their beat frequency is offset locked20 using a RedPitaya  FPGA-board running the Linien21 locking software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We generate Gaussian signal and control pulses with a full width at half  maximum (FWHM) of 25 ns and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='75 ns respectively using an arbitrary function generator (AFG, Tek AFG31152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' These  pulses are modulated onto cw laser beams with electro-optic amplitude modulators (EOMs, Jenoptik AM905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The experiment  is designed such that the signal and control pulses have linear and orthogonal polarization to each other to reduce the control  light leaking into the detection path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The atomic storage medium is confined to a cylindrical, 25x75 mm anti-reflection coated cesium vapor cell filled with 5 torr  N2 of buffer gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' It is kept at 60°C and shielded from ambient magnetic fields by a double-layered mu-metal housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Spatial  addressing of different rails within the cell is performed by one acousto-optic deflector (AOD) in front and one behind the vapor  cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The AODs (AA MT200-B100A0,5-800) have an aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5x2 mm2 and a measured deflection of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 mrad/MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Each of them is driven by the frequency sum of an arbitrary function generator (AFG, Tek AFG31152) and a local oscillator  (Mini-Circuits, ZOS-300+) resulting in 200±50 MHz of carrier frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We refer to the position of memory rails by the  AOD driving frequency used to deflect the beam to that position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' the distance between rails is thus expressed as the difference  in driving frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Changing the AOD driving frequency by 8 MHz changes the lateral position of the deflected beam by  270 µm, equaling one signal beam radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  While the signal pulses enter the memory unmodified after their generation, the control laser pulses are amplified by a self-  made tapered amplifier (TA, see22) and then spectrally filtered by a dielectric bandpass interference filter (IF) with a 1 nm  FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This results in 200 mW of coupled cw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To increase the control laser’s on-off ratio, the TA diode’s driving current  is only switched on for 120 ns, centered on the optical pulse, using a 4 A dc-coupled input follower driver with 2 ns rise/fall  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This reduces the unwanted interaction between control laser and atoms in times when no control pulse is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The beam paths of the cross-polarized signal and control lasers are overlapped at a polarizing beam splitter (PBS) on one side  of the memory and then propagate collinearly through the cesium cell, and both AODs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' At the position of the cell, control and  signal beam have a 1/e2-level radius of 350 µm and 270 µm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This yields an atomic transit time of ∆t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='7 µs for  one signal beam radius when using a 2D diffusion model of ∆x =  √  4D∆t for the diffusion length and an assumed diffusion  constant of D0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='24 cm2s−123 at T0 = 0 K and P0 = 760 torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' After traversing the second AOD, the signal and control  beams are split by a second PBS and are then individually coupled to fibers and detected by either a Si photodiode (Thorlabs  DET10A2) or a Si avalanche photodiode (Menlo Systems APD210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Optical pumping of the Cs atoms into the 62S1/2F=3 state is performed by a third ECDL laser locked to the 62S1/2F=4 →  a)  b)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content="  SIGNAL  62P1/ 2  F'= 4  F'= 3  [arb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=']  CONTROL  Intensity  m  m  dm  DETECTOR  F= 4  62S1/ 2  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 GHz  F= 3  PUMP  W  R  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0  Time [μs]3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Sketch of the experiment with A) laser sources, spectroscopy and optical pulse shaping, B) TA based pulse amplification, C) multi-rail  storage system and D) CCD image of the four used rails, with the camera at the place of the Cs cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' HWP: half-wave plate, QWP: quarter-wave  plate, DET: detector, (P)BS: (polarizing) beam splitter, L1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='L2/L3: aspheric/cylindrical lens, AOD: acousto-optic deflector, EOM: electro-optic  modulator, AFG: arbitrary function generator, IF: interference filter, OL: offset lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  62P1/2F=4 transition by saturated absorption spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The pump light power is controlled via transmission through an  electrically pulsed semiconductor optical amplifier (SOA) and illuminates each memory rail with 20 mW of optical power for  900 ns prior to the memory experiment sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Figure 1(b) shows a typical time trace for a single-rail storage experiment and a sketch of the experimental setup can be seen  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Several features of storage within an EIT medium can be observed in the time trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' At t = 0 µs a signal pulse enters  the atomic medium and is partly mapped to an atomic spin wave by the control laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The portion of that signal pulse that  is transmitted through the atomic vapor is detected by the photodiode as leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' After 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs the control laser field is switched  on again and retrieves the spin-wave excitation back into the signal beams optical mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A third pulse of the control laser at  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='8 µs serves to determine if all the excitation has been retrieved and also allows to estimate the signal noise induced by  the control laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Not having a significant detection event during this last pulse, we conclude that nearly all the spin-wave  excitation is mapped backed to optical and signal to noise ratio is not a limiting factor for this experiment with aforementioned  laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  RESULTS  Prior to performing multi-rail storage, we first identify optimal operating conditions for the multi-rail memory by assessing  the influence of rail separation on the interactions between two memory rails, and subsequently minimizing the cross-talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For  comparison of the single and multi-rail operation, we measure the 1/e lifetime and memory efficiency per rail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  To assess the influence of rail separation on their interaction, multiple storage experiments at different rail separations are  conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For effective operation of a memory, we require that operations on a given memory rail do not affect its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  To determine the minimal separation that shows no cross-talk, we write into a rail fixed at 190 MHz, read from a neighboring rail,  and then read from the 190 MHz rail again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This write/read/read sequence is depicted in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The rail separation  is varied from 0 to 25 MHz at steps of 1 MHz, and the retrieval peak intensities after the first and second read are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The  results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Below 5 to 8 MHz of separation no excitation is left for the second retrieval pulse and both read  pulses address the same ensemble of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' At a separation of 20 MHz the influence of the read operation on the neighboring  rail is no longer visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The AOD device used has a 100 MHz bandwidth and a 25% reduced diffraction efficiency at the edges of the frequency range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  consequently we chose to limit this experiment to four memory rails spaced by 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A CCD image of the four rails, taken  at the position of the Cs cell, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A straightforward method to increase the number of rails, is to use AODs  with a higher number of resolvable spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The 1/e storage lifetime per rail is determined by performing storage experiments with increasing time delay between the  memory write and read operation, for each rail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The delay was varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs in steps of 400 ns, and for each  A)  duwnd  EOM  C)  signal  HWP  AOD  L2  AOD  HWP  PBS  HwP  L1  QWP  DET  LASER  to AWG  BS  PBS  PBS  to OL  control  HWP  PBS  LASER  EOM  D)  B)  BP  HWP  1nm HWP  PBS  20 MHz  『』  TA-diode L2L3  opt, isolator  PD  625 um4  Peak 1  Peak 2  0  5  10  15  20  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='10  Rail separation [MHz]  Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=']  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='0  Time [µs]  W  R  R  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Cross-talk estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Intensities detected in the first (Peak 1, orange rail) and second (Peak 2, green rail) read peak depending on the  rail separation for the experiment sequence depicted in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For a difference of 20 MHz in AOD driving frequency (rail separation), the  influence between neighboring rails vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The inset shows the used experiment sequence consisting of a write on the green rail at t = 0, a  read on a neighboring orange rail at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs and finally a read on the green rail at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='8 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  delay, we measure the retrieved peak intensities, averaged over 500 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Uncertainties are given by the standard deviation  of the intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The intensities were fitted with an exponential function to extract the 1/e lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Measured retrieval peak  intensities and fit function are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The resulting lifetime values per rail are shown in Table I together with the achieved internal memory efficiencies ηmem at  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Since the measured lifetimes are consistent with the estimate using the simple diffusion model presented earlier, it is reasonable  to assume that diffusion is the most important lifetime-limiting process for the beam diameters chosen in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Independent investigation on a single rail setup also showed that spin polarization lifetimes at least on the order of several  hundred microseconds are possible with larger beam diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The memory efficiencies are calculated by extrapolating the pulse energy of a retrieved pulse after t = 0 µs of storage from a  retrieved pulse after t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs of storage using the memory lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This is then divided by the energy of a normalization pulse  to yield the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' To obtain the normalization pulse, we set the signal laser frequency 2 GHz below the F=3 → F ′=3  transition frequency, block the control beam and record the transmitted signal pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Under these conditions, we assume the  pulses not to be absorbed by the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Using the insights and results from the measurements on lifetime, efficiency and rail separation, we now explore the possibility  of random-access operation in the memory setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' For this purpose an experimental sequence was designed that highlights  important criteria for use as a random-access quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Figure 5 illustrates this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The bottom panel depicts the operation performed on each specific rail and the top panel shows the intensity detected by  the APD over a time span of about 5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The experimental sequence contains 12 operations, either read (r) or write (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We  define three features which are necessary for use as a memory: a) that reading or writing to a rail should not affect its neighbors  (interaction-free), b) rails which have not been written to should not return a retrieved pulse (empty state) and c) a read should  leave the memory empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' a subsequent read pulse should yield no excitation (full retrieval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Rail (MHz)  170  190  210  230 Mean  Lifetime (µs)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='3(5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4(7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='3(3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6(3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2(2)  Efficiency (%)  32  35  39  36  36  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Measured 1/e-lifetime and retrieval efficiency for each rail and weighted mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Measured retrieval amplitudes for storage times between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs together with an exponential fit to the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The inset shows  the per rail 1/e storage lifetime deduced from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='20  1  2  3  4  5  6  7  8  9  10  11  12  Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=']  Operation #  0  1  2  3  4  230MHz  210MHz  190MHz  170MHz  W  W  R  R  W  R  W  R  R  R  W  R  Time [µs]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Storage experiment in the random-access memory using four rails with detected intensity in the signal path in the top and performed  operations (read/write) in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' A total of 12 operations are performed over a span of 5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='20  1/e Lifetime [μus]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='15  4  Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=']  3  王  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='10  2  170  190  210  230  Rail [MHz]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='00F  0  2  4  6  8  10  12  Time [μs]6  Interaction-free operation is ensured by choosing an adequate rail separation and then confirmed by looking at the storage  performance of a specific rail while there are operations performed on the neighboring rails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The rails at 230 MHz (blue) and  190 MHz (green) can be used to show that operations are interaction free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Between the write (t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs) and read (t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs)  operation on the 190 MHz rail, four operations are performed on the neighboring rails and there is no visible impact on the read  peak shape or height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' On the 230 MHz rail a pulse is written at t = 0 and then retrieved during the last operation on the memory  at t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Taking into account the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs lifetime of this rail, the high remaining intensity of the retrieval peak clearly shows  that there is no significant detrimental influence from multi-rail operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Reading an empty rail should not result in a significant amount of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' We verify this by reading the 210 MHz and  170 MHz rail in a state that should not have excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In the 210 MHz rail a pulse is written to the memory at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='4 µs  and then this rail is immediately read twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The second read operation at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='8 µs yields negligible intensity compared to  the first read operation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='6 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Additionally the same rail is read again at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs to observe the amount of noise, which is  found to be comparable to the read at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' The first operation on the 190 MHz rail at t = 2 µs is a read of a rail that has  not been used before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This allows us to determine how well the memory was initialized by the pumping that is performed prior  to the experimental sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Observing a larger intensity peak would point to insufficient polarization of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' As the  observed peak is similar to the other reads of an empty rail mentioned above, we conclude that pumping is sufficient and reading  an empty rail, regardless of its history, does not lead to the detection of a significant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' In combination with the measurements  on lifetime and rail interaction it follows that this setup allows random-access storage and retrieval of optical pulses for times  comparable to the mean rail lifetime of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  CONCLUSION  We have presented a multiplexed optical random-access memory, realised within a single vapor cell at a temperature of 60°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  Using an EIT based storage scheme in a Λ-system on the cesium hyperfine transitions, we achieved a mean storage lifetime  and internal efficiency of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='2(2) µs and 36% respectively in multi-rail operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' According to the chosen rail separation of  20 MHz, we performed random-access storage and retrieval in four parallel rails without observing reciprocal influence between  the different rails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  The time between successive operations was chosen to be 400 ns for the sake of simplifying experiment control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This time  could be reduced considerably with the lower bound determined by the AODs switching time of 48 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Increasing the storage  lifetime and number of addressable rails is possible by increasing the beam diameters and using AODs that have a higher  time-bandwidth product respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' This step will be important for applications in quantum communication and repeater  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Reaching beyond the threshold number of 1000 individually addressable modes is possible by using 2-axis AODs and  multiplexing into a two dimensional grid of parallel storage modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  FUNDING  This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number  445183921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' acknowledges funding through the Helmholtz Einstein International Berlin Research School in Data Science  (HEIBRiDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  DISCLOSURES  The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content='  DATA AVAILABILITY  The data presented in this paper is available from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' Grosse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' Herr, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Israelsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' Krutzik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' L¨ammerzahl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' List,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' L¨udtke, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' M¨uller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' M¨untinga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' Oberschulte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
+page_content=' Papakonstantinou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQfFQzj/content/2301.04885v1.pdf'}
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+Draft version January 30, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Strong Variability in AzV 493, an Extreme Oe-Type Star in the SMC
+M. S. Oey,1 N. Castro,2 M. Renzo,3 I. Vargas-Salazar,1 M. W. Suffak,4 M. Ratajczak,5 J. D. Monnier,1
+M. K. Szymanski,5 G. D. Phillips,1 N. Calvet,1 A. Chiti,6, 7 G. Micheva,1, 8 K. C. Rasmussen,1, 9 and
+R. H. D. Townsend10
+1Astronomy Department, University of Michigan, 1085 South University Ave., Ann Arbor, MI, 48109, USA
+2Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany
+3Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA
+4Department of Physics and Astronomy, Western University, London, ON N6A 3K7, Canada
+5Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
+6Department of Astronomy & Astrophysics, University of Chicago, 5640 S Ellis Avenue, Chicago, IL 60637, USA
+7Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
+8Present address: Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany
+9Present address: Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA
+10Astronomy Department, University of Wisconsin, Madison, WI 53706, USA
+(Accepted January 23, 2023; to appear in the Astrophysical Journal)
+ABSTRACT
+We present 18 years of OGLE photometry together with spectra obtained over 12 years, revealing that
+the early Oe star AzV 493 shows strong photometric (∆I < 1.2 mag) and spectroscopic variability with
+a dominant, 14.6-year pattern and ∼40-day oscillations. We estimate stellar parameters Teff = 42000
+K, log L/L⊙ = 5.83 ± 0.15, M/M⊙ = 50 ± 9, and v sin i = 370 ± 40 km s−1. Direct spectroscopic
+evidence shows episodes of both gas ejection and infall. There is no X-ray detection, and it is likely
+a runaway star. AzV 493 may have an unseen companion on a highly eccentric (e > 0.93) orbit.
+We propose that close interaction at periastron excites ejection of the decretion disk, whose variable
+emission-line spectrum suggests separate inner and outer components, with an optically thick outer
+component obscuring both the stellar photosphere and the emission-line spectrum of the inner disk at
+early phases in the
+photometric cycle. It is plausible that AzV 493’s mass and rotation have been
+enhanced by binary interaction followed by the core-collapse supernova explosion of the companion,
+which now could be either a black hole or neutron star. This system in the Small Magellanic Cloud
+can potentially shed light on OBe decretion disk formation and evolution, massive binary evolution,
+and compact binary progenitors.
+Keywords: early-type stars — Oe stars — Be stars — high-mass X-ray binary stars — circumstellar
+disks — stellar pulsations — interacting binary stars — compact objects — runaway stars
+— variable stars — Small Magellanic Cloud
+1. INTRODUCTION
+Binary interactions are now understood to be a funda-
+mental component of massive star evolution, and they
+are the progenitors of a wide variety of energetic phe-
+nomena including high-mass X-ray binaries (HMXBs),
+ultra-luminous X-ray sources (ULXs), stripped-envelope
+core-collapse supernovae (SNe), and gravitational wave
+events. A consensus is emerging that classical OBe stars
+appear to originate from close massive binary systems,
+wherein they have spun up through mass and angular
+momentum transfer from their mass donors (e.g, Pols
+et al. 1991; Vinciguerra et al. 2020; Bodensteiner et al.
+2020, see also Rivinius et al. 2013 for a review). When
+donor stars subsequently explode as supernovae, result-
+ing post-explosion bound binaries are more likely to
+be eccentric, since they result from tight binaries (e.g.,
+Brandt & Podsiadlowski 1995; Tauris & Takens 1998;
+Renzo et al. 2019). Thus, a substantial subset of classi-
+cal OBe stars are likely to have eccentric orbits. In this
+paper, we present photometric and spectrocopic time-
+series data showing that the star AzV 493 exhibits dra-
+matic variability and may be an eccentric binary system.
+AzV 493 (Azzopardi et al. 1975) or [M2002]SMC-
+77616 (Massey 2002) was identified as an extreme, clas-
+arXiv:2301.11433v1  [astro-ph.SR]  26 Jan 2023
+
+2
+sical Oe star by Golden-Marx et al. (2016).
+In that
+work, it was found to be the earliest classical Oe star
+in our sample of field OB stars in the Small Magel-
+lanic Cloud (SMC), based on a spectrum obtained in
+2009 that shows double-peaked emission, not only in
+the Balmer lines, but also in He i and He ii λ4686, the
+latter feature being rarely observed in other Oe stars
+(Conti & Leep 1974). Specifically, it is classified as an
+Ope star, indicating that the He i absorption lines show
+infilled emission (Sota et al. 2011).
