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+DESY 23-001
+UWThPh-2023-1
+Investigation of the scale dependence in the MSR and MS top
+quark mass schemes for the tt invariant mass differential cross
+section using LHC data
+Toni M¨akel¨a∗a,b, Andr´e H. Hoang†c,d, Katerina Lipka‡a,e, and Sven-Olaf Moch§f
+aDeutsches Elektronen-Synchrotron, Notkestr. 85, 22607 Hamburg, Germany
+bNational Centre for Nuclear Research, Pasteura 7, PL-02-093 Warsaw, Poland
+cFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
+dErwin Schr¨odinger Institute for Mathematics and Physics, University of Vienna, Boltzmanngasse 9,
+A-1090 Vienna, Austria
+eFakult¨at f¨ur Mathematik und Naturwissenschaften, Bergische Universit¨at Wuppertal, Gaußstrassse 20,
+D-42119 Wuppertal, Germany
+fII. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 Hamburg,
+Germany
+January 10, 2023
+Abstract
+The computation of the single-differential top quark-antiquark pair (tt) production cross
+section at NLO in the fixed-order expansion is examined consistently using the MSR and MS
+short-distance top quark mass schemes. A thorough investigation of the dependence of the tt
+invariant mass spectrum on the renormalization scales R and µm of the MSR mass mMSR
+t
+(R) and
+MS mass mt(µm), respectively, is carried out. We demonstrate that a scale choice of R ∼ 80 GeV
+is important for the stability of the cross-section predictions for the low tt invariant mass range,
+which is important for a reliable extraction of the top quark mass. Furthermore, a choice of semi-
+dynamical renormalization and factorization scales is preferred. These findings are expected to
+remain valid once non-relativistic quasi-bound state effects are included in the low invariant
+mass region.
+∗[email protected]
+†[email protected]
+‡[email protected]
+§[email protected]
+1
+arXiv:2301.03546v1  [hep-ph]  9 Jan 2023
+
+1
+Introduction
+The top quark mass mt is a fundamental parameter of the Standard Model and has an important
+role in many predictions, both directly and via higher-order corrections. For instance, together with
+the values of the strong coupling constant αs and the mass of the Higgs boson, it determines the
+stability of the electroweak vacuum [1–4]. Yet, the formal definition of quark masses makes them
+renormalization scheme dependent quantities. The frequently used pole mass mpole
+t
+, which is based
+on the picture that real and virtual radiation can be resolved at arbitrarily small energy scales,
+suffers from the renormalon ambiguity, a spurious linear infrared (IR) sensitivity of the order of the
+QCD scale ΛQCD [5–7].1 In contrast, short-distance mass schemes such as the modified minimal
+subtraction (MS) scheme [10, 11] mass mt(µm), or the MSR scheme [12, 13] mass mMSR
+t
+(R), do not
+have this issue, and their renormalization scales µm and R, respectively, act as a finite resolution
+scale. This means that real and virtual radiation are treated inclusively for scales below µm and
+R, which provides a more suitable description for realistic physical observables. The absence of the
+O(ΛQCD) renormalon problem and the additional freedom to adopt suitable scale choices can be
+very useful to achieve higher precision. Moreover, the MSR scheme can be related to quark mass
+definitions used in parton shower Monte Carlo programs, as worked out conceptually in Refs. [14–
+16], see also Refs. [13, 17] for details. For small, but still perturbative R values at around 2 GeV
+the MSR mass serves as a viable and renormalon-free proxy for the pole mass concept.
+The sensitivity of an observable to mt is always associated to a dynamical physics scale, such as
+the inverse Bohr radius ⟨1/rB⟩ ∼ mtαs for the impact of the top quark-antiquark (tt) quasi-bound
+state on the tt cross section at the threshold, or the top quark width Γt for the single top resonance
+mass distribution. Thus, the scale dependence of mt(µm) and mMSR
+t
+(R) allows to properly adapt
+to these dynamical scales for an observable under consideration. The respective renormalization
+group equations (RGEs) and matching relations provide the tool to unambiguously relate the top
+quark mass extracted at different dynamical scales. This concept is well known for the running
+strong coupling αs and applies to the quark masses as well, particularly for increasing precision.
+In this work, the dependence of the invariant mass of the tt pair, mtt, on the MSR mass scale
+R and the MS mass scale µm is investigated concurrently for the first time accounting for QCD
+corrections. Using experimental measurements of tt production at the LHC at √s = 13 TeV [18],
+the next-to-leading order (NLO) prediction of the mtt differential cross section from Refs. [19, 20]
+and the scheme implementation procedure of Refs. [21, 22], we demonstrate that the proper scheme
+choice is of key importance and affects the size of higher-order corrections as well as the resulting
+value of the extracted top quark mass. In Sec. 2, we review the MS and MSR top quark mass
+schemes and the formulae to implement them, and in Sec. 3 we carry out a detailed investigation
+concerning the best choice of the MSR renormalization scale R. In Sec. 4 we quote the results for
+1We note that linear IR sensitivities arise in cross sections whenever cuts on soft radiation are imposed, see e.g.
+Ref. [8]. These are associated to nonperturbative corrections in contrast to the pole mass, where the IR sensitivity
+arises purely from the choice of scheme [9].
+2
+
+mMSR
+t
+(R = 1 GeV) and higher R values from the fits to the LHC measurements, demonstrating the
+impact of the renormalization scale choice. We close in Sec. 5 with a summary and an outlook on
+future improvements.
+2
+Running mt and the tt pair production cross section at NLO
+In terms of a general mass renormalization scale µm, the pole and MS masses are related in
+perturbative QCD as
+mpole
+t
+= mt(µm)
+�
+1 +
+�
+n=1
+dMS
+n (µm)
+�
+a(6)
+s (µm)
+�n
+�
+,
+(2.1)
+where as ≡ αs/π. Here and everywhere else in this study, we explicitly indicate by the superscript
+whether we use the strong coupling α(5)
+s
+in the 5-flavor or α(6)
+s
+in the 6-flavor scheme. For the
+parton distribution functions (PDFs) only the 5-flavor scheme is employed. All quarks except for
+the top quark are treated as massless. The coefficients dMS
+n (µm) in Eq. (2.1) are known up to four
+loops [23] and the first few orders read [24–26]
+dMS
+1 (µm) = 4/3 + L ,
+dMS
+2 (µm) = 7.1952 + 4.6806L + 1.4167L2 ,
+dMS
+3 (µm) = 54.161 + 21.776L + 9.2026L2 + 1.7940L3 ,
+(2.2)
+where the expansion uses α(6)
+s
+in the 6-flavor scheme and L = log((µm/m(µm))2). The running of
+the MS mass is described by the RGE
+µ2
+m
+dmt(µm)
+dµ2m
+= − mt(µm)
+�
+i=0
+γm
+i
+�
+a(6)
+s (µ)
+�i+1
+,
+(2.3)
+where the anomalous dimensions γm
+i
+are known to five loops [27, 28]. The first few orders [29–34]
+are given by
+γm
+0 = 1 ,
+γm
+1 = 3.3750 ,
+γm
+2 = 4.8387 ,
+γm
+3 = −4.5082 .
+(2.4)
+Electroweak corrections (see, e.g. [35, 36]) are not considered.
+The RGE in Eq. (2.3) has the
+solution
+mt(µ1) = mt(µ0) exp
+�
+−2
+�
+i=0
+� µ1
+µ0
+dµ
+µ γm
+i
+�
+a(6)
+s (µ)
+�i+1
+�
+,
+(2.5)
+yielding the MS mass at a scale µ1 via evolution from the known mass at a reference scale µ0.
+Here and below we quote relations at O(α3
+s) and evolution equations at O(α4
+s).
+We have also
+3
+
+used these relations in our analysis for determining numerical values for the quark masses (and the
+strong coupling), even though our cross section analysis is based on a fixed-order theory description
+at NLO. Since the mass (and strong coupling) matching relations and RGE equations are well
+convergent series and no subtle cancellations between the different ingredients need to be taken
+care of (which would be the case for the PDFs) this approach is fully consistent and has the
+advantage that the theoretical uncertainties in the numerical values of the masses (and the strong
+coupling) are eliminated entirely from our analysis. We recommend this approach also for future
+phenomenological analyses. For implementing different mass schemes in the analytic expression for
+the differential mtt cross sections at NLO, see Eq. (2.14) below, only the O(αs) coefficients from
+Eqs. (2.1) and (2.6) are used.
+The MS mass is by construction a 6-flavor quantity and should only be used in observables
+where the dynamical scale of the top-quark mass sensitivity is of order mt or larger, i.e. µm ≳ mt.
+The MSR mass is, like the MS mass mass, determined from top-quark self-energy corrections [13,
+17], but designed such that all virtual and off-shell top-quark quantum fluctuations are integrated
+out in the on-shell limit.2 The MSR mass mMSR
+t
+(R) is therefore a 5-flavor quantity and its R-
+dependence properly captures all radiation off the top quark that is soft in the top quark rest
+frame, which is not the case for the MS mass. The MSR mass is the proper choice if the dynamical
+scale of the top quark mass sensitivity is below mt, i.e. R ≲ mt.
+The pole and MSR masses are related as
+mpole
+t
+= mMSR
+t
+(R) + R
+∞
+�
+n=1
+dMSR
+n
+�
+a(5)
+s (R)
+�n
+,
+(2.6)
+where the coefficients dMSR
+n
+read [13]
+dMSR
+1
+= 4/3 ,
+dMSR
+2
+= 8.1330
+dMSR
+3
+= 71.602 .
+(2.7)
+In the limit R → mt(mt), mMSR
+t
+(R) approaches the MS mass mt(mt) and matches on it in analogy
+to the 5-flavor and 6-flavor strong coupling, see below. In contrast to the logarithmic µm evolution
+of mt(µm), the R-evolution of mMSR
+t
+(R) is linear and captures the correct physical logarithms
+for observables with mt dependence, generated at dynamical scales R < mt, such as resonances,
+thresholds, and low-energy endpoints [37]. The mass renormalization constant of the MSR mass
+only contains the on-shell self-energy corrections for scales larger than R in contrast to the pole
+mass which contains self-energy corrections at all scales. So while the MSR mass is numerically
+close to the pole mass for small R at low orders, it is free of the pole mass renormalon problem.
+Formally the MSR mass approaches the pole mass for R → 0, but the Landau pole prevents taking
+2We are using the natural MSR mass definition (MSRn), where virtual top-quark loops are integrated out consis-
+tently, see [13].
+4
+
+this limit in practice. For small R values in the range of 1 to 2 GeV the MSR mass captures the
+kinematic particle mass interpretation commonly associated of the pole mass. Within perturbative
+uncertainties at NLO, where we can still ignore the pole mass renormalon problem, the scheme
+choice mMSR
+t
+(R = 1 GeV) is therefore a proxy for the pole mass scheme. The matching of the
+5-flavor MSR mass to the 6-flavor MS mass at the scale R = mt(mt) reads [13]
+mMSR
+t
+(mt)
+=
+mt(mt)
+�
+1 + 0.10357
+�
+a(5)
+s (mt)
+�2
++ 1.8308
+�
+a(5)
+s (mt)
+�3
+�
+,
+(2.8)
+and the inverse at the scale R = mMSR
+t
+(mMSR
+t
+) reads [13]
+mt(mt)
+=
+mMSR
+t
+�
+mMSR
+t
+� �
+1 − 0.10357
+�
+a(5)
+s (mMSR
+t
+)
+�2
+− 1.6927
+�
+a(5)
+s (mMSR
+t
+)
+�3 �
+.
+(2.9)
+The matching starts at O(α2
+s), where virtual top quark loops first appear.3 These relations are in
+close analogy to the corresponding strong coupling matching relation which reads
+a(6)
+s (mt) = a(5)
+s (mt)
+�
+1 − 0.15278
+�
+a(5)
+s (mt)
+�2
+− 0.54881
+�
+a(5)
+s (mt)
+�3 �
+.
+(2.10)
+The MSR mass at an arbitrary scale R is then obtained from a given MS mass, applying Eq. (2.8),
+and evolving the scale R from mt(mt) to the desired value by solving the RGE
+R d
+dRmMSR
+t
+(R) = −R
+�
+n
+γR
+n
+�
+a(5)
+s (R)
+�n+1
+,
+(2.11)
+where the anomalous dimensions γR
+n are given by [17]
+γR
+0 = 4/3
+γR
+1 = 3.0219 ,
+γR
+2 = 2.8047 ,
+γR
+3 = −73.257 .
+(2.12)
+The solution of Eq. (2.11) yields
+mMSR
+t
+(mt) − mMSR
+t
+(R) = −
+�
+n=0
+γR
+n
+� mt
+R
+dR′ �
+a(5)
+s (R′)
+�n+1
++ O
+�
+a4
+s
+�
+≡ ∆m ,
+(2.13)
+so that the MSR mass at R is obtained as mMSR
+t
+(R) = mMSR
+t
+(mt)−∆m. As far as QCD corrections
+are concerned, the formulae above allow to relate MSR and MS top quark mass values at any
+(perturbative) scale with a precision of better than 20 MeV. The REvolver library [37] provides
+this functionality in user-friendly software package.
+In the present work, the MCFM program (version 6.8) [19, 20] is extended to include the im-
+plementation of the MSR scheme in the computation of the hadronic tt production cross section for
+3In the matching relations in Eqs. (2.8) and (2.9) we have not indicated the 5- or 6-flavor schemes for the strong
+coupling, since at the order shown the coefficients are identical in both schemes.
+5
+
+single-differential kinematics. Based on the procedure presented in Refs. [21, 22], the tt production
+cross section differential with respect to an observable X at NLO reads
+dσ
+dX = (as(µr))2 dσ(0)
+dX
+�
+m, µr, µf
+�
++ (as(µr))3 dσ(1)
+dX
+�
+m, µr, µf
+�
++ (as(µr))3 ˜R d1
+d
+dmt
+�
+dσ(0)(mt, µr, µf)
+dX
+� ����
+mt=m
+,
+(2.14)
+where σ(0) is the leading order (LO) and σ(1) the NLO cross section in the pole mass scheme. At
+NLO, the derivative term (the third summand in Eq. (2.14)) implements the MS or MSR top quark
+mass schemes. In the present work, the observable of interest is the invariant mass of the tt system,
+and X = mtt. In particular, we have the following set of parameters in Eq. (2.14))
+�
+as(µr), m, d1, ˜R
+�
+=
+�
+�
+�
+�
+a(5)
+s (µr), mMSR
+t
+(R), dMSR
+1
+, R
+�
+,
+R < mt(mt) (MSR regime) ,
+�
+a(5)
+s (µr), mt(µm), dMS
+1 (µm), mt(µm)
+�
+,
+µm > mt(mt) (MS regime) .
+(2.15)
+It is important to note that the choice of the renormalization and factorization scales µr and µf
+is independent of the mass renormalization scales R or µm in this implementation. We empha-
+size that it is essential that the mass scheme correction proportional to d1 is consistently used
+at the renormalization scale µr, which yields logarithms ln(R/µr) or ln(µm/µr) beyond NLO to
+consistently cancel the pole mass renormalon. Since MCFM is based on renormalization with 5
+dynamical flavors, one has to consistently expand a(6)
+s (µr) for the MS top mass scheme corrections
+of Eq. (2.1) in powers of a(5)
+s (µr) in the cross section formula of Eq. (2.14). At NLO this leads to
+Eq. (2.15).
+We note that the fixed-order perturbative corrections for the differential cross section in the
+pole mass scheme are known at next-to-next-to-leading order (NNLO) accuracy in QCD [38] and at
+NLO in the electroweak theory [39, 40]. In addition, an implementation of the MS mass scheme at
+NNLO has been provided in Ref. [41]. The conversion of the mass renormalization scheme from the
+pole mass to the running or the MSR mass beyond NLO accuracy in QCD (and LO for electroweak
+effects as presented here) needs to be performed numerically and requires theory predictions for
+differential cross sections with the pole mass at NNLO accuracy for a large array of pole mass
+values (typically in a range 150 GeV < m < 180 GeV)), which are currently not readily available in
+the literature.
+Non-relativistic quasi-bound state QCD corrections are important for the region mtt ∼ 340-
+360 GeV, where the strongest top quark mass sensitivity arises in the mtt distribution. In this
+threshold region the produced top quarks attain small non-relativistic velocities v ≪ 1 in the tt
+center-of-mass frame, and the dynamics of the tt system are hence governed by the mass mt, the
+relative momentum mtv, and the kinetic energy mtv2 of the top quark.
+Since mt ≫ mtv ≫
+mtv2, the appearance of ratios involving the masses, momenta and kinetic energy of the top quark
+renders the standard fixed-order expansion in powers of αs unreliable in this mtt range.
+The
+6
+
+most pronounced quasi-bound state effects arise from the Coulomb corrections due to the exchange
+of gluons between the produced t and t yielding a dependence of the prediction on the ratio
+mt/(mtv). This leads to a singular (αs/v)n behavior in the fixed-order perturbative QCD correction
+at n-loops [42]. These quasi-bound state effects have been considered in Refs. [43, 44], and more
+recently again in [45]. These predictions, however, do not provide an adequate description of the
+lowest mtt bin in the region between 300 GeV and the quasi-bound state region around 350 GeV,
+where the imaginary energy approach and the use of the optical theorem [46] predict a sizeable
+and unphysical finite tt production rate, see the results shown in Ref. [45].
+In this region the
+differential cross section depends on the experimental cuts on the top and antitop quark decay
+products [47, 48], which complicates the theoretical prediction as well as the experimental analysis,
+but any sensible choice of cuts leads to a strongly suppressed rate for mtt close to 300 GeV. This
+latter aspect is actually better described by the fixed-order predictions for stable top quarks where
+the rate vanishes identically for mtt < 2mt (for a correct top mass scheme choice as discussed
+below). Furthermore, a systematic treatment of the intermediate region, where the non-relativistic
+and relativistic calculations need to be matched, is currently not available with a reliable matching
+error estimate.4
+We also mention that for the electroweak corrections different scheme choices
+for the MS mass are available related to the definition of the vacuum expectation value [35, 36].
+Their effects concerning the MSR mass and their impact on the use of different mass schemes in
+experimental observables is unknown. Overall, there is currently no complete and reliable theory
+prediction for the low mtt distribution available for experimental analysis. For the study of the tt
+differential cross section as a function of mtt and its dependence on the MSR mass scale R, the NLO
+fixed order prediction for stable top quarks based on the MCFM program is appropriate, since it
+properly describes the generic size of subleading QCD corrections and vanishes for mtt < 2mt. For
+a reliable measurement of the MSR top quark mass, however, a more complete code including the
+features mentioned above has to be made available.
+3
+First investigation of the R scale dependence
+In this section we examine the dependence of the mtt distribution in different representative bins
+in the range between 300 and 700 GeV on the scales µr, µf, and R in the MSR mass scheme as
+well as µm in the MS scheme using as input the results of the ABMP16 PDF fit at NLO [50] with
+α(5)
+s (mZ) = 0.11905 at mZ = 91.19 GeV. For the MS mass value mt(mt) = 160.68 GeV has been
+chosen close to the fit of Ref. [51]. The latter value corresponds to a MSR masses at R = 1 GeV
+and R = 80 GeV of mMSR
+t
+(1 GeV) = 170.48 GeV and mMSR
+t
+(80 GeV) = 164.98 GeV, respectively.
+In Fig. 1, the cross section for the bin mtt ∈ [300, 333] GeV, i.e. the region below the tt pro-
+duction threshold, is shown for different scale choices at LO and NLO. The cross section is zero
+for R < 60 GeV, which corresponds to 2mMSR
+t
+(R) > 333 GeV. Non-zero contributions to the cross
+section in the mtt ∈ [300, 333] GeV range appear only at large values of R or when using the MS
+4Such a treatment is available only for top quark production in e+e− annihilation, see Ref. [49].
+7
+
+mass, which correspond to smaller values of mMSR
+t
+(R) or mt(µm). The LO contribution to the
+cross section is zero or positive throughout the probed range of R and µm. At NLO, however, the
+quick decrease of the derivative terms in Eq. (2.14) in comparison to the increase of the positive
+contributions would lead to unphysical negative values of the NLO cross section in this kinematic
+range, as was also pointed out in Ref. [41], where the MS mass scheme was examined.
+Since tt production in the range mtt ∈ [300, 333] GeV is impossible, the results in Fig. 1 also
+show that R values above 80 GeV must be avoided. This also implies that the MS mass cannot be
+used if the tt cross section in this mtt range is included in the experimental analysis. This conclusion
+holds even in the presence of quasi-bound state effects, since these provide a more precise prediction
+of the tt production threshold, which is, however, located at mtt values above 333 GeV. A further
+feature of the mtt ∈ [300, 333] GeV range, shown in Fig. 1, is the rapid increase of the cross section
+at µm ≳ 410 GeV. This occurs when mt(µm) is so small, such that LO tt production is even possible
+below 300 GeV.
+In Fig. 2, the cross section for the bin mtt ∈ [333, 366] GeV, i.e. the region where the tt produc-
+tion threshold is located, is shown as a function of R and µm at NLO in the left panel. The right
+panel displays the relative size of the NLO corrections with respect to the LO description. Here, the
+quasi-bound state effects are sizeable and our NLO result only provides a qualitative description.
+Similar as in the lowest bin, we observe a quite strong dependence on the mass renormalization
+scale. We see that for very small values of R the size of the NLO correction increases significantly,
+particularly for large µr and µf values, making the use of fixed-order perturbation theory unreliable
+for these choices. This shows that the impact of the higher-order QCD corrections, including the
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+6
+−
+4
+−
+2
+−
+0
+2
+4
+6
+8
+ [pb/GeV]
+tt
+dm
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+ < 333 GeV
+tt
+300 GeV < m
+)
+t
+m
+(
+t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+Total
+LO
+1
+Figure 1: The mtt ∈ [300, 333] GeV range of the mtt distribution. There is no tt production at
+R ≲ 60 GeV, but the region above it suffers from the lack of Coulomb corrections. The discontinuity
+at µm ≳ 410 GeV is due to the tt production threshold becoming artificially low, and such high
+values of the scale µm should be avoided.
+8
+
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+ [pb/GeV]
+tt
+dm
+NLO
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+pp, 
+ < 366 GeV
+tt
+333 GeV < m
+)t
+m
+(t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+1
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+tt
+dm
+ LO
+σ
+d
+ / 
+tt
+dm
+NLO
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+ < 366 GeV
+tt
+333 GeV < m
+)
+t
+m
+(
+t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+1
+Figure 2: The NLO cross section (left) and the ratio of the LO and NLO cross sections (right) for
+mtt ∈ [333, 366] GeV. The transition from a region suffering from the missing Coulomb corrections
+to a more stable region where the threshold effects become less important is seen at R ≳ 60 GeV
+(dashed blue). Further, predictions obtained using small values of µr, µf are observed to stabilize
+the prediction quickly as a function of R or µm.
+quasi-bound state corrections, is particularly sizeable and essentially maximized in the pole mass
+scheme. This is closely mimicked by the result for R = 1 GeV.
+We see that the most stable predictions are obtained and that the NLO corrections are signif-
+icantly smaller for R in the range of 60 to 80 GeV. This is not accidental, but expected from the
+fact that the smaller value of the MSR mass at these R values accounts for the reduced mass of
+the tt system due to the Coulomb-binding effects. So also the impact of the (missing) Coulomb
+corrections can be expected to be moderate and in particular much smaller than in the pole mass
+scheme. Adopting values for µr and µf below the top quark mass further diminishes the size of
+the NLO corrections. This is because for this R-range and for these µr and µf values mMSR
+t
+(R)
+captures a sizeable part of the non-relativistic bound state dynamics relevant in this bin.5
+At this point it is also instructive to examine mtt far above threshold. In Figs. 3 and 4, the
+results for mtt ∈ [465, 498] GeV and mtt ∈ [663, 696] GeV, respectively, are shown. Here the NLO
+predictions provide an appropriate theoretical description. In contrast to the low mtt bins discussed
+above, the mass renormalization scale behavior is very smooth. This is partly related to the much
+smaller top quark mass sensitivity, but also means that none of the top quark mass schemes (and
+values for R or µm) provide any advantage concerning capturing essential QCD corrections. Here,
+only the choices of the scales µr and µf are essential for the prediction showing a preference for
+values of around mt. This observation applies also to other invariant mass bins covering large mtt
+5Due to the integration over the bin range, the R and µr values are expected to be larger than for a description
+on the bound state resonance peak, where even lower scale choices are appropriate [46].
+9
+
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+ [pb/GeV]
+tt
+dm
+NLO
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+pp, 
+ < 498 GeV
+tt
+465 GeV < m
+)t
+m
+(t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+tt
+dm
+ LO
+σ
+d
+ / 
+tt
+dm
+NLO
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+ < 498 GeV
+tt
+465 GeV < m
+)
+t
+m
+(
+t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+Figure 3: The NLO cross section (left) and the ratio of the LO and NLO cross sections (right) for
+mtt ∈ [465, 498] GeV.
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [pb/GeV]
+tt
+dm
+NLO
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+pp, 
+ < 696 GeV
+tt
+663 GeV < m
+)t
+m
+(t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)t
+m
+(t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+1
+100
+200
+300
+400
+500
+600
+ [GeV]
+m
+µ
+R, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+tt
+dm
+ LO
+σ
+d
+ / 
+tt
+dm
+NLO
+σ
+d
+)=160.68 GeV
+t
+m
+(
+t
+m
+ABMP16_5_nlo
+ = 13 TeV
+s
+ < 696 GeV
+tt
+663 GeV < m
+)
+t
+m
+(
+t
+m
+ = 1/4 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 1/2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 2 
+f
+µ
+ = 
+r
+µ
+)
+t
+m
+(
+t
+m
+ = 4 
+f
+µ
+ = 
+r
+µ
+Figure 4:
+Same as Fig. 3 for the bin mtt ∈ [663, 696] GeV.
+values, see Ref. [52].
+Overall, our examination suggests that the MSR top quark mass mMSR
+t
+(R) and the choice for the
+central value of R = 80 GeV provide the most reliable theoretical predictions for all mtt bins. For the
+scales µr and µf the central values mt(mt) and, in particular mt(mt)/2 for the mtt range containing
+the tt threshold, are adequate choices. We note that these findings are also in line with the optimal
+scale choices for the total cross section for tt hadro-production, when using the top quark mass in
+the MS scheme. In this case, central values for µr and µf of the order mt(mt)/2 ≈ 80 GeV are in the
+10
+
+region of fastest apparent convergence considering perturbative QCD corrections through NNLO
+and also minimize the scale sensitivity of the total cross section [22]. Settings for PDF factorization
+scale µf different from µr have been explored in Refs [41, 53], corroborating these findings. On the
+other hand, for the total cross section with the top quarks in the pole mass scheme, which is well
+modeled by the MSR scheme mass mMSR
+t
+(1 GeV), the preferred central values for µr and µf, which
+minimize scale sensitivity and optimize perturbative convergence through NNLO, are of the order
+mpole
+t
+/4 ≈ 45 GeV, see e.g. Ref. [22]. This is also visible in the ratio plots on the right in Figs. 2–4.
+In the following, we demonstrate the impact of the mass scheme and the scale setting on the value
+of the top quark mass obtained in fits to the experimental data of Ref. [18].
+4
+Extraction of the top quark MSR mass
+The MSR mass mMSR
+t
+(R) is extracted from the differential tt production cross section measured by
+the CMS Collaboration in pp collisions at the LHC at √s = 13 TeV, corresponding to an integrated
+luminosity of 35.9 fb−1 [18]. The tt cross section is measured as a function of mtt in the ranges:
+mtt < 420 GeV, mtt ∈ [420, 550] GeV, mtt ∈ [550, 810] GeV and mtt > 810 GeV.
+The theoretical predictions are obtained using the ABMP16 5-flavor PDF set [51] at NLO.
+According to the preferred MSR mass scale settings described in the previous section, the initial
+value of the scale R is set to 80 GeV in Eq. (2.14), and the cross section is calculated for a range of
+assumed values of mMSR
+t
+(80 GeV). The function
+χ2 =
+�
+i,j
+(σexp
+i
+− σth
+i )C−1
+ij (σexp
+j
+− σth
+j ),
+(4.1)
+is computed for each mMSR
+t
+(80 GeV). The indices i, j in Eq. (4.1) run over the bins of the mtt
+distribution, while σexp
+i
+are the experimental data and σth
+i
+the theoretical predictions. The inverse
+covariance matrix C−1
+ij
+provided in Ref. [18] is used.
+The scales µr and µf are set to mMSR
+t
+(80 GeV) for all 4 bins of the mtt distribution or, alter-
+natively, to mMSR
+t
+(80 GeV)/2 for mtt < 420 GeV, to stabilize the prediction against the missing
+quasi-bound state corrections, and to mMSR
+t
+(80 GeV) for the remainder. Fig. 5 shows a 4th order
+polynomial fit to the χ2 values resulting from each configuration.
+The fit uncertainties are obtained via the ∆χ2 = 1 tolerance criterion, while the µr and µf scale
+uncertainties are evaluated by varying their central values in each bin up and down by a factor of
+2, avoiding the cases where one scale is multiplied by 1/2 and the other by 2, and constructing
+an envelope. For comparison with previous analyses, the extracted values of mMSR
+t
+(80 GeV) are
+evolved to the reference scales R of 1 and 3 GeV. Note that determining mMSR
+t
+(1 GeV) requires
+evaluating αs(1 GeV) rather close to the Landau pole, which is expected to lead to an increased
+perturbative uncertainty in the MSR mass at R = 1 GeV due to missing higher order corrections.