+As an extreme object, AzV 493 offers unique oppor-
+tunities to study massive binary evolution and decre-
+tion disk formation, structure, and dynamics. Section 2
+presents the unusual light curve and periodicity, and
+Section 3 presents our multi-epoch spectroscopy with
+resulting derived stellar parameters and individual spec-
+tral features. We then present two possible models for
+the AzV 493 system in Sections 4 and 5, one based on
+ejection of an optically thick disk near periastron; and
+another based on disk growth and disruption. Section 6
+discusses the likely binary origin of the system, and Sec-
+tion 7 summarizes our findings.
+2. PHOTOMETRIC LIGHT CURVE
+2.1. Long-term light curve
+The I and V -band light curves of AzV 493 from the
+OGLE Project (Udalski et al. 2008, 2015) are presented
+in Figure 1. The I-band shows a short eruption with
+the peak of the light curve on
+MJD 52212, followed
+by an abrupt decline of approximately 1.2 mag, to a
+minimum on MJD 52303 in early 2002. After this, the
+star eventually recovers its original luminosity. Another
+photometric minimum is seen in 2016 on MJD 57626,
+followed by the same brightening pattern.
+The gray
+symbols in Figure 1 show the I-band photometry from
+the 2016 cycle overplotted on the data from 2002 cy-
+cle. This shows that the minimum luminosity and subse-
+quent increase are quantitatively identical, although the
+photometry immediately preceding the minimum differs.
+Cross-correlating these segments yields a long-cycle pe-
+riod of 5311 days (14.55 years). There is no evidence of
+a similar eruption preceding the minimum in the 2016
+cycle on the same 91-day timescale, although the pho-
+tometry is incomplete in this range.
+After the minimum, the brightness increases and then
+starts to gradually decrease again, over a period of sev-
+eral years. Approximately in 2008, AzV 493 appears to
+go into a multiple outburst event. After this, the light
+curve drastically changes, showing a multi-mode pulsa-
+tion behavior that evolves with time (Section 2.2). The
+pulsation ends with another 0.2 – 0.3 mag drop, followed
+by a steady increase, repeating the light curve cycle that
+started in 2002, 14.55 years before.
+2.2. Photometric Oscillations
+Figure 2 shows short-term variability on the order of
+30 – 45 days. We quantify the evolution of these oscil-
+lations seen in the I-band light curve using Generalized
+Lomb-Scargle periodograms (Zechmeister & K¨urster
+2009) for the six contiguous OGLE datasets from 2010 –
+2016 (Figure 1). The individual fits to these six ranges
+are shown in Appendix A. Comparison of the periods
+shown in Figure 3 with the light curve (Figure 1) shows
+that they qualitatively appear to correlate with stellar
+brightness.
+The OGLE survey provides V -band magnitudes for
+a subset of the survey epochs, which are shown in red
+in Figure 1. Figure 4 displays the color-magnitude dia-
+gram (CMD) in V vs V − I for those days where both
+bands were observed.
+Figure 4a compares AzV 493’s
+color variations with data for the remainder of the RI-
+OTS4 sample stars (Lamb et al. 2016). The latter corre-
+spond to single-epoch photometry from the OGLE cat-
+alog of Poleski et al. (2012). Those stars classified as
+OBe stars by Lamb et al. (2016) are marked in the plot.
+The blue plume of non-emission-line stars is clearly sepa-
+rated from the cloud of OBe stars at redder colors in the
+CMD, a phenomenon already known from different pho-
+tometric bands (e.g., Bonanos et al. 2010; Castro et al.
+2018). The color variation of AzV 493 spans almost the
+entire range of V − I colors covered by the emission-line
+stars.
+Figure 4b shows a zoom in the CMD with the path of
+AzV 493 traced out. The star appears red during the
+broad peak of the light curve around 2006 (Figure 1),
+and then moves to bluer colors reaching the bluest V −I
+color during the pulsation phase.
+Approximately in
+2017, when the light curve is brightening after the mini-
+mum, AzV 493 shows redder colors again, moving to the
+original position observed in 2005 with V − I ∼ 0.18.
+Similar, semi-periodic variability with timescales on
+the order of weeks to months is seen in many other OBe
+stars, and their origin is unknown (e.g., Labadie-Bartz
+et al. 2017).
+Proposed explanations include forms of
+non-radial pulsations of the star and transitory or orbit-
+ing density enhancements in the disk, which may be the
+most likely scenario. The associated cyclical variation in
+the CMD (Figure 4) is also consistent with some kind of
+stellar radial pulsation. This is supported by the corre-
+lation between period and luminosity (cf. Figures 3 and
+4). In that case, the relatively long period implies that
+they could be an induced gravity mode or pulsational
+instability. However, there are many other possible ex-
+
+3
+Figure 1. AzV 493 OGLE light curves in I (black) and V (red) bands. The last segment of the I-band curve is overplotted
+(light grey dots) on the beginning of the dataset phase 14.6 years (5311 days) earlier. V − I is shown in the lower panel. The
+dashed lines mark the epochs for the observed spectra, assigned alphabetically in chronological sequence. The green shaded
+regions show consecutive 2656-day segments starting with the light curve maximum in 2001.
+Figure 2. Zoom on light curve (top) showing ∼ 40-day oscillations, and color variation (bottom).
+
+Time [year]
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+2022
+D
+H
+13.65
+E
+F
+13.80
+B
+C
+G
+13.95
+@ 14.10
+nitude
+14.25
+Magr
+14.40
+14.55
+14.70-
+I band
+14.85
+V band
+I band shifted -14 years
+C
+0.2
+:
+0.1
+io
+0.0
+-0.1 -
+53000
+54000
+55000
+56000
+57000
+58000
+59000
+MJD [days]Time [year]
+2010.4
+2010.6
+2010.8
+2011.0
+2011.2
+2011.4
+2011.6
+2011.8
+2012.0
+2012.2
+I band
+14.24-
+V band
+14.32
+.
+3
+14.40
+.
+.
+.
+.
+::
+.
+.
+:
+i
+14.64
+.
+.
+14.72
+14.80
+0.025
+0.000
+.
+-0.050
+.
+-0.075
+C
+-0.100
+55300
+55400
+55500
+55600
+55700
+55800
+55900
+56000
+MjD[days]4
+Figure 3.
+Fitted periods for the six contiguous OGLE
+datasets between ∼ 2010 – 2016, as a function of time.
+planations, perhaps including interactions with another
+star in a close orbit. We note that de Wit et al. (2006,
+see also Rivinius et al. 2013) reported similar loop-like
+excursions in the CMD of other OBe stars, and ascribed
+the anti-clockwise variation to the formation and dissi-
+pation of the circumstellar decretion disks in those ob-
+jects.
+2.3. Light curve period
+It is possible that the multiple-outburst event in 2008
+– 2009 may represent another periastron. Figure 1 shows
+the 5311-day cycle initiated at the light-curve peak at
+MJD 52212 instead of at the minima. We see that the
+mid-cycle occurs during this multiple-outburst event, al-
+though due to the OGLE observing cadence, it is unclear
+whether it occurs near the end or near the middle. In
+Section 3 below, we show that the spectrum obtained
+around this time, Epoch A (Figure 1), shows an un-
+usually strong emission-line spectrum, consistent with
+maximum disk activation and flaring. However, the light
+curve does not repeat the cycle minimum seen in 2002
+and 2016, and OBe stars are known to show temporary
+outbursts of activity (e.g., Labadie-Bartz et al. 2017;
+Baade et al. 2018).
+Thus, it is not clear whether 2008 – 2009 corresponds
+to the mid-cycle or not. The light curve does not repeat
+regularly in detail, and we caution that the period, if the
+system is a binary, is uncertain. Assuming that there is
+indeed a fundamental physical period, the same phases
+may not all generate the same observational signatures,
+which may depend on other factors such as disk orienta-
+tion and/or varying physical processes. In what follows,
+we adopt a system period of 5311 (2656) days, or 14.55
+(7.28) years, where the values in parentheses allow for
+the possibility that the period may be half of the long
+cycle.
+3. SPECTROSCOPY
+Spectroscopic observations of AzV 493 were obtained
+in the course of the RIOTS4 spectroscopic survey of
+field OB stars in the SMC (Lamb et al. 2016), and
+follow-up radial velocity monitoring of the SMC Wing
+region (Vargas-Salazar et al. 2023, in preparation). The
+observations were carried out using the Magellan tele-
+scopes at Las Campanas, Chile. Three different spectro-
+graphs were used: IMACS (Bigelow & Dressler 2003),
+MIKE (Bernstein et al. 2003) and M2FS (Mateo et al.
+2012). Table 1 gives details of our spectroscopic observa-
+tions, including the modified Julian day (MJD), signal-
+to-noise, spectral resolution, spectral range, phase in the
+light curve cycle, radial velocity, Hβ peak separation
+(Section 3.2), and instrument used. Figure 5 displays
+the 11 spectra in chronological sequence, labeled A – K
+as shown.
+IMACS was operated by default in multi-slit mode
+with the f/4 camera and 1200 lines/mm grating, which
+provides a resolving power of R ∼ 3000 and a wave-
+length coverage spanning ∼3800 – 5200 ˚A. One obser-
+vation (Epoch I) was observed with the f/2 camera, re-
+sulting in lower resolution (Table 1). The reduction was
+performed using the cosmos pipeline1. MIKE data were
+obtained using a 1′′ slit width for a spectral resolution
+of R ∼ 28000, covering the wavelength range ∼3600
+– 10000 ˚A. The spectra were processed with the the
+Carnegie Python (CarPy2) pipeline software (Kelson
+et al. 2000; Kelson 2003), except for Epoch B, which was
+extracted using IRAF3. M2FS data were observed us-
+ing a custom filter yielding ∼4080 – 4470 ˚A wavelength
+coverage at R ∼ 28000. The data were processed follow-
+ing the standard steps in fiber spectroscopic reduction
+using IRAF/PyRAF tasks implemented within python
+and designed for this instrument (see Walker et al. 2015).
+Figure 5 shows strong variability in the spectrum of
+AzV 493. The weaker epochs show a typical OBe spec-
+trum, with only Hβ showing double-peaked emission,
+and Hγ and Hδ absorption features showing evidence
+of infill; whereas Epochs A, B, and K show stronger
+emission-line spectra, with Hγ and He i often in emis-
+sion. Epoch F shows strong, high-order Balmer emis-
+sion and inverse P-Cygni features. These epochs will be
+discussed in Sections 3.3 – 3.4.
+1 http://code.obs.carnegiescience.edu/cosmos.
+2 http://code.obs.carnegiescience.edu/mike
+3 IRAF was distributed by the National Optical Astronomy Obser-
+vatory, which was managed by the Association of Universities for
+Research in Astronomy (AURA) under a cooperative agreement
+with the National Science Foundation.
+
+44
+42 
+40
+[days]
+38
+Period
+36
+34 
+32
+30
+55500 55750 56000 56250 56500 56750 57000 57250
+MJD [days]5
+Figure 4. Color-magnitude diagram (CMD) based on available V - and I-band OGLE photometry (see Figure 1). The variation
+of AzV 493 in the CMD is colored according to the MJD, and compared to single-epoch OGLE photometry (Poleski et al. 2012)
+for the RIOTS4 OB-star sample (Lamb et al. 2016) (grey dots). Objects classified as OBe by Lamb et al. (2016) are highlighted
+with circles. The right panel is a zoom of the same data around the track of AzV 493.
+Table 1. Spectroscopic Observations of AzV 493
+Epoch
+Date [UTC]
+MJD
+S/N
+R
+Wavelength
+Phasea
+RV
+∆v(Hβ)b
+Instrument
+Range [˚A]
+(km s−1)
+(km s−1)
+A
+2009-08-26T01:43:36.0
+55069.071944
+140
+3000
+3825–5422
+0.538 (0.076)
+152 ± 200
+279
+IMACS
+B
+2015-01-14T02:12:03.0
+57036.091701
+120
+28000
+3362–9397
+0.908 (0.817)
+192 ± 18
+(213)c
+MIKE
+C
+2016-06-15T07:47:54.3
+57554.324935
+130
+3000
+3879–5479
+0.006 (0.012)
+171 ± 60
+346
+IMACS
+D
+2016-09-08T01:42:08.0
+57639.070926
+60
+28000
+4079–4466
+0.022 (0.044)
+217 ± 50
+· · ·
+M2FS
+E
+2016-09-11T02:49:33.0
+57642.117743
+90
+28000
+4080–4465
+0.022 (0.045)
+239 ± 46
+· · ·
+M2FS
+F
+2016-09-22T05:36:51.0
+57653.233924
+150
+28000
+3538–9397
+0.024 (0.049)
+192 ± 29
+334
+MIKE
+G
+2016-12-04T04:09:41.5
+57726.173397
+110
+3000
+3862–5458
+0.038 (0.076)
+243 ± 38
+319
+IMACS
+H
+2017-06-05T06:35:11.2
+57909.274435
+50
+3000
+3871–5471
+0.073 (0.145)
+235 ± 54
+322
+IMACS
+I
+2017-06-07T08:08:18.9
+57911.339108
+130
+1300
+3900–8000
+0.073 (0.146)
+231 ± 83
+295
+IMACSd
+J
+2017-07-10T09:05:00.5
+57944.378478
+190
+3000
+3854–5468
+0.079 (0.159)
+181 ± 39
+303
+IMACS
+K
+2021-09-25T07:38:18.0
+59482.318264
+210
+28000
+3362–9397
+0.369 (0.738)
+183 ± 17
+289
+MIKE
+aPhase relative to the light curve peak at MJD 52212 (54868), adopting a period of 5311 (2655.5) days.
+b Hβ peak separation obtained by fitting two gaussians with fixed width of 2 ˚A (∼ 120 km s−1).
+c Epoch B does not show a double-peaked profile (see Figure 7 and Section 3.4); the value for ∆v(Hβ) assumes that two
+components exist, as they do for other epochs.
+dEpoch I was observed with the f/2 camera while the other IMACS observations were obtained with the f/4 camera.
+
+13.0
+O
+13.5
+O
+O
+14.0
+O
+> 14.5
+O
+O
+O
+0
+15.0
+8
+O
+0
+O
+0
+15.5
+O
+16.0
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+V-I14.0
+57600
+14.1
+57000
+14.2
+56400
+14.3
+MJD [days]
+55800
+> 14.4
+55200
+Q
+14.5
+00
+54600
+14.6
+00
+54000
+14.7
+O
+53400
+14.8
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+V-I6
+Figure 5. AzV 493 multi-epoch spectroscopic observations sorted by MJD and normalized to the continuum. Epoch I is low
+resolution (Table 1).
+
+M
+H
+normalized flux
+G
+F
+E
+D
+C
+B
+A
+1
+He lI
+He llI
+He ll
+4600
+4000
+4200
+4400
+4800
+5000
+wavelength [A]7
+3.1. Stellar fundamental parameters
+The photospheric He ii lines λ4200, λ4541, and λ5411
+lines at all epochs confirm the early O spectral type
+assigned by Golden-Marx et al. (2016). To improve S/N
+in the He ii λ4541 absorption line, we combine epochs
+C, G, H and J, which are all IMACS spectra obtained
+in 2016 – 2017.
+We use this composite spectrum to
+estimate the projected rotational velocity (υ sin i) using
+the iacob-broad code (Sim´on-D´ıaz & Herrero 2014,
+2007). We obtain υ sin i = 370±40 km s−1. As discussed
+in Section 4, the angle of inclination i is likely high,
+based on the amount of obscuration from the disk, and
+so the rotational velocity might be ≲ 450 km s−1.
+The combined spectrum was modelled using the stellar
+atmosphere code fastwind (Santolaya-Rey et al. 1997;
+Puls et al. 2005; Rivero Gonz´alez et al. 2012), using
+the same technique and stellar grid described in Cas-
+tro et al. (2018).
+The cores of the Balmer lines are
+omitted from the fit to ameliorate contamination from
+disk emission. Our best model yields effective tempera-
+ture Teff = 42000 K and surface gravity log g = 3.4 dex,
+which reproduce the main He i and He ii lines (Figure 6).
+Since He i photospheric features are not detected, this
+Teff may be a lower limit. The derived temperature is
+consistent with an O3-5 spectral type (Martins & Pala-
+cios 2021), matching the early O-type classification of
+AzV 493 (Lamb et al. 2016). However, we caution that
+the wings of the Balmer lines, which are the main spec-
+troscopic anchors for deriving the surface gravity, may
+be affected by the circumstellar emission, resulting in an
+underestimate of log g, as found for OBe stars by Castro
+et al. (2018).
+The stellar luminosity was calculated using the optical
+and IR photometry for AzV 493 (Massey 2002; Skrut-
+skie et al. 2006), adopting a distance to the SMC of
+62.1 kpc (Graczyk et al. 2014) and the synthetic fast-
+wind spectral energy distribution (SED) derived above.
+We explored the extinction curves published by Fitz-
+patrick & Massa (2007) until the observed photometry
+was reproduced by the fastwind synthetic SED. We
+obtain a luminosity log L/L⊙ = 5.83 ± 0.15 and radius
+R⋆/R⊙ = 15±3, in agreement with the expected values
+for an early O-type star of luminosity class III – V (e.g.