+Reporting the mass value also at R = 3 GeV thus ensures the stability of the result, and the use of
+reference scales R > 1 GeV will become increasingly important in future extractions of mMSR
+t
+(R).
+11
+
+160
+162
+164
+166
+168
+170
+172
+174
+176
+178
+(80 GeV) [GeV]
+MSR
+t
+m
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+2
+χ
+ = 1.86 / 3
+dof
+ / N
+2
+χ
+Min. 
+ [GeV]
+tt
+   m
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,
+r
+µ
+        < 420 : 
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,r
+µ
+ [420, 550] : 
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,r
+µ
+ [550, 810] : 
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,
+r
+µ
+        > 810 : 
+ = 1.86 / 3
+dof
+ / N
+2
+χ
+Min. 
+162
+164
+166
+168
+170
+172
+174
+176
+178
+180
+(80 GeV) [GeV]
+MSR
+t
+m
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+2
+χ
+ = 3.03 / 3
+dof
+ / N
+2
+χ
+Min. 
+ [GeV]
+tt
+   m
+(80 GeV)
+MSR
+t
+m
+2
+1
+ = 
+f
+µ
+,r
+µ
+        < 420 : 
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,
+r
+µ
+ [420, 550] : 
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,
+r
+µ
+ [550, 810] : 
+(80 GeV)
+MSR
+t
+ = m
+f
+µ
+,r
+µ
+        > 810 : 
+ = 3.03 / 3
+dof
+ / N
+2
+χ
+Min. 
+Figure 5: A 4th order polynomial fitted to the χ2 resulting from comparing the experimental
+data to theory predictions assuming different values of mMSR
+t
+(80 GeV).
+The scales µr and µf
+are set to mMSR
+t
+(80 GeV) considering the whole mtt distribution (left), or to mMSR
+t
+(80 GeV)/2 for
+mtt < 420 GeV and to mMSR
+t
+(80 GeV) for the remainder (right). The number of degrees of freedom
+in the fits is denoted by Ndof.
+Table 1: The values of mMSR
+t
+(R) obtained at different scales R (given in brackets below mMSR
+t
+),
+and the corresponding mt(mt), the χ2 divided by the number of degrees of freedom Ndof in the
+fit, along with the fit and scale uncertainties for the mMSR
+t
+(R) extracted at R = 80 GeV. The
+results are shown for the constant µr, µf setting, where the central µr and µf values are set to
+mMSR
+t
+(80 GeV) in the whole mtt distribution, and for the semi-dynamical (SD) setting where they
+are set to mMSR
+t
+(80 GeV)/2 for mtt < 420 GeV and to mMSR
+t
+(80 GeV) for higher mtt. The fit and
+µr, µf uncertainties correspond to the MSR mass extracted at R = 80 GeV. Within the reported
+accuracy, the uncertainty in the initial choice of R agrees in all cases when the extracted mMSR
+t
+(R)
+is evolved to the reference R.
+mMSR
+t
+mMSR
+t
+mMSR
+t
+mt
+Fit
+µr, µf
+R
+µr, µf
+χ2/Ndof
+(80 GeV)
+(1 GeV)
+(3 GeV)
+(mt)
+unc.
+unc.
+unc.
+setting
+[ GeV]
+[ GeV]
+[ GeV]
+[ GeV]
+[ GeV]
+[ GeV]
+[ GeV]
+Const.
+1.86/3
+167.7
+173.2
+172.9
+163.3
++0.6
+−0.6
++0.4
+−0.6
++0.4
+−0.5
+SD
+3.03/3
+169.3
+174.8
+174.5
+164.8
++0.5
+−0.5
++0.2
+−0.4
++0.2
+−0.3
+Furthermore, the results are translated into the standard MS mass mt(mt) by iteratively finding
+mMSR
+t
+(mMSR
+t
+) via the condition R = mMSR
+t
+(R), and applying the matching formula in Eq. (2.9)
+up to O(a3
+s). The uncertainty related to the initial choice of R is assessed by repeating the fits at
+R = 60 GeV and R = 100 GeV, and the difference in the resulting masses at the reference scales
+to the respective values obtained in the R = 80 GeV fit is taken as the R scale uncertainty. The
+resulting values for the top quark mass are listed in Table 1.
+In particular, setting the central µr and µf to mMSR
+t
+(80 GeV) and considering the complete mtt
+12
+
+distribution yields
+mMSR
+t
+(1 GeV) = 173.2 ± 0.6 (fit)+0.4
+−0.6 (µr, µf)+0.4
+−0.5 (R) GeV .
+(4.2)
+The value for mMSR
+t
+(80 GeV) in this fit translates into mt(mt) = 163.3+0.8
+−1.0 GeV. This is compatible
+within uncertainties with the value of mt(mt) = 162.1+1.0
+−1.0 GeV obtained at NLO in the ABMP16
+5-flavor PDF set [50].
+In accordance with the results shown in Fig. 1, multiplying the scales µr and µf by 1/2 within
+mtt < 420 GeV is observed to increase the NLO cross section at R = 80 GeV. To compensate for
+this effect, the fit for mMSR
+t
+(80 GeV) leads to a somewhat larger value for the top quark MSR mass,
+reducing the predicted cross section especially in the vicinity of the tt production threshold. This
+results in the value
+mMSR
+t
+(1 GeV) = 174.8 ± 0.5 (fit)+0.2
+−0.4 (µr, µf)+0.2
+−0.3 (R) GeV.
+(4.3)
+It is expected that the impact of the choices for µr and µf, i.e. the shift of 1.6 GeV in the cen-
+tral values between Eqs. (4.2) and (4.3), will be reduced at NNLO accuracy and once a reliable
+description of the quasi-bound state effects is available. Nonetheless, as already expected from the
+observations in Sec. 3, the scale setting in Eq. (4.3) already increases the robustness against scale
+variations, yielding somewhat smaller uncertainties than Eq. (4.2).
+In order to illustrate the main conceptual novelty and the phenomenological importance of the
+mass scheme choice, we perform the following variant of the fit: Instead of determining the top
+quark MSR mass at R = 80 GeV and evolving the extracted mMSR
+t
+(80 GeV) value to R = 1 GeV, as
+in Eqs. (4.2) and (4.3), we perform the fit to data directly with the initial scale set to R = 1 GeV
+166
+168
+170
+172
+174
+(1 GeV) [GeV]
+MSR
+t
+m
+0
+20
+40
+60
+80
+100
+120
+2
+χ
+ = 2.16 / 3
+dof
+ / N
+2
+χ
+Min. 
+ [GeV]
+tt
+   m
+(1 GeV)
+MSR
+t
+ = m
+f
+µ
+,
+r
+µ
+        < 420 : 
+(1 GeV)
+MSR
+t
+ = m
+f
+µ
+,r
+µ
+ [420, 550] : 
+(1 GeV)
+MSR
+t
+ = m
+f
+µ
+,r
+µ
+ [550, 810] : 
+(1 GeV)
+MSR
+t
+ = m
+f
+µ
+,
+r
+µ
+        > 810 : 
+ = 2.16 / 3
+dof
+ / N
+2
+χ
+Min. 
+Figure 6:
+Same as Fig. 5, now fitting mMSR
+t
+(1 GeV) and with the scales µr and µf set to
+mMSR
+t
+(1 GeV) in the whole mtt distribution.
+13
+
+in NLO cross section of Eq. (2.14). Using also the central scales µr, µf set to mMSR
+t
+(1 GeV), this
+results in
+mMSR
+t
+(1 GeV) = 170.1 ± 0.6 (fit)+1.1
+−0.9 (µr, µf) GeV ,
+(4.4)
+where the corresponding fit to χ2 is shown in Fig. 6. In Eq. (4.4) the µr and µf scale uncertainties
+are twice as large as those of Eq. (4.2). The sizeable discrepancy to the results of Eqs. (4.2) and (4.3)
+indicates that scale variation does not provide a proper estimate of the theoretical uncertainties
+due to the missing higher order and quasi-bound state corrections for the result quoted in Eq. (4.4).
+Since using mMSR
+t
+(1 GeV) closely approximates the outcome using pole mass scheme, this confirms
+our conclusions drawn in Sec. 3 that the use of the pole mass scheme (or a very small initial R value
+for the MSR mass) leads to less reliable results in a fixed order QCD description at NLO accuracy,
+where the quasi-bound state effects are missing. The significant difference of 4.7 GeV between the
+central values in Eqs. (4.3) and (4.4) demonstrates the phenomenological relevance of this issue.
+This underpins the importance of proper scale setting in future phenomenological analyses.
+Let us now comment on other recent extractions of the top-quark mass, which have employed
+different methodologies.
+Data from the CMS collaboration for the tt production cross section
+collected in pp collisions at the LHC at √s = 13 TeV has been used previously for a determination
+of the top-quark mass using both, the pole and the MS mass scheme [54, 55]. The emphasis of
+those analyses has been on keeping the correlations of the top-quark mass with the strong coupling
+αs(mZ) and the PDFs. In a different thread of analyses, the running of top quark MS mass mt(µm)
+has been studied at NLO [18] and NNLO [56] with dynamical scales, using data from the CMS
+collaboration for the mtt distributions.6
+Of these analyses, the results of Ref. [55] can be compared to the present work, since they are
+obtained from normalized multi-differential cross sections which also include the low mtt region
+discussed here, and the theoretical predictions have also been based on the NLO MCFM cross
+section description. Ref. [55] quotes mpole
+t
+= 170.5±0.8 GeV, which, if interpreted as the asymptotic
+pole mass [37], translates into mMSR
+t
+(1 GeV) = 170.2±0.8 GeV. This is compatible with the variant
+of the present study in Eq. (4.4) obtained by directly fitting mMSR
+t
+(1 GeV) to data, although the
+combined fit of mpole
+t
+, αs(mZ) and PDFs in Ref. [55] reports a smaller value of αs(mZ) than used
+in Eq. (4.4) on the basis of the ABMP16 PDF set, and a somewhat different gluon PDF.
+The ATLAS collaboration has derived a value for the top quark MSR mass at the reference scale
+R = 1 GeV in Ref. [57] by comparing QCD predictions at next-to-leading logarithmic accuracy for
+the soft-drop groomed top quark jet mass distribution to parton shower Monte Carlo simulations
+for a Monte-Carlo top quark mass mMC
+t
+= 172.5 GeV. Obtained in the Monte Carlo calibration
+(following [58]), the result of Ref. [57] is not based on experimental data and hence cannot be
+directly compared to the results of the present study.
+The value for the top quark MSR mass of mMSR
+t
+(3 GeV) = 169.6+0.8
+−1.1 GeV has been extracted in
+6See also http://cms-results.web.cern.ch/cms-results/public-results/publications/TOP-19-007/index.
+html#Figure-aux_001.
+14
+
+Ref. [53], using the CMS data of Ref. [55] and the same methodology, i.e. using fixed-order QCD
+perturbation theory at NLO accuracy, so that mMSR
+t
+(3 GeV) has been fitted simultaneously with
+the PDFs and strong coupling constant. Evolving the result of the present study in Eq. (4.3) to
+R = 3 GeV yields
+mMSR
+t
+(3 GeV) = 174.5 ± 0.5 (fit)+0.2
+−0.4 (µr, µf)+0.2
+−0.3 (R) GeV ,
+(4.5)
+which indicates some tension.7 Part of this difference is due to the direct fitting of mMSR
+t
+(3 GeV)
+in Ref. [53] compared to mMSR
+t
+(80 GeV) in Eq. (4.5). In addition, Ref. [53] has obtained αs(mZ) =
+0.1132+0.0023
+−0.0018, which is two standard deviations away from the value of the ABMP16 fit at NLO [50]
+used in the extraction of Eq. (4.5).
+Notably, neither any of the cited previous top quark mass extractions nor the present work have
+included the aforementioned corrections for the quasi-bound state effects. However, the extraction
+of the top quark MSR mass using predictions in the MSR scheme at the scale R = 80 GeV profits
+from the smaller size of these effects and thus from an improved stability of the cross section.
+5
+Summary and Conclusions
+We have presented the first comprehensive study of the mtt distribution in its dependence on the
+mass renormalization scales R and µm of the MSR and MS top quark mass schemes. Our findings
+suggest that the scale setting of R close to 80 GeV improves the robustness of the predictions for
+the mtt distribution against scale variations in general and, in particular, against the impact of
+quasi-bound state corrections in the region of mtt close to the tt threshold. The theory predictions
+are based on the NLO fixed order QCD description provided by the MCFM program, adapted to
+the MSR and MS top quark mass schemes. The optimized scale choices for those mass schemes are
+characterized by low values of the renormalization and factorization scales µr and µf. This holds
+in particular in the vicinity of the tt production threshold region in the mtt distribution, where
+values µr ≃ µf ≃ mt/2 are observed to stabilize cross section predictions and to decrease the scale
+uncertainty in the determination of the MSR mass.
+These settings have been applied in an extractions of the top quark MSR mass at R = 80 GeV,
+using tt pair production cross section, measured as a function of mtt in pp collisions at √s = 13 TeV
+at the LHC by the CMS collaboration, using fixed-order perturbative QCD predictions at NLO
+accuracy and also the semi-dynamical scales for µr, µf in the low-mtt regime. The fitted value of
+mMSR
+t
+(80 GeV) has then been evolved to various low reference scales R, rather than computing the
+cross sections directly at low R as performed in earlier analyses. This procedure yields the value
+mMSR
+t
+(3 GeV) = 174.5+0.6
+−0.7 GeV, which is discussed in the context of other recent extractions of the
+top quark mass from LHC data. The observed differences are explained in part by the scale choice
+7The computations in Ref. [53] rely on the practical MSR (pMSR) definition [13] instead of the natural MSR
+(nMSR) scheme used in this work. The difference is at the level of 10 MeV [13] and thus negligible for the uncertainties
+quoted.
+15
+
+of R = 80 GeV for the top quark MSR mass, advocated by the present study. Other reasons for
+differences are due to the choice of the value for the strong coupling αs(mZ), which directly affects
+the normalisation of the cross section and is anti-correlated with the top quark mass, and, to a
+lesser extent, due to the particular PDF sets used.
+While we have argued that the implementation of the MSR mass scheme in the tt cross section
+calculation and the optimal scale choice for R of 80 GeV provide more robust predictions even at
+NLO accuracy, the findings should be corroborated by extending the analysis to NNLO accuracy. In
+addition, the proper treatment of both, the quasi-bound state effects, together with a matching to
+the relativistic tt region, and the mtt region below the threshold are further important improvements
+to be implemented. A final reliable measurement of the top quark MSR mass needs to address those
+issues as well as the correlation of the top quark mass with the other theoretical parameters, which
+control the cross section predictions. We leave these aspects for future studies.
+Acknowledgements
+The work of A.H.H was supported in part by FWF Austrian Science Fund under the Project No.
+P32383-N27, the work of S.M. in part by the Bundesministerium f¨ur Bildung und Forschung under
+contract 05H21GUCCA, the work by T.M. and K. L. is supported by the Helmholtz Association
+under the contract W2/W3-123, and T.M. is also supported by the National Science Centre, Poland,
+research grant No. 2021/42/E/ST2/00031.
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+Spectrum Monitoring and Analysis in Urban and
+Rural Environments at Different Altitudes
+Amir Hossein Fahim Raouf∗, Sung Joon Maeng∗, Ismail Guvenc∗, ¨Ozg¨ur ¨Ozdemir∗, and Mihail Sichitiu∗
+∗Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC
+[email protected], {smaeng,iguvenc,oozdemi,mlsichit}@ncsu.edu
+Abstract—Due to the scarcity of spectrum resources, the emer-
+gence of new technologies and ever-increasing number of wireless
+devices operating in the radio frequency spectrum lead to data
+congestion and interference. In this work, we study the effect of
+altitude on sub-6 GHz spectrum measurement results obtained at
+a Helikite flying over two distinct scenarios; i.e., urban and rural
+environments. Specifically, we aim at investigating the spectrum
+occupancy of various long-term evolution (LTE), 5th generation
+(5G) and citizens broadband radio service (CBRS) bands utilized
+in the United States for both uplink and downlink at altitudes
+up to 180 meters. Our results reveal that generally the mean
+value of the measured power increases as the altitude increases
+where the line-of-sight links with nearby base stations is more
+available. SigMF-compliant spectrum measurement datasets used
+in this paper covering all the bands between 100 MHz to 6 GHz
+are also provided.
+Index Terms—5G, C-Band, CBRS, helikite, LTE, spectrum
+monitoring, unmanned aerial vehicles (UAV).
+I. INTRODUCTION
+Wireless communication services and the emergence of new
+technologies have created a huge demand for radio frequency
+spectrum [1]. One prominent problem is the availability of the
+spectrum and the increase in interference in the current wire-
+less networks [2]. In addition, more aggressive frequency reuse
+is gaining interest recently for achieving higher link capacity
+in networks without introducing additional spectrum [3]. It
+is necessary to conduct occupancy studies using spectrum
+sensing techniques to understand and characterize interference
+problems and identify spectrum sharing opportunities.
+There are various recent examples that highlight the im-
+portance of understanding spectrum occupancy characteristics,
+including non-terrestrial scenarios, for developing effective
+spectrum sharing mechanisms. The launch of 5th generation
+(5G) cellular service in the United States was a concern for
+the commercial airline and private aircraft communities who
+used the radar altimeters of the aircraft industry. Although
+the assigned spectrum band for the altimeters is between
+4.2-4.4 GHz, due to their poor design the current versions
+suffer from out-of-band leakage problem; i.e., they ignore their
+assigned spectrum boundaries [4]. More specifically, Verizon
+and AT&T have recently begun operating in the 3.7 GHz to
+3.8 GHz spectrum range which is 400 MHz away from the
+altimeter band. However, this gap may not be sufficient for
+some aircraft to land safely. Moreover, while both Verizon
+and AT&T have been delaying switching on portions of their
+This research is supported in part by the NSF award CNS-1939334 and its
+supplement for studying NRDZs.
+respective 5G C-band wireless networks until July 2023, it is
+expected after that day that the whole 3.7-3.98 GHz C-band
+may be used for 5G transmissions [5], introducing additional
+concerns. There is a similar coexistence concern for spectrum
+sharing between the 5G networks to be deployed in the 3.1-
+3.55 GHz band in the future and the existing airborne radars
+using the same spectrum. In another recent debate, there is a
+concern in using terrestrial nationwide network in the L-Band
+(i.e., 1-2 GHz) and its potential interference with GPS [6].
+Some existing academic studies on spectrum occupancy
+are summarized in [7]. In more recent works, [8] presents a
+framework that captures and models the short-time spectrum
+occupancy to determine the existing interference for Internet-
+of-things (IoT) applications. In another study [9], current
+state-of-the-art artificial intelligence techniques are reviewed
+for channel forecasting, spectrum sensing, signal detection,
+network optimization, and security in mega-satellite networks.
+In [10], authors investigate and characterize the performance
+of coexisting aerial radar and communication networks for
+spectrum overlay and time-division multiple access by uti-
+lizing stochastic geometry. In [11], the effect of interference
+coming from coexisting ground networks on the aerial link
+is studied, which could be the uplink (UL) of an aerial cell
+served by a drone base station. By considering a Poisson field
+of ground interferers, they characterize aggregate interference
+experienced by the drone.
+In this paper, by post-processing the measurements from
+the experiments conducted by the NSF AERPAW platform in
+Raleigh, NC [12] at urban and rural environments, we analyze
+the spectrum occupancy in different U.S. cellular network
+bands as well as the citizens broadband radio service (CBRS)
+band. In addition, we study the effect of Helikite altitude
+on the signal strength pattern. In Section II, we describe the
+data structure and the overall information of the measurement
+campaign. Section III and Section IV present the spectrum
+monitoring results for various sub-6 Ghz bands in the urban
+and rural environments, respectively. Section V studies the
+time dependency of the spectrum occupancy for the frequency
+bands under consideration. Finally, Section VI highlights the
+conclusions of this work.
+II. DATA STRUCTURE
+The experiment for the urban environment was conducted
+by a Helikite flying up to 140 m on August 27, 2022. For
+the rural environment, the Helikite flew up to 180 m altitude
+on May 5, 2022. An NI USRP B205mini SDR was mounted
+arXiv:2301.02380v1  [eess.SP]  6 Jan 2023
+
+13:00
+14:00
+15:00
+16:00
+17:00
+18:00
+19:00
+20:00
+Measurement time
+0
+50
+100
+150
+Height (m)
+(a) Experiment scenario in NC State Main Campus (urban).
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+140
+Measurement time (min)
+0
+50
+100
+150
+200
+Height (m)
+(b) Experiment scenario in NC State Lake Wheeler Field (rural).
+Fig. 1: Helikite altitude and experiment scenario for: (a) urban
+environment, and (b) rural environment.
+TABLE I: Summary of LTE and 5G bands in United States.
+Technology
+Band
+No
+Duplex
+Mode
+Uplink Band
+(MHz)
+DL Band
+(MHz)
+Operators
+LTE
+12
+FDD
+698 - 716
+728 - 746
+AT&T, Verizon,
+T-Mobile
+13
+FDD
+777 - 787
+746 - 756
+Verizon
+14
+FDD
+788 - 798
+758 - 768
+AT&T, FirstNet
+411
+TDD
+2496 - 2690
+2496 - 2690
+T-Mobile
+5G
+n5
+FDD
+824 - 849
+869 - 894
+AT&T, Verizon
+n71
+FDD
+663 - 698
+617 - 652
+T-Mobile
+n77
+TDD
+3700 - 3980
+3700 - 3980
+AT&T, Verizon,
+T-Mobile
+CBRS
+n48
+TDD
+3550 - 3700
+3550 - 3700
+North America
+on the Helikite which enables executing a Python script to
+collect samples at the desired center frequency with the desired
+sampling rate. The datasets are SigMF compliant and include
+information on spectrum usage in frequency bands ranging
+from 89 MHz up to 6 GHz for different altitudes [13], [14].
+The data consist of time, altitude, power and Helikite location.
+A detailed description of the measurement setups can be found
+in [15]. Fig. 1 illustrates the height of the Helikite during the
+operation time.
+III. URBAN SPECTRUM OCCUPANCY RESULTS
+In this section, we present the spectrum occupancy results
+for several LTE, 5G and CBRS bands. Table I summarizes the
+spectrum allocations for some major cellular providers based
+on the technology exploited in the United States [16]. In this
+work, we investigate the aggregate in-band power for UL and
+downlink (DL) spectrum of various bands.
+A. LTE Bands - Uplink
+Fig. 2 presents the measured power for LTE bands 13, 14,
+15 and 41 considering the UL frequency spectrum ranges.
+As it can be seen, the spectrum of LTE 12 and LTE 41
+bands are more crowded compared with LTE 13 and LTE 14
+bands. It is worth mentioning that, unlike other LTE bands
+1It is worth mentioning that T-Mobile 5G n41 also uses the same spectrum.
+700
+705
+710
+715
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) LTE band 12 (UL).
+778
+780
+782
+784
+786
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) LTE band 13 (UL).
+788
+790
+792
+794
+796
+798
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(c) LTE band 14 (UL).
+2500
+2550
+2600
+2650
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(d) LTE band 41 (TDD UL/DL).
+Fig. 2: Measured LTE UL power for urban environment.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-30
+-20
+-10
+0
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(a) Mean.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+0
+50
+100
+150
+200
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(b) Variance.
+Fig. 3: Spectrum occupancy versus altitude in LTE bands 12,
+13, 14 and 41 (UL) for urban environment.
+under consideration, LTE 41 works in time-division duplexing
+(TDD) mode and includes both UL and DL transmissions.
+The mean and variance of the measured power for various
+LTE bands are presented in Fig. 3. As it can be observed
+from Fig. 3a, generally the mean value of the measured power
+increases as the altitude increases. The mean power value for
+LTE bands 12 and 41 are almost identical and much higher
+than the other two bands under consideration. Note that band
+41 has significantly larger bandwidth than band 12 and it
+includes both UL and DL transmission. From Fig. 3b, it can
+be observed that the fluctuation of variance for LTE band 13
+is much lower than the other ones. Although the mean value
+of LTE 12 and 41 show similar behaviour, the variance of LTE
+41 is lower than LTE band 12.
+B. LTE Bands - Downlink
+Considering the DL frequency range for different LTE
+bands, Fig. 4 illustrates the measured power for the bands un-
+der consideration. It can be readily checked that the spectrum
+of DL frequency ranges are more crowded compared with the
+UL ones. Although the occupied spectrum for LTE 13 and 14
+expand the whole range, the main frequency usage of LTE 12
+is between 735 - 745 MHz.
+Fig. 5 shows the mean and variance of the measured power
+versus altitude. As it can be observed from Fig. 5a, the mean
+
+730
+735
+740
+745
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) LTE band 12 (DL).
+746
+748
+750
+752
+754
+756
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) LTE band 13 (DL).
+758
+760
+762
+764
+766
+768
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(c) LTE band 14 (DL).
+2500
+2550
+2600
+2650
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(d) LTE band 41 (TDD UL/DL).
+Fig. 4: Measured LTE DL power for urban environment.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-30
+-20
+-10
+0
+10
+20
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(a) Mean.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+0
+100
+200
+300
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(b) Variance.
+Fig. 5: Spectrum occupancy versus altitude in LTE bands 12,
+13, 14 and 41 (DL) for urban environment.
+value of the measured power increases as the altitude increases
+up to almost 80 m. This is due to the fact that at high
+altitudes the probability of receiving signal from neighbor
+cells increases as the obstacles decrease, which results in the
+availability of the line of sight (LoS). For higher altitudes
+(i.e., higher than 80 m), the mean values for LTE bands
+under consideration remain almost constant. As it is shown
+in Fig. 5b, the variance of the measured power for LTE bands
+13, 14 and 41 show relatively smaller variation over different
+altitudes compared to LTE band 12. The main reason for this
+behavior can be found by observing the measured power for
+LTE band 12 shown in Fig. 4a. It seems that some portion of
+the LTE band 12 is not fully utilized.
+C. 5G Bands - Uplink
+Fig. 6 presents the measured power for 5G bands n5, n71
+and n77 considering the UL frequency spectrum ranges. This
+result reveals that the spectrum of n77 is mainly occupied
+between 3700-3800 MHz. One should also note that 5G band
+n5 and n71 utilize the frequency-division duplexing (FDD),
+while 5G band n77 exploit TDD mode. The performance
+of mean and variance of the measured power for 5G bands
+(uplink) are presented in Fig. 7. As it can be observed from
+Fig. 7a, the mean value of the measured power increases as the
+altitude increases up to almost 80 m due to the same argument
+825
+830
+835
+840
+845
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) 5G band n5 (UL).
+670
+680
+690
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) 5G band n71 (UL)
+3750 3800 3850 3900 3950
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(c) 5G band n77 (TDD UL/DL).
+Fig. 6: Measured 5G UL power for urban environment.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-35
+-30
+-25
+-20
+-15
+-10
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(a) Mean.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+0
+50
+100
+150
+200
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(b) Variance.
+Fig. 7: Spectrum occupancy versus altitude in 5G n5, n71 and
+n77 bands (UL) for urban environment.
+870
+875
+880
+885
+890
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) 5G band n5 (DL).
+620
+630
+640
+650
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) 5G band n71 (DL).
+Fig. 8: Measured 5G DL power for urban environment.
+mentioned earlier. The mean value of 5G band n5 shows higher
+value compared with n71 and n77. As it is shown in Fig. 7b,
+the variance of the measured power for 5G bands n5 and n77
+intersect with each other around the altitude of 60 m. The
+variance of n77 band keeps increasing as the altitude increases.
+D. 5G Bands - Downlink
+Fig. 8 illustrates the measured power for 5G n5 and n71
+bands by considering the DL frequency range. It can be seen
+that the measured power for 870 - 880 MHz and 885-894 MHz
+are higher than the rest of spectrum. Fig. 9 shows the mean and
+variance of the measured power versus altitude. As it can be
+observed from Fig. 9a, the mean value of the measured power
+for n5 and n71 are similar and significantly higher than n77.
+
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(a) Mean.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+0
+50
+100
+150
+200
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(b) Variance.
+Fig. 9: Spectrum occupancy versus altitude in 5G bands n5
+and n77 (DL) for urban environment.
+(a)
+3550
+3600
+3650
+Frequency (MHz)
+20
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b)
+Fig. 10: (a) CBRS spectrum and tiers; and (b) Measured CBRS
+band n48 power for urban environment (TDD UL/DL).
+40
+60
+80
+100
+120
+140
+Altitude (m)
+-34
+-32
+-30
+-28
+-26
+-24
+Power (dB)
+CBRS Band-n48 (3550-3600 MHz)
+CBRS Band-n48 (3600-3650 MHz)
+CBRS Band-n48 (3650-3700 MHz)
+(a) Mean.
+40
+60
+80
+100
+120
+140
+Altitude (m)
+0
+10
+20
+30
+40
+50
+Power (dB)
+CBRS Band-n48 (3550-3600 MHz)
+CBRS Band-n48 (3600-3650 MHz)
+CBRS Band-n48 (3650-3700 MHz)
+(b) Variance.
+Fig. 11: Spectrum occupancy versus altitude in CBRS band
+for urban environment.
+For the bands under consideration, the mean value increases
+as the altitude increases up to almost 80 m. As it is shown in
+Fig. 9b, the variance of the measured power for n77 starts with
+a small value, while it climes up to near those of n5 values
+as the altitude increases. The variance of n71 band depicts
+a higher value for all the measured altitudes compared with
+those others 5G bands.
+E. CBRS Band
+Fig. 10a illustrates the CBRS spectrum which it lays out
+three tiers of users. Fig. 10b presents the measured power
+for CBRS n48 band. Similar to LTE 41 and 5G n77 bands,
+n48 also exploits TDD mode. As it can be seen, the spectrum
+is mainly occupied within the range of 3610-3690 MHz. In
+Fig. 11, we study the mean and variance of the measured
+power versus altitude whereas the CBRS band is divided into
+three equal portions. As it can be observed, the mean and
+variance of the measured power for the first portion (i.e.,
+3550-3600 MHz) are lower than the other parts. The mean
+value of the third portion (i.e., 3650-3700 MHz) increases
+700
+705
+710
+715
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) LTE band 12 (UL).
+778
+780
+782
+784
+786
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) LTE band 13 (UL).
+788
+790
+792
+794
+796
+798
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(c) LTE band 14 (UL).
+2500
+2550
+2600
+2650
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(d) LTE band 41 (TDD UL/DL).
+Fig. 12: Measured LTE UL power for rural environment.
+50
+100
+150
+Altitude (m)
+-30
+-20
+-10
+0
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(a) Mean.
+50
+100
+150
+Altitude (m)
+0
+100
+200
+300
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(b) Variance.
+Fig. 13: Spectrum occupancy versus altitude in LTE bands 12,
+13, 14 and 41 (UL) for rural environment.
+as the altitude increases up to 60 m and then it drops
+afterwards. However, the man value of the second part (i.e.,
+3600-3650 MHz) keeps increasing as the altitude increases.
+IV. RURAL SPECTRUM OCCUPANCY RESULTS
+In this section, we study the spectrum occupancy and its
+characteristic for the similar bands as previous section by
+considering the experimental results for the rural environment.
+A. LTE Bands - Uplink
+Fig. 12 illustrates the measured power for for LTE bands
+13, 14, 15 and 41 considering the UL frequency spectrum.
+As it can be seen, LTE bands 12 and 41 show more crowded
+spectrum compared with LTE bands 13 and 14. The mean and
+variance of the measured power for various LTE bands are
+presented in Fig. 13. As opposed to the urban environment
+(cf. Fig. 3a), the mean value for LTE bands 13 and 14 are
+much higher than the other two bands under consideration.
+B. LTE Bands - Downlink
+Considering the DL frequency range for different LTE
+bands, Fig. 14 illustrates the measured power for the bands
+under consideration. Same as the urban results, the spectrum
+of DL frequency range are more crowded compared with the
+UL ones in the rural environment. Fig. 15 shows the mean
+
+3550 MHz
+3600 MHz
+3650 MHz
+3700 MHz
+Tier 1
+Incumbent Users
+(e.g. the Navy)
+Tier 2
+Priority Access Licensees
+(e.g. private organizations)
+Tier 3
+General Authorized Access
+(e.g. unlicensed users)730
+735
+740
+745
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) LTE band 12 (DL).
+746
+748
+750
+752
+754
+756
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) LTE band 13 (DL).
+758
+760
+762
+764
+766
+768
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(c) LTE band 14 (DL).
+2500
+2550
+2600
+2650
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(d) LTE band 41 (TDD UL/DL).
+Fig. 14: Measured LTE DL power for rural environment.
+50
+100
+150
+Altitude (m)
+-30
+-20
+-10
+0
+10
+20
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(a) Mean.
+50
+100
+150
+Altitude (m)
+0
+100
+200
+300
+Power (dB)
+LTE Band-12 (AT&T, T-Mobile)
+LTE Band-13 (Verizon)
+LTE Band 14 (AT&T, FirstNet)
+LTE Band 41 (T-Mobile)
+(b) Variance.
+Fig. 15: Spectrum occupancy versus altitude in LTE bands 12,
+13, 14 and 41 (DL) for rural environment.
+and variance of the measured power versus altitude. As it can
+be observed from Fig. 15a, the mean value of the measured
+power increases as the altitude increases up to 80 m and it
+remains almost constant for the higher altitudes. The variance
+of LTE bands 13, 14, and 41 show similar behaviour, while
+the corresponded plot for LTE band 12 starts with increasing
+for the altitude up to 40 m and then it drops afterwards.
+C. 5G Bands - Uplink
+Fig. 16 illustrates the measured power for 5G bands n5, n71
+and n77 considering the UL frequency spectrum ranges. This
+result reveals that the spectrum of n77 is less crowded than
+those of n5 and n71. The performance of mean and variance
+of the measured power for 5G bands (uplink) are presented in
+Fig. 17. As it can be observed from Fig. 17a, while the mean
+value of the measured power for n77 is almost independent of
+the altitude, it increases for n5 and n71 bands as the altitude
+increases. As it is shown in Fig. 17b, the variance of the
+measured power for n71 depicts higher value compared with
+the other 5G bands.
+D. 5G Bands - Downlink
+Fig. 18 illustrates the measured power for 5G n5 and n71
+bands by considering the DL frequency range. Similar to the
+urban case, it can be seen that the measured power for 870
+825
+830
+835
+840
+845
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) 5G band n5 (UL).
+670
+680
+690
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) 5G band n71 (UL).
+3750 3800 3850 3900 3950
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(c) 5G band n77 (TDD UL/DL).
+Fig. 16: Measured 5G UL power for rural environment.
+50
+100
+150
+Altitude (m)
+-30
+-25
+-20
+-15
+-10
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(a) Mean.
+50
+100
+150
+Altitude (m)
+0
+50
+100
+150
+200
+250
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(b) Variance.
+Fig. 17: Spectrum occupancy versus altitude in 5G n5 and n77
+bands (UL) for rural environment.
+870
+875
+880
+885
+890
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(a) 5G band n5 (DL).
+620
+630
+640
+650
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+(b) 5G band n71 (DL).
+Fig. 18: Measured 5G DL power for rural environment.
+- 880 MHz and 885-894 MHz are higher than the rest of
+spectrum in the rural environment. Fig. 19 depicts the mean
+and variance of the measured power versus altitude. As it can
+be observed from Fig. 19a, the mean value of the measured
+power for n77 band remains almost constant for different
+altitudes, while it increases as the altitude increases up to
+almost 80 m for n5 and n71 bands. As it is shown in Fig. 19b,
+the variance of the measured power for 5G band n71 shows
+higher values compared with n5 and n77.
+E. CBRS Band
+Fig. 20 present the measured power for CBRS n48 band
+for rural environment. As it can be seen, the spectrum is less
+
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(a) Mean.
+50
+100
+150
+Altitude (m)
+0
+100
+200
+300
+400
+Power (dB)
+5G Band-n5 (AT&T, Verizon)
+5G Band-n71 (T-Mobile)
+5G Band-n77 (AT&T, Verizon, T-Mobile)
+(b) Variance.
+Fig. 19: Spectrum occupancy versus altitude in 5G bands n5
+and n77 (DL) for rural environment.
+3550
+3600
+3650
+Frequency (MHz)
+50
+100
+150
+Altitude (m)
+-40
+-20
+0
+20
+40
+dB
+Fig. 20: Measured power during Helikite operation over rural
+environment for CBRS band n48 (TDD UL/DL).
+50
+100
+150
+Altitude (m)
+-30
+-25
+-20
+-15
+-10
+Power (dB)
+CBRS Band-n48 (3550-3600 MHz)
+CBRS Band-n48 (3600-3650 MHz)
+CBRS Band-n48 (3650-3700 MHz)
+(a) Mean.
+50
+100
+150
+Altitude (m)
+0
+1
+2
+3
+4
+Power (dB)
+CBRS Band-n48 (3550-3600 MHz)
+CBRS Band-n48 (3600-3650 MHz)
+CBRS Band-n48 (3650-3700 MHz)
+(b) Variance.
+Fig. 21: Spectrum occupancy versus altitude in CBRS band
+for rural environment.
+crowded compared with the rural environment. In Fig. 21, we
+study the mean and variance of the measured power versus
+altitude. As it can be observed, the mean value of the measured
+power for all three considered portions are almost similar
+and remain constant as the altitude increases. In addition, the
+variance also shows slight fluctuations compared to the other
+bands under consideration.
+V. TIME DOMAIN ANALYSIS OF SPECTRUM OCCUPENCY
+In this section, we focus on the spectrum occupancy of
+LTE and NR signals in time, while we describe the altitude
+dependency of the spectrum in the previous section. For
+around 8 hours of measurement duration by the Helikite in
+the urban environment, we observe signal strength changes.
+This section focuses exclusively on those urban environment
+measurements.
+Fig. 22 shows the spectrum monitoring results by the
+Helikite. The x-axis is the monitored spectrum range and the
+y-axis is the measured time stamp, which is indicated by hours
+and minutes. In Fig. 22a, we capture the frequency range from
+700 MHz to 800 MHz, which contains LTE FDD bands 12,
+13, 14 (see Table I). First of all, we can clearly observe a
+series of occupied 10 MHz bandwidth 12, 13, and, 14 DL
+bands. On the other hand, the signal strength of UL bands is
+lower than DL bands, and UL bands 13 and 14 are scarcely
+occupied. We also observe that there are time periods when
+signal strength becomes low for the whole observed frequency
+range, which coincides with the periods where the altitude of
+the Helikite stays low in Fig. 1. It implies that received signal
+strength is abruptly reduced by the blockage when the altitude
+of the Helikite is lower than a certain height. In addition,
+this tendency is observed in other frequency bands as well in
+Fig. 22b and Fig. 22c. In Fig 22b, we capture the frequency
+range 2500 MHz - 2700 MHz, which contains LTE TDD
+41 band. Since carrier frequency is higher than Fig. 22a, we
+observe that this LTE band covers wider bandwidth: 20 MHz,
+40 MHz, and 100 MHz. It is also observed that the received
+signal strength is lower than the frequency range in Fig. 22a.
+This is due to the fact that as carrier frequency increases a
+received signal suffers higher path loss, which is also observed
+in a much higher carrier frequency range in Fig. 22c. In
+particular, Fig. 22c shows spectrum occupancy of NR TDD
+n77 band, 3700 MHz - 3800 MHz. We can observe 40 MHz
+and 60 MHz bandwidth signals.
+Fig. 23 shows the received signal strength changes during
+the measurement time for the captured LTE and NR bands. In
+Fig. 23a, we observe the LTE FDD UL/DL 12 band shown
+in Fig. 22a. Mean value of the received signal strength across
+the frequency band is represented by lines and half of the
+standard deviation (std) of signal strength is described by the
+shaded area around lines. It is observed that the signal strength
+of UL is lower than DL, while the variation of the signal
+strength of UL inside the band is higher than DL, which can
+be observed from higher std values. Fig. 23b and Fig. 23b
+show the received signal strength changes of LTE TDD 41
+and NR TDD 77 bands which can be shown in Fig. 22b and
+Fig. 22c. It is observed that the signal strength fluctuation of
+NR TDD 77 band is higher than other bands such as LTE 12
+and 41 bands.
+VI. CONCLUSION
+Using the data measured by a Helikite flying over an urban
+and rural environments, in this paper we studied spectrum
+measurements in various sub-6 GHz 4G, 5G and CBRS bands.
+Both UL and DL spectrum occupancy has been investigated.
+Our results revealed that generally the mean value of measured
+power tends to increase as the altitude increases due to higher
+probability of line-of-sight, at least for the considered max-
+imum altitude range. Further, the spectrum of DL frequency
+ranges showed to be more crowded compared with the uplink
+ones for both environments. It has been also seen that for the
+rural environment the mean value for LTE bands 13 and 14
+are much higher than the other two bands under considera-
+tion, as opposed to the urban environment. Furthermore, the
+performance of CBRS band for urban environment indicates
+more activity compared with the rural condition.
+
+(a) 700 MHz - 800 MHz.
+(b) 2500 MHz - 2700 MHz.
+(c) 3700 MHz - 3800 MHz.
+Fig. 22: Spectrum monitoring during the measurement time. We observe different LTE and NR bands’ occupancy and the
+received signal strength is strong when the Helikite floats at a high altitude.
+12:00
+14:00
+16:00
+18:00
+20:00
+Time
+-30
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+Power (dBm)
+std/2 | LTE DL 12
+mean | LTE DL 12
+std/2 | LTE UL 12
+mean | LTE UL 12
+(a) LTE FDD 12 band.
+12:00
+14:00
+16:00
+18:00
+20:00
+Time
+-30
+-25
+-20
+-15
+-10
+-5
+0
+Power (dBm)
+std/2 | LTE TDD 41
+mean | LTE TDD 41
+(b) LTE TDD 41 band.
+14:00
+16:00
+18:00
+20:00
+Time
+-35
+-30
+-25
+-20
+-15
+-10
+Power (dBm)
+std/2 | NR TDD n77
+maen | NR TDD n77
+(c) NR TDD 77 band.
+Fig. 23: Received power of different LTE and NR bands during the measurement time. The solid lines represent the mean value
+of signal power and shaded areas indicate half of the standard deviation (std) of signal strength, which shows the variation of
+signal strength inside the specific bands.
+REFERENCES
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+R. Dutta, and M. Mushi, “Spectrum Activity Monitoring and Analysis
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+Syst. Netw. (COMSNETS), Bengaluru, India, Jan. 2023.
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+Communications Workshops (ICC Workshops), 2022, pp. 687–692.
+
+18:00
+17:00
+Time
+16:00
+15:00
+111
+14:00
+12
+13
+14
+DL
+DL
+DL
+13:00
+12:00
+710
+720
+730
+740
+750
+760
+770
+780
+790
+Freq (MHz)20
+10
+0
+dBr
+-10
+-20
+-30
+-4020:00
+19:0040
+3018:00
+17:00
+Time
+16:00
+15:00
+14:00
+LTETDD.41
+13:00
+12:00
+2520
+25402560258026002620
+2640
+2660
+2680
+Freg (MHz)20
+10
+0
+dBr
+-10
+-20
+-30
+-4020:00
+19:0040
+3018:00
+17:00
+Time
+16:00
+15:00
+14:00
+NRTDD
+0n77
+13:00
+12:00
+3790
+Freg (MHz)20
+10
+0
+dBi
+-10
+-20
+-30
+-4020:00
+19:0040
+30

2NE2T4oBgHgl3EQfjAcm/content/tmp_files/2301.03963v1.pdf.txt

@@ -0,0 +1,787 @@
+Genetic optimization of Brillouin scattering gain in
+subwavelength-structured silicon membrane
+waveguides
+Paula Nuño Ruano∗, Jianhao Zhang1, Daniele Melati,
+David González-Andrade, Xavier Le Roux, Eric Cassan,
+Delphine Marris-Morini, Laurent Vivien, Daniel Lanzillotti-Kimura,
+Carlos Alonso-Ramos∗
+aCentre de Nanosciences et de Nanotechnologies, Université Paris-Saclay, CNRS, 10
+boulevard Thomas Gobert, 91120, Palaiseau, France
+Abstract
+On-chip Brillouin optomechanics has great potential for applications in com-
+munications, sensing, and quantum technologies. Tight confinement of near-
+infrared photons and gigahertz phonons in integrated waveguides remains a
+key challenge to achieving strong on-chip Brillouin gain. Here, we propose
+a new strategy to harness Brillouin gain in silicon waveguides, based on the
+combination of genetic algorithm optimization and periodic subwavelength
+structuration to engineer photonic and phononic modes simultaneously. The
+proposed geometry is composed of a waveguide core and a lattice of anchoring
+arms with a subwavelength period requiring a single etch step. The waveguide
+geometry is optimized to maximize the Brillouin gain using a multi-physics
+genetic algorithm. Our simulation results predict a remarkable Brillouin gain
+exceeding 3300 W−1m−1, for a mechanical frequency near 15 GHz.
+Keywords:
+Brillouin scattering, subwavelength, genetic optimization
+∗Corresponding author
+Email addresses: [email protected] (Paula Nuño Ruano),
+[email protected] (Carlos Alonso-Ramos)
+1Present address: National Research Council Canada, 1200 Montreal Road, Bldg. M50,
+Ottawa, Ontario K1A 0R6, Canada
+arXiv:2301.03963v1  [physics.optics]  10 Jan 2023
+
+1. Introduction
+Brillouin scattering (BS) refers to the nonlinear interaction between opti-
+cal and mechanical fields inside a material. BS has been widely exploited in
+optical fibers to implement a wide range of devices, including optical ampli-
+fiers, ultra-narrow linewidth lasers, radio-frequency (RF) signal generators,
+and distributed sensors [1].
+Brillouin scattering was for long thought to be mediated by electrostric-
+tive forces only. Thus, its spectrum was considered to be governed by ma-
+terial properties [2]. In 2006, microstructuration of optical fibers enabled
+shaping the BS spectrum [3], opening a new path for geometric control of
+this effect [4]. In 2012, a new theory [5] predicted that Brillouin interactions
+could be greatly magnified by strong radiation pressure on the boundaries
+of suspended silicon waveguides with nanometric-scale core sizes [6, 7]. The
+simultaneous confinement of optical and mechanical modes is challenging in
+silicon-on-insulator (SOI) waveguides due to a strong phonon leakage towards
+the silica cladding [8–10]. However, this limitation can be circumvented by
+isolating the silicon waveguide core by complete or partial removal of the silica
+cladding [5, 11, 12]. Suspended or quasi-suspended structures such as silicon
+membrane rib waveguides [13] and fully suspended silicon nanowires [12] have
+demonstrated large Brillouin gain. These results generated a great scientific
+interest for its potential for laser sources [14], microwave signal generation
+[15] and processing [16], sensing applications [17, 18] and non-reciprocal op-
+tical devices [19]. In particular, pedestal waveguides [20] yield an experi-
+mental Brillouin gain of 3000 W−1m−1. However, the need for narrow-width
+pedestals to optimize the Brillouin gain complicates the fabrication process
+and may compromise the mechanical stability of the structures. On the other
+hand, a lower experimental Brillouin gain (1000 W−1m−1) was obtained for
+silicon membrane rib waveguides due to the very different confinement of
+optical and mechanical modes [13]. Still, this comparatively modest Bril-
+louin gain was compensated by achieving ultra-low optical propagation loss,
+allowing the demonstration of lasing effect [14]. The use of photonic crystals
+with simultaneous photonic and phononic bandgaps [21] (also referred to as
+phoxonic crystals) has been proposed to maximize the Brillouin gain in silicon
+membrane waveguides, achieving calculated values up to 8000 W−1m−1. Yet,
+the narrow bandwidth and high optical propagation loss, typically linked to
+bandgap confinement [22], may compromise the performance of these phox-
+onic crystals.
+2
+
+Subwavelength grating silicon waveguides, with periods shorter than half
+of the wavelength of the guided light, exploit index-contrast confinement to
+yield low optical loss and wideband operation [23, 24]. Interestingly, near-
+infrared photons and GHz phonons in nanoscale Si waveguides have compara-
+ble wavelengths (near 1 µm) [10]. Thus, the same periodic structuration could
+operate in the subwavelength regime for both, photons and phonons. In addi-
+tion, forward Brillouin scattering (FBS), used to demonstrate Brillouin gain
+in Si, relies on longitudinally propagating photons and transversally propa-
+gating phonons [8–10]. Hence, engineering the longitudinal and transversal
+subwavelength geometries would allow independent control of photonic and
+phononic modes. Brillouin optimization in silicon membranes has been pro-
+posed based on index-contrast confinement of photons (longitudinal subwave-
+length grating) and bandgap confinement of phonons (transversal phononic
+crystal) [25], achieving a calculated gain of 1750 W−1m−1. More recently,
+the combination of subwavelength index-contrast and subwavelength soften-
+ing has been proposed to optimize Brillouin gain in suspended Si waveguides,
+achieving a calculated value of 3000 W−1m−1, for a minimum feature size of
+50 nm [26]. Still, these two approaches require several etch steps of the silicon
+core, complicating the device’s fabrication. In this work, we propose a novel
+subwavelength-structured Si membrane, illustrated in Fig. 1, requiring only
+one etch step of silicon. We develop an optimization method to design the
+waveguide geometry, combining multi-physics optical and mechanical simu-
+lations with a genetic algorithm (GA) capable of handling a large number of
+parameters [27]. The optimized geometry yields a calculated Brillouin gain
+of 3300 W−1m−1, with a minimum feature size of 50 nm, compatible with
+electron-beam lithography.
+2. Design and Results
+The proposed optomechanical waveguide geometry, depicted in Fig. 1,
+comprises a suspended central strip of width Wg = 400 nm that is anchored
+to the lateral silicon slabs by a lattice of arms with a longitudinal period
+(z-direction) of Λ = 300 nm. This period is shorter than half of the optical
+wavelength, ensuring optical operation in the subwavelength regime. The
+anchoring arms are symmetric with respect to the waveguide center. We
+split the arms into five different sections with widths and lengths of Wi (x-
+direction) and Li (z-direction), respectively. The index i = 1 refers to the
+section adjacent to the waveguide core, while the index i = 5 refers to the
+3
+
+outermost section (see Fig. 1, inset). The fifth section has a fixed width
+of W5 = 500 nm and length of L5 = 50 nm to ensure proper guidance and
+localization of the optical mode. The widths and lengths of sections 1 to 4
+are optimized using the genetic algorithm. The whole waveguide has a fixed
+silicon thickness of t = 220 nm, allowing fabrication in a single-etch step.
+Figure 1: Proposed optomechanical waveguide. In the inset, the different sections of the
+anchoring arms are numbered from 1 to 5. The width of the waveguide core (Wg = 400
+nm), the period (Λ = 300 nm), and the dimensions of the outermost section (L5 = 50 nm,
+W5 = 500 nm) remain fixed throughout the optimization process. The thickness of the
+silicon slab is t = 220 nm.
+We focus on FBS, where only near-cut-off acoustic modes are involved.
+In the absence of optical absorption, which is the case of silicon at near-
+infrared wavelengths, the optical and mechanical mode equations describing
+FBS decouple and can be solved separately [10].
+We use here COMSOL
+Multiphysics software for the optomechanical simulations. For the calculation
+of optical and mechanical modes in the optimization process, we reduce the
+3D structure to an equivalent 2D geometry. The effective index method [28]
+is considered for the computation of the transverse-electric (TE) polarized
+4
+
+Wg
+Wi
+Anchoring arms: sections
+2
+3
+4
+5optical modes while the in-plane mechanical modes are calculated assuming
+the plane stress approximation [29]. We compute the Brillouin gain, GB, as
+[9]
+GB(Ωm) = Qm
+2ωp
+meff Ω2
+m
+����
+�
+fMB dℓ +
+�
+fPE dA
+����
+2
+,
+(1)
+where ωp is the frequency of the optical pump, Ωm is the mechanical fre-
+quency, Qm is the mechanical quality factor, meff =
+�
+ρ |um|2/ max |um|2 dA
+is the effective linear mass density of the mechanical mode with displacement
+profile um, and fMB and fPE are the linear and surface overlap of optical force
+density and deformation representing the moving boundaries effect (MB) and
+the photoelastic effect (PE), respectively,
+fMB = u∗
+m · n
+�
+δεMB E∗
+p,t · Es,t − δε−1
+MB D∗
+p,n · Ds,n
+�
+max |um| Pp Ps
+and
+fPE = E∗
+p · δε∗
+PE · Es
+max |um| Pp Ps
+,
+(2)
+where the permittivity differences due to the moving boundaries effects are
+given by δεMB = ε1 − ε2 and δε−1
+MB = 1/ε1 − 1/ε2, with εi = ε0n2
+i being
+the permittivities of the silicon (i = 1) and air (i = 2). The photoelastic
+tensor perturbation in the material permittivity is δεPE = −ε0 n4 p : S, with
+n being the material refractive index, p the photoelastic tensor, and S the
+mechanical stress tensor induced by the mechanical mode. The term um · n
+is the normal component of the mechanical displacement and Ej,t and Dj,n
+are the tangential electric field and normal dielectric displacement for the
+pump (j = p) and the scattered field (j = s). The denominator represents
+the power normalization given by Pj = [2ℜ(
+�
+[Ej × H∗
+j] · z dA)]1/2.
+The symmetry directions [100], [010], and [001] of the crystalline silicon
+are set to coincide with the x, y, and z simulation axis, respectively. With this
+orientation, the photoelastic tensor [6, 30] is [p11, p12, p44] = [−0.094, 0.017, −0.051].
+The refractive index of silicon is n = 3.45 and its density ρ = 2329 kg m−3
+while the corresponding values for the air are n = 1 and ρ = 1.293 kg m−3.
+The quality factor of the mechanical mode, Qm, is related to the full width
+at half maximum (FWHM) of the gain spectrum, γm, through Qm = Ωm/γm
+and it is limited by different loss mechanisms,
+1
+Qm
+=
+1
+QTE
++ 1
+QL
++
+1
+Qair
+.
+(3)
+5
+
+Here, we consider the thermoelastic loss (QTE), the mechanical leakage to-
+wards the silica under-cladding (QL), and the viscous loss from surround-
+ing air (Qair). The thermoelastic loss yields mechanical quality factors of
+QTE ∼ 6 · 105 [31] for silicon nanostructures while the leakage loss is mainly
+governed by the geometries of the waveguide and the arms anchoring it to
+the lateral silicon slab.
+These two effects are directly considered in the
+mechanical-mode simulations performed in COMSOL Multiphysics. The vis-
+cous loss induced by the surrounding air is considered here by imposing a
+limiting value to the mechanical quality factor of Qm = 4 · 103, which is the
+highest expected value at atmospheric pressure and room temperature for
+phonon frequency in the order of GHz [32].
+Based on the resulting optomechanical coupling calculations, a genetic
+algorithm [33] is used to maximize the FBS gain. Starting with randomly
+generated combinations of parameters Wi and Li (individuals), optomechan-
+ical simulations are carried out and the individuals are ranked according to
+their Brillouin gain. Recombination is used to produce a successor set of
+individuals, the next generation. The best-performing individuals directly
+become part of the next generation (elitism). A large number of individuals
+of the new generation is obtained by combining the parameter of pairs of
+individuals from the current generation (crossover). Finally, the remaining
+individuals of the new generation are produced by randomly modifying the
+parameters of single individuals of the current generation (mutation). This
+process continues until the convergence criterion has been reached.
+In our particular optimization problem, an individual is a possible geome-
+try, represented by a set of 8 parameters (width and length of each of the arm
+sections). Each generation is composed of 50 individuals and the successive
+generations are obtained applying a rate of elitism and crossover of 6% and
+80%, respectively, with the remaining elements obtained through mutation.
+The convergence criterion was defined in terms of the difference between the
+best and the average performance, GB − ⟨GB⟩ < 10 W−1m−1, over 10 gener-
+ations. For this work, we have used a standard computer with the following
+specifications: a 64-bit operating system with an x64-based processor Intel®
+Core™ i7-4790 (4 total cores, 8 total threads, base-frequency of 3.60 GHz),
+and an installed RAM of 8.00 GB. Under these conditions, the optimiza-
+tion process was completed in 12h 35 min, comprising 1500 optomechanical
+simulations of 30 seconds each.
+The method we propose here relies on a defined geometry whose pa-
+rameters are allowed to vary within a specific range of values. Hence, the
+6
+
+optimized structure will depend strongly on our initial guess.
+In Fig. 2, we present the optimization process. Figures 2a and 2b show
+the Brillouin gain and mechanical frequency, respectively, as a function of
+the generation number. As a result of the evolution of the geometry, we
+observe an increase in the gain and a variation in the mechanical frequency.
+This result should be expected as the Brillouin shift in FBS is particularly
+sensitive to the waveguide dimensions. The optimum performance is achieved
+after 10 generations while 30 generations are required for convergence. The
+optimized geometry, whose dimensions are listed in Table 1, is characterized
+by a Brillouin gain of GB = 3350 W−1m−1 for a mechanical mode with
+frequency of Ωm = 14.357 GHz and mechanical quality factor of Qm ≈ 3.2 ·
+103. The optical mode has a mode effective index of 2.36 and wavelength in
+vacuum of λ = 1556.5 nm (ωp = 2π · 192.6 THz in (1)).
+Figure 2: Optimization process. a) Best (in blue) and average (in orange) Brillouin gain
+as a function of the number of generations during genetic optimization.
+b) Evolution
+of the mechanical frequency as a function of the number of generations.
+During the
+optimization process, all possible mechanical losses are considered, including thermoelastic
+loss, mechanical leakage, and viscous loss due to air (operation in air ambient at room
+temperature).
+In terms of geometry, the first and fourth sections, with considerably
+larger widths, generate reflections that help localize the mechanical mode
+in the waveguide core. The frequency of the mechanical mode is governed
+by the interplay between the waveguide width and the length of the partial
+cavity formed by the fourth section on each side.
+Full 3D simulations are realized to verify the performance of the optimized
+geometry. This structure provides a Brillouin gain of GB = 3310 W−1m−1 for
+a mechanical mode with a frequency of Ωm = 14.579 GHz. The optical mode
+7
+
+a)
+b)
+4000
+16.0
+[GHz]
+Brillouin Gain [(Wm)-1]
+3000
+Frequency
+15.5
+2000
+15.0
+Mechanical
+1000
+14.5
+Best
+Average
+0
+14.0
+0
+5
+10
+15
+20
+25
+30
+35
+0
+5
+10
+15
+20
+25
+30
+35
+Number of
+ generation
+Number of
+ generationTable 1: Dimensions for the GA-optimized geometry when operating in air ambient at
+room temperature. In the table above, Si stands for section i in Fig. 1.