+Martins et al. 2005). We compare the position of AzV
+493 in the Hertzsprung–Russell diagram with the rotat-
+ing evolutionary tracks by Brott et al. (2011) for SMC
+metallicity. Based on the Teff and L/L⊙ and their re-
+spective uncertainties, we estimate that the stellar mass
+is M/M⊙ = 50±9. If the observed luminosity is overes-
+timated by the inferred log g, or includes a contribution
+from a non-compact binary companion and/or the disk
+continuum, then the stellar mass may be somewhat over-
+estimated; for reference, a factor of two overestimate in
+luminosity implies M/M⊙ ∼ 40.
+3.2. Hβ emission-line profile
+Variability in the emission lines is a common charac-
+teristic of the Be phenomenon (e.g., Rivinius et al. 2013;
+Richardson et al. 2021). One effect is the violet-to-red
+(V/R) variations, which are cycles that can last weeks
+or decades. The V/R variations describe changes in the
+dominant peak strength for double-peaked emission lines
+observed in some stars. These cycles are attributed to
+variation in the morphology and density of the circum-
+stellar disks (Poeckert 1982; Okazaki 1991).
+Figure 7 shows Hβ profiles in the spectroscopic epochs
+where this line is available, and Gaussian models used to
+disentangle the V and R components. The two peaks are
+clearly resolved in all our observations of Hβ, except for
+Epoch B, which instead shows a P-Cygni profile (Fig-
+ures 5, 7; see Section 3.4). Table 1 gives the peak separa-
+tions ∆Hβ fitted in Figure 7. The V peak is usually more
+prominent than R. There may be a long-timescale V/R
+cycle, but further spectroscopic monitoring is needed to
+determine whether V/R indeed oscillates, or whether
+there is any trend in ∆Hβ with phase.
+3.3.
+Epochs A and K: Evidence of disk evolution
+Epoch A is observed at a phase of 0.54 (0.08), soon
+after the apparent eruption event in 2009 (Figure 1, Ta-
+ble 1). This spectrum shows the strongest helium line
+emission (Figure 5), although we have no other spec-
+troscopic observations within several years of this data
+point. Only photospheric He ii is seen in absorption in
+this spectrum; the H i and He i lines are all in emission
+or filled in. Moreover, He ii λ4686 is also in emission,
+which prompted Golden-Marx et al. (2016), to identify
+this spectrum as the hottest-known observation of the
+OBe phenomenon. Nebular He ii is only generated by
+the very hottest O stars (e.g., Martins & Palacios 2021).
+All of the emission lines in Epoch A are double peaked.
+Hβ and Hγ show larger peak separations than the He i
+and He ii emission lines. For a Keplerian disk, this would
+imply that the higher-temperature species is dominated
+by larger radii than the Hβ and Hγ emission. Figure 5
+shows that the emission is slightly redshifted relative to
+the photospheric Balmer absorption.
+Epoch K, observed at phase 0.37 (0.74) (Figure 5; Ta-
+ble 1) shows the opposite relation between ionization
+and disk radius. Here, the He i lines have larger peak
+separations than Hβ, implying that the hotter species
+dominates at smaller radii, unlike Epoch A. We also
+see that the Hβ and Hγ line profiles show high-velocity
+wings that are not observed at other epochs, consistent
+
+8
+Figure 6. Spectroscopic analysis of the composite IMACS spectrum from epochs C, G, H and J (black; cf. Fig. 5). The
+best fastwind (Santolaya-Rey et al. 1997; Puls et al. 2005; Rivero Gonz´alez et al. 2012) stellar atmosphere synthetic model is
+overplotted (red). The main transitions used in the analysis are marked.
+Figure 7. Hβ emission-line profiles from our spectra of AzV 493. The best-fit photospheric model (Figure 6) is subtracted,
+after which the violet and red components are fitted by two Gaussian profiles having fixed widths of 2 ˚A. The figure shows
+the data overplotted by these summed fitted Gaussians. The resulting peak values are shown by the vertical lines, and their
+separations are given in Table 1. Epoch I has low spectral resolution and is not included in this figure.
+with high-velocity gas at smaller orbital radii. Epoch K
+is similar in emission-line strength
+to Epoch A and
+shows He i in emission, but He ii λ4686 is in absorption
+in this observation, as it is in all the other observations
+of this line.
+3.4. Epochs B and F: Gas Outflow and Infall
+Epoch B shows P-Cygni emission-line profiles in Hβ
+and Hγ (Figures 5, 7), suggesting an outflow episode.
+This is also the only spectrum obtained during the pe-
+riod where the strong pulsations dominate the flux (Fig-
+ure 1), and it is observed at the latest phase, 0.91 (0.82).
+Figure 13 shows that the observation coincides with a
+local minimum in the light curve. Thus the P-Cygni fea-
+tures could suggest that the pulsations may be directly
+linked to mass ejection, since it coincides with the star
+reaching its smallest radius.
+The spectrum of Epoch F is dramatically different
+from most of the other spectra (Figure 5).
+It shows
+strong, asymmetric Balmer and He i emission that show
+remarkable, inverse P-Cygni line profiles, with red-
+shifted absorption and blue-shifted emission. Figure 8
+
+1.3
+He lI
+He lI
+He ll
+He l
+Hel
+Hel
+Hel
+1.2
+1.1
+1
+1.0
+0.9
+0.8-
+0.7
+0.6.
+4000
+4100
+4200
+4300
+4400
+4500
+4600
+4700
+1.3
+Hel
+He llI
+1.2
+≤ 1.1
+1.0
+0.9
+0.8
+0.7
+0.6
+4800
+4900
+5000
+5100
+5200
+5300
+5400
+5500
+wavelength [A]1.3
+3
+1.3
+1.3
+B
+C
+A
+1.2
+1.2
+1.2
+1.1
+1.1
+1.1
+1.1
+1.0
+1.0
+1.0
+1.0
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+-500
+500
+-500
+0
+500
+-500
+0
+500
+-500
+0
+500
+0
+velocity - 200 [kms-1]
+1.3
+1.3
+1.3
+1.3
+G
+H
+K
+1.2
+1.2
+1.2
+1.2
+1.1
+1.1
+1.1
+1.0
+1.0
+1.0
+1.0
+0.9
+0.9
+0.9
+0.9
+0.8
+0.8
+0.8
+0.8
+-500
+0
+500
+-500
+0
+500
+-500
+0
+500
+-500
+0
+5009
+1.8 
+1.6 
+1.4 
+1.2 
+1 
+-2000
+-1000
+0 
+Velocity (km/s) 
+1000 
+2000 
+Hα
+1.2 
+1.15 
+1.1 
+1.05 
+1 
+.95 
+_._ _ _ _ _  
+___._ 
+_ _ _ _  
+__. _ _ _ _ _  
+.__ _ _ _ _  
+..__ _ _ _ _  
+_.___. 
+-3000
+-2000
+-1000
+0 
+1000 
+Velocity (km/s) 
+2000 
+3000 
+Hβ
+1.2 
+1.15 
+1.1 
+1.05 
+1 
+.95 
+-3000
+-2000
+-1000
+0 
+Velocity (km/s) 
+1000 
+2000 
+3000 
+Hγ
+1.2 
+1.1 
+1 
+.  9
+-7500
+-5000
+-2500
+0 
+Velocity (km/s) 
+2500 
+5000 
+Hδ
+He I  4026
+1.15 
+1.1 
+1.05 
+1 
+.95 
+-3000
+-2000
+-1000
+0 
+1000 
+Velocity (km/s) 
+2000 
+3000 
+He I 4471
+Figure 8. Epoch F line profiles for Balmer and He i emission lines, as shown, centered at the systemic velocity obtained from
+the He ii absorption. This Magellan/MIKE observation was obtained on 2016 September 22 (Table 1).
+shows the line profiles relative to the systemic velocity of
+the He ii photospheric lines. Such observations are usu-
+ally interpreted as infall of matter (e.g., Hartmann et al.
+2016), which appears to imply a re-absorption of decre-
+tion disk material. The free-fall velocity at the stellar
+surface for our adopted stellar parameters (Section 3.1)
+is ∼ 800 km s−1, which is consistent with the red edge
+of the absorption trough seen in Hδ and He i λ4471.
+The Balmer emission-line intensities do not follow the
+Balmer decrement and are almost uniform (Figures 6
+and 8), indicating optically thick emission. This sug-
+gests that the infalling material is also likely dense, and
+thus has high emissivity.
+Although Epochs D and E are taken only 14 and 11
+days before Epoch F, respectively, Epochs D and E show
+most lines in absorption with no sign of these features.
+Similarly, Epoch G is obtained only 73 days after Epoch
+F, and also shows primarily an absorption spectrum.
+Thus, this infall episode corresponds to a short-lived
+event, which we fortuitously captured with this MIKE
+observation. In the spectra observed before and after
+Epoch F, the Balmer emission, which presumably origi-
+nates from the disk, does not seem substantially different
+in intensity. This suggests that the reabsorbed material
+corresponds to a negligible fraction of the disk mass.
+The timing of Epoch F is at a very early phase, 0.024
+(0.05), only 27 days after the light curve minimum on
+
+10
+MJD 57626. There is no significant feature in the pho-
+tometry near the time of Epoch F, and the light curve is
+gradually brightening during this phase. This similarly
+implies that the continuum luminosity is dominated by
+the star and/or disk sources unrelated to the P-Cygni
+event.
+4. DISK EJECTION SCENARIO
+The distinctive shape of the light curve seen in 2002
+– 2004, and again in 2016 – 2018, showing a strong
+drop in brightness followed by gradual increase (Fig-
+ure 1), is seen in some other emission-line stars (Riv-
+inius et al. 2013). We suggest that this may be due to
+the repeated ejection of an optically thick circumstel-
+lar decretion disk, perhaps related to interaction with
+a binary companion.
+The exact reproduction of this
+part of the light curve across two cycles, starting with
+a 1.2-magnitude drop in brightness, suggests a geomet-
+ric extinction effect caused by an optically thick disk.
+This event’s pattern in photometry and Hβ line profile
+is consistent with a disk ejection outburst, similar to,
+e.g., HD 38708 (Labadie-Bartz et al. 2017).
+Assuming that an optically thick disk is indeed ex-
+pelled to generate the deep light-curve mimima (I ∼
+14.85) in 2002 and 2016, we can estimate the geometric
+obscuration by considering the maximum flux following
+these minima, which peaks around I ∼ 14.0. The differ-
+ence of 0.85 mag corresponds to reduction in flux by a
+factor of ∼ 0.46, or over half, assuming that all of this
+difference is due to obscuration. This suggests not only
+a fairly high angle of inclination, but also a thick, or in
+particular, a geometrically flared disk, which is consis-
+tent with spectroscopic evidence (Section 3.3).
+In this model, most of the emission lines originate from
+an inner disk region that experiences variable obscura-
+tion to our line of sight from a thicker outer disk or
+torus. The weaker spectroscopic epochs in Figure 5 with
+the typical OBe spectrum are the most obscured, while
+Epochs A, B, and K are less obscured and therefore show
+stronger emission-line spectra. Epoch C is observed in
+2016 at a phase of 0.01 (0.01), and thus near the same
+phase as the light curve peak in late 2001 (2009) (Fig-
+ure 1; Table 1). However, as noted above (Section 2.1),
+although the light curve repeats the disk ejection pat-
+tern, there is no evidence of a corresponding peak pre-
+ceding this sequence on the same timescale as that in
+2002.
+The Epoch C Hβ profile (Figure 7) is consis-
+tent with the optically thick disk already having formed.
+Epochs D and E, observed immediately after this min-
+imum, are similarly unremarkable, although they cover
+a much shorter spectral range. Since we see that a pu-
+tative disk ejection apparently occurred in 2016, it may
+be that the system has precessed such that an associ-
+ated photometric outburst is obscured by the ejection
+process.
+The emission lines in Epoch A are dominated by
+higher temperature species at larger radii, whereas
+Epoch K shows the opposite effect (Section 3.3).
+Epoch A is consistent with very dense, optically thick
+disks that have extended vertical flaring, as shown in
+models by, e.g., Sigut et al. (2009), where the emission,
+including from harder radiation, is dominated by this
+outer region. In contrast, the disk geometry at Epoch K
+is dominated by high-density gas near the center and
+no flaring, thus differing significantly from Epoch A.
+Epoch A is observed at a phase of 0.54 (0.08), and
+Epoch K shows the system at a phase of 0.37 (0.74;
+Table 1, Figure 1). This suggests that the disk changes
+between having a large, flared outer region at Epoch A
+that contributes significantly to the emission, and a con-
+figuration where flaring is insignificant and emission is
+dominated by a dense central region at Epoch K, per-
+haps also reaccreting material onto the star. The exis-
+tence of two different components dominated by inner
+and outer regions, respectively, could also be due to disk
+tearing, resulting in an inner disk and outer, expanding
+annulus with different inclinations (Suffak et al. 2022;
+Marr et al. 2022).
+The decreasing Hβ peak separations seen from Epoch
+C (346 km s−1) to Epoch J (303 km s−1) and to Epoch
+K (289 km s−1; Table 1) suggest that the emission is
+weighted toward increasing radii over this period, which
+is consistent with the inner disk dissipating or forming
+an annular disk with an expanding inner radius. How-
+ever, this scenario does not explain the strong line emis-
+sion in Epochs A and K (Figure 5), which have the min-
+imum Hβ peak separations. If the inner radius is indeed
+expanding, then the emitting region either must become
+dense, or the disk must precess to lower inclination an-
+gles to reveal stronger line emission. The latter could
+also contribute to a model in which the decreasing peak
+separation is due to decreasing obscuration of the disk,
+allowing emission at larger radii to dominate. This is
+consistent with the system’s increasing brightness over
+this period (Figure 1). The extinction may result from
+the outer component, or optically thick torus or flare in
+the disk which either precesses or dissipates. However,
+we caution that such a fast precession rate may not be
+feasible. Moreover, if the long-term photometric cycle is
+due to precession, the light curve should be symmetric
+around the minima, whereas the observed strong, sud-
+den drops (Figure 1) are difficult to explain with such a
+model.
+
+11
+The outflow and inflow episodes described in Sec-
+tion 3.4 apparently are not significant in mass relative to
+the entire disk. If the minima of the 14-year light curve
+indeed correspond to the bulk of disk ejection, followed
+by gradual disk dissipation, then the mass ejection as-
+sociated with the P-Cygni features in Epoch B are not
+likely to be a dominant source of disk material. How-
+ever, we note that pulsations have been suggested to be
+important in replenishing the disk in other OBe systems
+(e.g., Baade et al. 2016, 2018).
+The timing of Epoch F is 27 days after the light curve
+minimum on MJD 57626. Although there are 3 other
+intermediate spectroscopic epochs between the putative
+disk ejection and Epoch F, this still takes place dur-
+ing what we assume is the heavily obscured phase in
+the light curve. The lack of any photometric event near
+the appearance of inverse P-Cygni features in epoch F
+suggests that the reabsorbed material is an insignificant
+portion of the disk material. The disk is therefore sub-
+stantial and can plausibly provide material that may fall
+back to the star. This is consistent with the optically
+thick conditions indicated by the Balmer decrement in
+Epoch F.
+Thus, this model is driven by repeated ejection of a
+flared, optically thick disk whose outer region gradually
+dissipates, revealing the inner, line-emitting region. A
+flared disk is most clearly implied by the ionization and
+emission-line peak separation in Epoch A (Section 3.3),
+and is also consistent with a maximum geometric ob-
+scuration that may be > 50% implied by this model.
+The spectroscopic variation could also be caused by
+disk tearing or precession of the system. The decreas-
+ing trend in Hβ peak separations with increasing flux
+suggests that more light from larger radii can be seen
+(Section 3.2). Additionally, the high-amplitude, semi-
+regular pulsations with the ∼month-long period become
+visible at low extinction (Figure 1). Other photometric
+and spectral variations may be due to contributions from
+the inner disk’s radial expansion, reabsorption, or evap-
+oration/ionization, and possible geometric distortion or
+warping of the disk system.
+5. DISK GROWTH SCENARIO
+However, some observations seem inconsistent with a
+disk ejection model. For example, the system is bluest
+when faintest (Figure 1), contrary to expectations for
+reddening.
+As noted above, the strong emission-line
+spectra at Epochs A and K seem inconsistent with a
+dissipating inner disk scenario implied by the trend in
+∆Hβ. If the long-period cycle is attributed to disk pre-
+cession, it would require an additional mechanism to
+explain the assymmetric light curve, and also a third,
+external massive star that is not seen, to torque the
+disk. Thus, alternative models for the AzV 493 system
+should also be considered.
+Some other Be stars such as δ Sco (Suffak et al. 2020)
+and ω CMa (Ghoreyshi et al. 2018) show long-term pho-
+tometric variability in which the increasing flux is due to
+contributions from a growing disk, while the minima rep-
+resent episodes of disk destruction by the secondary at
+periastron. Such a model is therefore opposite to the one
+presented above. In this alternative scenario, the light
+curve minima of AzV 493 in 2002 and 2016 (Figure 1)
+correspond to episodes with the lowest disk contribu-
+tion. The disk then grows and brightens, recovering its
+full size around 2005. In this case, the decreasing trend
+in Hβ peak separation with increasing flux is simply due
+to the disk growth itself. This scenario is consistent with
+the blue color at the light curve minimum in 2016 (Fig-
+ure 1), and the weak emission-line spectra near the 2016
+minimum (epochs C – J; Figure 5).