+S1
+S2
+S3
+S4
+Width
+170 nm
+320 nm
+330 nm
+100 nm
+Length
+130 nm
+60 nm
+60 nm
+190 nm
+has a mode effective index of 2.23 and wavelength in vacuum of λ = 1557.2
+nm (ωp = 2π · 192.52 THz in (1)).
+Figure 3 shows the calculated field
+distribution for the mechanical and optical modes in the optimized geometry.
+Figure 3: Optical and mechanical modes of the optimized geometry operating in air ambi-
+ent and room temperature (table 1): a) Approximated 2D structure. The upper structure
+corresponds to the normalized mechanical displacement at 14.357 GHz and the lower fig-
+ure to the x-component of the electric field at 1556.5 nm (mode effective index 2.36). b)
+Full 3D device. On the bottom left, x-component of the electric field at 1557.2 nm (mode
+effective index 2.23), and on the top right, normalized mechanical displacement at 14.579
+GHz.
+These results show a good agreement between the approximated 2D ge-
+ometry used for the optimization and the full 3D structure. The small dis-
+crepancies in the optical mode index and mechanical frequency are due to
+the influence of the thickness.
+Finally, we study the fabrication tolerance of the proposed structure us-
+ing again 3D simulations. We consider under- and over-etching errors that
+we model by a variation of all the waveguide lengths and widths by a factor
+∆, measured in nm (Fig. 4a). Figure 4c shows the variation of the Bril-
+louin gain (in blue) and mechanical frequency (in orange) as a function of ∆.
+8
+
+b)
+a)
+[ul / max|ul
+u/max|u
+E.The Brillouin gain remains above 2000 W−1m−1 for geometry variations of
+±10 nm. It should be noted that for the over-etch case (∆ < 0 in Fig. 4c),
+the Brillouin gain is larger than the optimized case due to the larger optome-
+chanical coupling resulting from a better overlap of the mechanical mode
+with the optical field. However, these smaller structures are incompatible
+with the target minimum feature size of 50 nm that was chosen to guarantee
+fabrication reliability. The mechanical frequency varies less than 2% (Fig.4c,
+in orange) and the mechanical profile is not modified significantly.
+We also study the effect of stitching errors, modeled by a deviation ζ (in
+nm) of the arm axis at both sides of the waveguide core, hence breaking the
+symmetry of the structure (Fig. 4b). Figure 4d shows the variation of the
+Brillouin gain (in blue) and mechanical frequency (in orange) as a function of
+ζ. A non-perfectly symmetric structure is slightly detrimental to the Brillouin
+gain but does not affect the mechanical frequency or profile. Interestingly,
+both parameters (Brillouin gain and mechanical frequency) remain constant
+over a large range of stitching errors.
+Lastly, we examine the effect of random fabrication errors affecting each
+section independently (Table 2). We consider deviations of 5 to 20 nm, both
+in positive (enlargement) or negative (shrinking) directions. Our geometry
+exhibits a robust performance despite these errors with Brillouin gains above
+2000 W−1m−1 (Fig. 4e, blue) and mechanical frequencies between 14 and 15
+GHz (Fig. 4e, orange). It should be noted that the period remains constant,
+Λ = 300 nm since it is controlled with high precision (±2 nm) in terms of
+fabrication.
+3. Conclusions
+In summary, we have proposed a new approach to optimizing Brillouin
+gain in silicon membrane waveguides. We exploit genetic optimization to
+maximize Brillouin gain in subwavelength-structured Si waveguides, requir-
+ing only one etch step.
+Genetic algorithm is a well-known optimization
+technique capable of handling design spaces of moderate dimension [33].
+It has the main advantage over gradient-based algorithms in its capabil-
+ity to search the design space in many directions simultaneously. On the
+other hand, the genetic algorithms cannot guarantee a global optimum so-
+lution, being the final result strongly dependent on the initial population.
+Based on this strategy, a calculated Brillouin gain up to 3310 W−1m−1 is
+achieved for air environment. This result compares favorably to previously
+9
+
+Figure 4: Fabrication tolerance of the optimized geometry. a) and b) Variation of the
+geometry due to fabrication errors. The solid black line corresponds to optimized geometry,
+dotted (solid) blue depicts a positive deviation from the nominal design, and dotted orange
+refers to a negative deviation from the expected design. c) and d) Evolution of the Brillouin
+gain (in blue, left axis) and the mechanical frequency (in orange, right axis) for different
+values of under- and over-etching (c), different values of stitching errors (d), and different
+structures with randomized geometrical parameters (e). In e), N stands for the nominal
+design obtained after the optimization problem and i for the different geometries listed in
+Table 2.
+10
+
+a)
+b)
+Wg/2
+Si Slab
+Si Slab
+W.
+Si Slab
+c)
+d)
+6000
+15.0
+3500
+15.0
+[(Wm)-
+(Wm)
+14.8
+ZH
+14.8
+3250
+GH
+4000
+Brillouin Gain
+'requency
+requency
+3000
+14.4
+14.4
+2000
+2750
+14.2
+14.0
+2500
+14.0
+-10
+-5
+0
+5
+10
+0
+5
+10
+15
+20
+25
+30
+35
+Fabrication error, △ [nm
+Stiching error, S[nm]
+e)
+6000
+16
+4000
+15
+[2H)]
+Brillouin Gain
+2000
+14 
+13
+N
+1
+2
+3
+4
+5
+6
+7
+8
+9
+GeometryTable 2: Dimensions for the different geometries used for studying the effect of random-
+ization of the design parameters. In the table, Si stands for section i in Fig. 1, N stands
+for the nominal design as obtained from the optimization (Table 1), and i stands for the
+different geometries in Fig. 4e. In all cases, the period, Λ = 300 nm, remains constant.
+Geometry
+S1
+S2
+S3
+S4
+S5
+Wg
+N
+Width
+170 nm
+320 nm
+330 nm
+100 nm
+500 nm
+400 nm
+Length
+130 nm
+60 nm
+60 nm
+190 nm
+50 nm
+1
+Width
+165 nm
+305 nm
+345 nm
+90 nm
+510 nm
+405 nm
+Length
+130 nm
+45 nm
+65 nm
+180 nm
+60 nm
+2
+Width
+165 nm
+320 nm
+340 nm
+115 nm
+495 nm
+400 nm
+Length
+110 nm
+45 nm
+55 nm
+170 nm
+35 nm
+3
+Width
+155 nm
+340 nm
+340 nm
+100 nm
+485 nm
+405 nm
+Length
+150 nm
+40 nm
+70 nm
+200 nm
+55 nm
+4
+Width
+185 nm
+300 nm
+325 nm
+95 nm
+480 nm
+385 nm
+Length
+140 nm
+65 nm
+75 nm
+185 nm
+60 nm
+5
+Width
+160 nm
+320 nm
+330 nm
+95 nm
+510 nm
+390 nm
+Length
+140 nm
+65 nm
+55 nm
+210 nm
+60 nm
+6
+Width
+185 nm
+340 nm
+315 nm
+120 nm
+520 nm
+420 nm
+Length
+135 nm
+40 nm
+50 nm
+190 nm
+35 nm
+7
+Width
+185 nm
+340 nm
+340 nm
+110 nm
+480 nm
+410 nm
+Length
+140 nm
+55 nm
+65 nm
+175 nm
+40 nm
+8
+Width
+150 nm
+300 nm
+345 nm
+110 nm
+510 nm
+395 nm
+Length
+120 nm
+80 nm
+40 nm
+175 nm
+65 nm
+9
+Width
+170 nm
+340 nm
+325 nm
+105 nm
+520 nm
+410 nm
+Length
+120 nm
+70 nm
+50 nm
+190 nm
+70 nm
+11
+
+reported subwavelength-based Brillouin waveguides requiring several etch-
+ing steps [25, 26], with calculated Brillouin gain of 1750 W−1m−1 and 3000
+W−1m−1. Our results show the potential of optimization for obtaining novel
+designs with improved performance in the context of Brillouin scattering.
+Moreover, they show the reliability of computationally efficient optimizations
+based on approximated 2D simulations.
+Declaration of Competing Interest
+The authors declare that they have no known competing financial inter-
+ests or personal relationships that could have appeared to influence the work
+reported in this paper.
+Author Statement
+Paula Nuño Ruano, Jianhao Zhang, and Carlos Alonso Ramos proposed
+the concept. Paula Nuño Ruano, Jianhao Zhang, and Daniele Melati devel-
+oped the simulation framework. Paula Nuño Ruano, Jianhao Zhang, Daniele
+Melati, David González Andrade, and Carlos Alonso Ramos optimized and
+analyzed the results. All authors contributed to the manuscript.
+Data Availability Statement
+The data supporting this study’s findings are available from the corre-
+sponding author upon reasonable request.
+Acknowledgements
+The authors want to thank the Agence Nationale de la Recherche for sup-
+porting this work through BRIGHT ANR-18-CE24-0023-01 and MIRSPEC
+ANR-17-CE09-0041. P.N.R. acknowledges the support of Erasmus Mundus
+Grant: Erasmus+ Erasmus Mundus Europhotonics Master program (599098-
+EPP-1-2018-1-FR-EPPKA1-JMD-MOB) of the European Union. This project
+has received funding from the European Union’s Horizon Europe research and
+innovation program under the Marie Sklodowska-Curie grant agreement Nº
+101062518.
+12
+
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+

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+Exit options sustain altruistic punishment and decrease the second-order free-riders,
+but it is not a panacea
+Chen Shen1,2, Zhao Song3, Lei Shi2,∗ Jun Tanimoto1, and Zhen Wang3,4†
+1. Faculty of Engineering Sciences, Kyushu University, Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan
+2. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
+3. School of Mechanical Engineering,Northwestern Polytechnical University, Xi’an 710072, China
+4. School of Artifcial Intelligence, OPtics and ElectroNics (iOPEN),
+Northwestern Polytechnical University, Xi’an 710072, China
+(Dated: January 13, 2023)
+The emergence and maintenance of altruistic punishment remains an open question and this
+conundrum is shared across diverse fields. In this study, we evaluated the evolution of altruistic
+punishment in a two-stage prisoner’s dilemma game in which cooperators and defectors interact with
+another two actors called altruistic punishers and exiters. Traditionally cooperators and defectors,
+in the first stage, choose to cooperate and defect with their opponent, respectively, but they do not
+punish in the second stage; the altruistic punishers cooperate in the first stage and punish defectors
+in the second stage, and the exiters who simply exit the game in favor of a small payoff.
+We
+found that exiters did not provide any substantial assistance to altruistic punishment in well-mixed
+populations, they destabilize defection and finally replace them. In the finite population, although
+the exit option enables the coexistence of altruistic punishers, defectors, and exiters through cyclic
+dominance. Altruistic punishers never dominate the finite population and the exit option provides
+another alternative cyclic dominance route for the emergence of non-punishing cooperators.
+In
+networked populations, however, adding the exit option allows for the establishment of altruistic
+punishment, and enables the coexistence of altruistic punishers, defectors, and exiters through cyclic
+dominance. However, this type of cyclic dominance is not always stable, with adjustments to the
+exit payoff, this type of cyclic dominance is replaced by the cyclic dominance of non-punishing
+cooperators, defectors, and exiters or a bi-stable state between these two types of cyclic dominance.
+Our results indicate that although the exit option can help explain altruistic punishment, it is
+certainly not a panacea.
+Keywords: Evolutionary game theory; Cooperation; Coexistence; Cyclic dominance; Bi-stable
+INTRODUCTION
+Costly punishment is ubiquitous in many animal
+species including humans [1–3].
+Unlike other animals,
+humans often show altruistic traits, i.e., humans punish
+other individuals who have harmed others even at the ex-
+pense of their own interest [3, 4], however, the emergence
+and maintenance of altruistic punishment is an evolu-
+tionary conundrum as costly punishment is unlikely to
+evolve according to natural selection. Costly punishment
+reduces the payoff for both the punisher and the pun-
+ished. If it is the fittest who survive, the second-order
+free riders that cooperate but do not punish are better
+off than punishers, and defectors should eventually take
+over the whole population. Therefore, the understanding
+of whether and how costly punishment can evolve is a
+crucial issue in the study of human cooperation. Fehr
+and G¨achter pointed out that the evolutionary study of
+human cooperation in large groups of unrelated individ-
+uals should include a focus on explaining altruistic pun-
+ishment [4]. In addition, they argued that negative emo-
+tions may be a potential explanation for the emergence
+of costly punishment.
+∗ shi [email protected]
+† [email protected]
+To resolve this evolutionary puzzle, many scholars have
+explored how and why costly punishment can emerge in
+humans both from a theoretical and experimental per-
+spective. Egas Martijn and Riedl Arno experimentally
+explored the boundary conditions that altruistic punish-
+ment can promote cooperation.
+They found that the
+maintenance of cooperation is subject to the cost-to-
+effect ratio of altruistic punishment, and cooperation is
+maintained if the conditions for altruistic punishment are
+relatively favorable [5]. It has been well established that
+voluntary participation plays a vital role in sustaining the
+prevalence of costly punishment both in finite and infinite
+populations [6–11]. The main idea behind established al-
+truistic punishment is that a loner itself is sufficient to
+maintain cooperation through cyclic dominance even in a
+one-shot game. Other reciprocity mechanisms including
+indirect reciprocity [12–16], group selection [17–19], spa-
+tial interaction [20–23], prior commitment [24–27], and
+so on [28], that can explain the emergence of cooperation
+have been applied to explain costly punishment, and its
+effect on costly punishment has previously been widely
+explored.
+To avoid the exploitation of defectors, exiters simply
+exit the game in favor of a small-but-positive payoff and
+generate nothing for their opponent. While loners can
+receive a small-but-positive payoff by opting out but gen-
+erates the same payoff for its opponent. Although these
+arXiv:2301.04849v1  [q-bio.PE]  12 Jan 2023
+
+2
+two mechanisms seem materially similar, such a subtle
+difference leads to completely different outcomes [29, 30].
+On one hand, exit means a potential punishment for their
+opponent, although the exiters can avoid being exploited
+by the defectors through opting out, they also hurt the
+cooperators. However, loners enable the coexistence with
+cooperators and defectors through cyclic dominance in a
+one-shot game [31], while exiters allow cooperation to
+flourish only if they adhere to either direct, indirect or
+network reciprocity [32]. Given these differences, an in-
+teresting question arises: to what extent do exiters help
+explain altruistic punishment. To this end, we introduce
+the exit option and altruistic punishment in a two-stage
+prisoner’s dilemma game, and we start our analysis in
+well-mixed populations in which the extended prisoner’s
+dilemma game in both finite and infinite populations are
+considered. Then, we turn our attention to a networked
+population.
+In doing so, we found that the exit op-
+tion does not bring any substantial benefit to altruistic
+punishment in well-mixed populations, but enables the
+existence of altruistic punishment in networked popula-
+tions.
+In addition, multiple dynamical phenomena in-
+cluding cyclic dominance and a bi-stable state can be
+observed in networked populations.
+METHODS
+We studied the evolution of altruistic punishment in
+a two-stage prisoner’s dilemma game by introducing two
+other action types, altruistic punishment and exit. In the
+first stage, each individual must make a choice simulta-
+neously between cooperation (C), defection (D), and exit
+(E). In the second stage, cooperators decide whether to
+punish the defectors at a personal cost to themselves γ.
+To the defectors, this means an imposed fine β.
+This
+process results in four possible actions:
+• AP, cooperate and punish defectors. Those who
+cooperate and punish are altruistic punishers be-
+cause they punish free riders even at the expense
+of its own interests .
+• NC,
+cooperate but do not punish defectors.
+These non-punishing cooperators are also known
+as second-order free riders because by free-riding
+on punishment save the the cost of punishing the
+defectors.
+• D, defect but do not punish. These are also known
+as first-order free riders.
+• E, exit the game in favor of a small but positive
+payoff ϵ irrespective of whom they encounter. They
+do not participate in these two stages.
+In a typical prisoner’s dilemma game, mutual cooper-
+ation (defection) generates the reward (punishment) R
+(P). If one player cooperates and the other defects, the
+cooperative player gets the sucker’s payoff S, and the de-
+fected player obtains the temptation to defect T.
+For
+simplicity, we choose the weak prisoner’s dilemma game
+as our base model by setting R = 1, P = S = 0, T = b.
+TABLE I.
+Payoff matrix for the weak prisoner’s dilemma
+game with altruistic punishment and an exit option.
+AP
+NC
+D
+E
+AP
+1
+1
+−γ
+0
+NC
+1
+1
+0
+0
+D
+b − β
+b
+0
+0
+E
+ϵ
+ϵ
+ϵ
+ϵ
+The extended weak prisoner’s dilemma game contains four
+competing action types: altruistic punishers who cooperate
+and punish defectors(AP), non-punishing cooperators who
+cooperate but do not punish defectors (NC), defectors who
+free ride on the non-punishing cooperators and do not punish
+(D), and exiters who exit the game irrespective of whom they
+encounter (E). The first row indicates that when an altruis-
+tic punisher, AP, meets another altruistic punisher AP, non-
+punishing cooperator NC, defector D, or exiter E, they earn
+a payoff equal to 1, 1, −γ, or 0, respectively. When a non-
+punishing cooperator meets another altruistic punisher, non-
+punishing cooperator, defector, or exiter, they earn a payoff
+equal to 1, 1, 0, or 0, respectively. Analogously, when a de-
+fector meets an altruistic punisher, non-punishing cooperator,
+defector, or exiter, they earn a payoff equal to b−β, b, 0, or 0,
+respectively. Finally, exiters earn a payoff equal to ϵ ∈ [0, 1),
+irrespective of whom they meet, and their opponent receives
+nothing.
+To make exiting less valuable than cooperating, and to
+ensure that the weak prisoner’s dilemma game satisfied
+the payoff ranking of the strict prisoner’s dilemma game,
+T > R > P > S was used. Additional limits placed on
+the parameters were 1 ≤ b < 2 and ϵ < 1. The described
+above is summarized in table I. Altruistic punishment
+maintains cooperation only when its effectiveness is rel-
+atively large [5, 33], thus to investigate the effect of the
+exit option on the explanation of altruistic punishment,
+throughout this study, the cost of punishment γ and the
+fine of the defectors was set as 0.1 and 0.3, respectively.
+Finite population
+We first considered a finite and well-mixed population
+of N individuals.
+Each individual adopted the Moran
+process, also known as frequently dependent process, to
+select their action. At each time step, a randomly se-
+lected player i with fitness fi = esΠi (Πi is the actual
+payoff of the individual i obtained through their interac-
+tion) updates its action by imitating the action of player
+j with fitness fj = esΠj who is selected with a proba-
+bility proportional to its fitness. Here, s is the selection
+strength, the condition of s → 0 corresponds to the weak
+selection and evolution proceeds as neutral drift.
+We
+further assumed that with a small probability µ, players
+randomly select their action from the rest of the other
+actions.This small mutation ensures that the population
+is homogeneous most of the time.
+Suppose that there are only two actors in the popula-
+tion, i.e., action A and B, and these actions can be one
+
+3
+of the four actions among the full action set {a, b, c, d}.
+Here, the symbols a, b, c, d represent AP, NC, D and E,
+respectively. In a finite population of size N with x A
+and y = N − x B actions, the average payoff of Πxy and
+Πyx to players with A and B actions are the following:
+ΠAB = (x−1)PAA+(N−x)PAB
+N−1
+ΠBA = xPBA+(N−x−1)PBB
+N−1
+,
+(1)
+where PAB is the payoff obtained from the single en-
+counter of actors A and B, and so does payoffs PAA, PBA,
+and PBB. This allows us to describe the evolutionary dy-
+namics of the population in terms of a reduced Markov
+Chain of size 4 [34–37]. Given the above assumptions,
+the probability to change the number of x individuals
+with action A in a population of y = N − x individuals
+with action B by ±1, T ±
+AB is:
+T +
+AB =
+xfi
+xfi+yfj
+y
+N
+T −
+AB =
+yfj
+xfi+yfj
+x
+N
+,
+(2)
+and hence the fixation probability ρAB of a single mutant
+actor A within a population of N − 1 B actors can be
+derived as [38, 39]:
+ρAB =
+1
+N−1
+�
+k=0
+k�
+x=1
+T −
+AB
+T +
+AB
+=
+1
+N−1
+�
+k=0
+k�
+x=1
+esΠBA
+esΠAB
+.
+(3)
+The fixation probabilities ρAB define the transition prob-
+abilities of the reduced Markov Chain, with the following
+associated transition matrix:
+�
+�
+�
+AP
+NC
+D
+E
+AP
+ρaa
+ρab
+ρac
+ρad
+NC
+ρba
+ρbb
+ρbc
+ρbd
+D
+ρca
+ρcb
+ρcc
+ρcd
+E
+ρda
+ρdb
+ρdc
+ρdd
+�
+�
+�.
+(4)
+Here, ρAA = 1 − �
+A̸=B
+ρAB, A, B ∈ {a, b, c, d}. The nor-
+malized right eigenvector to the largest eigenvalue deter-
+mines the stationary distribution of each strategy. For
+any pair of strategies A and B in the finite population,
+natural selection favors B replacing A only if ρAB > 1
+N .
+Infinite population
+We then employed replicator dynamics to analyze the
+evolutionary outcomes in an infinite and well-mixed pop-
+ulation. Let x, y, z, w denote the fractions of altruistic
+punishers (AP), non-punishing cooperators (NC), de-
+fectors (D), and exiters (E) in the population. Where
+0 ≤ x, y, z, w ≤ 1, and x + y + z + w = 1. The replicator
+equations are:
+˙x = x
+�
+ΠAP − Π
+�
+,
+˙y = y
+�
+ΠNC − Π
+�
+,
+˙z = z
+�
+ΠD − Π
+�
+,
+˙w = w
+�
+ΠE − Π
+�
+.
+(5)
+The symbols ΠAP , ΠNC, ΠD, and ΠE denote the average
+payoff of altruistic punishers, non-punishing cooperators,
+defectors, and exiters. Whereas Π = xΠAP + yΠNC +
+zΠD+wΠE is the average payoff of the whole population.
+According to the defined payoffs in table I, we obtained
+the following equation:
+ΠAP = x + y − zγ
+ΠNC = x + y
+ΠD = x(b − β) + yb
+ΠE = ϵ
+.
+(6)
+Using the constraint w = 1 − x − y − z, we obtained:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+˙x = f (x, y, z)
+= x [(1 − x) (ΠAP − ΠE) − y (ΠNC − ΠE) − z (ΠD − ΠE)]
+˙y = g (x, y, z)
+= y [(1 − y) (ΠNC − ΠE) − x (ΠAP − ΠE) − z (ΠD − ΠE)]
+˙z = h (x, y, z)
+= z [(1 − z) (ΠD − ΠE) − y (ΠNC − ΠE) − x (ΠAP − ΠE)]
+(7)
+For the detailed stability analysis of each equilibria,
+please refer to the Appendix.
+Networked population
+Different with well-mixed populations, global interac-
+tions in which an individual can interact with any other
+individual are no longer possible in the networked pop-
+ulation. Instead, networks only allow local interactions,
+which means that individuals can only interact with their
+direct neighbors. Our basic network structure is a two
+dimensional regular lattice with periodic boundary con-
+ditions, each node was occupied by one individual, and
+each individual can only interact with its neighbors along
+its links. Our simulation contained the following steps.
+Initially, each individual was designed as either an altru-
+istic punisher (AP), a non-punishing cooperator (NC),
+a defector (D), or an exiter (E) with equal probability.
+Each player acquires their total payoff by playing with
+all their direct neighbors according to the payoff matrix
+defined in table I. A randomly selected player i decides
+to imitate the strategy of player j who is also randomly
+selected from all the direct neighbors of player i by com-
+paring their payoff difference with the following proba-
+bility:
+Wi←j =
+1
+1 + exp ((Πi − Πj) /K),
+(8)
+where Πi and Πj is the acquired total payoff of the focal
+player i and its randomly selected neighbor j, respec-
+tively. K denotes the noise in the imitation process, and
+we fixed the value of K to be 0.1 throughout the study.
+A full Monte Carlo step is to repeat the above proce-
+dure L2 times, and L2 is the number of nodes in the given
+network. Each individual update their strategy once on
+
+4
+FIG. 1.
+Exiters establish altruistic punishment in a
+finite population, but altruistic punishers struggle to
+dominate the population. A. Stationary probability dis-
+tributions of each actors independence on the exiters’ payoff
+ϵ.
+B. Transition probabilities for each pair of actors when
+the exiters’ payoff is negative (left) and positive (right). The
+parameter values are b = 1.5, β = 0.3, γ = 0.1, s = 0.2,
+N = 100.
+average. To subside the transient dynamics and avoid the
+finite-size effect, we ran simulations for 50,000 steps on a
+regular lattice with size ranging from 200*200 to 800*800.
+The final fraction of each strategy was obtained after up
+to 45,000 steps. The presented data was averaged over
+20 independent runs.
+RESULTS
+Well-mixed populations
+We started our analysis in a well-mixed and finite pop-
+ulation, then we turned our attention to a well-mixed and
+infinite population, and finally, we investigated the evo-
+lution of altruistic punishment in networked population.
+Finite population.
+In the prisoner’s dilemma game
+with altruistic punishment, cooperation can only be
+maintained if the cost-to-fine ratio of altruistic punish-
+ment is relatively small [5].
+The favorable conditions
+for altruistic punishment imply the small enough pun-
+ishment cost or high enough punishment fine. Although
+altruistic punishment can establish the cooperation even
+in a one-shot game, punishment reduces the social wel-
+fare [40, 41]. If the cost-to-fine ratio of altruistic pun-
+ishment is high, altruistic punishment does not support
+the survival of cooperation, and thus defectors take over
+the whole population. As previously mentioned, nega-
+tive values of exiters’ payoff revert the extended model
+to the traditional weak prisoner’s dilemma game with al-
+truistic punishment, and in this case, selection favors the
+dominance of the defectors (refer to the left panel in fig-
+ure.1B and figure.A1A). A small but positive exiters’ pay-
+off enables the coexistence of altruistic punishers through
+cyclic dominance with defectors and exiters (refer to the
+right panel in figure.1B and figure.A1B). However, ex-
+iters also enable the survival of non-punishing cooper-
+ators, and allows the coexistence of non-punishing co-
+operators, defectors, and exiters through an alternative
+route of cyclic dominance. With increasing ϵ, the fac-
+tion of altruistic punishers first reaches its peak, where
+the maximum faction of altruistic punishers is less than
+0.2, and then decreases until its extinction. (figure.1A).
+The exiters facilitate the evolution of altruistic punishers
+in a finite population, but also allow for the survival of
+second-order free riders. Importantly, altruistic punish-
+ers never dominate the whole population.
+Infinite population.
+The situation changes greatly
+when the finite population is replaced by the infinite pop-
+ulation. Stability analysis shows that, (i), when b−β > 1
+and ϵ < 0, the monomorphic defecting equilibrium is sta-
+ble, and the others are unstable (figure.A2A); (ii), when
+b−β > 1 and ϵ > 0, the monomorphic exiting equilibrium
+is stable, and the others are unstable (figure.A2B); (iii),
+when b − β < 1 and ϵ < 0, the evolutionary dynamics
+result in either the mixed equilibrium of altruistic pun-
+ishers and non-punishing cooperators or the monomor-
+phic defecting equilibrium (figure.A2C); and (iv), when
+b − β < 1 and ϵ > 0, the evolutionary dynamics result in
+either the mixed equilibrium of altruistic punishers and
+non-punishing cooperators or the monomorphic exiting
+equilibrium (figure.A2D). In other words, exiters support
+the emergence of altruistic punishment only when the
+cost-to-fine ratio of punishment is favorable for coopera-
+tors in the infinite population. Nevertheless, the exiters
+destabilize the defection and eventually replace them re-
+gardless of whether altruistic punishment can establish
+cooperation.
+In a word, our results show that when the exit option
+was introduced in well-mixed populations, there was little
+additional benefit to the dominance of altruistic punish-
+ment. Rather by adding the exit option the equilibrium
+was either monomorphic exiting in the infinite popula-
+tion or joint dominance between the defectors and ex-
+iters in the finite population. Given the above conclusion,
+the natural question arises: does a networked population
+support the dominance of altruistic punishment in the
+extended model?
+Networked population
+Figure.2 shows the full ϵ − b phase diagram obtained
+by the extensive Monte Carlo simulations. It is noted
+
+1.0
+A
+AP
+0.8
+NC
+么
+Fractions
+D
+0.6
+A
+E
+V
+0.4
+7
+0.2
+0.0
++
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Exit pay-off, E
+B
+AP
+NC
+NC
+AP
+P=0.01
+p=0.01
+4%
+1%
+8%
+16%
+p=0.015
+=0.07
+=0.
+=0.
+=0.04
+=0.04
+=0.07
+=0.015
+E
+D
+E
+D
+p=0.04
+p=0.04
+0%
+95%
+44%
+32%
+E =-0.2
+E =0.25
+FIG. 2. Adding exit option establishes altruistic pun-
+ishment in networked population. Presented is the full
+ϵ − b phase diagram obtained by Monte Carlo simulations of
+the extended weak prisoner’s dilemma game on a regular lat-
+tice. Exiters dominate the whole population when the incen-
+tives to the exiters are large, ϵ ≳ 0.51. Fewer exit option in-
+centives lead to six possible outcomes. If b is relatively small,
+b ≲ 1.19, the effectiveness of altruistic punishment ensures
+the dominance of cooperators, and, altruistic punishers can
+coexist with defectors when 1.19 ≲ b ≲ 1.29. For large temp-
+tation b, b ≳ 1.29, negative ϵ leads to full defection, whereas,
+positive ϵ ensures the coexistence of altruistic punishers with
+defectors and exiters, the coexistence of second-order free rid-
+ers with defectors and exiters, or the bi-stable state of these
+two coexistence types.
+that the addition of the simple exit option leads to com-
+plicated evolutionary outcomes. Initially, when the in-
+centives to exiters are sufficiently large, ϵ ≳ 0.51, the
+exiters outcompete other action types and dominate the
+whole population (the E phase in figure.2), and this is
+consistent with previous findings [32]. Less incentives to
+exiters, ϵ ≲ 1.51, lead to six different possible outcomes.
+In detail, if the temptation to defect is relatively small,
+b ≲ 1.29, altruistic punishment together with network
+reciprocity are sufficient to maintain prosocial behavior
+(the All C phase and the AP + D phase in figure.2).
+When b ≲ 1.19, defectors can be completely eliminated
+by altruistic punishers, and thus altruistic punishers and
+non-punishing cooperators can coexist in a regular lat-
+tice.
+In the absence of defectors, non-punishing coop-
+erators and altruistic punishers cannot be distinguished,
+and whether the evolutionary dynamics lead to the full
+AP state, the full NC state or the mixed AP +NC state
+are determined by the initial conditions (the All C phase
+in figure.2). With increasing b, 1.19 ≲ b ≲ 1.29, the effec-
+tiveness of altruistic punishment is greatly reduced, and
+defectors cannot be completely eliminated by altruistic
+punishers, and they coexist with the altruistic punish-
+ers in the population (the AP + D phase in figure.2).
+It is well established that altruistic punishment together
+with network reciprocity promotes cooperation even in
+the presence of antisocial punishment or second-order
+free-riders when the cost-to-fine ratio of punishment is
+low
+[14, 21, 22].
+The results of this study confirmed
+this conclusion. If b is sufficiently large, altruistic pun-
+ishment loses its effectiveness in sustaining prosocial be-
+havior, and defectors dominate the entire population for
+negative ϵ (the D phase in figure.2). When exit options
+are added, this undesirable outcome is solved and leads
+to three possible outcomes. These outcomes can be either
+(i) the coexistence of AP, D and E (the AP +D+E phase
+in figure.2), (ii) the coexistence of NC, D, and E (the
+NC +D +E phase in figure.2), or (iii) the bi-stable state
+between these two types of coexistences (the B phase in
+figure.2). When the cost-to-fine ratio of punishment is
+relatively large, the exiters sustain cooperation in a net-
+worked population in that it facilitates its coexistence
+of two different routes for altruistic punishers and non-
+punishing cooperators, but interestingly, these two types
+of cooperators cannot coexist in the networked popula-
+tion.
+To gain a better understanding of how these actors
+coexist in the population, the evolution features of the
+fractions of each actors was examined and the results
+are presented in figure.3.
+In the bi-stable phase, it is
+the cooperators (altruistic punishers or non-punishing
+cooperators) start giving way to the defectors and with
+fewer cooperators around, defectors then giving way to
+the exiters. With large numbers of exiters, both the al-
+truistic punishers and non-punishing cooperators com-
+pete for the exiters as they can only survive by adhering
+to the exiters. The described phenomenon is the cyclic
+dominance in which these actors dominate one another.
+Here, the cyclic dominance routes can be either (i) al-
+truistic punishers that dominate exiters, who dominate
+defectors, who in turn dominate the altruistic punishers;
+or (ii) non-punishing cooperators that dominate the ex-
+iters, who dominate the defectors, who then dominate
+the non-punishing cooperators.
+As a key mechanism,
+researchers have verified the efficiency of cyclic domi-
+nance in sustaining bio-diversity or promoting cooper-
+ation [42, 43]. Although we started with random initial
+conditions, the evolutionary outcomes are different by
+implementing more independent simulations under same
+parameter combinations. For example, in the NC+D+E
+attractor (figure.3A), the fraction of altruistic punishers
+is temporarily much larger than that of non-punishing co-
+operators at around 100th step, then the faction of altru-
+istic punishers gradually decreases until it is eliminated
+and the fraction of second-order free riders increases un-
+til it reaches a stable state. However, in the AP + D + E
+attractor (figure.3B), the fraction of altruistic punishers
+is always comparable to that of non-punishing cooper-
+ators up to around 1000th step, after this critical time
+step, the fraction of non-punishing cooperators gradually
+decreases until it is eliminated, and altruistic punishers
+gradually increase to reach a stable state. Thus, it is the
+initial distributions of the actors which determines the
+
+1.0
+0.8
+E
+3
+Exit's payoff,
+0.6
+0.4
+NC+D+E
+All C
+AP+D+E
+0.2
+AP+D
+B
+0.0
+D
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+Temptation, b6
+FIG. 3.
+Time dependence of actor abundances exhibits complicated evolutionary dynamics.
+In the bi-stable
+phase, starting from random initial conditions, small incentives to exit option lead the system to either NC + D + E or
+AP + D + E attractor but the coexistence of these four actors is not possible. During the evolution, if the abundance of
+altruistic punishers in the initial stage is much larger than that of the non-punishing cooperators, then altruistic punishers are
+eliminated and non-punishing cooperators coexist with defectors and exiters through cyclic dominance (figure.3A). However,
+if the abundance of altruistic punishers in the initial stage is comparable to that of non-punishing cooperators, then the
+non-punishing cooperators are eliminated and altruistic punishers coexist with defectors and exiters through cyclic dominance
+(figure.3B). Larger incentives to exiters turn the bi-stability to monostability and the evolutionary outcomes are determined
+by the incentives that were presented to exiters. The parameters were fixed as b = 1.8, ϵ = 0.05 (top rows), ϵ = 0.2 (bottom
+left), and ϵ = 0.4(bottom right).
+fate of altruistic punishers and non-punishing coopera-
+tors.
+The phenomenon of bi-stability disappears by increas-
+ing the incentives for exiters.
+Evolutionary dynamics
+lead to either a NC + D + E phase or a AP + D + E
+phase depending on the incentives for exiters. Although
+both altruistic punishers and non-punishing cooperators
+can dominate exiters when the fraction of exiters reaches
+its peak. However, it is non-punishing cooperators who
+dominate the exiters when the incentives for exiters are
+intermediate, ϵ = 0.2.
+Altruistic punishers lose when
+in indirect competition with the non-punishing cooper-
+ators and it is eliminated with simulation proceeds. Fi-
+nally, the non-punishing cooperators coexist with the de-
+fectors and exiters through cyclic dominance in the net-
+worked population(figure.3C). If the incentives for exiters
+are larger, ϵ = 0.4, it is the altruistic punishers start to
+dominate the exiters, and the non-punishing cooperators
+cannot exceed the exiters and is eventually eliminated.
+Finally, the altruistic punishers coexist with defectors
+and exiters through cyclic dominance in the system (fig-
+ure.3D).
+To understand the quantitative power relationships at
+the equilibria abundances of these actors, we present the
+two representative cross sections of the phase diagram in
+figure.4. Along the vertical transect of the ϵ − b phase
+plane, figure.4A shows the stationary fractions of the four
+competing actors in dependence on the exit payoff ϵ at
+b = 1.8. In the traditional weak prisoner’s dilemma game
+with only cooperators and defectors, a high temptation
+leads to the complete dominance of defectors and the net-
+work reciprocity loses its efficiency to support the coexis-
+tence of cooperators and defectors [44]. Although adding
+altruistic punishment in the weak prisoner’s dilemma
+game can avoid this unfavorable outcome, its efficiency
+to decrease defection is at the expanse of social welfare.
+
+A
+B
+B phase: NC+D+E attractor
+B phase: AP+D+E attractor
+1.0
+1.0
+= 0.05 
+0.8
+0.8
+= 0.05 -
+Fractions
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+10-2 10-1 100101 102 103
+104
+10-2
+ 10-1 100 101102 103104
+C
+D
+NC+D+E phase
+AP+D+E phase
+1.0
+1.0
+AP
+= 0.2
+8= 0.4
+0.8
+0.8
+NC
+Fractions
+0.6
+D
+0.6
+E
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+2 10-1 100 101 102 103 104
+10-2 10-1 100 101 102 103 104
+10-2
+time steps
+time steps7
+FIG. 4. Power relations between altruistic punishers, second-order free riders, defectors, and exiters exhibiting
+complicated equilibra. A. Along the vertical transect of ϵ − b phase at b = 1.8. When ϵ ≲ 0.06, the networked population
+falls into the bi-stable state between the coexistence type of altruistic punishers, defectors, and exiters and the coexistence
+type of second-order free riders, defectors and exiters. In the range 0.06 ≲ ϵ ≲ 0.16, altruistic punishment outcompetes the
+second-order free riders, and coexists with the defectors and exiters. Whereas, in the range 0.16 ≲ ϵ ≲ 0.17, there is narrow
+dominance of the bi-stable state. In the range 0.17 ≲ ϵ ≲ 0.25, the second-order free riders outcompete the altruistic punishers,
+and coexist with the defectors and exiters. When 0.25 ≲ ϵ ≲ 0.51, the coexistence of altruistic punishers, defectors, and exiters
+again dominates the population. Finally the eixters dominate the population when 0.51 ≲ ϵ. B. Along the horizontal transect
+of ϵ−b phase plane at ϵ = 0.2, the effectiveness of altruistic punishment together with network reciprocity is sufficient to secure
+prosocial behavior when b ≲ 1.29. With increasing b, altruistic punishment loses its efficiency to sustain prosocial behavior,
+and adding exit option enables the networked population to first enter a coexistence state of altruistic punishers, defectors, and
+exiters in the temptation range of 1.29 ≲ b ≲ 1.73, and reaches a coexistence state between second-order free riders, defectors,
+and exiters when b ≳ 1.73.
+That is the decreasing defection can be realized only if
+the cost-to-fine ratio of altruistic punishment is relatively
+low, i.e., small punishment cost γ or large punishment
+fine β [5, 20–22]. If the cost-to-fine ratio of altruistic pun-
+ishment is relatively large, the altruistic punishment to-
+gether with network reciprocity cannot provide sufficient
+benefit for cooperators, and the complete dominance of
+defectors is still as per the Nash equilibrium. Adding the
+exit option to the weak prisoner’s dilemma game with
+altruistic punishment changes the equilibrium dramati-
+cally even if the conditions to support cooperation for
+altruistic punishment are unfavorable.
+When exit is costly (ϵ < 0), the defectors dominate the
+whole population (the D phase in figure.2). As shown
+in figure.4A, if the incentives to exiters are small but
+positive, the D phase gives way to the B phase, where
+the system converges to either the AP +D +E attractor
+or the NC +D +E attractor depending on the results of
+the indirect competition between the altruistic punishers
+and non-punishing cooperators.
+By further increasing
+the ϵ, the NC + D + E phase is reached at ϵ ≈ 0.17, and
+there are two narrow strips that AP + D + E phase and
+B phase can dominate separately during this increment.
+The AP + D + E phase dominates in the range 0.06 ≲
+ϵ ≲ 0.16, and the B phase is short-lived again in the
+range 0.16 ≲ ϵ ≲ 0.17. As ϵ continues to increase, the
+NC + D + E phase gives way to AP + D + E phase
+via discontinuous phase transition at ϵ ≈ 0.25. When
+incentives to exiters are sufficiently large, the AP +D+E
+phase is finally replaced by the E phase at the critical
+point ϵ ≈ 0.51.
+Figure.4B shows the horizontal transect of ϵ − b at
+ϵ = 0.2, it also reveals the power relations between these
+competing actors, but it is dependent on the temptation
+level, b. When b is small, 1 ≤ b ≲ 1.29, the altruistic pun-
+ishment together with the network reciprocity are able to
+support prosocial behavior. When 1 ≤ b ≲ 1.23, the al-
+truistic punishers can completely eliminate the defectors,
+the elimination of the defectors also negatively affects the
+exiters, and thus altruistic punishers coexist with non-
+punishing cooperators as they cannot be distinguished in
+the absence of defectors. The All C phase gives way to
+the AP + D phase through continuous phase transition.
+Although the advantages of cooperators decreases with
+
+A
+B
+NC+D+E
+B
+E
+AP+D+E
+NC+D+E
+1.0
+1.0
+0.8
+0.8
+AP
+口
+ractions
+NC
+0.6
+0.6
+ID
+-E
+0.4
+0.4
+M
+0.2
+0.2
+0.0
+0.0
+15888888888
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+Exit pay-off, 
+temptation, b8
+FIG. 5. Evolutionary snapshots reveal the detailed dominance modes between all actors. Shown are evolutionary
+snapshots at different time steps (column) and for different temptations for defection (rows). When the temptation is small
+(top row), both altruistic punishers and second-order free riders dominate the exiters, who take over the defectors. However,
+the decrease of exiters is much fast than its increase, and they are eliminated first. The defectors are then eliminated by the
+altruistic punishers, and finally the altruistic punishers coexist with second-order free riders in the population, and these two
+actors cannot separately be distinguished. When the temptation is larger (second row), the fate of exiters is the same as in the
+first row, however, the larger temptation leads more competitive defectors. Therefore, instead of completely dominating the
+defectors, the altruistic punishers coexist with defectors who replace the second-order free-riders until second-order free-riders
+they are eliminated. When the temptation is even larger (third row), more competitive defectors can encroach on both, the
+altruistic punishers and second-order free riders can only survive when they adhere to exiters. The indirect competition between
+altruistic punishers and second-order free riders with exiters determine the outcome for these two actors. Compared with non-
+punishing cooperators, altruistic punishers have greater fitness when compared to defectors and have greater probability to
+endure, therefore non-punishing cooperators are eliminated, and altruistic punishers coexist with defectors and exiters through
+cyclic dominance. When the temptation is at its largest (bottom row), exiters dominate and non-punishing cooperators have
+a larger probability to endure than altruistic punishers as it avoids the cost of punishment. Finally altruistic punishers are
+eliminated and non-punishing cooperators coexist with defectors and exiters. Results were obtained with ϵ = 0.2 after the
+30000th step to generate the final snapshots (rightmost column). The intermediate snapshots (second to fourth columns) were
+taken at different time steps across rows to ensure that the figure as illustrative as possible.
+increasing b, the cooperators who punish defectors gain
+a greater advantage when compared against defectors,
+and thus network reciprocity supports the coexistence of
+altruistic punishers and defectors in this instance. If the
+conditions to support cooperation with altruistic punish-
+ment are unfavorable, adding an exit option can promote
+the system to the AP +D+E phase when b ≲ 1.73. How-
+ever, with increasing b, the AP + D + E phase gives way
+to the NC + D + E phase through discontinuous phase
+transition at the critical point, b ≈ 1.73.
+To reexamine the evolutionary dynamics and further
+check the indirect competition between altruistic punish-
+ers and non-punishing cooperators in both spatial and
+temporal dimensions,. We plotted the evolutionary snap-
+shots for varying b at ϵ = 0.2, and the results are pre-
+sented in figure.5. When the temptation is small (top row
+in figure.5), the exiters were eliminated first by altruistic
+punishers and non-punishing cooperators, and the de-
+fectors experienced the same fate shortly after. The al-
+truistic punishers coexist with non-punishing cooperators
+eventually as they cannot be distinguished and the sys-
+tem falls into frozen state. A larger temptation makes the
+defectors more competitive (second row in figure.5), and
+instead of being eliminated by the altruistic punishers,
+
+Temptation, b
+NC
+D
+b=1.04
+AP
+E
+b=1.25
+b=1.5
+b=1.8
+timesteps9
+FIG. 6. Initial conditions determine the outcome of
+altruistic punishers and non-punishing cooperators in
+the bi-stable phase. Shown are the evolutionary outcomes
+after implementing 100 independent simulations for each pa-
+rameter combination under four different initial conditions.
+The initial conditions were (i) 97% of players were initially
+assigned as AP, (ii) 97% of players were initially assigned as
+NC, (iii) 97% of players were initially assigned as D, and (iv)
+97% of players were initially assigned as E. The rest of the
+other action types were assigned to the other players with
+equal probability in these different initial conditions. Param-
+eters were fixed as b = 1.8, from left to right, ϵ = 0.05, 0.4, 0.6,
+respectively.
+they can coexist. However, the coexistence of defectors
+cannot ensure the survival of exiters, who are eliminated
+in situations with small temptation. The non-punishing
+cooperators are eliminated by defectors and finally, the
+altruistic punishers coexist with defectors in the popula-
+tion. When the temptation is even larger, the coexistence
+of defectors and altruistic punishers was no longer pos-
+sible, instead, defectors can invade both altruistic pun-
+ishers and non-punishing cooperators. The competitive
+defectors allow for the survival of exiters. In turn, altru-
+istic punishers and non-punishing cooperators can sur-
+vive by adhering to the survived exiters. It is therefore,
+both altruistic punishers and non-punishing cooperators
+can coexist with defectors and exiters through different
+cyclic dominance routes.
+However, these two types of
+cyclic dominance cannot coexist in the population, and
+the indirect competition to the territories of exiters be-
+tween altruistic punishers and non-punishing cooperators
+determine the outcome of the competitors. Competitive
+defectors more easily negatively affexct non-punishing co-
+operators than altruistic punishers (third row and second
+column in figure.5), and therefore, non-punishing cooper-
+ators are eliminated first, and the altruistic punishers, de-
+fectors, and exiters coexist within the population. When
+the temptation is the largest (bottom row in figure.5),
+defectors are the most competitive, altruistic punishers
+and non-punishing cooperators are exploited by defectors
+at almost the same speed, and the exiters dominate by
+eliminating the defectors. In the indirect competition of
+exiters with the non-punishing cooperators, the altruis-
+tic punishers loses its advantages due to the existence of
+punishment cost, and non-punishing cooperators coexist
+with defectors and exiters.
+Our results have shown that by adding an exit option
+results in the bi-stable dynamics and it is the initial dis-
+tribution of actors determines the outcome of altruistic
+punishers and non-punishing cooperators. It is generally
+accepted that the initial conditions are crucial for evolu-
+tionary outcomes in agent-based models [45]. We further
+assessed whether the initial fractions of actors is a poten-
+tial reason that the system exhibits bi-stability. Figure.6
+presents the evolutionary outcomes with ϵ = 0.05, 0.4,
+and 0.6 under four different initial conditions. The four
+different conditions are: (i) 97% of players were initially
+assigned as AP, (ii) 97% of players were initially assigned
+as NC, (iii) 97% of players were initially assigned as D,
+and (iv) 97% of players were initially assigned as E. The
+other players were assigned one of the other three actions
+with equal probability in these conditions. The results
+were obtained by implementing 100 independent simula-
+tions. We found that when ϵ = 0.05 (left column in fig-
+ure.6), the evolutionary outcome was always AP +D+E
+if the majority of players initially had action AP or action
+D. However, if the majority of players initially had NC
+action, then the system reached the attractor NC+D+E
+with 95% probability. If the majority of players were E,
+then the system reached the attractor AP + D + E or
+NC +D+E with 36% and 62% probability, respectively.
+Larger incentives to exiters switched the bi-stability to
+monostability (middle and right column in figure.6). In
+the monostability state, evolutionary dynamics lead to
+either the AP + D + E or the E phase depending on the
+incentives to the exiters, and evolutionary outcomes are
+independent on the initial conditions. The finite-size ef-
+fects are a potential pitfall that may generate misleading
+results when implementing agent-based models in struc-
+tured populations [45]. Thus, it is crucial to choose a
+sufficiently large network size or to employ the method of
+subsystem solutions to avoid this potential issue [46, 47].
+It is noteworthy that the system has 23% probability to
+fall into the full E phase when most players initially had
+D action at ϵ = 0.4 (middle column in figure.6).
+We
+do believe that the counterintuitive E phase is the prod-
+uct of the finite-size effect, and the pure AP + D + E
+phase can be expected as long as a larger network size
+was implemented.
+DISCUSSION
+To discuss, we have shown that by adding an exit op-
+tion to the two-stage prisoner’s dilemma game results in
+complicated dynamics. Particularly, in the infinite and
+well-mixed population, it was observed that exiters pro-
+
+NC+D+E
+E
+AP+D+E
+62%
+100%
+100%
+E
+36%
+1
+1
+1
+2%
+/
+1
+1
+-
+I
+100%
+77%
+100%
+D
+23%
+I
+1
+1
+1
+/
+95%
+1
+100%
+100%
+NC
+/
+5%
+1
+1
+1
+1
+/
+100%
+100%
+I
+100%
+AP
+1
+I
+1
+--
+1
+= 0.05
+=0.4
+=0.610
+vide little benefit to cooperation. When the effectiveness
+of altruistic punishment is sufficient to support cooper-
+ation, adding an exit option turns the bi-stable equilib-
+rium between the mixed AP −NC and pure D to another
+bi-stable equilibrium, whereby the mixed AP − NC and
+pure E coexists (panel C and D in the figure.A2). When
+altruistic punishment itself cannot establishes coopera-
+tion, the monomorphic defecting equilibrium is replaced
+by the monomorphic exiting equilibrium (panel A and B
+in the figure.A2). In the finite and well-mixed popula-
+tion, although the availability of exit options maintains
+the survival of both altruistic punishers and second-order
+free riders through two types of cyclic dominance, the
+altruistic punishers never dominate the population. In
+contrast with the well-mixed populations, combining the
+exit option with network reciprocity produces greatly dif-
+ferent outcomes. We determined that the domination of
+altruistic punishment is possible in a networked popula-
+tion. Altruistic punishers can coexist with defectors and
+exiters through cyclic dominance in a majority of the ϵ−b
+phase plane. When the temptation is large, b ≳ 1.71, ex-
+iters enable the survival of second-order free riders. De-
+pending on the incentives to exiters, the system also fall
+into a bi-stable phase or single NC+D+E phase. There-
+fore the exit option is certainly not a panacea in solving
+social dilemmas.
+Previous studies have shown that introduced voluntary
+participation is capable of establishing altruistic punish-
+ment in both finite and infinite populations [6–10]. In the
+infinite population, evolutionary dynamics can result in
+either a Nash equilibrium of punishing and non-punishing
+cooperators or to an oscillating state without punish-
+ers [6].
+If a single cooperator (either a non-punishing
+cooperator or a punisher) can participate in the game,
+and a punisher can punish the non-punishing cooperator
+even in the absence of defectors, the evolutionary dynam-
+ics result in the stable coexistence of altruistic punishers
+and non-punishing cooperators [9]. In a finite population,
+with the assistance of loners, altruistic punishers can pre-
+vail and even dominate the whole population for most
+of the time when mutations are rare [8]. If loners can
+escape punishment, altruistic punishment prevails even
+under the threat of anti-social punishment [11]. Exiters
+produce outcomes that differ greatly from these in lon-
+ers. In the infinite and well-mixed population, adding
+an exit option can also result in a bi-stable outcome, in
+which the Nash equilibrium can be either the coexistence
+of altruistic punishers and non-punishing cooperators or
+a monomorphic exiting equilibrium.
+However, this bi-
+stable outcome is only possible when the punishment it-
+self is sufficient to maintain cooperation, otherwise, the
+bi-stable outcome can be replaced with a monomorphic
+exiting equilibrium. In other words, exiters just simply
+destabilize the defectors and eventually replaces them in
+the infinite population. In the finite population, although
+exiters allow the survival of altruistic punishment when
+the exiter’s payoff is moderate, altruistic punishers never
+dominate the whole population (e.g. figure.1A). The di-
+rect comparison between exiters and loners in a finite
+and infinite population lead us to conclude that loners
+are more effective than exiters in supporting the preva-
+lence of altruistic punishment.
+The effectiveness of altruistic punishment is not only
+challenged by second-order free riders but also by anti-
+social punishment. It has been experimentally reported
+that the existence of antisocial punishment is widespread
+in different human cultures [48–50]. Recent theoretical
+studies have shown that the existence of antisocial pun-
+ishment can prevent the successful coevolution of pun-
+ishment and cooperation [51, 52]. Furthermore, if pun-
+ishment is available for loners, punishment does not in-
+creases cooperation and altruistic punishment becomes a
+self-interested tool for protecting itself against potential
+competitors [53]. As discussed above, exiters are a po-
+tential spiteful punishment as it harms both cooperators
+and defectors, while loners generate a small-but-positive
+payoff for its opponent. This tiny difference leads to to-
+tally different equilibrium in a one-shot game. Exiters
+can destabilize defectors and finally replace them, while
+loners can sustain cooperation through cyclic dominance.
+If we extend our model by considering all punishment sets
+where actors can be punished by each other and exiters
+cannot escape potential punishment by both cooperators
+and defectors. By restricting the analysis to a one-shot
+game, we determine how this setup influenced the stabil-
+ity of punishment, and whether and how this setup gen-
+erates outcomes that differ from that of loners. These
+undoubtedly invite future considerations.
+Exiters established the prevalence of altruistic punish-
+ers and eliminated the second-order free riders when it
+adheres to network reciprocity in a certain parameter
+range (e.g. figure.2 and figure.4). However exiters allow
+for the survival of second-order free riders, who can not
+only survive, but also dominate the population in some
+certain areas of the phase plane. The robustness of this
+finding needs to be verified in a human behavior exper-
+iment. Human behavior experiments may generate con-
+trasting or surprising outcomes with theories on many is-
+sues. Scale-free topology, for example, is often recognized
+theoretically as an optimal structure for the survival of
+cooperation, however, this argument cannot be verified
+by experiment and the cooperation level among humans
+cannot exceed the level established in the lattice [54].
+Similarly, although strong reciprocity theorists believe
+that humans are inherently altruistic and cooperators
+will sacrifice their personal interests to (i). achieve fair
+outcomes and to (ii). punish non-cooperators[55, 56], this
+theory cannot be confirmed by experiments. Yamagishi
+et al. performed large scale human behavior experiments
+and found that there was no correlation between the ten-
+dencies to reject unfair offers in the ultimate game and
+tendencies to exhibit prosocial behavior in other games
+[57]. Although Yamagishi’s finding was challenged due
+to its insufficient sample size, Egloff et al. further con-
+firmed that there was indeed no correlation between pos-
+itive and negative reciprocity through analyzing the pri-
+
+11
+vate household data from the Socio-Economic Panel of
+the German Institute for Economics Research [58].
+A
+recent experimental work is of direct relevance for our
+model. Introducing punishment into networks has been
+proven to be an efficient method to promote cooperation
+theoretically [14, 21–23]. However, in a recent large-scale
+human behavior experiment, it was concluded that the
+introduced peer punishment did not promote coopera-
+tion in structured populations, and instead diminished
+the benefits of network reciprocity [59].
+Although we
+have shown that exiters support the dominance of altru-
+istic punishment when it adheres to network reciprocity,
+human behavior experiments are needed to further verify
+our theory.
+ARTICLE INFORMATION
+Acknowledgements.
+We thank Prof.
+Dr.
+Marko
+Jusup for valuable discussions. This research was sup-
+ported by the National Science Fund for Distinguished
+Young Scholars (grants no. 62025602). We also acknowl-
+edge support from (i) a JSPS Postdoctoral Fellowship
+Program for Foreign Researchers (grant no.
+P21374),
+and an accompanying Grant-in-Aid for Scientific Re-
+search from JSPS KAKENHI (grant no. JP 22F31374),
+and the National Natural Science Foundation of China
+(grant no. 11931015) to C. S. as a co-investigator, (ii) the
+National Natural Science Foundation of China (grants
+no. 11931015, 12271471 and 11671348) to L. S., (iii) Na-
+tional Natural Science Foundation of China (grants no.
+U22B2036, 11931015), Key Technology Research and De-
+velopment Program of Science and Technology-Scientific
+and Technological Innovation Team of Shaanxi Province
+(Grant No. 2020TD-013) and the XPLORER PRIZE.
+to Z. W, and (iv) the grant-in-Aid for Scientific Research
+from JSPS, Japan, KAKENHI (grant No. JP 20H02314)
+awarded to J. T.
+Author contributions.
+C. S. and L. S. conceived re-
+search.
+C. S. and Z. S. performed simulations.
+All co-
+authors discussed the results and wrote the manuscript.
+Conflict of interest.
+Authors declare no conflict of in-
+terest.
+Appendix
+STABILITY ANALYSIS OF THE EQUILIBRIA IN
+INFINITE AND WELL-MIXED POPULATION
+Solving
+Eq.7,
+we
+obtain
+12
+equilibrium
+points:
+(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (ϵ, 0, 0, 1 −
+ϵ),
+(0, ϵ, 0, 1 − ϵ),
+(x, 1 − x, 0, 0),
+(x, ϵ − x, 0, 1 −
+ϵ), ( −1+b
+β
+, 1−b+β
+β
+, 0, 0), (
+ϵ
+b−β , 0, ϵ−β−ϵβ
+(βϵ)γ , 1 − ϵ(γ+1+β−b)
+(b−β)γ
+),
+( (−1+b)ϵ
+β
+, ϵ−β+βϵ
+β
+, 0, 1−ϵ), (
+γ
+1−b+β+γ , 0,
+−1+b−β
+−1+b−β+γ , 0). To
+examine the stability of these equilibria, we calculate the
+0.0
+5.0x104
+1.0x105
+1.5x105
+2.0x105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractions
+time steps
+ AP
+ NC
+ D
+ E
+A
+B
+Fractions
+FIG. A1. Numerical simulation further demonstrates
+that the survival of altruistic punishment is due to the
+cyclic dominance between altruistic punishers, defec-
+tors, and exiters. A. Defectors take over the whole popu-
+lation even if altruistic punishers initially dominate the pop-
+ulation when the exiters’ payoff is negative.