+If the disk is responsible for the factor of 2.2 increase in
+flux, then the equivalent width (EW) of stellar absorp-
+tion features should decrease proportionately. Figure 9
+shows the EW of He ii λ4200 and λ4540 as a function of
+V and I magnitude. A slight trend is indeed apparent,
+although not as large as a factor of two in amplitude.
+These lines are in the B-band, and thus not in the range
+of our photometry. Figure 1 shows that the amplitude
+of the photometric variations may be smaller at bluer
+wavelengths, although with the given V -band sampling
+it is not entirely clear. It may be challenging for the
+alternative model to produce and maintain the viscous
+disk necessary to generate continuum luminosities that
+compete with those of the star, given the harsh circum-
+stellar environment of an extreme, early-type O-star.
+The extinction-dominated model is supported by the
+lack of correlation between the strength of the emission-
+line spectrum and photometric flux from the sys-
+tem. There is no significant variation between spectral
+Epochs C – J (Figure 5), which should correspond to the
+period of strong disk growth in this model, whereas the
+obscuration-dominated model implies dissipation (Sec-
+tion 3.3). The one exception showing spectral variation,
+Epoch F, has P-Cygni emission and stronger emission-
+line features, yet it is photometrically unremarkable
+(Section 3.4).
+Another issue is that the photometric
+minimum corresponds to the bluest color (Figure 1),
+which is more consistent with the alternative model.
+However, the star itself may be changing substantially in
+magnitude and color. Blueing is also caused by scatter-
+ing in high-extinction conditions, as seen in the UXOR
+class of Herbig Ae stars (Natta & Whitney 2000).
+
+12
+Figure 9.
+Equivalent width of He ii λ4200 (red) and λ4540
+(blue) as a function of V (bottom) and I (top) magnitude.
+A constant value of 0.5 ˚A is shown for reference.
+The overall shape of the light curve for AzV 493
+is rather different from those of δ Sco and ω CMa,
+which show extended minima with more top-hat-like
+light curves (Ghoreyshi et al. 2018; Suffak et al. 2020).
+In contrast, AzV 493 shows sharp minima (Figure 1),
+implying very rapid disk destruction and immediate,
+regular regrowth in the alternative model. It seems hard
+to explain such sudden dissipation of a several-AU dense,
+viscous disk by a neutron star or black hole (Section 6)
+during the brief periastron passage. Moreover, the exact
+reproduction of the photometric cycle’s initial segment
+(Section 2.1) is unusual and may be harder to explain
+with a disk-growth model.
+Overall, the fundamental nature of the light curve and
+disk evolution remain unclear. Tailored modeling of this
+system and further multimode observational monitoring
+is needed to clarify the relationship between the decre-
+tion disk and interaction with a secondary star.
+6. AN EXTREME INTERACTING BINARY
+The fast surface rotation for this evolved O star is
+a natural signature of accretion during a mass transfer
+event (e.g., Packet 1981; Cantiello et al. 2007; Renzo
+& G¨otberg 2021), consistent with an interacting binary
+scenario.
+If the disk is induced by a periastron pas-
+sage of an undetected companion, then this may imply
+a long, 14.6 (7.3)-year period, and hence a large and
+highly eccentric orbit. For the AzV 493 stellar parame-
+ters obtained in Section 3.1, a neutron star companion
+of mass 1.4 M⊙ would require e ∼ 0.95 (0.93) and apas-
+tron of ∼ 43 (27) AU for a typical OBe star periastron
+distance of 40R⋆. These orbital parameters are similar
+to those of the Be star δ Sco (e.g., Che et al. 2012).
+The unseen companion could also be a somewhat more
+massive main-sequence or stripped star, or a black hole.
+The eccentricity may be lower, but if a binary compan-
+ion is responsible for disk ejection, then periastron must
+be small and the eccentricity high. The nominal peri-
+astron value used here would likely be an upper limit,
+since δ Sco showed no disk ejection at periastron (Che
+et al. 2012).
+6.1. Neutron star or black hole?
+Thus, if a binary companion excites disk ejection or
+is otherwise responsible for the observed properties of
+AzV 493, then it is probably an eccentric system, and
+the most likely explanation for such an orbit is that the
+companion has already experienced core collapse, receiv-
+ing a strong kick. Large natal kicks are routinely invoked
+in core-collapse events that form neutron stars (e.g., Ar-
+zoumanian et al. 2002; Podsiadlowski et al. 2004; Ver-
+bunt et al. 2017; Janka 2017). Natal kicks during black
+hole formation are still highly debated (e.g., Dray et al.
+2005; Janka 2013; Mandel 2016; Repetto et al. 2017;
+Atri et al. 2019; Renzo et al. 2019; Callister et al. 2020),
+but not excluded. Assuming a large 450 km s−1 kick,
+Brandt & Podsiadlowski (1995) found a broad correla-
+tion between eccentricity and orbital period of binaries
+surviving the first core-collapse. This is in agreement
+with the high e and long period we find for AzV 493.
+The present-day mass of AzV 493 can be used to con-
+strain the nature of a putative compact object. Assum-
+ing a flat distribution in initial mass ratio, the average
+initial binary mass ratio q = M2/M1 ≃ 0.5 (e.g., Moe &
+Di Stefano 2017). Without any accretion during mass
+transfer, the present-day mass of AzV 493, M2 ≃ 50 M⊙,
+would suggest M1 ≃ 100 M⊙, which at SMC metallicity
+implies that the compact object should be a black hole
+(e.g., Sukhbold et al. 2016; Couch et al. 2020; Zapartas
+et al. 2021). In this case, however, the rapid rotation of
+AzV 493 would need to be primordial.
+Instead, it is more likely that mass transfer has oc-
+curred, in which case M1 is likely to be quite different,
+depending on the mass transfer efficiency. A mass trans-
+fer phase during the donor’s main sequence (Case A) is
+expected to be slower and more conservative, possibly
+causing significant mass growth of the accretor with-
+out extreme chemical pollution. This scenario has been
+invoked to explain the formation of low-mass compact
+objects in very young regions (Belczynski et al. 2008),
+and in particular, the origin of very massive companions
+(van der Meij et al. 2021), such as we have for AzV 493.
+In this case, the zero-age-main-sequence (ZAMS) mass
+of M1 ∼ 30 − 40 M⊙ for the adopted q, also accounting
+for the final donor core mass.
+However, mass trans-
+fer is far more likely to occur after the donor main se-
+
+1.2
+1.0
+0.8
+[y] 
+O
+EW
+0.6
+0.4
+:
+0.2
+0.0
+14.0
+14.2
+14.4
+14.6
+14.8
+I[mag]
+1.2
+1.0
+0.8
+EW
+0.6
+0.4
+8
+0.2
+0.0
+14.35
+14.40
+14.45
+14.50
+14.55
+14.60
+14.65
+V [mag]13
+quence (Case B), due to the star’s expansion (e.g., van
+den Heuvel 1969). It then takes place rapidly, on the
+thermal or He core-burning nuclear timescale (Klencki
+et al. 2022), and system mass loss is far more likely,
+implying a higher ZAMS mass for M1.
+Although post-SN outcomes are stochastic, black hole
+production is expected to dominate for Z⊙ progenitors
+with initial masses ≳ 20 M⊙. This nominal threshold
+ZAMS mass is expected to decrease for lower metallicity
+(e.g., Zhang et al. 2008; O’Connor & Ott 2011; Sukhbold
+et al. 2016), which in principle enhances the likelihood
+that the compact object should be a black hole. The
+high eccentricity in AzV 493 strongly suggests that a SN
+occurred. While this implies that the companion is more
+likely to be a neutron star, black holes can form from
+fall-back if the SN is insufficient to unbind the ejecta,
+which is more likely to happen at low metallicity (e.g.,
+Zhang et al. 2008). There are multiple mechanisms to
+produce core-collapse black holes, and if mass-loss oc-
+curs, a SN and/or kick to the system may result (e.g.,
+Janka 2013).
+We note that M1 ∼ 20 − 40 M⊙ is a
+range that has been extensively simulated and where
+explodability and fallback are uncertain (e.g., O’Connor
+& Ott 2011; Sukhbold et al. 2016; Janka 2013; Zhang
+et al. 2008). Establishing that a neutron star or black
+hole resulted from this ZAMS range, with some kind
+of kick, would provide an important empirical reference
+for theoretical models of the explosion and the binary
+interactions preceding it.
+Follow-up observations at subsequent periastra could
+more firmly establish whether AzV 493 has a compan-
+ion, and whether it is a black hole vs a neutron star. A
+74.33 ksec Chandra/HRC observation on 2012 February
+12 (MJD 55969) of a field including AzV 493 (ObsID
+14054) shows no detection. Given the tiny orbital inter-
+val during which the two stars interact, no significant ac-
+cretion onto the compact object is expected, explaining
+why the system is not a known high-mass X-ray binary.
+However, well-timed X-ray observations near periastron
+may be able to catch a brief flare event.
+6.2. Radial Velocities
+We also measure the radial velocity (RV) for the ob-
+tained spectra to search for evidence of a companion.
+This is challenging, since AzV 493 is a luminous, fast-
+rotating, early-type O-star, with few photospheric fea-
+tures, several of which are often in emission. We carried
+out cross-correlations against the FASTWIND model
+spectra for the entire observed spectral range using the
+iSpec code (Blanco-Cuaresma et al. 2014), as well as de-
+terminations based on cross-correlations against PoWR
+model spectra (Hainich et al. 2019) for only the He ii
+lines (λ4200, λ4540 lines, and λ4686), which are the
+only clean features appearing in all epochs.
+The lat-
+ter are carried out with the Markov Chain Monte Carlo
+code of Becker et al. (2015), and since they yield better
+results, we adopt these RV measurements (Table 1).
+We find that the mean systemic radial velocity is
+202 ± 9 km s−1, weighted inversely by the errors. We
+caution that the quoted standard error on this value
+underestimates the uncertainty if there is true variation.
+Given the difficulty of these measurements, with median
+error on individual epochs of 46 km s−1, it is difficult to
+evaluate any variability (Figure 10). There is possible
+evidence for very short-term RV variations; however, the
+data are ambiguous.
+We compute RV models for a possible periastron sug-
+gested in Section 2.3 at MJD 57523, which is near the
+second minimum in the light curve (Figure 1). For this
+7.3-year period, and the above, nominal periastron dis-
+tance of 40R⋆, the eccentricity e ∼ 0.93 and apastron
+∼ 28 AU. For this scenario, Figure 10 demonstrates that
+the RV signature of a neutron-star companion at perias-
+tron is very brief, on the order of 0.01 in orbital phase,
+and moreover, the observational uncertainties are larger
+than the expected amplitude. This is the case even for
+e = 0.99. Thus, our existing RV measurements do not
+strongly constrain whether MJD 57523 corresponds to
+a periastron, nor the existence and properties of a com-
+panion,
+6.3. Proper Motion
+A post-SN bound system can be expected to have been
+accelerated from its original rest frame. Relative to the
+blue stars from Massey (2002) within a 5′ radius, the
+Gaia EDR3 (Gaia Collaboration et al. 2021) residual
+proper motions of AzV 493 show two potential velocity
+vectors. Figure 11 provides the velocity histograms of
+these local field stars, showing strong bimodality in the
+R.A. components. These define two possible local ve-
+locity fields implying R.A. and Dec residual velocity for
+AzV 493 of either (vα, vδ) = (53 ± 11, 3 ± 12) km s−1;
+or (vα, vδ) = (−11 ± 11, 12 ± 13) km s−1. These yield
+total projected transverse velocities of 54±11 km s−1 or
+16 ± 12 km s−1.
+Figure 12 shows a wide-field view of the surround-
+ing environment, with the two possible proper motion
+vectors superposed.
+We see that the nearest massive
+star-forming region is the N84 complex (Henize 1956)
+about 15′ − 20′ or ∼ 300 pc to the west. If the velocity
+measurements are correct, the faster, east-bound veloc-
+ity is consistent with AzV 493 originating in N84 and
+traveling for ≳ 5 Myr. The lifetime itself of a 50 M⊙
+star with v sin i ∼ 500 km s−1 is about 5 Myr (Brott
+
+14
+Figure 10. Left: Heliocentric radial velocities measured from He ii photospheric absorption vs MJD for all epochs. Epoch A
+has only one available line of low quality, and hence has a very large uncertainty. The vertical dashed lines show the possible
+periastra at MJD 54686 and 57523. Right: Zoom for the same data showing RV models for eccentricities of 0.93 (dashed lines)
+and 0.99 (solid lines), assuming a periastron occurs at MJD 57523; and for inclination angles of 90◦ (black) and 45◦ (blue), for
+the 50 M⊙ primary and assuming a 3 M⊙ secondary. If a periastron is closer to the light curve minimum at MJD 57626, the
+models would shift to 103 days later.
+Figure 11. Distribution of Gaia proper motion velocities in R.A. (left) and Dec (right) for stars from Massey (2002) within 5′
+of AzV 493. The bimodal R.A. distribution defines two kinematic groups. The first group has 13 stars with median velocity
+(vα, vδ) = (254 ± 7, −378 ± 9) km s−1 and the second has 10 stars with (vα, vδ) = (318 ± 6, −386 ± 11) km s−1. The one star
+between the two groups in vα is included in both. The median velocities for these groups are shown with the vertical green and
+blue lines, together with the velocity of AzV 493 (red).
+et al. 2011), and for a SN ejection, its travel time would
+only be the post-SN lifetime. However, since the star
+presumably acquired its total mass and spin later in
+life, the system may have been ejected earlier by dy-
+namical processes as a tight, non-compact binary. If so,
+it would have been reaccelerated by the SN explosion,
+therefore implying that it may be a two-step ejection
+(Pflamm-Altenburg & Kroupa 2010). Supernova accel-
+erations are typically weaker than dynamical ejections
+(e.g., Renzo et al. 2019), and so the dominant velocity
+component could still be due to a dynamical ejection
+from N84. A dynamically active past in a dense stellar
+environment of N84 may also help to explain the eccen-
+tricity (e.g., Sim´on-D´ıaz et al. 2015), although it would
+seem unlikely that the system could maintain its highly
+eccentric configuration for 5 Myr. On the other hand,
+we note that the inferred runaway velocity, orbital ec-
+centricity, and period are still consistent with being due
+
+AzV 493
+350
+ (km/sec)
+300
+E
+250
+-
+B
+D
+200
+--
+K
+RV
+A
+Heliocentric
+150
+100
+-
+50
+0
+50
+--
+55000
+56000
+57000
+58000
+59000
+60000
+Date (MJD)Example RV Curves
+260
+e=0.99
+e=0.94
+240
+Primary RV (km/s)
+220
+200
+180
+160
+140
+-200
+0
+200
+400
+600
+Days around Periastron = MJD 5752377616 field v RA plot
+77616 field v DEC plot
+6
+8
+7
+5
+6
+4
+(stars)
+3
+4
+N
+N
+2
+2
+1
+1
+0
+0
+200
+250
+300
+350
+-450
+-400
+-350
+-300
+Velocity (km/s)
+Velocity (km/s)15
+Figure 12. Location of AzV 493 in the SMC field, with the green and blue proper motion vectors corresponding to the two
+field velocities indicated with the same color coding in Figure 11, superposed on an Hα images from Smith et al. (2005). The
+nearest massive star-forming region is the N84 complex (Henize 1956), indicated. For the adopted SMC distance, 10′ = 181 pc.
+solely to SN acceleration (e.g., Brandt & Podsiadlowski
+1995). Thus, in order to explain both the long travel
+time and high eccentricity, the most plausible scenario
+may be the two-step ejection.
+There is also a small possibility that the slow, alterna-
+tive proper motion vector (Figure 12) is correct. How-
+ever, this would mean that the AzV 493 system formed
+in isolation since there is no corresponding young clus-
+ter whence it could have originated (Figure 11). Vargas-
+Salazar et al. (2020) find that < 5% of OB stars, if any,
+formed in the field, and this is especially unlikely for
+AzV 493, given its high mass.
+We caution that the ve-
+locity errors do not include unknown systematic errors,
+and so these measurements need to be confirmed. Thus,
+although AzV 493 indeed appears to be a runaway star,
+this does not provide especially useful information to
+constrain its binary interaction history.
+6.4. Similar systems
+A comprehensive study by Marr et al. (2022) shows
+that the B8 Vpe star Pleione (HD 23862) has a light
+curve with a similar long-term pattern of slow growth
+with sudden drops, and similar variations in the Balmer
+emission-line profiles. It is a triple system with a close
+companion on a 218-day orbit (Katahira et al. 1996;
+Nemravov´a et al. 2010). Marr et al. (2022) suggest that
+the photometric drops correspond to the decretion disk
+tearing into two components, where one remains aligned
+with the star’s equatorial plane and the other is mis-
+aligned due to tidal torque from the close companion.
+Pleione’s long-term photometric cycle is 34 years, simi-
+lar in magnitude to that of AzV 493. Nemravov´a et al.