+B. Small but
+positive exiters’ payoff enables the coexistence of altruistic
+punishers, non-punishing cooperators, defectors, and exiters
+through cyclic dominance. If defectors initially dominate the
+population, the mutated exiters invade the defectors, and af-
+ter transient dynamics, the defectors finally give way to the
+exiters.
+When exiters dominate the population, altruistic
+punishment is less costly and cooperating is more valuable
+than exiting, and thus altruistic punishers take over the whole
+population. Thereafter, non-punishing cooperators dominate
+altruistic punishers and take over the whole population since
+altruistic punishers are less valuable than non-punishing coop-
+erators. This proceeds until the dominance of non-punishing
+cooperators gives way to defectors again. The Parameter val-
+ues are b = 1.5, β = 0.3, γ = 0.1, µ = 0.001, s = 0.2, ϵ = −0.2
+(A) and ϵ = 0.2 (B).
+eigenvalues of Jacobian matrix:
+J =
+�
+��
+∂f(x,y,z)
+∂x
+∂f(x,y,z)
+∂y
+∂f(x,y,z)
+∂z
+∂g(x,y,z)
+∂x
+∂g(x,y,z)
+∂y
+∂g(x,y,z)
+∂z
+∂h(x,y,z)
+∂x
+∂h(x,y,z)
+∂y
+∂h(x,y,z)
+∂z
+�
+�� .
+(A1)
+Then we have the following conclusion.
+Theorem 1. When b < 1 + β, and ϵ < 0, the equi-
+librium points (x∗, 1 − x∗, 0, 0) and (0, 0, 1, 0) are stable,
+while the rest of others are unstable; When b < 1 + β,
+and ϵ > 0, the equilibrium points (x∗, 1 − x∗, 0, 0) and
+(0, 0, 0, 1) are stable, and the others are unstable; When
+b ≥ 1 + β, only the equilibrium point (0, 0, 1, 0) is stable,
+and the rest of others are unstable. When ϵ > 0, only
+the equilibrium point (0, 0, 0, 1) is stable, and the rest of
+others are unstable.
+Proof. (1). For K1: (x, y, z, w) = (1, 0, 0, 0), the Jacobian
+
+12
+FIG. A2. Adding exit option destabilizes defection regardless of whether altruistic punishment can establish
+cooperation in an infinite population. When the cost-to-effect ratio of altruistic punishment is insufficient to establish
+cooperation (top row), b − β > 1, the monomorphic defecting equilibrium is replaced by the monomorphic exiting equilibrium
+for positive values of ϵ. When the cost-to-effect ratio of altruistic punishment is capable of establishing cooperation (bottom
+row), b − β < 1, the bi-stable equilibrium of the mixed altruistic punisher and non-punishing cooperator equilibrium and the
+monomorphic defecting equilibrium is replaced by the other bi-stable equilibrium between the mixed altruistic punisher and
+non-punishing cooperator equilibrium and the monomorphic exiting equilibrium for positive values of ϵ. The dashed line on the
+AP − NC edge indicates that all the points on this edge are unstable. The filled black circles, filled gray circles, and unfilled
+circles represent stable fixed points, saddle points, and unstable points, respectively.
+The parameters values are β = 0.3,
+γ = 0.1, ϵ = −0.2 (left column), ϵ = 0.2 (right column), b = 1.5 (top row), and b = 1.2 (bottom row).
+matrix J1 is
+J1 =
+�
+�
+−1 + ϵ −1 + ϵ −b + β + ϵ
+0
+0
+0
+0
+0
+−1 + b − β
+�
+�
+(A2)
+and its corresponding eigenvalues are
+{λ1, λ2, λ3} = {0, −1 + b − β, −1 + ϵ}.
+(A3)
+When b > β + 1, K1 is unstable because −1 + b − β is a
+positive eigenvalue. Otherwise, there is at least one zero
+eigenvalue. Thus, we use the center manifold theorem to
+analyze the stability of K1. Using b < β + 1 as an ex-
+ample. First, there is an invertible matrix whose column
+elements are the eigenvectors of J1
+P =
+�
+�
+−1 −1 1
+1
+0
+0
+0
+1
+0
+�
+�
+(A4)
+and J1 can be diagonalized as
+P −1J1P =
+�
+�
+0
+0
+0
+0 −1 + b − β
+0
+0
+0
+−1 + ϵ
+�
+� .
+(A5)
+Then change of variable:
+�
+�
+x1
+y1
+z1
+�
+� = P −1
+�
+�
+x
+y
+z
+�
+� =
+�
+�
+y
+z
+x + y + z
+�
+�
+(A6)
+
+A
+B
+D
+E
+D
+D
+E
+D
+O
+AP
+NC
+AP
+NC
+D
+D
+c
+D
+D
+E
+D
+D
+E
+D
+C
+Q
+C
+AP
+NC
+AP
+NC
+D
+D13
+and the system becomes
+˙x1 =g(z1 − x1 − y1, x1, y1)
+=x1((1 − x1)(−ϵ − y1 + z1)−
+x1(−ϵ + bx1 + (b − β)(−x1 − y1 + z1))−
+y1(−ϵ + bx1 + (b − β)(−x1 − y1 + z1)))
+˙y1 =h(z1 − x1 − y1, x1, y1)
+=y1(−x1(−ϵ − y1 + z1) − y1(−ϵ − y1 − x1y1 + z1)+
+(1 − y1)(−ϵ + bx1 + (b − β)(−x1 − y1 + z1)))
+˙z1 =f(z1 − x1 − y1, x1, y1) + g(z1 − x1 − y1, x1, y1)
++ h(z1 − x1 − y1, x1, y1)
+=x1(ϵ(−1 + 2x1 + y1) + (−1 + x1 + bx1 + by1)(y1 − z1)−
+β(x1 + y1)(x1 + y1 − z1))+
+y1((−1 + y1)(ϵ + b(y1 − z1) − β(x1 + y1 − z1))+
+x1(ϵ + y1 − z1) + y1(ϵ + y1 + x1y1 − z1))+
+(x1 + y1 − z1)(ϵ + (1 − b + β + x1)y12−
+ϵz1 + (−1 + z1)z1 + y1(1 + x2
+1 + x1(1 + β − z1)+
+(−2 + b − β)z1))
+.
+(A7)
+Put the system into the form
+˙X = AX + F(X, Y )
+˙Y = BY + G(X, Y ) ,
+(A8)
+where X = [x1], Y
+=
+�
+y1
+z1
+�
+, and A = [0], B =
+�
+−1 + b − β
+0
+0
+−1 + ϵ
+�
+, whose eigenvalues have zero and
+negative real parts, respectively. F and G are the func-
+tions of X and Y . They satisfy the condition F (0, 0) =
+0, F ′(0, 0) = O. According to the existence theorem of
+the center manifold, the system has the center manifold
+S = {(X, H(X))|H : R1 → R2}. We define a mapping
+(Mϕ)(X) =ϕ′(X)(AX + F (X, ϕ(X))
+− Bϕ(X) − G(X, ϕ(X))
+(A9)
+Set ϕ(Y ) = O(X2), we obtain
+˙x1 = x1(−ϵ(1 − x1) − x1(−ϵ + bx1 − x1(b − β))) + O(x4
+1)
+(A10)
+Then we define m(x1) = x1(−ϵ(1 − x1) − x1(−ϵ + bx1 +
+−x1(b−β))), and m(x1)′ = 2ϵx1−3x2
+1β−ϵ. Since m(0) <
+0, then x1 = 0 is asymptotically stable. Accordingly, we
+can conclude the point K1 is stable when b < β+1. When
+b = β + 1, K1 is unstable in accordance with the center
+manifold theorem whose derivation process is similar to
+the above analysis.
+(2). For K2: (x, y, z, w) = (0, 1, 0, 0), the correspond-
+ing eigenvalues of J are
+{λ1, λ2, λ3} = {0, −1 + b, −1 + ϵ}.
+(A11)
+K2 is unstable since −1 + b > 0.
+(3). For K3: (x, y, z, w) = (0, 0, 1, 0). Its correspond-
+ing eigenvalues of J are
+{λ1, λ2, λ3} = {0, ϵ, −γ}.
+(A12)
+When ϵ < 0, K3 has an eigenvalue with zero real part and
+other eigenvalues with negative real part. According to
+the center manifold theorem, K3 is stable. When ϵ > 0,
+K3 is unstable because the eigenvalue ϵ has a positive
+real part.
+(4). For K4 : (x, y, z, w) = (0, 0, 0, 1). Its correspond-
+ing eigenvalues of J are
+{λ1, λ2, λ3} = {−ϵ, −ϵ, −ϵ}.
+(A13)
+K4 is stable when ϵ > 0 because all eigenvalues have
+negative real parts. K4 is unstable when ϵ < 0 because
+all eigenvalues have positive real parts.
+(5). For K5 : (x, y, z, w) = (ϵ, 0, 0, 1 − ϵ). Its corre-
+sponding eigenvalues of J are
+{λ1, λ2, λ3} = {0, ϵ(−1 + b − β), ϵ(1 − ϵ)}.
+(A14)
+When 0 < ϵ < 1 or ϵ < 0 and b < 1 + β, K5 is unstable
+because one of its eigenvalues has a positive real part.
+When ϵ < 0 and b ≥ 1+β, K5 has at least one eigenvalue
+with a zero real part and the others have negative real
+parts. According to the center manifold theorem, K5 is
+unstable.
+(6). For K6 : (x, y, z, w) = (0, ϵ, 0, 1 − ϵ). Its corre-
+sponding eigenvalues of J are
+{λ1, λ2, λ3} = {0, ϵ(−1 + b), ϵ(1 − ϵ)}.
+(A15)
+When ϵ > 0, K6 is unstable because eigenvalue ϵ(−1 +
+b) >. When ϵ < 0, there is one eigenvalue with a zero
+real part and two eigenvalues with negative real parts.
+According to the center manifold theorem, K6 is unsta-
+ble.
+(7). For K7 : (x, y, z, w) = (x∗, 1 − x∗, 0, 0). Its corre-
+sponding eigenvalues of J are
+{λ1, λ2, λ3} = {0, ���1 + ϵ, −1 + b − βx∗}.
+(A16)
+When x∗ > b−1
+β , namely b < 1 + β, there is one eigen-
+value with a zero real part and others with negative real
+parts. According to the center manifold theorem, K7 is
+stable. When x∗ < b−1
+β , K7 is unstable because one of
+its eigenvalues has a positive real part.
+(8). For K8 : (x, y, z, w) = (x∗, ϵ − x∗, 0, 1 − ϵ + x∗).
+Its corresponding eigenvalues of J are
+{λ1, λ2, λ3} = {0, ϵ − ϵ2, −ϵ + β − βx∗}.
+(A17)
+When ϵ > 0, K8 is unstable because ϵ − ϵ2 > 0. When
+ϵ < 0, K8 is unstable because −ϵ + β − βx∗ > 0.
+(9).
+For K9 : (x, y, z, w) = ( −1+b
+β
+, 1−b+β
+β
+, 0, 0).
+Its
+corresponding eigenvalues of J are
+{λ1, λ2, λ3} = {0, 0, −1 + ϵ}.
+(A18)
+
+14
+K9 exists only when b < 1 + β. When K9 exists, there is
+one eigenvalue with a negative real part and two eigenval-
+ues with zero real parts. According to the center manifold
+theorem, k9 is unstable.
+(10).
+For K10 : (x, y, z, w) = (
+ϵ
+b−β , 0, ϵ−β−ϵβ
+(b−β)γ , 1 −
+ϵ(γ+1+β−b)
+(b−β)γ
+). Its corresponding eigenvalues of J are
+{λ1, λ2, λ3} =
+{−ϵ(−1 + b − β)
+b − β
+,−ϵ(−1 + b − β)
+b − β
+,ϵ+ ϵ2(−1 + b − β + γ)
+(b − β)γ
+}
+.
+(A19)
+K10 exists when 1 − ϵ(γ+1+β−b)
+(b−β)γ
+) < 1, namely b > β +
+ϵ 1−γ
+γ+ϵ . Then its eigenvalue ϵ + ϵ2(−1+b−β+γ)
+(b−β)γ
+> 0. Thus,
+K10 is unstable.
+(11). For K11 : (x, y, z, w) = ( (−1+b)ϵ
+β
+, ϵ−β+βϵ
+β
+, 0, 1−ϵ).
+Its corresponding eigenvalues of J are
+{λ1, λ2, λ3} = {0, 0, ϵ(1 − ϵ)}.
+(A20)
+K11 exists when ϵ > 0, then eigenvalue ϵ(1 − ϵ) > 0.
+Thus, K11 is unstable.
+(12). For K12 : (x, y, z, w) = (
+γ
+1−b+β+γ , 0,
+1−b+β
+1−b+β+γ , 0).
+Its corresponding eigenvalues of J are
+{λ1, λ2, λ3} =
+{ (1 − b + β)γ
+1 − b + β + γ , (1 − b + β)γ
+1 − b + β + γ , ϵ +
+(−b + β)γ
+1 − b + β + γ }.
+(A21)
+K12 exists when b < 1+β, then eigenvalue (1−b+β)γ
+1−b+β+γ > 0.
+Thus K12 is unstable.
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4dE4T4oBgHgl3EQfBAsA/content/tmp_files/2301.04847v1.pdf.txt

@@ -0,0 +1,630 @@
+Real-time FPGA implementation of the
+Semi-Global Matching stereo vision algorithm
+for a 4K/UHD video stream
+Mariusz Grabowski
+and Tomasz Kryjak
+Embedded Vision Systems Group, Computer Vision Laboratory,
+Department of Automatic Control and Robotics,
+AGH University of Science and Technology, Krakow, Poland
+[email protected], [email protected]
+Abstract. In this paper, we propose a real-time FPGA implementation
+of the Semi-Global Matching (SGM) stereo vision algorithm. The de-
+signed module supports a 4K/Ultra HD (3840 × 2160 pixels @ 30 frames
+per second) video stream in a 4 pixel per clock (ppc) format and a 64-
+pixel disparity range. The baseline SGM implementation had to be mod-
+ified to process pixels in the 4ppc format and meet the timing constrains,
+however, our version provides results comparable to the original design.
+The solution has been positively evaluated on the Xilinx VC707 devel-
+opment board with a Virtex-7 FPGA device.
+Keywords: SGM · FPGA · 4K · Ultra HD · real-time processing · stereo
+vision.
+1
+Introduction
+Information on the 3D structure (depth) of a scene is very important in many
+robotic systems, including self-driving cars and unmanned aerial vehicles (UAVs),
+as it is used in object detection and navigation modules. The depth map can
+be estimated using several different approaches, active: LiDAR (Light Detec-
+tion and Ranging), Time of Flight (ToF) cameras, stereo vision with structured
+lighting; and passive: stereo vision. Stereo vision uses two or more cameras that
+acquire the same scene, but from slightly different points in space. A detailed
+discussion of the advantages and disadvantages of different sensors can be found
+in the work of Jamwal, Jindal, and Singh [1].
+Stereo vision, in its passive variant, is an often used solution in embedded
+systems due to the low price of the equipment, its small size and weight (no
+need for a laser light source, rotating elements or projectors). The accuracy
+of the results obtained with this technology strictly depends on the algorithm
+used to process the acquired images. The methods used can be divided into two
+groups: local and global [2]. In both cases, the key element is to find the same
+pixels in the image captured by the left (usually considered as the base) and
+arXiv:2301.04847v1  [cs.CV]  12 Jan 2023
+
+2
+M. Grabowski et al.
+right camera (reference). Their offset expressed in pixels is referred to as the
+disparity. This value can be easily converted to the distance from the sensors
+using the vision system parameters.
+Global methods introduce appropriate discontinuity penalties in order to
+smooth the disparity map. Their aim is to optimise the energy function de-
+fined over the whole image. By means of global algorithms, much more reliable
+and accurate disparity maps are determined, but the smoothing task is NP-hard
+and algorithms are very computationally demanding, and for this reason they
+are not suitable for implementation in real-time systems.
+It should be also noted that the current dominant trend is depth estimation
+using deep neural networks [3]. However, due to the high computational com-
+plexity, especially for high-resolution video streams, this topic remains outside
+the focus of our present work.
+The SGM (Semi-Global Matching) algorithm was introduced by Hirshm¨uller
+in 2005 [4] and 2008 [5]. It is based on two components: (1) matching at a single
+pixel level with the use of mutual information and (2) approximation of a global,
+two-dimensional smoothness constraint (obtained by combining multiple 1D con-
+straints). The SGM algorithm is an example of an intermediate method between
+local and global approaches for determining disparity maps and is a compromise
+between accuracy and computational complexity. However, using SGM for high-
+resolution images is still challenging. For example, for a resolution of 1920×1080
+pixels at 30 frames per second, an execution of about 2 TOPS (Tera Operations
+Per Second) with memory bandwidth of 39 Tb/s is required to process all pixels
+(2 million) [6].
+In this paper we present an architecture of a stereo vision system with a mod-
+ified SGM algorithm to process a 4K/Ultra HD (3840 × 2160 pixels @ 30 frames
+per second) video stream in 4ppc (pixel per clock) format and its implemen-
+tation in an FPGA (Field Programmable Gate Array) device. The proposed
+modification solves the data dependency problem while not affecting the algo-
+rithm’s accuracy. To the authors’ knowledge, this is the only verified hardware
+implementation of the SGM method for 4K/Ultra HD resolution.
+The reminder of this paper is organised as follows. In Section 2 we present
+basic information about the SGM algorithm. In Section 3 we review the previous
+work on SGM implementation on FPGAs. We describe the proposed method and
+architecture, as well as the evaluation of the algorithm and the hardware im-
+plementation in Section 4. The paper ends with conclusions and future research
+directions.
+2
+The SGM algorithm
+As mentioned in the introduction, the SGM algorithm is an intermediate ap-
+proach between local and global methods for determining disparity maps. Fur-
+thermore, it is possible to implement it in an FPGA, in a pipelined vision system.
+The input to the algorithm is a pair of rectified images. It consists of the
+following steps: calculation of the matching cost, aggregation of the cost (cal-
+
+Real-time FPGA implementation of the SGM stereo vision in 4K
+3
+Fig. 1: Matching cost calculation with the Census transform and the Hamming
+distance metric, with example values.
+culation of the smoothness constraint) and determination of the final disparity
+map.
+In this work, the cost of matching C(p, d) between a pixel p = [px, py]T
+from the base image Ib, and the potentially corresponding pixel (shifted by the
+disparity d in a horizontal line) in the reference image Im, is calculated using
+the Census transform and the Hamming distance measure, as shown in Figure
+1.
+Determining the correspondence between pixels using only the matching cost
+alone can lead to ambiguous and incorrect results. Therefore, an additional global
+condition is proposed in the SGM algorithm, which adds a “penalty” for changing
+the disparity value (i.e, supports the smoothness of the image), by aggregating
+the costs along independent paths.
+Let Lr denote the path in the direction r. The path cost Lr(p, d) is defined
+recursively as:
+Lr(p, d) = C(p, d) + min[Lr(p − r, d),
+Lr(p − r, d − 1) + P1,
+Lr(p − r, d + 1) + P1,
+min
+i
+Lr(p − r, i) + P2]
+− min
+k Lr(p − r, k)
+(1)
+where: C(p, d) is the matching cost, and the second part of the equation is
+the minimum path cost for the previous pixel p − r on the path, taking into
+account the corresponding discontinuity penalty. Two penalties were applied in
+the algorithm, P1 for a 1-level change in disparity and P2 for a larger change.
+Finally, the matching cost is given as:
+S(p, d) =
+�
+r
+Lr(p, d)
+(2)
+The author of SGM recommend aggregation along at least 8 paths, i.e, ver-
+tically, horizontally and diagonally in both directions (cf. Figure 3), although he
+suggests that good results are achieved for the number 16. The disparity map
+Db corresponding to the base image Ib is obtained by selecting for each value p
+the disparity d that corresponds to the minimum cost i.e, mindS(p, d). Optional
+element of the algorithm is the final post-processing: median filtering and map
+consistency check (so called left-right consistency check).
+
+Ib (pxPy)
+Cntx
+Census
+4
+Gen.
+Transform
+1314
+0010
+C(p,d)
+Hamming
+3
+Distance
+(px+d,py)
+11111
+6
+7
+6
+Cntx
+Census
+10
+3
+4
+3
+Gen.
+Transform
+1
+1100
+5 
+24
+M. Grabowski et al.
+Due to the reasonable trade-off between computational complexity and the
+quality of the resulting disparity map, the SGM algorithm has become very
+popular. It is a basic method in the popular OpenCV library and the Computer
+Vision Toolbox of the Matlab software. It also provides an attractive solution
+for hardware implementations in FPGAs.
+3
+Previous work
+The topic of implementing stereo correspondence using FPGAs is very extensive,
+and hence this review is narrowed only to selected articles describing the SGM
+algorithm. Interested readers are referred to the review [7].
+The paper written by Gehrig, Eberli, and Meye in 2009 [8] described an
+SGM architecture for processing images with a resolution of 750 × 480 pixels
+(effectively 340 × 200) @ 27 fps at 64 levels of disparity. It is worth noting that
+this was the first implementation of the SGM method in an FPGA.
+The paper of Hofmann, Korinth, and Koch from 2016 [9] also proposes a hard-
+ware implementation of the SGM algorithm. The architecture features scalability
+and combines coarse-grain and fine-grain parallelisation capabilities. The authors
+performed tests for different configurations and resolutions. For 1920×1080 pix-
+els @ 30 fps and 128 disparity levels, real-time processing was achieved at a clock
+of 130 MHz (VC709 board with Virtex-7 FPGA device).
+In the paper of Zhao et al. from 2020 [10], the authors presented the FP-
+Stereo library, which uses the HLS language and allows the creation of SGM
+disparity calculation modules. The module has been designed in the form of
+an accelerator interfacing with a DMA controller, rather than directly with the
+video stream. For a 300 MHz clock, a resolution of 1242 × 374 pixels and 128
+disparity range, 161 fps were achieved on the ZCU 102 board with the Xilinx
+Zynq UltraScale+ MPSoC device.
+In the latest publications by Shrivastava et al. in 2020 [11] and Lee with Kim
+in 2021 [6], the support for parallel pixel processing has been added to increase
+throughput. In this approach, the main challenge is the presence of an inherent
+data dependency. In the paper from 2020 [11], it is addressed by dependency
+relaxation, i.e, the aggregation is performed on the basis of the pixel k earlier,
+where k is the number of pixels processed simultaneously. The author points out
+that such a solution represents a trade-off between accuracy and throughput.
+In the work from 2021 [6], on the other hand, a different approach is pre-
+sented, in which operations involving the inherent data dependency are per-
+formed not on a single pixel, but on a vector of pixels. This allows the genera-
+tion of disparity maps with very close accuracy to the original SGM algorithm.
+In both solutions, the matching costs are determined based on the Census trans-
+form. In the first publication [11], for images at a resolution of 1280 × 960 pixels
+and disparity range of 64, 322 fps, and in the second [6] for a resolution of
+1920 × 1080 pixels and disparity range of 128, 103 fps were obtained.
+We also propose a solution to the inherent data dependency problem. Our
+architecture is based on estimating the previous pixel aggregation cost on a path
+
+Real-time FPGA implementation of the SGM stereo vision in 4K
+5
+Fig. 2: A general scheme of the proposed SGM disparity estimation system.
+with minimal additional logic needed. That allows us to process images with a 4K
+resolution and also to obtain comparable results to the original SGM algorithm
+without parallelism.
+4
+The proposed hardware implementation
+The aim of our work was to implement a hardware architecture capable of pro-
+cessing a video stream with a resolution of 3840×2160 pixels in real-time (i.e pro-
+cessing 30 frames per second with no pixel dropping). That stream transmitted
+in a 1 pixel per clock format requires a pixel clock frequency of approximately
+250 MHz. Adding to this value i.e, the vertical and horizontal blanking fields,
+the required clock equals about 300 MHz, which is too high for the rather com-
+plicated SGM algorithm. At the bottleneck, cost aggregation calculations take
+more than 10 ns on our platform. So, in order to process the data in the de-
+sired resolution, it is necessary to introduce parallelisation. In this work, a 4ppc
+(pixel per clock) format is used, in which 4 pixels are processed in parallel. This
+allows the pixel clock to be lowered to approximately 75 MHz. However, the use
+of such format has significant implications on the implementation of the SGM
+algorithm, due to the inherent data dependency.
+A general scheme for the proposed vision system is shown in Figure 2. The
+module accepts a synchronised video stream of rectified images, the base IB(p)
+and the reference IM(p) one. Further processing consists of several steps: de-
+termination of the matching cost C(p, d) using the Census transform based
+matching method, calculation of the cost aggregation Lr(p, d), summation of
+the aggregation costs from all directions S(p, d) and disparity determination
+D(p).
+4.1
+Determination of the matching cost
+The 4ppc format does not introduce major complications into the hardware
+architecture of the matching cost determination module, but only increases the
+hardware resource requirements. First, 5×5 contexts are created for both images.
+For the base image, in a given cycle, 4 contexts are created (as implied by the
+4ppc format [12]), and for the reference image this number is increased by the
+disparity range (4 + disp range − 1), so that it is possible to simultaneously
+compare each of the 4 contexts of the base image with all the contexts in the
+disparity range of the reference image. A Census transform is performed on the
+generated contexts, and the contexts are then compared accordingly using the
+Hamming distance metric. The output consists of matching cost vectors.
+
+Ib(p)
+C(p, d)
+Lr(p, d)
+S(p,d)
+Disparity
+D(p)
+Matching Costs
+Costs
+Im(p)
+Sum
+Selection
+Determination
+Aggregation6
+M. Grabowski et al.
+Fig. 3: Cost aggregation paths in SGM.
+4.2
+Cost aggregation
+In the next step, a quasi-global optimisation is performed by aggregating the
+costs for the whole image according to the SGM algorithm. In the current version
+of the module, this is implemented on four paths in the directions 0°, 45°, 90°,
+135°, as shown in Figure 3, which can be processed directly (without additional
+video stream buffering).
+Theoretically, it is also possible to realise the other four directions (180°,
+225°, 270°, 315°), but this would require storing the entire image in external
+RAM, using additional resources of the FPGA device, complex control logic and
+introducing additional latency in image processing.
+In order to calculate the aggregation cost for a given pixel, it is necessary to
+know the value of the aggregation cost for the previous pixel on the path (cf.
+Equations (1) and (2)). For the 45°, 90°, 135° paths, the aggregation costs for the
+pixels in a given line are stored in Block RAM and read out accordingly during
+the processing of the next image line to calculate the costs for the subsequent
+pixels on these paths. The hardware architecture of this computation is shown in
+Figure 4 and follows Equation (1). The grey part is replicated for the entire range
+of disparities (disp range) and performs in parallel and one block of finding the
+minimum value of aggregation costs of the previous pixel on the path minLr(p−
+r) is exploited to calculate the aggregation cost for the current pixel for each
+disparity value in the range.
+For the 4ppc format, the difficulty arises for the 0° path. Using the aggrega-
+tion cost of the previous pixel Lr(p − r, d), which for this path lies in the same
+image line and potentially in the same 4ppc format data vector, results in the
+need to process four pixels in the same clock cycle. In the worst case, for the
+last pixel in the vector, in one clock cycle the data would have to propagate
+through four serially connected aggregation cost calculation units, as in Figure
+4. The critical path would contain 4 minimum modules of size disp range, four
+minimum modules of size 4 and 12 adders/subtractors. For this reason, the cost
+aggregation based on a baseline architecture (i.e, as proposed by the authors of
+SGM) for the 0° path is not feasible for the considered 4K resolution, without
+violating timing constraints.
+It is therefore necessary to propose a new solution for the calculation of the
+aggregation cost for the 0° path. Time constraints require that the new architec-
+ture does not introduce significant additional propagation time and maintains
+
+video stream direction
+45°
+.06
+135°
+。0
+dReal-time FPGA implementation of the SGM stereo vision in 4K
+7
+Fig. 4: Hardware architecture of the aggregation cost calculation unit for path
+r, pixel p and disparity d.
+the approximation assumption of the global smoothness constraint of the SGM
+algorithm.
+In our work, we designed and implemented an architecture with a proposed
+estimation of the aggregation cost value for consecutive pixels based on the
+calculated aggregation cost for the last pixel of the previous 4ppc vector (the
+pixel processed in the previous clock cycle) and the matching costs of the previous
+pixels in the same 4ppc vector.
+For the first pixel in the 4ppc vector, the aggregation cost of the previous
+pixel is available during the calculation (it was calculated for the previous 4ppc
+vector), i.e:
+Lr(p1 − r, d) = Lr(plast, d)
+(3)
+where: Lr(p1 − r, d) is the aggregation cost of the previous pixel relative to the
+first pixel in the 4ppc vector (p1 − r), and Lr(plast − r, d) is the aggregation cost
+of the last pixel in the previous 4ppc vector.
+For the consecutive pixels, we propose an estimation, which is performed
+according to the following Equations:
+L′
+r(p2 − r, d) = Lr(plast, d) + 1
+λ(C(p1, d) − Lr(plast, d))
+L′
+r(p3 − r, d) = Lr(plast, d) + 1
+λ(C(p1, d) + C(p2, d)
+2
+− Lr(plast, d))
+L′
+r(p4 − r, d) = Lr(plast, d) + 1
+λ(
+C(p1, d) + C(p2, d)
+2
++ C(p3, d)
+2
+− Lr(plast, d))
+(4)
+where: L′
+r(p−r, d) is the estimated aggregation cost for the previous pixel relative
+to the pixel p, C(p, d) is the matching cost for a given pixel, and the coefficient
+
+Lr(p -r,d)
+Lr(p - r,d - 1)
+C(p, d)
+P1
+Minimum
+Lr(p,d)
+Lr(p -r,min disp)
+Lr(p -r,d + 1)
+(size: 4) 
+Lr(p - r,min disp + 1)
+P1
+Minimum
+min Lr(p - r
+(size: disp range)
+P2
+Lr(p -r,max disp - 1
+Lr(p - r,max disp )8
+M. Grabowski et al.
+Fig. 5: The architecture for estimating the aggregation cost of the previous pixel
+for each pixel in the 4ppc vector.
+λ may take a value which is a power of two (1, 2, 4, 8, 16, ...). The architecture of
+this solution is shown in Figure 5.
+The algorithm is based on the difference of the matching cost values of the
+previous pixels in a given 4ppc vector with the aggregation cost for the last pixel
+of the previous vector. The aggregation cost estimation architecture consists of
+basic components and introduces an additional delay only by the propagation
+time of the 3 adders/subtractors (critical path for Lr(p4−r, d). Note: multiplica-
+tion/division by a number that is a power of two is only a bit shift and requires
+no delay in the hardware implementation.
+The solution takes into account the matching cost values of all previous pixels
+with the possibility to adjust the impact of the matching cost of previous pixels
+in a given vector by a factor of λ.
+The estimated aggregation costs are then used to calculate the aggregation
+costs according to the architecture in Figure 4. In the work of Shrivastava et al.
+[11] the estimation has been omitted and in the work of Lee and Kim [6] it has
+been solved by the cluster-wise cost aggregation.
+The aggregation costs from all paths are then summed and the disparity is
+calculated. This involves finding the minimum matching cost.
+4.3
+Evaluation of the proposed method
+The accuracy evaluation of the proposed algorithm was performed on a set of
+stereo images from the Middlebury 2014 [13] dataset. We skipped the final post-
+processing to better highlight the differences between the base SGM algorithm
+and the modified version proposed in this paper (SGM 4ppc). The accuracy was
+
+Lr(p1 - r,d)
+Lr(piast -r,d)
+C(p1,d)
+1
+Lr(p2 -r,d)
+Lr(piast -r,d)
+C(p1, d)
+C(p2, d)
+L'r(p3 -r,d)
+Lr(plast -r,d)
+C(p1, d)
+4
+C(p2, d)
+1
+C(p3, d)
+1
+Lr(p4 -r,d)
+X
+-2
+Lr(piast -r, d)Real-time FPGA implementation of the SGM stereo vision in 4K
+9
+(a) Input image – left
+(b) Ground truth
+(c) SGM 4ppc
+(d) Local method based
+on CT
+(e) SGM – 3 paths
+(f) SGM – 4 paths
+Fig. 6: Comparison of output disparity maps for the Motorcycle image in Mid-
+dlebury 2014 dataset: (a) the left input image, (b) the ground truth disparity
+map, (c), (d), (e), (f) estimated disparity maps (on the top) and the error maps
+(on the bottom).
+measured by the ratio of pixels with incorrect disparity value to all pixels of the
+image (all) and also to the non-occluded (noc) pixels (occluded pixels should be
+filled with the Left/Right Check post-processing).
+We compared the proposed method (SGM 4ppc) with the conventional local
+block matching based on the Census transform and the SGM algorithm (also
+
+YAMRMA区X10
+M. Grabowski et al.
+Table 1: Comparison of error rates for the Middlebury 2014 dataset, based on
+all (all) and non-occluded (noc) pixels.
+all
+noc
+Local based on CT
+68.21%
+63,36%
+SGM 3 paths
+38.01%
+28.79%
+SGM 4 paths
+36.27%
+26.88%
+SGM 8 paths
+33.31%
+23.11%
+SGM 4ppc
+36.64%
+27.32%
+based on the Census transform) with 3 and 4 aggregation paths. Figure 6 shows
+sample evaluation results on the Motorcycle images from the Middlebury 2014
+dataset. Table 1 shows the average evaluation results for the entire dataset.
+The accuracy of the proposed method is comparable to the original SGM
+algorithm with 4 paths. The difference between error rates is about 0.4%.
+4.4
+Hardware implementation
+We implemented the proposed stereo vision system on a VC707 evaluation board
+with Xilinx’s Virtex-7 XC7VX485T-2FFG1761C device. We set up a test envi-
+ronment to evaluate the system, with test images sent directly from a PC do the
+board and later displayed on a 4K monitor.
+We compared our solution with previous FPGA implementations of the SGM
+algorithm in Table 2. We used the following metrics: Frames per Second (FPS),
+Million Disparity Estimates per second (MDE/s) and MDE/s per Kilo LUTs
+(Look-Up Tables) (MDE/s/KLUT). First of all, our solution is the only one ver-
+ified in hardware for a 4K/ Ultra HD resolution. We also would like to point out
+that the lower performance in FPS and MDE/s relative to previous work from
+2020 [11] and 2021 [6] is due to the use of an FPGA chip with fewer resources. For
+this work, it was necessary to select a suitable platform to enable image acquisi-
+tion in 4K resolution (i.e, having two high-bandwidth FMCs (FPGA Mezzanine
+Connectors) to which TB-FMCH-HDMI4K modules were attached).
+It is also worth mentioning that the used FPGA technology differs not only
+in the number of resources but also in the performance. To compare: the critical
+path propagation time for the technology used in this paper after synthesis
+is 12.967 ns, but for the Xilinx Virtex UltraScale+ XCVU9P-L2FLGA2104E
+FPGA with the same parameters, it is 8.240 ns (36.45% faster).
+5
+Conclusion
+In this paper, we presented a hardware architecture for an SGM algorithm to
+process a 4K/Ultra HD video stream in real-time. We proposed a solution to
+the inherent data dependency problem. It allowed us to maintain high accuracy
+of the depth map estimation, while making it possible to take advantage of the
+
+Real-time FPGA implementation of the SGM stereo vision in 4K
+11
+Table 2: Comparison with previous FPGA implementations of the SGM algo-
+rithm.
+Image
+Disparity
+Platform
+FPGA
+Throughput
+resolution
+range
+resources
+LUT FF BRAM FPS MDE/s MDE/s/KLUT
+[14]
+1920x1080
+128
+Virtex-7
+195k 217k
+368
+30
+7 963
+40.84
+[15]
+1600x1200
+128
+Stratix-V
+222k 149k
+N/A
+43
+10 472
+47.2
+[11]
+1280x960
+64
+Virtex-7 690T 211k N/A
+641
+322 25 056
+118.6
+[6]
+1920x1080
+128
+Zynq US+
+222k 135k
+252
+103 27 297
+123.0
+New 3840x2160
+64
+Virtex-7 485T 138k 65k
+197
+30
+15 925
+116.2
+4ppc vector format. We implemented the module on a Virtex-7 FPGA platform
+achieving 30 frames per second for a resolution of 3840 × 2160 pixels with 64
+disparity levels.
+In future work, we plan to add more aggregation paths to the algorithm. With
+that, it will be possible to get more accurate results, but at the cost of latency and
+resource usage. We also plan to implement a video stream rectification module.
+Acknowledgements The work presented in this paper was supported by: the
+National Science Centre project no. 2016/23/D/ST6/01389 entitled ”The de-
+velopment of computing resources organization in latest generation of hetero-
+geneous reconfigurable devices enabling real-time processing of UHD/4K video
+stream”, the AGH University of Science and Technology project no. 16.16.120.773
+and the program ”Excellence initiative — research university” for the AGH Uni-
+versity of Science and Technology.
+References
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+

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+page_content='Real-time FPGA implementation of the  Semi-Global Matching stereo vision algorithm  for a 4K/UHD video stream  Mariusz Grabowski  and Tomasz Kryjak  Embedded Vision Systems Group, Computer Vision Laboratory,  Department of Automatic Control and Robotics,  AGH University of Science and Technology, Krakow, Poland  grabowski@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='agh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='pl, tomasz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='kryjak@agh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='pl  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In this paper, we propose a real-time FPGA implementation  of the Semi-Global Matching (SGM) stereo vision algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The de-  signed module supports a 4K/Ultra HD (3840 × 2160 pixels @ 30 frames  per second) video stream in a 4 pixel per clock (ppc) format and a 64-  pixel disparity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The baseline SGM implementation had to be mod-  ified to process pixels in the 4ppc format and meet the timing constrains,  however, our version provides results comparable to the original design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The solution has been positively evaluated on the Xilinx VC707 devel-  opment board with a Virtex-7 FPGA device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Keywords: SGM · FPGA · 4K · Ultra HD · real-time processing · stereo  vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  1  Introduction  Information on the 3D structure (depth) of a scene is very important in many  robotic systems, including self-driving cars and unmanned aerial vehicles (UAVs),  as it is used in object detection and navigation modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The depth map can  be estimated using several different approaches, active: LiDAR (Light Detec-  tion and Ranging), Time of Flight (ToF) cameras, stereo vision with structured  lighting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' and passive: stereo vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Stereo vision uses two or more cameras that  acquire the same scene, but from slightly different points in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' A detailed  discussion of the advantages and disadvantages of different sensors can be found  in the work of Jamwal, Jindal, and Singh [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Stereo vision, in its passive variant, is an often used solution in embedded  systems due to the low price of the equipment, its small size and weight (no  need for a laser light source, rotating elements or projectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The accuracy  of the results obtained with this technology strictly depends on the algorithm  used to process the acquired images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The methods used can be divided into two  groups: local and global [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In both cases, the key element is to find the same  pixels in the image captured by the left (usually considered as the base) and  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='04847v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='CV]  12 Jan 2023  2  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Grabowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  right camera (reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Their offset expressed in pixels is referred to as the  disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' This value can be easily converted to the distance from the sensors  using the vision system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Global methods introduce appropriate discontinuity penalties in order to  smooth the disparity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Their aim is to optimise the energy function de-  fined over the whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' By means of global algorithms, much more reliable  and accurate disparity maps are determined, but the smoothing task is NP-hard  and algorithms are very computationally demanding, and for this reason they  are not suitable for implementation in real-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  It should be also noted that the current dominant trend is depth estimation  using deep neural networks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' However, due to the high computational com-  plexity, especially for high-resolution video streams, this topic remains outside  the focus of our present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The SGM (Semi-Global Matching) algorithm was introduced by Hirshm¨uller  in 2005 [4] and 2008 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' It is based on two components: (1) matching at a single  pixel level with the use of mutual information and (2) approximation of a global,  two-dimensional smoothness constraint (obtained by combining multiple 1D con-  straints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The SGM algorithm is an example of an intermediate method between  local and global approaches for determining disparity maps and is a compromise  between accuracy and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' However, using SGM for high-  resolution images is still challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' For example, for a resolution of 1920×1080  pixels at 30 frames per second, an execution of about 2 TOPS (Tera Operations  Per Second) with memory bandwidth of 39 Tb/s is required to process all pixels  (2 million) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In this paper we present an architecture of a stereo vision system with a mod-  ified SGM algorithm to process a 4K/Ultra HD (3840 × 2160 pixels @ 30 frames  per second) video stream in 4ppc (pixel per clock) format and its implemen-  tation in an FPGA (Field Programmable Gate Array) device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The proposed  modification solves the data dependency problem while not affecting the algo-  rithm’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' To the authors’ knowledge, this is the only verified hardware  implementation of the SGM method for 4K/Ultra HD resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The reminder of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In Section 2 we present  basic information about the SGM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In Section 3 we review the previous  work on SGM implementation on FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We describe the proposed method and  architecture, as well as the evaluation of the algorithm and the hardware im-  plementation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The paper ends with conclusions and future research  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  2  The SGM algorithm  As mentioned in the introduction, the SGM algorithm is an intermediate ap-  proach between local and global methods for determining disparity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Fur-  thermore, it is possible to implement it in an FPGA, in a pipelined vision system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The input to the algorithm is a pair of rectified images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' It consists of the  following steps: calculation of the matching cost, aggregation of the cost (cal-  Real-time FPGA implementation of the SGM stereo vision in 4K  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 1: Matching cost calculation with the Census transform and the Hamming  distance metric, with example values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  culation of the smoothness constraint) and determination of the final disparity  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In this work, the cost of matching C(p, d) between a pixel p = [px, py]T  from the base image Ib, and the potentially corresponding pixel (shifted by the  disparity d in a horizontal line) in the reference image Im, is calculated using  the Census transform and the Hamming distance measure, as shown in Figure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Determining the correspondence between pixels using only the matching cost  alone can lead to ambiguous and incorrect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Therefore, an additional global  condition is proposed in the SGM algorithm, which adds a “penalty” for changing  the disparity value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, supports the smoothness of the image), by aggregating  the costs along independent paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Let Lr denote the path in the direction r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The path cost Lr(p, d) is defined  recursively as:  Lr(p, d) = C(p, d) + min[Lr(p − r, d),  Lr(p − r, d − 1) + P1,  Lr(p − r, d + 1) + P1,  min  i  Lr(p − r, i) + P2]  − min  k Lr(p − r, k)  (1)  where: C(p, d) is the matching cost, and the second part of the equation is  the minimum path cost for the previous pixel p − r on the path, taking into  account the corresponding discontinuity penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Two penalties were applied in  the algorithm, P1 for a 1-level change in disparity and P2 for a larger change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Finally, the matching cost is given as:  S(p, d) =  �  r  Lr(p, d)  (2)  The author of SGM recommend aggregation along at least 8 paths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, ver-  tically, horizontally and diagonally in both directions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Figure 3), although he  suggests that good results are achieved for the number 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The disparity map  Db corresponding to the base image Ib is obtained by selecting for each value p  the disparity d that corresponds to the minimum cost i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, mindS(p, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Optional  element of the algorithm is the final post-processing: median filtering and map  consistency check (so called left-right consistency check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Ib (pxPy)  Cntx  Census  4  Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Transform  1314  0010  C(p,d)  Hamming  3  Distance  (px+d,py)  11111  6  7  6  Cntx  Census  10  3  4  3  Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Transform  1  1100  5  24  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Grabowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Due to the reasonable trade-off between computational complexity and the  quality of the resulting disparity map, the SGM algorithm has become very  popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' It is a basic method in the popular OpenCV library and the Computer  Vision Toolbox of the Matlab software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' It also provides an attractive solution  for hardware implementations in FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  3  Previous work  The topic of implementing stereo correspondence using FPGAs is very extensive,  and hence this review is narrowed only to selected articles describing the SGM  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Interested readers are referred to the review [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The paper written by Gehrig, Eberli, and Meye in 2009 [8] described an  SGM architecture for processing images with a resolution of 750 × 480 pixels  (effectively 340 × 200) @ 27 fps at 64 levels of disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' It is worth noting that  this was the first implementation of the SGM method in an FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The paper of Hofmann, Korinth, and Koch from 2016 [9] also proposes a hard-  ware implementation of the SGM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The architecture features scalability  and combines coarse-grain and fine-grain parallelisation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The authors  performed tests for different configurations and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' For 1920×1080 pix-  els @ 30 fps and 128 disparity levels, real-time processing was achieved at a clock  of 130 MHz (VC709 board with Virtex-7 FPGA device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In the paper of Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' from 2020 [10], the authors presented the FP-  Stereo library, which uses the HLS language and allows the creation of SGM  disparity calculation modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The module has been designed in the form of  an accelerator interfacing with a DMA controller, rather than directly with the  video stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' For a 300 MHz clock, a resolution of 1242 × 374 pixels and 128  disparity range, 161 fps were achieved on the ZCU 102 board with the Xilinx  Zynq UltraScale+ MPSoC device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In the latest publications by Shrivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' in 2020 [11] and Lee with Kim  in 2021 [6], the support for parallel pixel processing has been added to increase  throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In this approach, the main challenge is the presence of an inherent  data dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In the paper from 2020 [11], it is addressed by dependency  relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, the aggregation is performed on the basis of the pixel k earlier,  where k is the number of pixels processed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The author points out  that such a solution represents a trade-off between accuracy and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In the work from 2021 [6], on the other hand, a different approach is pre-  sented, in which operations involving the inherent data dependency are per-  formed not on a single pixel, but on a vector of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' This allows the genera-  tion of disparity maps with very close accuracy to the original SGM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In both solutions, the matching costs are determined based on the Census trans-  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In the first publication [11], for images at a resolution of 1280 × 960 pixels  and disparity range of 64, 322 fps, and in the second [6] for a resolution of  1920 × 1080 pixels and disparity range of 128, 103 fps were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  We also propose a solution to the inherent data dependency problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Our  architecture is based on estimating the previous pixel aggregation cost on a path  Real-time FPGA implementation of the SGM stereo vision in 4K  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 2: A general scheme of the proposed SGM disparity estimation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  with minimal additional logic needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' That allows us to process images with a 4K  resolution and also to obtain comparable results to the original SGM algorithm  without parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  4  The proposed hardware implementation  The aim of our work was to implement a hardware architecture capable of pro-  cessing a video stream with a resolution of 3840×2160 pixels in real-time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e pro-  cessing 30 frames per second with no pixel dropping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' That stream transmitted  in a 1 pixel per clock format requires a pixel clock frequency of approximately  250 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Adding to this value i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, the vertical and horizontal blanking fields,  the required clock equals about 300 MHz, which is too high for the rather com-  plicated SGM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' At the bottleneck, cost aggregation calculations take  more than 10 ns on our platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' So, in order to process the data in the de-  sired resolution, it is necessary to introduce parallelisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In this work, a 4ppc  (pixel per clock) format is used, in which 4 pixels are processed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' This  allows the pixel clock to be lowered to approximately 75 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' However, the use  of such format has significant implications on the implementation of the SGM  algorithm, due to the inherent data dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  A general scheme for the proposed vision system is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The  module accepts a synchronised video stream of rectified images, the base IB(p)  and the reference IM(p) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Further processing consists of several steps: de-  termination of the matching cost C(p, d) using the Census transform based  matching method, calculation of the cost aggregation Lr(p, d), summation of  the aggregation costs from all directions S(p, d) and disparity determination  D(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='1  Determination of the matching cost  The 4ppc format does not introduce major complications into the hardware  architecture of the matching cost determination module, but only increases the  hardware resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' First, 5×5 contexts are created for both images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  For the base image, in a given cycle, 4 contexts are created (as implied by the  4ppc format [12]), and for the reference image this number is increased by the  disparity range (4 + disp range − 1), so that it is possible to simultaneously  compare each of the 4 contexts of the base image with all the contexts in the  disparity range of the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' A Census transform is performed on the  generated contexts, and the contexts are then compared accordingly using the  Hamming distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The output consists of matching cost vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Ib(p)  C(p, d)  Lr(p, d)  S(p,d)  Disparity  D(p)  Matching Costs  Costs  Im(p)  Sum  Selection  Determination  Aggregation6  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Grabowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 3: Cost aggregation paths in SGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='2  Cost aggregation  In the next step, a quasi-global optimisation is performed by aggregating the  costs for the whole image according to the SGM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In the current version  of the module, this is implemented on four paths in the directions 0°, 45°, 90°,  135°, as shown in Figure 3, which can be processed directly (without additional  video stream buffering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Theoretically, it is also possible to realise the other four directions (180°,  225°, 270°, 315°), but this would require storing the entire image in external  RAM, using additional resources of the FPGA device, complex control logic and  introducing additional latency in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In order to calculate the aggregation cost for a given pixel, it is necessary to  know the value of the aggregation cost for the previous pixel on the path (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Equations (1) and (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' For the 45°, 90°, 135° paths, the aggregation costs for the  pixels in a given line are stored in Block RAM and read out accordingly during  the processing of the next image line to calculate the costs for the subsequent  pixels on these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The hardware architecture of this computation is shown in  Figure 4 and follows Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The grey part is replicated for the entire range  of disparities (disp range) and performs in parallel and one block of finding the  minimum value of aggregation costs of the previous pixel on the path minLr(p−  r) is exploited to calculate the aggregation cost for the current pixel for each  disparity value in the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  For the 4ppc format, the difficulty arises for the 0° path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Using the aggrega-  tion cost of the previous pixel Lr(p − r, d), which for this path lies in the same  image line and potentially in the same 4ppc format data vector, results in the  need to process four pixels in the same clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In the worst case, for the  last pixel in the vector, in one clock cycle the data would have to propagate  through four serially connected aggregation cost calculation units, as in Figure  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The critical path would contain 4 minimum modules of size disp range, four  minimum modules of size 4 and 12 adders/subtractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' For this reason, the cost  aggregation based on a baseline architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, as proposed by the authors of  SGM) for the 0° path is not feasible for the considered 4K resolution, without  violating timing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  It is therefore necessary to propose a new solution for the calculation of the  aggregation cost for the 0° path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Time constraints require that the new architec-  ture does not introduce significant additional propagation time and maintains  video stream direction  45°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='06  135°  。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='0  dReal-time FPGA implementation of the SGM stereo vision in 4K  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 4: Hardware architecture of the aggregation cost calculation unit for path  r, pixel p and disparity d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  the approximation assumption of the global smoothness constraint of the SGM  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In our work, we designed and implemented an architecture with a proposed  estimation of the aggregation cost value for consecutive pixels based on the  calculated aggregation cost for the last pixel of the previous 4ppc vector (the  pixel processed in the previous clock cycle) and the matching costs of the previous  pixels in the same 4ppc vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  For the first pixel in the 4ppc vector, the aggregation cost of the previous  pixel is available during the calculation (it was calculated for the previous 4ppc  vector), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e:  Lr(p1 − r, d) = Lr(plast, d)  (3)  where: Lr(p1 − r, d) is the aggregation cost of the previous pixel relative to the  first pixel in the 4ppc vector (p1 − r), and Lr(plast − r, d) is the aggregation cost  of the last pixel in the previous 4ppc vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  For the consecutive pixels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' we propose an estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' which is performed  according to the following Equations:  L′  r(p2 − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) = Lr(plast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) + 1  λ(C(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) − Lr(plast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d))  L′  r(p3 − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) = Lr(plast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) + 1  λ(C(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) + C(p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d)  2  − Lr(plast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d))  L′  r(p4 − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) = Lr(plast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) + 1  λ(  C(p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) + C(p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d)  2  + C(p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d)  2  − Lr(plast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d))  (4)  where: L′  r(p−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) is the estimated aggregation cost for the previous pixel relative  to the pixel p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' C(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d) is the matching cost for a given pixel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' and the coefficient  Lr(p -r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='d)  Lr(p - r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='d - 1)  C(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' d)  P1  Minimum  Lr(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='d)  Lr(p -r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='min disp)  Lr(p -r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='d + 1)  (size: 4)  Lr(p - r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='min disp + 1)  P1  Minimum  min Lr(p - r  (size: disp range)  P2  Lr(p -r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='max disp - 1  Lr(p - r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='max disp )8  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Grabowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 5: The architecture for estimating the aggregation cost of the previous pixel  for each pixel in the 4ppc vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  λ may take a value which is a power of two (1, 2, 4, 8, 16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The architecture of  this solution is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The algorithm is based on the difference of the matching cost values of the  previous pixels in a given 4ppc vector with the aggregation cost for the last pixel  of the previous vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The aggregation cost estimation architecture consists of  basic components and introduces an additional delay only by the propagation  time of the 3 adders/subtractors (critical path for Lr(p4−r, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Note: multiplica-  tion/division by a number that is a power of two is only a bit shift and requires  no delay in the hardware implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The solution takes into account the matching cost values of all previous pixels  with the possibility to adjust the impact of the matching cost of previous pixels  in a given vector by a factor of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The estimated aggregation costs are then used to calculate the aggregation  costs according to the architecture in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' In the work of Shrivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  [11] the estimation has been omitted and in the work of Lee and Kim [6] it has  been solved by the cluster-wise cost aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The aggregation costs from all paths are then summed and the disparity is  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' This involves finding the minimum matching cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='3  Evaluation of the proposed method  The accuracy evaluation of the proposed algorithm was performed on a set of  stereo images from the Middlebury 2014 [13] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We skipped the final post-  processing to better highlight the differences between the base SGM algorithm  and the modified version proposed in this paper (SGM 4ppc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=" The accuracy was  Lr(p1 - r,d)  Lr(piast -r,d)  C(p1,d)  1  Lr(p2 -r,d)  Lr(piast -r,d)  C(p1, d)  C(p2, d)  L'r(p3 -r,d)  Lr(plast -r,d)  C(p1, d)  4  C(p2, d)  1  C(p3, d)  1  Lr(p4 -r,d)  X  2  Lr(piast -r, d)Real-time FPGA implementation of the SGM stereo vision in 4K  9  (a) Input image – left  (b) Ground truth  (c) SGM 4ppc  (d) Local method based  on CT  (e) SGM – 3 paths  (f) SGM – 4 paths  Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 6: Comparison of output disparity maps for the Motorcycle image in Mid-  dlebury 2014 dataset: (a) the left input image, (b) the ground truth disparity  map, (c), (d), (e), (f) estimated disparity maps (on the top) and the error maps  (on the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  measured by the ratio of pixels with incorrect disparity value to all pixels of the  image (all) and also to the non-occluded (noc) pixels (occluded pixels should be  filled with the Left/Right Check post-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  We compared the proposed method (SGM 4ppc) with the conventional local  block matching based on the Census transform and the SGM algorithm (also  YAMRMA区X10  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Grabowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Table 1: Comparison of error rates for the Middlebury 2014 dataset, based on  all (all) and non-occluded (noc) pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  all  noc  Local based on CT  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='21%  63,36%  SGM 3 paths  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='01%  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='79%  SGM 4 paths  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='27%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='88%  SGM 8 paths  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='31%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='11%  SGM 4ppc  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='64%  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='32%  based on the Census transform) with 3 and 4 aggregation paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Figure 6 shows  sample evaluation results on the Motorcycle images from the Middlebury 2014  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' Table 1 shows the average evaluation results for the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  The accuracy of the proposed method is comparable to the original SGM  algorithm with 4 paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' The difference between error rates is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='4  Hardware implementation  We implemented the proposed stereo vision system on a VC707 evaluation board  with Xilinx’s Virtex-7 XC7VX485T-2FFG1761C device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We set up a test envi-  ronment to evaluate the system, with test images sent directly from a PC do the  board and later displayed on a 4K monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  We compared our solution with previous FPGA implementations of the SGM  algorithm in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We used the following metrics: Frames per Second (FPS),  Million Disparity Estimates per second (MDE/s) and MDE/s per Kilo LUTs  (Look-Up Tables) (MDE/s/KLUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' First of all, our solution is the only one ver-  ified in hardware for a 4K/ Ultra HD resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We also would like to point out  that the lower performance in FPS and MDE/s relative to previous work from  2020 [11] and 2021 [6] is due to the use of an FPGA chip with fewer resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' For  this work, it was necessary to select a suitable platform to enable image acquisi-  tion in 4K resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='e, having two high-bandwidth FMCs (FPGA Mezzanine  Connectors) to which TB-FMCH-HDMI4K modules were attached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  It is also worth mentioning that the used FPGA technology differs not only  in the number of resources but also in the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' To compare: the critical  path propagation time for the technology used in this paper after synthesis  is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='967 ns, but for the Xilinx Virtex UltraScale+ XCVU9P-L2FLGA2104E  FPGA with the same parameters, it is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='240 ns (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='45% faster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  5  Conclusion  In this paper, we presented a hardware architecture for an SGM algorithm to  process a 4K/Ultra HD video stream in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We proposed a solution to  the inherent data dependency problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' It allowed us to maintain high accuracy  of the depth map estimation, while making it possible to take advantage of the  Real-time FPGA implementation of the SGM stereo vision in 4K  11  Table 2: Comparison with previous FPGA implementations of the SGM algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Image  Disparity  Platform  FPGA  Throughput  resolution  range  resources  LUT FF BRAM FPS MDE/s MDE/s/KLUT  [14]  1920x1080  128  Virtex-7  195k 217k  368  30  7 963  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='84  [15]  1600x1200  128  Stratix-V  222k 149k  N/A  43  10 472  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='2  [11]  1280x960  64  Virtex-7 690T 211k N/A  641  322 25 056  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='6  [6]  1920x1080  128  Zynq US+  222k 135k  252  103 27 297  123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='0  New 3840x2160  64  Virtex-7 485T 138k 65k  197  30  15 925  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='2  4ppc vector format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We implemented the module on a Virtex-7 FPGA platform  achieving 30 frames per second for a resolution of 3840 × 2160 pixels with 64  disparity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  In future work, we plan to add more aggregation paths to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' With  that, it will be possible to get more accurate results, but at the cost of latency and  resource usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' We also plan to implement a video stream rectification module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='  Acknowledgements The work presented in this paper was supported by: the  National Science Centre project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 2016/23/D/ST6/01389 entitled ”The de-  velopment of computing resources organization in latest generation of hetero-  geneous reconfigurable devices enabling real-time processing of UHD/4K video  stream”, the AGH University of Science and Technology project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
+page_content='773  and the program ”Excellence initiative — research university” for the AGH Uni-  versity of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
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+page_content=' In: 2020 30th  International Conference on Field-Programmable Logic and Applications  (FPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE4T4oBgHgl3EQfBAsA/content/2301.04847v1.pdf'}
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5dE1T4oBgHgl3EQfBAJm/content/tmp_files/2301.02846v1.pdf.txt

@@ -0,0 +1,1687 @@
+Simulations of momentum correlation functions of light (anti)nuclei in relativistic
+heavy-ion collisions at √sNN = 39 GeV
+Ting-Ting Wang(王婷婷),1 Yu-Gang Ma(马余刚)
+ID ,1, 2, ∗ and Song Zhang(张松)
+ID 1, 2
+1Key Laboratory of Nuclear Physics and Ion-Beam Application (MOE),
+Institute of Modern Physics, Fudan University, Shanghai 200433, China
+2Shanghai Research Center for Theoretical Nuclear Physics，NSFC and Fudan University, Shanghai 200438, China
+(Dated: January 10, 2023)
+Momentum correlation functions of light (anti)nuclei formed by the coalescence mechanism of
+(anti)nucleons are calculated for several central heavy-ion collision systems, namely 10
+5 B +10
+5 B,
+16
+8 O +16
+8 O, 40
+20Ca +40
+20 Ca as well as 197
+79 Au +197
+79 Au in different centralities at center of mass energy
+√sNN = 39 GeV within the framework of A Multi-Phase Transport (AMPT) model complemented
+by the Lednick´y and Lyuboshitz analytical method. Momentum correlation functions for identical
+or nonidentical light (anti)nuclei are constructed and analyzed for the above collision systems. The
+Au + Au results demonstrate that emission of light (anti)nuclei occurs from a source with smaller
+space extent in more peripheral collisions. The effect of system-size on the momentum correlation
+functions of identical or nonidentical light (anti)nuclei is also explored by several collision system
+in central collisions. The results indicate that the emission source-size of light (anti)nuclei pairs
+deduced from their momentum correlation functions and system-size is self-consistent. Momentum
+correlation functions of nonidentical light nuclei pairs gated on velocity are applied to infer the
+average emission sequence of them. The results illustrate that protons are emitted in average on a
+similar time scale with neutrons but earlier than deuterons or tritons in the small relative momentum
+region. In addition, larger interval of the average emission order among them is exhibited for smaller
+collision systems or at more peripheral collisions.