+(2010) find that the close companion is on an eccentric
+orbit with e > 0.7.
+AzV 493’s initial peak brightness and subsequent drop
+in 2001 (Figure 1) qualitatively resemble the photomet-
+ric pattern characteristic of heartbeat stars. These are a
+rare class of interacting binary systems with high eccen-
+tricities such that the periastron passage tidally induces
+regular photometric outbursts. However, the observed
+pattern in AzV 493 cannot be induced by this type of
+tidal interaction; preliminary simulations using new ca-
+pabilities in the GYRE stellar oscillation code (Sun et al.
+2023) suggest that the combined amplitude and width of
+the periastron pulse cannot be reproduced by eccentric
+tidal models. Nevertheless, given that AzV 493 seems
+likely to be a massive eccentric binary system, massive
+heartbeat stars thus share some similarities with this
+
+53.5±10.9 km/s
+73°00'00"
+16.0±12.1 km/s
+10'00"
+Dec (J2000)
+N84
+20'00"
+30'00"
+20°00'00"
+19°00'00"
+18°00'00"
+RA (12000)16
+object if a companion indeed interacts with the primary
+and/or its disk.
+Examples include the non-Be binary
+system ι Ori (O9 III + B1 III/IV), which has orbital
+period 29 d and eccentricity e = 0.764, as determined by
+Pablo et al. (2017). They find that the two components
+have masses of 23.2 and 13.4 M⊙, respectively, generat-
+ing tidally excited oscillations with periods on the order
+of ∼ 1 day. MACHO 80.7443.1718 is another heartbeat
+system with two stars of type B0 Iae and O9.5 V and
+masses of 35 and 16 M⊙, respectively (Jayasinghe et al.
+2021).
+The B0.5 Ve star δ Sco is has a B2 V star companion
+in an eccentric (e = 0.94) orbit with period 10.7 years
+(e.g., Tango et al. 2009; Tycner et al. 2011). The two
+components have masses of 13.9 M⊙ and 6 M⊙ (Che
+et al. 2012). This system shows a long-term photomet-
+ric cycle somewhat similar to that of AzV 493, although
+much more poorly defined. There is no obvious link be-
+tween the disk properties and binary interaction (Suffak
+et al. 2020; Che et al. 2012), but the long-term pho-
+tometry has a timescale similar to that of the orbital
+period.
+H 1145–619 is a Be X-ray binary whose primary is a
+B0.2e III star estimated to be 18.5 M⊙ (Alfonso-Garz´on
+et al. 2017), and the secondary is an X-ray pulsar. As
+shown by Alfonso-Garz´on et al. (2017), H 1145–619 has
+a light curve with a ∼ 10-year cycle together with un-
+explained multiple modes of much shorter periods (∼ 1
+year), qualitatively similar to what we see for AzV 493,
+which has a long cycle of 14.6 (7.3) years and short oscil-
+lations of ∼ 40 days. While it remains unclear whether
+the light curves of H 1145–619 and AzV 493 have fun-
+damental similarities, both stars are massive OBe stars.
+If they are related, the fact that H 1145–619 has a con-
+firmed compact binary companion may suggest that the
+unusual variability of AzV 493 may have a similar origin.
+These objects provide a context for AzV 493 that sup-
+ports this object being a member of this broad class of
+binary, massive OBe systems with high eccentricities.
+At 50 M⊙, AzV 493 is more massive than any of these
+similar objects.
+It is also one of the earliest O stars
+in the SMC, since there is no photospheric He i. Thus,
+AzV 493 may be the most extreme such object known,
+in terms of its mass and effective temperature. Its vari-
+ability amplitudes are also among the largest known.
+We note that, based on only the Epoch A spectrum
+(Figure 5), Golden-Marx et al. (2016) suggested that
+AzV 493 is a normal, but extremely early, classical Oe
+star. Given the strong spectroscopic and photometric
+variability, the nature of this spectrum may be some-
+what different than inferred in that work, and the origin
+of the strong line emission seen in this particular spec-
+trum is unclear (Section 3.3). Still, its status as a post-
+SN binary where the observed star was likely spun up
+by mass transfer from the compact object progenitor, is
+consistent with the origin of classical OBe stars. Indeed,
+given that most of the massive OBe stars are post-SN
+systems (e.g., Dallas & Oey 2022; Dorigo Jones et al.
+2020), we can expect that more of them are likely to be
+high-eccentricity, compact-object binaries.
+6.5. Alternative Companion Scenarios
+We now consider alternative scenarios for a putative
+binary component. First, such a companion might be
+an unexploded former donor in an interacting binary. In
+this case, it could be a stripped star (e.g., Schootemeijer
+et al. 2018; G¨otberg et al. 2017), which can be elusive to
+detect. Wang et al. (2021) identified hot, stripped star
+companions to Be stars based on FUV spectral cross-
+correlations; however, the extremely hot temperature of
+AzV 493, which is commensurate with the hottest O
+stars, poses a serious challenge for this method. If the
+observed star has previously experienced accretion from
+binary mass transfer, then its surface might be He- and
+N-enriched (e.g., Blaauw 1993; Renzo & G¨otberg 2021),
+although whether this occurs depends on the accretion
+efficiency and mixing processes in the accretor’s enve-
+lope. Since early O stars have few metal lines, it is again
+difficult to evaluate any enrichment, especially in a fast
+rotator like AzV 493. There is no immediate evidence
+for any unusual abundances in this star. Moreover, a
+non-degenerate companion does not naturally explain
+the high observed eccentricity, which would then have
+to be primordial, avoiding tidal dissipation, or of dy-
+namical origin.
+Alternatively, the high rotation rate and variability of
+AzV 493 might be caused by a non-standard internal
+structure of the star because of a merger.
+These are
+common among massive stars, occurring in 22+26
+−9 % of
+isolated massive binaries (Renzo et al. 2019), with an
+even higher rate if accounting for the presence of further
+companions (e.g. Toonen et al. 2020). For example, η
+Car has been suggested to originate from a merger in
+a hierarchical triple system, resulting in a present-day
+eccentric binary (e.g., Hirai et al. 2021). However, η Car
+is a luminous blue variable star and has other substantial
+differences from AzV 493.
+Yet another possibility is that AzV 493 might be a
+triple system with a third, also invisible, star on a
+shorter-period orbit.
+This speculative scenario might
+help to explain how the strong, 40-day pulsations are
+maintained (Section 2.2). It also might help explain the
+apparently sporadic ejection and accretion events seen
+in Epochs B and F (Section 3.4). Such a system would
+
+17
+be unstable, but the brief interaction phase with the sec-
+ondary may enhance its longevity. We note that the sys-
+tem is unlikely to be a triple in which the third star has
+an even larger orbit than the secondary. Although high
+orbital eccentricities can be produced by Kozai-Lidov
+cycles in such a system, this high-eccentricity phase of
+the cycle is short in duration. Thus, such extreme eccen-
+tricity may require a triple or higher-order multiple-star
+interaction in the system’s birth cluster, and may be
+linked to a dynamical ejection of AzV 493 into the field.
+Overall, however, it is challenging to explain AzV 493 in
+terms of a triple-star scenario. Unfortunately, RV mon-
+itoring is complicated due to the technical difficulty and
+possible presence of varying stellar pulsations, so it will
+be hard to evaluate whether the system consists of more
+than two stars.
+7. SUMMARY
+We present 18 years of OGLE Project photometric
+data and spectroscopic data over 12 years, revealing the
+remarkable variability of AzV 493. This is perhaps the
+earliest known classical Oe star, with Teff = 42000 K,
+log L/L⊙ = 5.83 ± 0.15, and R⋆/R⊙ = 15 ± 3. These
+parameters imply a mass of 50 ± 9 M⊙.
+The domi-
+nant photometric pattern is reproduced after 14.6 years.
+There are also large, semi-regular ∼ 40-day pulsations of
+unknown origin, as well as other structure in the light
+curve. It is not a known HMXB. The observed v sin i
+= 370± 40 km s−1, with a high inferred sin i, suggesting
+a rotational velocity of 400 − 450 km s−1. The system
+is ∼ 300 pc from the nearest massive star-forming com-
+plex and its proper motion shows that it is likely a run-
+away star from that region, with a transverse velocity
+of 54 ± 11 km s−1, possibly having experienced two-step
+acceleration.
+Altogether, the data suggest that this object is likely
+an eccentric, interacting binary system with an unde-
+tected compact companion.
+If so, the orbital period
+could correspond to the 14.6 (7.3)-year period, imply-
+ing a high eccentricity of at least e ∼ 0.95 (0.93) and
+apastron ∼ 43 (28) AU. If this binary scenario is cor-
+rect, AzV 493 would be among the most extreme sys-
+tems known, in terms of its early spectral type, high
+mass, and extreme eccentricity.
+In our favored model, an optically thick decretion disk
+is regularly ejected, likely by a periastron encounter. A
+two-component disk system forms, with the outer re-
+gion responsible for the 0.85-magnitude drop in I-band
+flux, while the inner disk is the origin of most of the
+observed emission-line spectrum. The spectra appear to
+show varying relative contributions from the inner and
+outer regions, consistent with the optically thick outer
+region dissipating over the cycle. The outer region may
+correspond to a flared disk, torus, or possibly, a separate
+inclined annulus formed by tearing from the inner disk.
+We see direct spectroscopic evidence for episodes of both
+matter ejection and infalling reabsorption of dense disk
+material onto the star. The lack of exact regularity of
+photometric and spectroscopic variations in the cycle
+implies that the geometry and/or mechanics of the disk
+ejection may vary. An alternative, opposite model seen
+in some Be stars, in which the brightness increases due
+to contribution from growing disk emission (e.g., Suf-
+fak et al. 2020; Ghoreyshi et al. 2018), should also be
+considered.
+If AzV 493 indeed has a highly eccentric orbit, it would
+suggest that the system experienced a strong SN kick,
+implying that the unseen companion is a neutron star
+or black hole. The high v sin i also suggests that mass
+transfer occurred before this event.
+For conservative,
+Case A mass transfer, the progenitor donor’s ZAMS
+mass would be 30 − 40 M⊙ for a typical q ∼ 0.5, and
+larger for non-conservative Case B mass transfer. This
+mass range is well within that suggested by models to
+produce black holes, although the occurrence of strong
+natal kicks in cases of black hole formation is less clear.
+Alternatively, the donor could be a stripped star; how-
+ever, this scenario cannot explain the extreme eccentric-
+ity, which would have to be dynamical or primordial.
+The system could also be a merger, but the eruptions
+and long-term pulsations seem less consistent with this
+scenario.
+AzV 493 could possibly be a triple system,
+which might explain how the strong photometric oscil-
+lations are maintained (Section 6.5).
+Establishing the existence and nature of the unseen
+companion(s) can provide important constraints on bi-
+nary evolution, core explodability, and the origin of
+compact binaries. AzV 493 may offer an opportunity
+to directly observe the relationship between the binary
+companion’s dynamical interaction and the disk ejec-
+tion. Since many classical OBe stars are massive, post-
+SN objects, it suggests a likely link between OBe stars
+and massive, eccentric systems. Further study of this
+fascinating object can more definitively confirm its sta-
+tus and exploit the opportunities it offers to learn about
+massive binary evolution and disk ejection.
+
+18
+ACKNOWLEDGMENTS
+We benefited from useful discussions with many peo-
+ple, including Jon Bjorkman, Paul Crowther, Julian
+Deman, Jim Fuller, Jay Gallagher, Carol Jones, Max
+Moe, Megan Reiter, Steve Shore, and Drew Weisser-
+man.
+Many thanks to Juliette Becker for the use of
+her code, and to Traci Johnson, Mario Mateo, and the
+M2FS Team for help with observing runs.
+We also
+thank the anonymous referees for valuable comments
+that greatly improved this paper. This work was sup-
+ported by NSF grant AST-1514838 to M.S.O. and by the
+University of Michigan. N. Castro acknowledges funding
+from the Deutsche Forschungsgemeinschaft (DFG), CA
+2551/1-1; M.R. is supported by EUH2020 OPTICON
+RadioNet Pilot grant No.
+101004719; and R.H.D.T.
+is supported by NASA grant 80NSSC20K0515.
+This
+research made use of Astropy, a community-developed
+core Python package for Astronomy (Astropy Collabo-
+ration et al. 2013). M.S.O. acknowledges MDRS, LLC,
+for pandemic hospitality.
+Facilities: Magellan, OGLE, Gaia
+
+19
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+APPENDIX
+A. GENERALIZED LOMB-SCARGLE PERIODOGRAMS
+Figure 13 shows the individual generalized Lomb-Scargle periodograms (Zechmeister & K¨urster 2009) and ancillary
+information for the six, roughly contiguous, OGLE datasets during ∼ 2010 – 2016 (Section 2.2).
+
+22
+Figure 13. Top panels show the generalized Lomb-Scargle periodogram for light curves shown in the middle-left panels. The
+fitted light curves are shown in the middle-right panels, with each cycle superposed according to color from the middle-left
+panel. Residuals are shown in the bottom panels, as a function of MJD and phase, as shown. The middle and bottom panels
+have the same x-axes. The fitted period is shown in the top panel as the inverse of the frequency f. The observation time of
+spectroscopic epoch B is shown by the vertical dashed line in the plots for the fifth dataset.
+
+Period [day]
+100
+20
+(ZK)
+1/f=37.3±0.2[day]
+Power
+0.5
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+A
+14.4
+14.6
+Residuals
+0.1
+0.0
+55350
+55400
+55450
+55500
+55550
+0
+5
+10
+15
+20
+25
+30
+35
+MJD [day]
+PhasePeriod [day]
+100
+20
+(ZK)
+1/f=31.8±0.1[day]
+Power
+0.5
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+14.4
+[mag]
+14.6
+0.05
+Residuals
+08
+00
+0.00
+.
+0.05
+:
+55700
+55750
+5580055850
+55900
+55950
+0
+5
+10
+15
+20
+25
+30
+MJD [day]
+Phase23
+Figure 13. (Continued)
+
+Period [day]
+100
+20
+(ZK)
+1/f=30.8±0.3[day]
+0.5
+Power
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f [day-1]
+AA
+ 14.5
+ma
+:
+14.6
+14.7
+Residuals
+0.05
+.
+:
+.
+..
+0.00
+.
+.
+.
+.
+0.05
+:
+56100
+56150
+56200
+56250
+56300
+0
+5
+10
+15
+20
+25
+30
+MJD [day]
+PhasePeriod [day]
+100
+20
+(ZK)
+1/f = 34.5±1.2 [day]
+0.2
+Power
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+14.4
+14.5
+Residuals
+0.1
+0.0
+0.1
+56500
+56550
+56600
+56650
+0
+5
+10
+15
+20
+25
+30
+MJD [day]
+Phase24
+Figure 13. (Continued)
+
+Period [day]
+100
+20
+10
+(ZK)
+0.5
+1/f=43.6±0.8[day]
+Power
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+Frequency f[day-1]
+iB
+[ma
+6
+14.6
+0.1
+Residuals
+159
+!
+0.0
+b60
+iB
+0.1
+iB
+56800
+56850
+56900
+56950
+57000
+57050
+10
+0
+20
+30
+40
+MJD [day]
+PhasePeriod [day]
+100
+20
+(ZK)
+0.4
+1/f= 42.3±1.2[day]
+Power
+0.2
+0.0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Frequency f[day-1]
+[ma
+14.6
+Residuals
+0.1
+"
+0.0
+.
+000
+:
+U.
+57200
+57250
+57300
+57350
+57400
+0
+10
+20
+30
+40
+MJD [day]
+Phase

FtFLT4oBgHgl3EQfGS-3/content/tmp_files/2301.11991v1.pdf.txt

@@ -0,0 +1,904 @@
+Real-time non-perturbative dynamics of jet production:
+quantum entanglement and vacuum modification
+Adrien Florio,1, ∗ David Frenklakh,2, † Kazuki Ikeda,2, 3, ‡ Dmitri Kharzeev,1, 2, 3, §
+Vladimir Korepin,4, ¶ Shuzhe Shi,2, ∗∗ and Kwangmin Yu5, ††
+1Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
+2Center for Nuclear Theory, Department of Physics and Astronomy,
+Stony Brook University, Stony Brook, New York 11794-3800, USA
+3Co-design Center for Quantum Advantage, Department of Physics and Astronomy,
+Stony Brook University, Stony Brook, New York 11794-3800, USA
+4C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York, 11794-3840, USA
+5Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
+The production of jets should allow to test the real-time response of the QCD vacuum disturbed
+by the propagation of high-momentum color charges. Addressing this problem theoretically requires
+a real-time, non-perturbative method. As a step in developing such an approach, we report here on
+fully quantum simulations of a massive Schwinger model coupled to external sources representing
+quark and antiquark jets as produced in e+e− annihilation. It is well known that the Schwinger
+model [QED in (1 + 1) dimensions] shares many common properties with QCD, including confine-
+ment, chiral symmetry breaking and the existence of vacuum fermion condensate. This allows us to
+study, for the first time, the modification of the vacuum chiral condensate by the propagating jets,
+and the quantum entanglement between the fragmenting jets. Our results indicate strong entangle-
+ment between the fragmentation products of the two jets at rapidity separations ∆η ≤ 2 that can
+potentially be studied in experiment.
+Introduction:
+The discovery of jets played a crucial
+role in establishing Quantum Chromodynamics (QCD)
+as the theory of strong interactions, see [1, 2] for reviews.
+The production of the initial high momentum partons is
+a short-distance process that can be described in pertur-
+bative QCD due to asymptotic freedom. However, as the
+initial partons keep radiating gluons and quark-antiquark
+pairs as described by QCD evolution equations, the char-
+acteristic virtuality decreases, and non-perturbative phe-
+nomena should come into play.
+In particular, one expects that the propagating color
+charges will disturb the non-perturbative QCD vacuum,
+and the corresponding real-time response should contain
+valuable information about the vacuum structure. More-
+over, the initial partons should be entangled by the pro-
+duction process, but whether any trace of this entangle-
+ment can be found in fragmenting jets is not clear. The
+answers to these questions lie outside of the realm of per-
+turbative QCD, and finding them requires a real-time,
+non-perturbative method.
+Such an approach is enabled by the advent of quantum
+simulations. Unfortunately, the case of real (3+1) dimen-
+sional QCD is still out of reach for the existing quantum
+hardware, as well as for real-time simulations on classical
+computers. However one can start developing real-time
+non-perturbative methods using simpler models in lower
+number of space-time dimensions.
+In this respect QED in (1 + 1) dimensions (the
+Schwinger model [3]) holds a special place: just like QCD,
+it possesses confinement, chiral symmetry breaking, and
+fermion condensate [4]. In the massless fermion limit, the
+theory is exactly solvable by bosonization, and admits a
+dual description in terms of a free massive scalar theory.
+In 1974, Casher, Kogut, and Susskind [5] proposed to
+model quark-antiquark production in e+e− annihilation
+by coupling Schwinger model to external sources propa-
+gating along the light cone.
+An explicit analytical solution of this model has been
+found in [6, 7], where this setup was also used to de-
+scribe jet quenching in heavy ion collisions by introducing
+in-medium scattering of the sources, and the anomalous
+enhancement of soft photon production in jet fragmenta-
+tion [8] observed by the DELPHI Collaboration [9].
+A more realistic extension of this approach is based
+on a massive Schwinger model, which in the bosonized
+description is dual to an interacting meson theory. In this
+case, the model is no longer analytically solvable, and so
+a numerical approach is necessary. The first study of this
+setup was carried out in [10] using a numerical classical-
+statistical approach. Coupling the Schwinger model to an
+external Yukawa theory has also been used to mimic the
+propagation of jets through a thermal environment [11].
+Various other aspects of the Schwinger model have also
+been addressed using quantum simulations, see [12–17]
+for examples and [18] for a recent review of quantum
+simulations.
+In this work, using the massive Schwinger model cou-
+pled to external sources, we perform the first fully quan-
+tum simulation of jet production. In particular, we focus
+on real-time, non-perturbative effects that have not been
+studied before: the modification of the vacuum structure
+and the entanglement between the produced jets.
+The model:
+We use the massive Schwinger model
+Hamiltonian in temporal gauge A0 = 0 in the presence
+arXiv:2301.11991v1  [hep-ph]  27 Jan 2023
+
+2
+of an external current jµ
+ext describing the produced jets:
+HC = HC
+S + HC
+ext ,
+(1)
+HC
+S =
+�
+dx
+�1
+2E2 + ¯ψ(−iγ1∂1 + gγ1A1 + m)ψ
+�
+, (2)
+HC
+ext =
+�
+dx j1
+extA1 ,
+(3)
+where Aµ is the U(1) gauge potential, E = − ˙A1 is
+the corresponding electric field, ψ is a two-component
+fermionic field, m is the fermion mass, and γµ are two-
+dimensional γ-matrices satisfying Clifford algebra; we use
+ηµν = diag(1, −1) as our metric.
+The superscript C
+stands for “continuum”.
+The effect on the theory of the interaction with the
+external source Hext is to modify Gauss law to
+∂1E − j0 = j0
+ext .
+(4)
+with j0 = g ¯ψγ0ψ. In other words, the theory is gauge
+invariant up to the presence of the external charge j0
+ext;
+the external current is a “defect” of the U(1) gauge trans-
+formation.
+To mimic production of a pair of jets in e+e− annihila-
+tion, we choose the external current to represent charges
+of opposite sign flying apart along the light cone:
+j0
+ext(x, t) = g[δ(∆x − ∆t) − δ(∆x + ∆t)]θ(∆t) ,
+j1
+ext(x, t) = g[δ(∆x − ∆t) + δ(∆x + ∆t)]θ(∆t) ,
+(5)
+where (t0, x0) is the time and position of a point where
+the jet pair is produced, and ∆x ≡ x−x0 and ∆t ≡ t−t0
+are the space and time distance from this position.
+Note that in principle one could replace the external
+probe charges by “hard” dynamical fermions, which can,
+for instance, be produced by short lived pulses of electric
+fields. This has been done in [10], where it was found
+that, at least within the semiclassics, the use of exter-
+nal charges is a very good approximation to a pair of
+dynamical relativistic “hard” fermions. This motivates
+us to restrict ourselves to the simpler case of external
+currents.
+Our goal is to study the modification of the vacuum
+due to the presence of the external sources (5). To this
+end, we evolve the ground state of the massive Schwinger
+model with the time-dependent Hamiltonian (1). In or-
+der to solve this problem, we need to discretize space-
+time and approximate the theory by a finite-dimensional
+Hilbert space.
+Lattice model: We begin by discretizing space in a lat-
+tice of N points with lattice spacing a. We choose to
+work with staggered fermions χn [19, 20]. We use a non-
+compact formulation for the U(1) gauge fields, and in-
+troduce a lattice electric field operator Ln = E(an)/g, a
+lattice vector potential φn = ag A1(an), and a link opera-
+tor Un = e−iagA1(an). We further impose open-boundary
+conditions (OBC) χN+1 = LN = 0 on the fermion and
+gauge fields. Using the Dirac matrices γ0 = σz, γ1 = i σy,
+the Hamiltonian is
+HL(t) = HL
+S + HL
+ext(t) ,
+(6)
+HL
+S = − i
+2a
+N−1
+�
+n=1
+�
+U †
+nχ†
+nχn+1 − Unχ†
+n+1χn
+�
++ ag2
+2
+N−1
+�
+n=1
+L2
+n + m
+N
+�
+n=1
+(−1)nχ†
+nχn ,
+(7)
+HL
+ext(t) = 1
+g
+N−1
+�
+n=1
+j1
+ext(a n, t)φn ,
+(8)
+where the superscript L stands for “lattice”. Even in the
+presence of point charges, Gauss law is well defined when
+integrated over a lattice spacing and reads
+Ln − Ln−1 − Qn = 1
+g
+� (n+1/2)a
+(n−1/2)a
+dx j0
+ext(x, t) ,
+(9)
+with Qn = χ†
+nχn (1 − (−1)n) the lattice charge density
+operator. For the rest of this work, we insert the sources
+at the center of our lattice, x0 = a
+� N+1
+2
+�
+, at time t0
+a = 1.
+Before proceeding with the time evolution, we take
+advantage of the fact that the gauge fields are non-
+dynamical in (1+1) dimensions to express them in terms
+of fermionic operators through Gauss law. This has the
+advantage of drastically reducing the size of the discrete
+Hilbert space needed down to 2N, at the cost of intro-
+ducing non-localities. The former turns out to outweigh
+the latter for the method we use (direct diagonalization,
+or “exact diagonalization” of the Hamiltonian), see also
+the Supplementary Material.
+We then use the remaining freedom to perform a space-
+only dependent gauge transformation to set all gauge
+links to unity. The explicit gauge transformation which
+achieves this result is Ω1 = 1, Ωn = �n−1
+i=1 U †
+i [17]. Note
+that the existence of such a transformation is a pecu-
+liarity of (1 + 1) dimensions and is related to the fact
+that the gauge field is not dynamical. We then rewrite
+Ln = Ldyn,n + Lext,n and solve Gauss law (9) as follows:
+Ldyn,n =
+n
+�
+i=1
+Qi ,
+(10)
+Lext,n(t) = −θ
+�
+t − t0 −
+���x − x0 + a
+2
+���
+�
+.
+(11)
+The non-locality is contained in the dynamical gauge field
+and the external sources create a chain of electric fluxes
+between them.
+The Hamiltonian (6) is now directly suitable for di-
+agonalization.
+However, having in mind future quan-
+tum computing applications, we have used an equivalent
+form in terms of Pauli matrices X, Y, Z, or “spin” degrees
+
+3
+t/a
+A B
+2-
+3-
+4-
+5-
+6-
+7-
+8-
+9-
+10-
+source
+fermion
+anti-fermion
+electric charge
+dynamical
+electric field
+0
+5
+10
+15
+Eele,t - Eele,0
+a g2 / 2
+external only
+0.0
+0.5
+1.0
+1.5
+νt - ν0
+0.4
+0.6
+0.8
+1.0
+SEE
+0
+2
+4
+6
+8
+10
+0.0
+0.1
+t / a
+Qt
+FIG. 1.
+(Left) Time evolution of the local charge density (vertical bars) and of the electric field (arrows), with vacuum
+expectation values subtracted. Black(white) even(odd)-sites correspond to (anti)fermions. The position of the external sources
+is shown above each configuration. From top to bottom, the rows are for time values (in units of lattice spacing a) t/a = 2−
+to 10−, where n− ≡ n − ε with ε being an arbitrarily small positive number. (Right) (from top to bottom) Time evolution
+of electric energy, scalar fermion density, entanglement entropy, and electric charge. Dotted lines in the first panel show the
+electric energy generated by the external sources.
+of freedom. We employ the Jordan–Wigner transforma-
+tion [21]
+χn = Xn − iYn
+2
+n−1
+�
+j=1
+(−iZj),
+χ†
+n = Xn + iYn
+2
+n−1
+�
+j=1
+(iZj),
+(12)
+to obtain
+HL(t) = 1
+4a
+N−1
+�
+n=1
+(XnXn+1 + YnYn+1) + m
+2
+N
+�
+n=1
+(−1)nZn
++ ag2
+2
+N−1
+�
+n=1
+(Ldyn,n + Lext,n(t))2 .
+(13)
+Our simulations then proceed as follows.
+We start
+by finding the ground state |Ψ0⟩ of the usual massive
+Schwinger model HL(0).
+We then compute the state
+|Ψt⟩ = T e−i
+� t
+0 HL(t′)dt′ |Ψ0⟩ corresponding to the evolu-
+tion under the time-dependent Hamiltonian HL(t), with
+T being the time-ordering operator. The system is ef-
+fectively “quenched” at
+t
+a = t0
+a = 1, when the external
+sources are introduced. We then compute different time-
+dependent expectation values ⟨O⟩t ≡ ⟨Ψt| O |Ψt⟩ where
+O are the operators corresponding to observables of in-
+terest.
+Vacuum modification and quantum entanglement be-
+tween the jets: We measure the local electric charge den-
+sity, the total electric charge, the scalar fermion density
+⟨ ¯ψψ⟩, the local electric field strength, and the electric
+field energy, that are given respectively by
+qn,t ≡ ⟨ψ†(a n)ψ(a n)⟩t = ⟨Zn⟩t + (−1)n
+2a
+,
+(14)
+Qt ≡
+�
+⟨ψ†(x)ψ(x)⟩t dx = a
+N
+�
+n=1
+qn,t,
+(15)
+νn,t ≡ ⟨ ¯ψ(a n)ψ(a n)⟩t = (−1)n⟨Zn⟩t
+2a
+,
+(16)
+νt ≡
+�
+⟨ ¯ψ(x)ψ(x)⟩t dx = a
+N
+�
+n=1
+νn,t,
+(17)
+Πn,t ≡ ⟨E(a n)⟩t = g ⟨Ln⟩t,
+(18)
+Eele,t ≡ 1
+2
+�
+⟨E2(x)⟩t dx = a g2
+2
+N−1
+�
+n=1
+⟨L2
+n⟩t.
+(19)
+We also compute the entanglement entropy between the
+left- and the right-hand sides of the chain
+SEE(t) = −TrA(ρt,A log ρt,A),
+(20)
+with A = {1, · · · , N/2} and B = {N/2 + 1, · · · , N}. The
+operator ρt,A = TrBρt is the partial trace of the time
+dependent density matrix ρt ≡ |Ψt⟩ ⟨Ψt| over B [see il-
+lustration in Fig. 1(left)].
+In Fig. 1, we show the time evolution of local and
+global observables respectively, for parameters N = 20,
+m = 0.25/a, and g = 0.5/a. In the left panel, we show
+the full time evolution of our quantum state.
+We ob-
+serve that both the gauge fields and the fermion fields
+
+4
+are excited by the external sources, and their effects are
+constrained within the light cone spanned by them. In
+the right panel, we observe a step-like increase in electric
+field energy. The growth of νt − ν0 shown in Fig. 1 indi-
+cates destruction of the (negative) vacuum chiral conden-
+sate ν0 by the propagating jets [22]. This destruction is
+due to the pair production from the vacuum that also re-
+sults in the screening of the electric energy which appears
+smaller than the contribution from external sources.
+Since we can access the entire quantum state, we are
+able to compute also for the first time the entanglement
+entropy between the jets. The growth of this entangle-
+ment entropy (third panel) results from the pair creation.
+Lastly, as a consistency check, we also show in the lower
+panel the total electric charge, which remains zero, as
+expected.
+Observing quantum entanglement between the jets:
+With an eye towards possible experimental studies of
+quantum entanglement between the produced jets, we
+measure the two-point correlation of scalar fermion den-
+sity operators with the vacuum expectation value sub-
+tracted,
+⟨∆νN/2+ℓ ∆νN/2+1−ℓ⟩,
+(21)
+where ∆νn ≡ νn − ⟨νn⟩vac.
+The motivation behind this study is the following.
+In the bosonization dictionary of the massive Schwinger
+model, the correlation between the scalar fermion densi-
+ties translates into the correlation among the boson pairs
+(and higher order correlations). Therefore we hope that
+this correlation function may be used to infer informa-
+tion about quantum entanglement between the pion pairs
+produced in jet fragmentation. A concrete proposal of an
+observable correlation between pion pairs produced in jet
+fragmentation has been put forward in [23].
+To isolate the effect of entanglement between the jets,
+we measure the correlation function for the cases of cor-
+related and uncorrelated sources of fermion-antifermion
+pairs. Because the entanglement should stem from the
+correlation between the sources, the case of uncorrelated
+sources provides the classical baseline for the correlation
+functions.
+…
+…
+1
+2
+l=3
+(a) correlated:
+(b) left:
+(c) right:
+FIG. 2. Illustration of correlated and uncorrelated measure-
+ments of two point correlation functions. The uncorrelated
+setup is obtained as an uncorrelated linear superposition of
+jets created by a single (anti)fermion source moving to the
+(left)right.
+0
+1
+2
+3
+4
+l = 3
+5
+7
+9
+4
+6
+8
+correlated
+0
+2
+4
+6
+8
+10
+0.0
+0.1
+0.2
+0.3
+0.4
+t / a
+uncorrelated
+102 × 〈ΔνN/2+l ΔνN/2+1-l〉
+0.0
+0.5
+1.0
+1.5
+0
+1
+2
+3
+4
+ηs
+uncorrelated
+correlated
+t = 10 a
+102 × 〈Δν-ηs Δν+ηs〉
+FIG. 3.
+Time evolution of two-point correlation functions
+with various separations.
+The upper(lower) panel is for a
+correlated(uncorrelated) setup. The large difference between
+the two cases is a signature of quantum entanglement in the
+produced pairs. (Insert) Spatial-rapidity dependence of the
+two-point correlation at the end of the evolution.
+Our method of preparation of two uncorrelated quan-
+tum systems is illustrated in Fig. 2 (b, c).
+In one of
+these systems, there is only an antifermion source mov-
+ing to the left while the fermion source sits still at the
+origin.
+We denote the quantum state of such a sys-
+tem as |ψL⟩.
+We then define its counterpart, |ψR⟩,
+corresponding to the setup of Fig. 2(c), with fermion
+source moving to the right and the antifermion source
+fixed at the origin.
+The uncorrelated state is defined
+as the superposition of left and right state with a ran-
+dom phase, |ψuncorr⟩ =
+1
+√
+2 |ψL⟩ + eiϕ
+√
+2 |ψR⟩, and the ex-
+pectation value of any observable is obtained by aver-
+aging over this random phase, ⟨⟨ψuncorr|O|ψuncorr⟩⟩ ≡
+�
+⟨ψuncorr|O|ψuncorr⟩ dϕ
+2π = ⟨ψL|O|ψL⟩
+2
++ ⟨ψR|O|ψR⟩
+2
+.
+The correlation function (21) is designed to measure
+the points that are symmetric with respect to the jet
+production vertex. We measure the two-point correlation
+function with different separation distances as functions
+of time, and the results are presented in Fig. 3. We find
+that the correlation functions measured for the correlated
+state are an order of magnitude greater than those for the
+uncorrelated state. Note that it is non-zero in the latter
+case because of the classical correlation between the par-
+ticle production in left- and right-moving jets which is
+similar to the correlation that would be induced by the
+propagation of sound along the jets’ axes.
+Meanwhile, for the quantum correlated state, we ob-
+serve the propagation of a similar pattern for odd ℓ’s and
+
+5
+similarly for even ℓ’s, which is driven by the correlated
+moving sources. After a sufficiently large time, we take
+a snapshot and present the space dependence of the cor-
+relation functions in Fig. 3 (insert), where we have con-
+verted the site separation to spatial rapidity separation,
+ηs ≡ arctanh z
+t = arctanh (ℓ−1/2)a
+t
+.
+One can clearly see a big difference between the strong
+quantum correlation for the quantum state and the near
+absence of correlations for the uncorrelated baseline.
+This difference is especially pronounced for moderate ra-
+pidity separations ∆ηs = 2ηs ≤ 2. Using the approxi-
+mate equality of space-time and momentum space rapidi-
+ties in jet fragmentation, this suggests that one should
+look for quantum entanglement among the pions pro-
+duced in the fragmentation of the two jets at rapidity
+separation ∆η ≤ 2. An observation of correlations among
+these pion pairs would constitute a direct signature of en-
+tanglement between the jets.
+Specifically, it would be interesting to study the quan-
+tum correlations between the “handedness” of the pion
+pairs produced in the fragmentation of the quark and
+antiquark jets [23].
+Some hints of such correlations
+had been reported in the data from DELPHI Collabo-
+ration [24].
+To summarize, we have performed a real-time, non-
+perturbative study of jet fragmentation using a massive
+Schwinger model with external sources. Strong distortion
+of the vacuum chiral condensate by the propagating jets
+has been observed. We have also found strong quantum
+entanglement between the fragmenting jets for rapidity
+separation ∆η ≤ 2. We hope that this result will moti-
+vate dedicated experimental studies. Our work also paves
+the way for quantum simulations of jet fragmentation us-
+ing quantum hardware; we plan to address this problem
+in the near future.
+ACKNOWLEDGEMENT
+We thank Jo˜ao Barata, Fangcheng He, Yuta Kikuchi,
+Semeon Valgushev, Tzu-Chieh Wei, and Ismail Zahed
+for useful discussions and communications.
+This work
+was supported by the U.S. Department of Energy, Of-
+fice of Science, National Quantum Information Science
+Research Centers, Co-design Center for Quantum Ad-
+vantage (C2QA) under Contract No.DE-SC0012704 (AF,
+KI, VK), and the U.S. Department of Energy, Office of
+Science, Office of Nuclear Physics, Grants Nos.
+DE-
+FG88ER41450 (DF, DK, SS) and DE-SC0012704 (AF,
+DK, KY). This research used resources of the National
+Energy Research Scientific Computing Center, a DOE
+Office of Science User Facility supported by the Office of
+Science of the U.S. Department of Energy under Contract
+No. DE-AC02-05CH11231 using NERSC award NERSC
+DDR-ERCAP0022229.
+∗ afl[email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+¶ [email protected]
+∗∗ [email protected]
+†† [email protected]
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+arXiv:2002.11146 [quant-ph].
+[16] D. E. Kharzeev and Y. Kikuchi, “Real-time chiral
+dynamics from a digital quantum simulation,” Phys.
+Rev. Res. 2 no. 2, (2020) 023342, arXiv:2001.00698
+[hep-ph].
+[17] K. Ikeda, D. E. Kharzeev, and Y. Kikuchi, “Real-time
+dynamics of Chern-Simons fluctuations near a critical
+point,” Phys. Rev. D 103 no. 7, (2021) L071502,
+arXiv:2012.02926 [hep-ph].
+[18] C. W. Bauer et al., “Quantum Simulation for High
+Energy Physics,” arXiv:2204.03381 [quant-ph].
+[19] J. B. Kogut and L. Susskind, “Hamiltonian Formulation
+of Wilson’s Lattice Gauge Theories,” Phys. Rev. D 11
+(1975) 395–408.
+[20] L. Susskind, “Lattice Fermions,” Phys. Rev. D 16
+(1977) 3031–3039.
+[21] P. Jordan and E. P. Wigner, “About the Pauli exclusion
+principle,” Z. Phys. 47 (1928) 631–651.
+[22] For the case of static sources, partial destruction of the
+chiral condensate in Schwinger model was studied in [?
+].
+[23] A. Efremov and D. Kharzeev, “CP violating effect of
+QCD vacuum in quark fragmentation,” Phys. Lett. B
+366 (1996) 311–315, arXiv:hep-ph/9506412.
+[24] “A Measurement of Quark Spin Correlations in
+Hadronic Z Decays,”.
+[25] M. Bruno, The energy scale of the 3-flavour Lambda
+parameter. PhD thesis, Humboldt-Universit¨at zu Berlin,
+Mathematisch-Naturwissenschaftliche Fakult¨at, 2016.
+
+7
+Supplementary Material
+In the main text, we study the evolution of the Schwinger model Hamiltonian in the presence of external charges
+moving on the light-cone. In this supplemental material, we show that despite the relatively modest lattice sizes,
+the volume dependence and effect of open-boundary conditions are well under control for the quantities and set of
+parameters we studied.
+0.0
+0.5
+1.0
+1.5
+νt - ν0
+N=6, Λ=3
+N=20
+16
+12
+8
+0
+2
+4
+6
+8
+10
+12
+Eele,t - Eele,0
+a g2 / 2
+0
+2
+4
+6
+8
+10
+0.4
+0.6
+0.8
+1.0
+t / a
+SEE,t
+-0.32
+-0.30
+-0.28
+-0.26
+-0.24
+-0.22
+νn [1/a]
+m=0.25/a
+g=0.5/a
+0
+5
+10
+15
+20
+0.05
+0.10
+0.15
+0.20
+0.25
+n
+Eele,n [1/a]
+-0.47
+-0.46
+-0.45
+-0.44
+-0.43
+νn [1/a]
+m=1/a
+g=1/a
+periodic, dynamical
+open, Gauss' law
+0
+2
+4
+6
+8
+10
+1.7
+1.8
+1.9
+2.0
+2.1
+2.2
+n
+Eele,n [10-3/a]
+FIG. 4.
+(Left) Time evolution of total electric field energy, mass creation, and entanglement entropy for periodic boundary
+condition with dynamical gauge field with N = 6 and Λ = 3 (black dotted) versus open boundary condition with gauge field
+fixed by the Gauss’ law with lattice size from 8(red) to 20 (purple). (Middle) Comparison of local electric field energy and
+chiral condensate. Black dotted lines are determined by the bulk values. In both left and middle panels, parameters are set to
+be N = 20, m = 0.25/a, and g = 0.5/a. (Right) Same as middle but with parameters with parameters N = 10, m = 1/a, and
+g = 1/a. Red dots correspond to open boundary condition with gauge field fixed by the Gauss’ law, whereas black lines are for
+periodic boundary condition with dynamical gauge field.
+In the left-hand side of Fig. 4, plain colored lines show the time evolution of the chiral condensate, electric field
+energy and entanglement entropy for different lattice sizes. The maximal time until which a simulation is meaningful
+is set by half the lattice site plus one unit of time, as after this the point sources exit the system. As illustrated by
+the agreement of the different curves, finite size effects are minimal.
+We also assess the effect of using open-boundary conditions. We expect that the introduction of a physical boundary
+to have the same effect as the introduction of a defect. Excitations localize on the boundary and affect the system in a
+“boundary zone” of order the correlation length of the system, see for instance [25]. We can see in the middle panel of
+Fig. 4 that this is indeed what happens. We show in the upper(lower) panel the value of the chiral condensate(electric
+energy density) as a function of lattice sites in the ground state. In both cases, we can clearly observe a boundary
+zone extending over approximately 4-5 lattice sites.
+It also matches the naive estimate of the correlation length
+ξ ∼ 1
+m = 4a.
+To further crosscheck our results, we also decided to implement simulations with periodic-boundary conditions,
+χN+1 = χ1 and χ†
+N+1 = χ†
+1, and to keep the gauge field as independent operators. The Hamiltonian reads
+HPBC = 1
+8a
+N
+�
+n=1
+�
+(Un + U †
+n) ⊗ (XnXn+1 + YnYn+1) + i(Un − U †
+n) ⊗ (XnYn+1 − YnXn+1)
+�
++ m
+2
+N
+�
+n=1
+(−1)nZn + a g2
+2
+N
+�
+n=1
+L2
+n + 1
+g
+N
+�
+n=1
+j1
+ext(xn)φn ,
+(22)
+where XN+1 ≡ (−1)
+N
+2 X1
+�N−1
+m=2 Zm, and likewise for YN+1. We implement the electric-field operator and the link
+
+8
+operator as
+Ln =
+Λ
+�
+ϵ=−Λ
+ϵ |ϵ⟩n ⟨ϵ|n ,
+(23)
+Un = |Λ⟩n ⟨−Λ|n +
+Λ−1
+�
+ϵ=−Λ
+|ϵ⟩n ⟨ϵ + 1|n ,
+(24)
+where Λ is a cutoff [15], the eigenbasis |ϵ⟩n of electric field operator Ln.
+The size of the discrete Hilbert space for a truncation Λ is (2Λ + 1)N 2N, namely it is (2Λ + 1)N times larger than
+in the case of open boundary conditions after integrating out the gauge fields through Gauss law. This also means
+that only smaller lattices can be simulated in this set-up.
+We show results of the chiral condensate and electric field energy for N = 6 and Λ = 3 as black dotted lines in the
+left-hand side of Fig. 4. No deviations from the open-boundary conditions can be seen.
+We also investigated the space-dependence of observables. In particular, we expect the bulk value of the open-
+boundary conditions to equal the periodic boundary condition average. Unfortunately, we could not directly verify
+this for the parameters used in the main text as the lattice size required are not achievable not integrating out gauge
+fields. As an alternative, we verified it for a larger mass and larger coupling ma = ga = 1 such that the boundary
+zone is smaller. The results are shown in the right-hand side panel of Fig. 4. Again, the two lattice sites affected by
+the boundary is in agreement with naive expectations. And as expected, the bulk value of the open-boundary system
+matches the value of the periodic one.
+

FtFLT4oBgHgl3EQfGS-3/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf,len=477
+page_content='Real-time non-perturbative dynamics of jet production:  quantum entanglement and vacuum modification  Adrien Florio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ∗ David Frenklakh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' † Kazuki Ikeda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ‡ Dmitri Kharzeev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' §  Vladimir Korepin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ¶ Shuzhe Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ∗∗ and Kwangmin Yu5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' ††  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' New York 11973-5000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' USA  2Center for Nuclear Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Stony Brook University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Stony Brook,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' New York 11794-3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' USA  3Co-design Center for Quantum Advantage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Stony Brook University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Stony Brook,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' New York 11794-3800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' USA  4C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York, 11794-3840, USA  5Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973-5000, USA  The production of jets should allow to test the real-time response of the QCD vacuum disturbed  by the propagation of high-momentum color charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Addressing this problem theoretically requires  a real-time, non-perturbative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' As a step in developing such an approach, we report here on  fully quantum simulations of a massive Schwinger model coupled to external sources representing  quark and antiquark jets as produced in e+e− annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' It is well known that the Schwinger  model [QED in (1 + 1) dimensions] shares many common properties with QCD, including confine-  ment, chiral symmetry breaking and the existence of vacuum fermion condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This allows us to  study, for the first time, the modification of the vacuum chiral condensate by the propagating jets,  and the quantum entanglement between the fragmenting jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Our results indicate strong entangle-  ment between the fragmentation products of the two jets at rapidity separations ∆η ≤ 2 that can  potentially be studied in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Introduction:  The discovery of jets played a crucial  role in establishing Quantum Chromodynamics (QCD)  as the theory of strong interactions, see [1, 2] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The production of the initial high momentum partons is  a short-distance process that can be described in pertur-  bative QCD due to asymptotic freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' However, as the  initial partons keep radiating gluons and quark-antiquark  pairs as described by QCD evolution equations, the char-  acteristic virtuality decreases, and non-perturbative phe-  nomena should come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In particular, one expects that the propagating color  charges will disturb the non-perturbative QCD vacuum,  and the corresponding real-time response should contain  valuable information about the vacuum structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' More-  over, the initial partons should be entangled by the pro-  duction process, but whether any trace of this entangle-  ment can be found in fragmenting jets is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The  answers to these questions lie outside of the realm of per-  turbative QCD, and finding them requires a real-time,  non-perturbative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Such an approach is enabled by the advent of quantum  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Unfortunately, the case of real (3+1) dimen-  sional QCD is still out of reach for the existing quantum  hardware, as well as for real-time simulations on classical  computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' However one can start developing real-time  non-perturbative methods using simpler models in lower  number of space-time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In this respect QED in (1 + 1) dimensions (the  Schwinger model [3]) holds a special place: just like QCD,  it possesses confinement, chiral symmetry breaking, and  fermion condensate [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the massless fermion limit, the  theory is exactly solvable by bosonization, and admits a  dual description in terms of a free massive scalar theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In 1974, Casher, Kogut, and Susskind [5] proposed to  model quark-antiquark production in e+e− annihilation  by coupling Schwinger model to external sources propa-  gating along the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  An explicit analytical solution of this model has been  found in [6, 7], where this setup was also used to de-  scribe jet quenching in heavy ion collisions by introducing  in-medium scattering of the sources, and the anomalous  enhancement of soft photon production in jet fragmenta-  tion [8] observed by the DELPHI Collaboration [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  A more realistic extension of this approach is based  on a massive Schwinger model, which in the bosonized  description is dual to an interacting meson theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In this  case, the model is no longer analytically solvable, and so  a numerical approach is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The first study of this  setup was carried out in [10] using a numerical classical-  statistical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Coupling the Schwinger model to an  external Yukawa theory has also been used to mimic the  propagation of jets through a thermal environment [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Various other aspects of the Schwinger model have also  been addressed using quantum simulations, see [12–17]  for examples and [18] for a recent review of quantum  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In this work, using the massive Schwinger model cou-  pled to external sources, we perform the first fully quan-  tum simulation of jet production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In particular, we focus  on real-time, non-perturbative effects that have not been  studied before: the modification of the vacuum structure  and the entanglement between the produced jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The model:  We use the massive Schwinger model  Hamiltonian in temporal gauge A0 = 0 in the presence  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='11991v1  [hep-ph]  27 Jan 2023  2  of an external current jµ  ext describing the produced jets:  HC = HC  S + HC  ext ,  (1)  HC  S =  �  dx  �1  2E2 + ¯ψ(−iγ1∂1 + gγ1A1 + m)ψ  �  , (2)  HC  ext =  �  dx j1  extA1 ,  (3)  where Aµ is the U(1) gauge potential, E = − ˙A1 is  the corresponding electric field, ψ is a two-component  fermionic field, m is the fermion mass, and γµ are two-  dimensional γ-matrices satisfying Clifford algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' we use  ηµν = diag(1, −1) as our metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The superscript C  stands for “continuum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The effect on the theory of the interaction with the  external source Hext is to modify Gauss law to  ∂1E − j0 = j0  ext .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (4)  with j0 = g ¯ψγ0ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In other words, the theory is gauge  invariant up to the presence of the external charge j0  ext;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  the external current is a “defect” of the U(1) gauge trans-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  To mimic production of a pair of jets in e+e− annihila-  tion, we choose the external current to represent charges  of opposite sign flying apart along the light cone:  j0  ext(x, t) = g[δ(∆x − ∆t) − δ(∆x + ∆t)]θ(∆t) ,  j1  ext(x, t) = g[δ(∆x − ∆t) + δ(∆x + ∆t)]θ(∆t) ,  (5)  where (t0, x0) is the time and position of a point where  the jet pair is produced, and ∆x ≡ x−x0 and ∆t ≡ t−t0  are the space and time distance from this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Note that in principle one could replace the external  probe charges by “hard” dynamical fermions, which can,  for instance, be produced by short lived pulses of electric  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This has been done in [10], where it was found  that, at least within the semiclassics, the use of exter-  nal charges is a very good approximation to a pair of  dynamical relativistic “hard” fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This motivates  us to restrict ourselves to the simpler case of external  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Our goal is to study the modification of the vacuum  due to the presence of the external sources (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' To this  end, we evolve the ground state of the massive Schwinger  model with the time-dependent Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In or-  der to solve this problem, we need to discretize space-  time and approximate the theory by a finite-dimensional  Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Lattice model: We begin by discretizing space in a lat-  tice of N points with lattice spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We choose to  work with staggered fermions χn [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We use a non-  compact formulation for the U(1) gauge fields, and in-  troduce a lattice electric field operator Ln = E(an)/g, a  lattice vector potential φn = ag A1(an), and a link opera-  tor Un = e−iagA1(an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We further impose open-boundary  conditions (OBC) χN+1 = LN = 0 on the fermion and  gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Using the Dirac matrices γ0 = σz, γ1 = i σy,  the Hamiltonian is  HL(t) = HL  S + HL  ext(t) ,  (6)  HL  S = − i  2a  N−1  �  n=1  �  U †  nχ†  nχn+1 − Unχ†  n+1χn  �  + ag2  2  N−1  �  n=1  L2  n + m  N  �  n=1  (−1)nχ†  nχn ,  (7)  HL  ext(t) = 1  g  N−1  �  n=1  j1  ext(a n, t)φn ,  (8)  where the superscript L stands for “lattice”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Even in the  presence of point charges, Gauss law is well defined when  integrated over a lattice spacing and reads  Ln − Ln−1 − Qn = 1  g  � (n+1/2)a  (n−1/2)a  dx j0  ext(x, t) ,  (9)  with Qn = χ†  nχn (1 − (−1)n) the lattice charge density  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' For the rest of this work, we insert the sources  at the center of our lattice, x0 = a  � N+1  2  �  , at time t0  a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Before proceeding with the time evolution, we take  advantage of the fact that the gauge fields are non-  dynamical in (1+1) dimensions to express them in terms  of fermionic operators through Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This has the  advantage of drastically reducing the size of the discrete  Hilbert space needed down to 2N, at the cost of intro-  ducing non-localities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The former turns out to outweigh  the latter for the method we use (direct diagonalization,  or “exact diagonalization” of the Hamiltonian), see also  the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We then use the remaining freedom to perform a space-  only dependent gauge transformation to set all gauge  links to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The explicit gauge transformation which  achieves this result is Ω1 = 1, Ωn = �n−1  i=1 U †  i [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Note  that the existence of such a transformation is a pecu-  liarity of (1 + 1) dimensions and is related to the fact  that the gauge field is not dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then rewrite  Ln = Ldyn,n + Lext,n and solve Gauss law (9) as follows:  Ldyn,n =  n  �  i=1  Qi ,  (10)  Lext,n(t) = −θ  �  t − t0 −  ���x − x0 + a  2  ���  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (11)  The non-locality is contained in the dynamical gauge field  and the external sources create a chain of electric fluxes  between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The Hamiltonian (6) is now directly suitable for di-  agonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  However, having in mind future quan-  tum computing applications, we have used an equivalent  form in terms of Pauli matrices X, Y, Z, or “spin” degrees  3  t/a  A B  2-  3-  4-  5-  6-  7-  8-  9-  10-  source  fermion  anti-fermion  electric charge  dynamical  electric field  0  5  10  15  Eele,t - Eele,0  a g2 / 2  external only  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5  νt - ν0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  SEE  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1  t / a  Qt  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (Left) Time evolution of the local charge density (vertical bars) and of the electric field (arrows), with vacuum  expectation values subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Black(white) even(odd)-sites correspond to (anti)fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The position of the external sources  is shown above each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' From top to bottom, the rows are for time values (in units of lattice spacing a) t/a = 2−  to 10−, where n− ≡ n − ε with ε being an arbitrarily small positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Right) (from top to bottom) Time evolution  of electric energy, scalar fermion density, entanglement entropy, and electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Dotted lines in the first panel show the  electric energy generated by the external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We employ the Jordan–Wigner transforma-  tion [21]  χn = Xn − iYn  2  n−1  �  j=1  (−iZj),  χ†  n = Xn + iYn  2  n−1  �  j=1  (iZj),  (12)  to obtain  HL(t) = 1  4a  N−1  �  n=1  (XnXn+1 + YnYn+1) + m  2  N  �  n=1  (−1)nZn  + ag2  2  N−1  �  n=1  (Ldyn,n + Lext,n(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (13)  Our simulations then proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We start  by finding the ground state |Ψ0⟩ of the usual massive  Schwinger model HL(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We then compute the state  |Ψt⟩ = T e−i  � t  0 HL(t′)dt′ |Ψ0⟩ corresponding to the evolu-  tion under the time-dependent Hamiltonian HL(t), with  T being the time-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The system is ef-  fectively “quenched” at  t  a = t0  a = 1, when the external  sources are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We then compute different time-  dependent expectation values ⟨O⟩t ≡ ⟨Ψt| O |Ψt⟩ where  O are the operators corresponding to observables of in-  terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Vacuum modification and quantum entanglement be-  tween the jets: We measure the local electric charge den-  sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the total electric charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the scalar fermion density  ⟨ ¯ψψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' the local electric field strength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' and the electric  field energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' that are given respectively by  qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ ⟨ψ†(a n)ψ(a n)⟩t = ⟨Zn⟩t + (−1)n  2a  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (14)  Qt ≡  �  ⟨ψ†(x)ψ(x)⟩t dx = a  N  �  n=1  qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (15)  νn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ ⟨ ¯ψ(a n)ψ(a n)⟩t = (−1)n⟨Zn⟩t  2a  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (16)  νt ≡  �  ⟨ ¯ψ(x)ψ(x)⟩t dx = a  N  �  n=1  νn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (17)  Πn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ ⟨E(a n)⟩t = g ⟨Ln⟩t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (18)  Eele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='t ≡ 1  2  �  ⟨E2(x)⟩t dx = a g2  2  N−1  �  n=1  ⟨L2  n⟩t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (19)  We also compute the entanglement entropy between the  left- and the right-hand sides of the chain  SEE(t) = −TrA(ρt,A log ρt,A),  (20)  with A = {1, · · · , N/2} and B = {N/2 + 1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The  operator ρt,A = TrBρt is the partial trace of the time  dependent density matrix ρt ≡ |Ψt⟩ ⟨Ψt| over B [see il-  lustration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1(left)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1, we show the time evolution of local and  global observables respectively, for parameters N = 20,  m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25/a, and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In the left panel, we show  the full time evolution of our quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We ob-  serve that both the gauge fields and the fermion fields  4  are excited by the external sources, and their effects are  constrained within the light cone spanned by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In  the right panel, we observe a step-like increase in electric  field energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The growth of νt − ν0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 1 indi-  cates destruction of the (negative) vacuum chiral conden-  sate ν0 by the propagating jets [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This destruction is  due to the pair production from the vacuum that also re-  sults in the screening of the electric energy which appears  smaller than the contribution from external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Since we can access the entire quantum state, we are  able to compute also for the first time the entanglement  entropy between the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The growth of this entangle-  ment entropy (third panel) results from the pair creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Lastly, as a consistency check, we also show in the lower  panel the total electric charge, which remains zero, as  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Observing quantum entanglement between the jets:  With an eye towards possible experimental studies of  quantum entanglement between the produced jets, we  measure the two-point correlation of scalar fermion den-  sity operators with the vacuum expectation value sub-  tracted,  ⟨∆νN/2+ℓ ∆νN/2+1−ℓ⟩,  (21)  where ∆νn ≡ νn − ⟨νn⟩vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The motivation behind this study is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In the bosonization dictionary of the massive Schwinger  model, the correlation between the scalar fermion densi-  ties translates into the correlation among the boson pairs  (and higher order correlations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Therefore we hope that  this correlation function may be used to infer informa-  tion about quantum entanglement between the pion pairs  produced in jet fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' A concrete proposal of an  observable correlation between pion pairs produced in jet  fragmentation has been put forward in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  To isolate the effect of entanglement between the jets,  we measure the correlation function for the cases of cor-  related and uncorrelated sources of fermion-antifermion  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Because the entanglement should stem from the  correlation between the sources, the case of uncorrelated  sources provides the classical baseline for the correlation  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  …  …  1  2  l=3  (a) correlated:  (b) left:  (c) right:  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Illustration of correlated and uncorrelated measure-  ments of two point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The uncorrelated  setup is obtained as an uncorrelated linear superposition of  jets created by a single (anti)fermion source moving to the  (left)right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  0  1  2  3  4  l = 3  5  7  9  4  6  8  correlated  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4  t / a  uncorrelated  102 × 〈ΔνN/2+l ΔνN/2+1-l〉  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5  0  1  2  3  4  ηs  uncorrelated  correlated  t = 10 a  102 × 〈Δν-ηs Δν+ηs〉  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Time evolution of two-point correlation functions  with various separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The upper(lower) panel is for a  correlated(uncorrelated) setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The large difference between  the two cases is a signature of quantum entanglement in the  produced pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Insert) Spatial-rapidity dependence of the  two-point correlation at the end of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Our method of preparation of two uncorrelated quan-  tum systems is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2 (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In one of  these systems, there is only an antifermion source mov-  ing to the left while the fermion source sits still at the  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We denote the quantum state of such a sys-  tem as |ψL⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We then define its counterpart, |ψR⟩,  corresponding to the setup of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 2(c), with fermion  source moving to the right and the antifermion source  fixed at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The uncorrelated state is defined  as the superposition of left and right state with a ran-  dom phase, |ψuncorr⟩ =  1  √  2 |ψL⟩ + eiϕ  √  2 |ψR⟩, and the ex-  pectation value of any observable is obtained by aver-  aging over this random phase, ⟨⟨ψuncorr|O|ψuncorr⟩⟩ ≡  �  ⟨ψuncorr|O|ψuncorr⟩ dϕ  2π = ⟨ψL|O|ψL⟩  2  + ⟨ψR|O|ψR⟩  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The correlation function (21) is designed to measure  the points that are symmetric with respect to the jet  production vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We measure the two-point correlation  function with different separation distances as functions  of time, and the results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We find  that the correlation functions measured for the correlated  state are an order of magnitude greater than those for the  uncorrelated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Note that it is non-zero in the latter  case because of the classical correlation between the par-  ticle production in left- and right-moving jets which is  similar to the correlation that would be induced by the  propagation of sound along the jets’ axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Meanwhile, for the quantum correlated state, we ob-  serve the propagation of a similar pattern for odd ℓ’s and  5  similarly for even ℓ’s, which is driven by the correlated  moving sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' After a sufficiently large time, we take  a snapshot and present the space dependence of the cor-  relation functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 3 (insert), where we have con-  verted the site separation to spatial rapidity separation,  ηs ≡ arctanh z  t = arctanh (ℓ−1/2)a  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  One can clearly see a big difference between the strong  quantum correlation for the quantum state and the near  absence of correlations for the uncorrelated baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  This difference is especially pronounced for moderate ra-  pidity separations ∆ηs = 2ηs ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Using the approxi-  mate equality of space-time and momentum space rapidi-  ties in jet fragmentation, this suggests that one should  look for quantum entanglement among the pions pro-  duced in the fragmentation of the two jets at rapidity  separation ∆η ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' An observation of correlations among  these pion pairs would constitute a direct signature of en-  tanglement between the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Specifically, it would be interesting to study the quan-  tum correlations between the “handedness” of the pion  pairs produced in the fragmentation of the quark and  antiquark jets [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Some hints of such correlations  had been reported in the data from DELPHI Collabo-  ration [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  To summarize, we have performed a real-time, non-  perturbative study of jet fragmentation using a massive  Schwinger model with external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Strong distortion  of the vacuum chiral condensate by the propagating jets  has been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We have also found strong quantum  entanglement between the fragmenting jets for rapidity  separation ∆η ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We hope that this result will moti-  vate dedicated experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Our work also paves  the way for quantum simulations of jet fragmentation us-  ing quantum hardware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' we plan to address this problem  in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  We thank Jo˜ao Barata, Fangcheng He, Yuta Kikuchi,  Semeon Valgushev, Tzu-Chieh Wei, and Ismail Zahed  for useful discussions and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  This work  was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Energy, Of-  fice of Science, National Quantum Information Science  Research Centers, Co-design Center for Quantum Ad-  vantage (C2QA) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='DE-SC0012704 (AF,  KI, VK), and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Energy, Office of  Science, Office of Nuclear Physics, Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  DE-  FG88ER41450 (DF, DK, SS) and DE-SC0012704 (AF,  DK, KY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This research used resources of the National  Energy Research Scientific Computing Center, a DOE  Office of Science User Facility supported by the Office of  Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Department of Energy under Contract  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' DE-AC02-05CH11231 using NERSC award NERSC  DDR-ERCAP0022229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  ∗ aflorio@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='gov  † david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='frenklakh@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='edu  ‡ kazuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='ikeda@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='edu  § dmitri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='kharzeev@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='edu  ¶ vladimir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='korepin@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='edu  ∗∗ shuzhe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='shi@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='edu  †† kyu@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='gov  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Sterman, “QCD and jets,” in Theoretical  Advanced Study Institute in Elementary Particle  Physics: Physics in D ≧ 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 67–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 12, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='  [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Dokshitzer, “QCD and hadron dynamics,” Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' Susskind, “Charge  Shielding and Quark Confinement in the Massive  Schwinger Model,” Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' 7, (2021) L071502,  arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='02926 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content=' Kogut and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Susskind, “Hamiltonian Formulation  of Wilson’s Lattice Gauge Theories,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='  [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Susskind, “Lattice Fermions,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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+page_content='  [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Jordan and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Wigner, “About the Pauli exclusion  principle,” Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 47 (1928) 631–651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  [22] For the case of static sources, partial destruction of the  chiral condensate in Schwinger model was studied in [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Efremov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Kharzeev, “CP violating effect of  QCD vacuum in quark fragmentation,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' B  366 (1996) 311–315, arXiv:hep-ph/9506412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  [24] “A Measurement of Quark Spin Correlations in  Hadronic Z Decays,”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Bruno, The energy scale of the 3-flavour Lambda  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' PhD thesis, Humboldt-Universit¨at zu Berlin,  Mathematisch-Naturwissenschaftliche Fakult¨at, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  7  Supplementary Material  In the main text, we study the evolution of the Schwinger model Hamiltonian in the presence of external charges  moving on the light-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In this supplemental material, we show that despite the relatively modest lattice sizes,  the volume dependence and effect of open-boundary conditions are well under control for the quantities and set of  parameters we studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5  νt - ν0  N=6, Λ=3  N=20  16  12  8  0  2  4  6  8  10  12  Eele,t - Eele,0  a g2 / 2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  t / a  SEE,t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='22  νn [1/a]  m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25/a  g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5/a  0  5  10  15  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25  n  Eele,n [1/a]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content="43  νn [1/a]  m=1/a  g=1/a  periodic, dynamical  open, Gauss' law  0  2  4  6  8  10  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='2  n  Eele,n [10-3/a]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  (Left) Time evolution of total electric field energy, mass creation, and entanglement entropy for periodic boundary  condition with dynamical gauge field with N = 6 and Λ = 3 (black dotted) versus open boundary condition with gauge field  fixed by the Gauss’ law with lattice size from 8(red) to 20 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Middle) Comparison of local electric field energy and  chiral condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Black dotted lines are determined by the bulk values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In both left and middle panels, parameters are set to  be N = 20, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='25/a, and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='5/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' (Right) Same as middle but with parameters with parameters N = 10, m = 1/a, and  g = 1/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Red dots correspond to open boundary condition with gauge field fixed by the Gauss’ law, whereas black lines are for  periodic boundary condition with dynamical gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  In the left-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4, plain colored lines show the time evolution of the chiral condensate, electric field  energy and entanglement entropy for different lattice sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The maximal time until which a simulation is meaningful  is set by half the lattice site plus one unit of time, as after this the point sources exit the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' As illustrated by  the agreement of the different curves, finite size effects are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We also assess the effect of using open-boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We expect that the introduction of a physical boundary  to have the same effect as the introduction of a defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Excitations localize on the boundary and affect the system in a  “boundary zone” of order the correlation length of the system, see for instance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We can see in the middle panel of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4 that this is indeed what happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We show in the upper(lower) panel the value of the chiral condensate(electric  energy density) as a function of lattice sites in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In both cases, we can clearly observe a boundary  zone extending over approximately 4-5 lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  It also matches the naive estimate of the correlation length  ξ ∼ 1  m = 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  To further crosscheck our results, we also decided to implement simulations with periodic-boundary conditions,  χN+1 = χ1 and χ†  N+1 = χ†  1, and to keep the gauge field as independent operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The Hamiltonian reads  HPBC = 1  8a  N  �  n=1  �  (Un + U †  n) ⊗ (XnXn+1 + YnYn+1) + i(Un − U †  n) ⊗ (XnYn+1 − YnXn+1)  �  + m  2  N  �  n=1  (−1)nZn + a g2  2  N  �  n=1  L2  n + 1  g  N  �  n=1  j1  ext(xn)φn ,  (22)  where XN+1 ≡ (−1)  N  2 X1  �N−1  m=2 Zm, and likewise for YN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' We implement the electric-field operator and the link  8  operator as  Ln =  Λ  �  ϵ=−Λ  ϵ |ϵ⟩n ⟨ϵ|n ,  (23)  Un = |Λ⟩n ⟨−Λ|n +  Λ−1  �  ϵ=−Λ  |ϵ⟩n ⟨ϵ + 1|n ,  (24)  where Λ is a cutoff [15], the eigenbasis |ϵ⟩n of electric field operator Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  The size of the discrete Hilbert space for a truncation Λ is (2Λ + 1)N 2N, namely it is (2Λ + 1)N times larger than  in the case of open boundary conditions after integrating out the gauge fields through Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' This also means  that only smaller lattices can be simulated in this set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We show results of the chiral condensate and electric field energy for N = 6 and Λ = 3 as black dotted lines in the  left-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' No deviations from the open-boundary conditions can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content='  We also investigated the space-dependence of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' In particular, we expect the bulk value of the open-  boundary conditions to equal the periodic boundary condition average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Unfortunately, we could not directly verify  this for the parameters used in the main text as the lattice size required are not achievable not integrating out gauge  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' As an alternative, we verified it for a larger mass and larger coupling ma = ga = 1 such that the boundary  zone is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' The results are shown in the right-hand side panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
+page_content=' Again, the two lattice sites affected by  the boundary is in agreement with naive expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtFLT4oBgHgl3EQfGS-3/content/2301.11991v1.pdf'}
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