+I.
+INTRODUCTION
+In heavy-ion collisions (HICs), two-particle momentum
+correlation function is different from the original applica-
+tion in astronomy [1, 2], and has been normally utilized
+to extract space-time information of the emission source
+and probe the dynamical evolution of nuclear collisions in
+an extensive energy range [3–12]. Many different studies
+on the two-particle momentum correlation functions in
+intermediate energy HICs can be also found in literature,
+eg. Refs. [12–27], which include the momentum correla-
+tion functions of neutron, proton as well as light charged
+particle (LCP) pairs. Multi-variable dependences of the
+momentum correlation functions, such as impact param-
+eters, total momentum of particle pairs, isospin of the
+emission source, nuclear symmetry energy, nuclear equa-
+tion of state (EOS) as well as in-medium nucleon-nucleon
+cross section (NNCS) etc., contain a wealth of informa-
+tion about the space-time characteristics of intermediate
+energy HICs. In high energy HICs, two-hadron momen-
+tum correlation function，also called as Hanbury Brown-
+Twiss (HBT) interferometry, was also well extensively
+measured and some interesting properties on emission
+source were extracted [28, 29].
+Oscillations of the ex-
+tracted HBT radii versus emission angle indicate that
+emission source is elongated perpendicular to the reaction
+plane. The results indicate that the initial shape is more
+or less remained and could be identified even though the
+collision system undergoes the pressure and expansion.
+∗ Corresponding author: [email protected]
+Furthermore, interaction between antiprotons has been
+also measured with the momentum correlation functions
+and the equality of interactions between p-p and ¯p-¯p was
+proved by the STAR Collaboration [30].
+The interac-
+tion property of the particle pairs has been discussed for
+other particles, for instance Λ pairs [31], proton-Ω and
+proton-Ξ etc [32, 33], with the same momentum correla-
+tion technique. Furthermore, the measurements of mo-
+mentum correlation functions for nonidentical nucleons
+and light clusters can be used to characterize the mean
+emission sequence of them, which was firstly proposed in
+Ref. [34]. Theoretical study has been extended to differ-
+ent kinds of nonidentical particle pairs, for instance p-d,
+n-p [35–38], π-p [39], K+-K− [40], d-t [12, 22] as well as
+3He-α particles [41] in intermediate energy HICs.
+In this work we extend the studies, for the first time, on
+the momentum correlation functions of light (anti)nuclei
+to ultra-relativistic heavy-ion collisions simulated by A
+Multi-Phase Transport (AMPT) model [42, 43] coupled
+with a dynamical coalescence model [44–46], specifically
+at √sNN = 39 GeV. Different gating conditions such as
+centrality gate, system-size gate as well as velocity gate
+are applied to the momentum correlation functions of
+light (anti)nuclei pairs. In particular, we report on the in-
+dication of the emission chronology of protons, deuterons
+and tritons which can be deduced from their correspond-
+ing momentum correlation functions in ultra-relativistic
+HICs at √sNN = 39 GeV. The emission sequence of
+light clusters inferred from the correlation functions is
+expected measurable in future experiments to verify our
+deduction from the coalescence picture.
+The rest of this article is organized as follows. In Sec-
+tion II A and II B, we briefly describe A Multi-Phase
+arXiv:2301.02846v1  [hep-ph]  7 Jan 2023
+
+2
+Transport model [42, 43] and the coalescence model [44–
+46], then introduce how to calculate the momentum cor-
+relation functions of particle pairs by using the Lednick´y
+and Lyuboshitz analytical formalism [3, 47–50] in Sec-
+tion II C. In Section III, we summarize the simulated re-
+sults of the light (anti)nuclei momentum correlation func-
+tions gated on various parameters in relativistic heavy-
+ion collisions.
+Section III A compares the results of
+proton-proton and proton-antiproton momentum corre-
+lation functions with experimental data from the RHIC-
+STAR collaboration. From Section III B to III D, iden-
+tical and nonidentical light (anti)nuclei momentum cor-
+relation functions gated on different conditions are sys-
+tematically discussed. Finally, a summary and outlook
+are given in Section IV.
+II.
+MODELS AND FORMALISM
+A.
+AMPT model
+To obtain phase-space distributions of (anti)particles,
+A Multi-Phase Transport model [42, 43] is used as the
+event generator, which has been applied successfully for
+studying heavy-ion collisions at relativistic energies, eg.
+[45, 46, 51–59]. We briefly review the main components
+of the AMPT model used in the present work. In the
+version of AMPT, the initial phase-space information of
+partons is generated by the heavy-ion jet interaction gen-
+erator (HIJING) model [60, 61]. The interaction between
+partons is then simulated by Zhang’s parton cascade
+(ZPC) model [62].
+During the hadronization process,
+a quark coalescence model is used to combine partons
+into hadrons [63–65].
+Then, the hadronic rescattering
+evolution is described by a relativistic transport (ART)
+model [66].
+In this paper, the collisions of 10
+5 B +10
+5 B, 16
+8 O +16
+8 O,
+40
+20Ca +40
+20 Ca at 0 − 10 % centrality and mid-rapidity
+(|y| < 0.5) as well as 197
+79 Au+197
+79 Au at same mid-rapidity
+for five centralities of 0−10 %, 10−20 %, 20−40 %, 40−60
+%, and 60 − 80 % at √sNN = 39 GeV are simulated.
+Te phase-space distributions of (anti)particles are se-
+lected at the final stage in the hadronic rescattering pro-
+cess (ART model [66]) with considering baryon-baryon,
+baryon-meson, and meson-meson elastic and inelastic
+scatterings, as well as resonance decay or week decay.
+The transverse momentum spectra of light (anti)nuclei
+have been successfully reproduced by the AMPT model
+with the maximum hadronic rescattering time (MRT) of
+100 fm/c [46]. Therefore, the same maximum hadronic
+rescattering time is used for the most calculations in this
+work except for a quantitative comparison with the p-p
+and p-¯p data from the STAR collaboration in Sec. III A.
+B.
+Coalescence model
+The coalescence model has been used widely in de-
+scribing the production of light clusters in the interme-
+diate [67–71] and high [72, 73] energy heavy-ion colli-
+sions. The detailed definitions of the probability for pro-
+ducing a cluster of nucleons is in Ref. [44]. In our model
+calculations, light (anti)clusters such as (anti)deuterons
+and tritons are constructed by using the coalescence
+model as follows [74, 75]. The probability for producing
+M-nucleon cluster is determined by its Wigner phase-
+space density and the nucleon phase-space distribution
+at the freeze-out stage [44]. The multiplicity of an M-
+nucleon cluster in transport model simulations for heavy-
+ion collisions is given by,
+NM = G
+�
+�
+i1>i2>···>iM
+d⃗ri1d⃗ki1 · · · d⃗riM−1d⃗kiM−1
+�
+ρW
+i
+�
+⃗ri1,⃗ki1, · · · ,⃗riM−1,⃗kiM−1
+��
+(1)
+where ⃗ri1,⃗riM−1 and ⃗ki1,⃗kiM−1 are the relative coordi-
+nates and momentum in the M-nucleon rest frame, and
+spin-isospin statistical factor G is 3/8 for (anti)deuteron
+and 1/3 for triton [44]. In addition, ρW is the Wigner
+density function, which is different for all kinds of parti-
+cles. Therefore, we will calculate separately the Wigner
+phase-space density of (anti)deuteron and triton in de-
+tail. The Wigner phase-space density of (anti)deuteron
+is constructed by,
+ρW
+d (⃗r,⃗k) = 8
+15
+�
+i=1
+c2
+i exp
+�
+−2ωir2 − k2
+2ωi
+�
++ 16
+15
+�
+i>j
+cicj
+�
+4ωiωj
+(ωi + ωj)2
+� 3
+4
+exp
+�
+− 4ωiωj
+ωi + ωj
+r2
+�
+× exp
+�
+−
+k2
+ωi + ωj
+�
+cos
+�
+2ωi − ωj
+ωi + ωj
+⃗r · ⃗k
+�
+(2)
+where ⃗k =
+�
+⃗k1 − ⃗k2
+�
+/2 is the relative momentum and
+⃗r = (⃗r1 − ⃗r2) is the relative coordinate of (anti)proton
+and (anti)neutron. The Wigner phase-space density of
+triton is constructed by a spherical harmonic oscilla-
+tor [44, 45, 76],
+ρW
+t
+�
+ρ, λ,⃗kρ,⃗kλ
+�
+=
+�
+ψ
+�
+ρ +
+⃗R1
+2 , λ +
+⃗R2
+2
+�
+ψ∗
+�
+ρ −
+⃗R1
+2 , λ −
+⃗R2
+2
+�
+× exp
+�
+−i⃗kρ · ⃗R1
+�
+exp
+�
+−i⃗kλ · ⃗R2
+�
+3
+3
+2 d⃗R1d⃗R2
+= 82 exp
+�
+−ρ2 + λ2
+b2
+�
+exp
+�
+−
+�
+⃗k2
+ρ + ⃗k2
+λ
+�
+b2�
+(3)
+
+3
+where ρ and λ are relative coordinates, ⃗kρ and ⃗kλ are the
+relative momenta in the Jacobi coordinate.
+The above parameters of the Gaussian fit coefficient
+ci and wi for (anti)deuteron as well as b for triton are
+given in Ref. [44].
+Based on the phase-space informa-
+tion of light (anti)cluster obtained by the above coa-
+lescence model, the momentum correlation functions of
+(non)identical light (anti)cluster pairs can be discussed
+in the following.
+C.
+Lednik´ynd Lyuboshitz technique
+Next, we briefly review the technique of the two-
+particle momentum correlation function proposed by
+Lednick´y and Lyuboshitz [47–49]. The method is based
+on the principle as follows: when two particles are emit-
+ted at small relative momentum, their momentum corre-
+lation function is determined by the space-time charac-
+teristics of the production processes owing to the effects
+of quantum statistics (QS) and final-state interactions
+(FSI) [3, 50]. The details on the formalism of the two-
+particle momentum correlation function can be found in
+Ref. [36].
+Here, comparing with our previous literature [36], more
+particle pairs are considered in the article. Therefore, the
+final-state interaction of different particle pairs can be
+known well by introducing fc (k∗) particularly as follows:
+fc (k∗) =
+�
+Kc (k∗) − 2
+ac
+h (λ) − ik∗Ac (λ)
+�−1
+(4)
+fc (k∗) is the s-wave scattering amplitude renormalizied
+by the long-range Coulomb interaction, with h (λ) =
+λ2 �∞
+n=1
+�
+n
+�
+n2 + λ2��−1−C −ln [λ] where C = 0.5772 is
+the Euler constant. Kc (k∗) =
+1
+f0 + 1
+2d0k∗2 +Pk∗4 +· · · is
+the effective range function, where d0 is the effective ra-
+dius of the strong interaction, f0 is the scattering length
+and P is the shape parameter. The parameters of effec-
+tive range function are important to characterize the es-
+sential properties of the final-state interactions, and can
+be extracted from the correlation function measured ex-
+perimentally [30, 36, 77, 78]. Table I shows the param-
+eters of the effective range function for different particle
+pairs in the present work.
+In the table I, for n-n (¯n-¯n) and n-p (¯n-¯p) momentum
+correlation functions which include uncharged particle,
+the Coulomb penetration factor (Ac (λ)) is not consid-
+ered and only the short-range particle interaction works.
+For the momentum correlation functions of charged parti-
+cles such as p-¯p, p-p (¯p-¯p), d-d ( ¯d- ¯d), t-t, p-d (¯p- ¯d), p-t and
+d-t, both the Coulomb interaction and the short-range in-
+teraction dominated by the s-wave interaction are taken
+into account. The momentum correlation function of p-
+p (¯p-¯p) particle pairs is dominantly contributed by only
+TABLE I. Experimental determination of the effective range
+function parameters for n-n (¯n-¯n), p-p (¯p-¯p), t-t, n-p (¯n-¯p),
+p-d (¯p- ¯d), p-t and d-t systems [30, 77, 78].
+System
+Spin f0 (fm) d0 (fm) P
+�
+fm3�
+n-n (¯n-¯n)
+0
+17
+2.7
+0.0
+p-p (¯p-¯p)
+0
+7.8
+2.77
+0.0
+t-t
+0
+1 × 10−6
+0.0
+0.0
+n-p (¯n-¯p)
+0
+23.7
+2.7
+0.0
+p-d (¯p- ¯d)
+1/2
+-2.73
+2.27
+0.08
+3/2
+-11.88
+2.63
+-0.54
+p-t
+0
+1 × 10−6
+0.0
+0.0
+d-t
+0
+1 × 10−6
+0.0
+0.0
+the singlet (S = 0) s-wave final-state interactions while
+both spins 1/2 and 3/2 contribute in the case of p-d (¯p-
+¯d) system. Moreover, for (anti)deuteron-(anti)deuteron
+momentum correlation function, a parametrization of the
+s-wave phase shifts δ has been used from the solution of
+Kc (k∗) = cot δ for each total pair spin S = 0, 1, 2. Note
+that the effective range function for the total spin S = 1
+is irrelevant, since it does not contribute due to the quan-
+tum statistics symmetrization.
+III.
+ANALYSIS AND DISCUSSION
+A.
+Comparison between our p-p and p-¯p correlation
+functions with experimental data
+Fig. 1 presents results of p-p and p-¯p correlation func-
+tions for three different centrality classes of 0 − 10 %,
+10−30 %, and 30−70 % calculated by the AMPT model
+in Au + Au collisions at √sNN = 39 GeV. Within the
+cut of transverse momentum pt and rapidity y, we con-
+front the experimental data with the predictions of the
+AMPT model combined with Lednick´y and Lyuboshitz
+code. When the phase-space information of nucleons at
+the maximum rescattering time among hadrons of 700
+fm/c is selected from the AMPT model, it is found that
+the results can well describe the experimental data for
+the momentum correlation functions of p-p and p-¯p from
+the RHIC-STAR collaboration [79, 80], especially in more
+central collisions. Considering that the preliminary ex-
+perimental results were not corrected by feed-down effect
+corrections [79, 80], the real correlation functions for pri-
+mary p-p and p-¯p could be much more stronger. In this
+case, using much longer MRT of 700 fm/c in the AMPT
+model might be a reasonable choice for making quanti-
+tative comparison with feed-down uncorrected data since
+the system will become more expanded and weakly corre-
+lated among particles after longer MRT in AMPT. How-
+ever, the quantitative reproduction is not our main con-
+cern in the present work. In the following calculations, we
+fixed the MRT at 100 fm/c and presented systematic re-
+sults among different light (anti)nuclei. However, as one
+can notice that the results for p-p and p-¯p change substan-
+
+4
+tially when changing the MRT by comparing Fig. 1 and
+2. To estimate this uncertainty, we also check some re-
+sults for light nuclei correlations with different MRT. For
+example, d-d or p-d correlations for MRT equal 700fm/c.
+It is found that the correlation becomes slightly weaker
+at smaller q (i.e. a little larger value of Cdd or Cpd close
+to 1 at MRT = 700 fm/c), which has the similar trend as
+p-p and p-¯p cases. But the uncertainty is less than 20%
+at the lowest relative momentum and tends to vanish at
+q > 50 MeV/c for light nuclei correlations (d-d or p-d)
+when changing the MRT from 100fm/c to 700fm/c, which
+can be essentially understood by weak feed-down effects
+for light nuclei. In addition, we also check the p-d cor-
+relation with different velocity selection. Only less than
+10% uncertainty is found for lower q between the case
+of MRT equal 700 fm/c to the one at 100fm/c. By this
+comparison of results at MRT equal 700 and 100 fm/c,
+we conclude that nucleon-(anti)nucleon correlations are
+much influenced by the MRT but light nuclei correla-
+tions only change slightly. Overall, the MRT = 100 fm/c
+is basically safe choice for such light nuclei correlations.
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+0
+20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
+0.6
+0.8
+1.0
+1.2
+1.4
+  Cpp(q)
+   STAR data     LL-model
+    0-10%   
+    0-10%                 
+  10-30%   
+  10-30%  
+  30-70%   
+  30-70% 
+  
+ 
+      
+ 
+Au+Au@√sNN=39GeV, |y|<0.5, 0.4<pt<2.5GeV/c 
+ 
+ 
+(a)
+(b)
+  Cpp(q)
+ q (MeV/c) 
+FIG. 1.
+Proton-proton (a) and proton-antiproton (b) mo-
+mentum correlation functions for different centrality classes in
+√sNN = 39 GeV Au + Au collisions. Solid markers represent
+the preliminary experimental data from the RHIC-STAR col-
+laboration [79, 80], and lines represent our model calculation
+results from the AMPT model plus the Lednick´y and Lyu-
+boshitz code. Note that the longer hadronic rescattering time
+of 700 fm/c is used in this specific calculation for comparing
+with the data.
+B.
+Centrality and system-size dependence of
+identical light (anti)nuclei momentum correlation
+functions
+The centrality dependence of the two-particle mo-
+mentum correlation function can systematically investi-
+gate the contributions from the system-size and parti-
+cle interactions on the correlations. Fig. 2 (a) and (c)
+present the momentum correlation functions of identi-
+cal (anti)particle pairs (n-n (¯n-¯n) and p-p (¯p-¯p) ) for
+197
+79 Au +197
+79 Au collisions at different centralities of 0 − 10
+%, 10 − 20 %, 20 − 40 %, 40 − 60 %, and 60 − 80 % at
+√sNN = 39 GeV. The momentum correlation functions
+of (anti)neutron pairs exhibit more than unity in Fig. 2
+(a), which is caused by the attractive s-wave interaction
+between the two (anti)neutrons. In Fig. 2 (c), the shape
+of the (anti)proton−(anti)proton momentum correlation
+functions looks as expected from the interplay between
+the quantum statistical (QS) and final state interactions
+(FSI) and is consistent with previous results [13, 30, 36].
+The (anti)proton−(anti)proton momentum correlation
+functions exhibit less than unity at low relative mo-
+mentum q in Fig. 2 (c), which is mainly caused by the
+Coulomb repulsion between the (anti)proton pairs. With
+increasing relative momentum, the attractive s-wave in-
+teraction between the two (anti)protons gives rise to a
+maximum of the (anti)proton−(anti)proton momentum
+correlation functions at q around 0.020 GeV in Fig. 2
+(c). The antiproton−antiproton momentum correlation
+functions show a similar structure with proton pairs, re-
+sulting from the same attractive interaction between two
+antiprotons
+[30]. Fig. 2 (a) and (c) compare five cen-
+tralities of 0 − 10 %, 10 − 20 %, 20 − 40 %, 40 − 60
+%, and 60 − 80 % of the two-(anti)particle momentum
+correlation functions. The enhanced strength of the n-n
+(¯n-¯n) and p-p (¯p-¯p) momentum correlation functions is
+observed in peripheral collisions. These results indicate
+that (anti)particle emission occurs from a source with
+smaller space extent in peripheral collision. In addition,
+the effect of system−size on the momentum correlation
+functions of (anti)particles is also investigated by four dif-
+ferent systems, namely 10
+5 B+10
+5 B, 16
+8 O+16
+8 O, 40
+20Ca+40
+20Ca
+and 197
+79 Au+197
+79 Au, in central collisions. In Fig. 2 (b) and
+(d), the n-n (¯n-¯n) and p-p (¯p-¯p) momentum correlation
+functions appear strong sensitivity to system-size and an
+enhanced strength is observed when particle pairs are
+emitted from smaller system collisions. This enhanced
+strength of the momentum correlation functions for par-
+ticle pairs is a physical effect stemming from the smaller
+space extent of the emission source [8]. Therefore, the
+emission source-size of particle pairs obtained by their
+momentum correlation functions and system-size is self-
+consistent.
+Figure 3 shows the centrality and system-size depen-
+dences of the momentum correlation functions for light
+(anti)cluster in similar condition as in Fig. 2. Figure 3
+(a) and (c) present the momentum correlation functions
+of d-d ( ¯d- ¯d) and t-t for 197
+79 Au +197
+79 Au collisions at dif-
+ferent centralities of 0 − 10 %, 10 − 20 %, 20 − 40 %,
+40 − 60 %, and 60 − 80 % at √sNN = 39 GeV. The
+d-d ( ¯d- ¯d) momentum correlation functions exhibit less
+than unity at lower relative momentum q in Fig. 3 (a)
+and (b), which is caused by the Coulomb repulsion. The
+two-triton momentum correlation functions are less than
+unity with increasing relative momentum q as shown in
+Fig. 2 (c) and (d), which is caused by only the Coulomb
+potential in the Lednick´y and Lyuboshitz code [47–49].
+
+5
+0
+50
+100
+150
+200
+250
+300
+0
+1
+2
+3
+0
+50
+100
+150
+200
+250
+300
+0
+1
+2
+3
+4
+5
+ C(q)
+  nn       nn 
+   
+    0-10%
+   
+  10-20%
+   
+  20-40%
+   
+  40-60%
+   
+  60-80%
+ C(q)
+(a) 39GeV AuAu 
+(b) 39GeV 0-10% 
+  nn       nn
+   
+ B+B
+   
+ O+O
+   
+ Ca+Ca
+   
+ Au+Au
+(c) 39GeV AuAu 
+  pp       pp 
+   
+    0-10%
+   
+  10-20%
+   
+  20-40%
+   
+  40-60%
+   
+  60-80%
+q (MeV/c) 
+(d) 39GeV 0-10% 
+  pp       pp
+   
+ B+B
+   
+ O+O
+   
+ Ca+Ca
+   
+ Au+Au
+ q (MeV/c) 
+FIG. 2.
+The momentum correlation functions at mid-
+rapidity (|y| < 0.5) of (anti)neutron-pairs and (anti)proton-
+pairs as a function of five different centralities for 197
+79 Au +197
+79
+Au reaction at √sNN = 39 GeV are presented in (a) and
+(c), respectively.
+The momentum correlation functions of
+(anti)neutron-pairs and (anti)proton-pairs at mid-rapidity
+(|y| < 0.5) for 0-10% central collisions of 10
+5 B+10
+5 B, 16
+8 O+16
+8 O,
+40
+20Ca +40
+20 Ca as well as 197
+79 Au +197
+79 Au systems at √sNN = 39
+GeV are presented in (b) and (d), respectively. The p-p and
+n-n momentum correlation functions (solid symbols) and the
+anti-one (open symbols) are shown in each panel.
+0.0
+0.5
+1.0
+0
+50
+100
+150
+200
+250
+300
+  dd       dd 
+   
+    0-10%
+   
+  10-20%
+   
+  20-40%
+   
+  40-60%
+   
+  60-80%
+(a) 39GeV AuAu 
+ C(q)
+  dd       dd
+   
+ B+B
+   
+ O+O
+   
+ Ca+Ca
+   
+ Au+Au
+(b) 39GeV 0-10% 
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+  tt       
+     0-10%
+   10-20%
+   20-40%
+   40-60%
+   60-80%
+(c) 39GeV AuAu 
+ Ctt(q)
+q (MeV/c) 
+  tt       
+  B+B
+  O+O
+  Ca+Ca 
+  Au+Au
+(d) 39GeV 0-10% 
+ q (MeV/c) 
+FIG. 3.
+Same as Fig. 2 but for the light (anti)cluster pairs.
+(a) and (b) for d − d momentum correlation functions (solid
+symbols) and the anti-one (open symbols), (c) and (d) for t−t
+momentum correlation functions (solid symbols).
+The antideuteron−antideuteron momentum correlation
+function also shows an exact similar shape with deuteron
+pairs due to the similar phase-space distributions be-
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+1.5
+0
+50
+100
+150
+200
+250
+300
+0
+1
+2
+3
+4
+5
+  np       np 
+   
+    0-10%
+   
+  10-20%
+   
+  20-40%
+   
+  40-60%
+   
+  60-80%
+(a) 39GeV AuAu 
+ C(q)
+  np       np
+   
+ B+B
+   
+ O+O
+   
+ Ca+Ca
+   
+ Au+Au
+(d) 39GeV 0-10% 
+(b) 39GeV 0-10% 
+   pp       
+     0-10%
+   10-20%
+   20-40%
+   40-60%
+   60-80%
+(c) 39GeV AuAu 
+ Cpp(q)
+q (MeV/c) 
+  pp      
+    B+B
+    O+O
+    Ca+Ca
+    Au+Au
+ q (MeV/c) 
+FIG. 4.
+Same as Fig. 2 but for the nonidentical particle
+pairs. (a) and (b) for n-p momentum correlation functions
+(solid symbols) and the anti-one (open symbols), (c) and (d
+for) p-¯p momentum correlation functions (solid symbols).
+tween deuteron and antideuteron.
+Due to significant
+less yields of tritons which induce too large error, the
+antitriton−antitriton momentum correlation function is
+not shown in the present work, which should be observed
+as the same trend with triton pairs. Fig. 3 (a) and (c) also
+compare five centralities of 0 − 10 %, 10 − 20 %, 20 − 40
+%, 40 − 60 %, and 60 − 80 % for the momentum correla-
+tion functions of two light (anti)clusters. The larger sup-
+pression of the d-d ( ¯d- ¯d) and t-t correlation functions is
+clearly visible in peripheral collisions. These results also
+indicate that light (anti)cluster emission occurs from a
+source with smaller space extent for peripheral collision,
+which is similar to Fig. 2 (a) and (c). In Fig. 3 (b) and
+(d), an enhanced strength of the momentum correlation
+function for d-d ( ¯d- ¯d) and t-t is also observed when light
+(anti)cluster pairs emitted from smaller systems, such as
+in B + B and O + O collisions. However, the sensitivity
+seems disappear in these small systems.
+C.
+Nonidentical light (anti)nuclei momentum
+correlation functions gated on centrality and
+system-size
+Now we investigate centrality and system-size depen-
+dence of the nonidentical (anti)particle momentum corre-
+lation functions, such as n-p (¯n-¯p), p-¯p, p-d (¯p- ¯d), p-t and
+d-t. Fig. 4 (a) and (c) show results for the momentum
+correlation functions of n-p (¯n-¯p) and p-¯p for the same
+centrality classes as Fig. 2. The same centrality depen-
+dence is also clearly seen in Fig. 4 (a) and (c). Because
+of the strong attractive final state interaction between
+n and p, the n-p (¯n-¯p) momentum correlation functions
+show a strong positive correlation at small values of the
+
+6
+0.0
+0.5
+1.0
+0
+50
+100
+150
+200
+250
+300
+  pd       pd 
+   
+    0-10%
+   
+  10-20%
+   
+  20-40%
+   
+  40-60%
+   
+  60-80%
+(a) 39GeV AuAu 
+ C(q)
+  pd       pd
+   
+ B+B
+   
+ O+O
+   
+ Ca+Ca
+   
+ Au+Au
+(b) 39GeV 0-10% 
+0.0
+0.5
+1.0
+   pt       
+     0-10%
+   10-20%
+   20-40%
+   40-60%
+   60-80%
+(c) 39GeV AuAu 
+ Cpt(q)
+  pt      
+    B+B
+    O+O
+    Ca+Ca
+    Au+Au
+(d) 39GeV 0-10% 
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+   dt       
+     0-10%
+   10-20%
+   20-40%
+   40-60%
+   60-80%
+(e) 39GeV AuAu 
+ Cdt(q)
+q (MeV/c) 
+  dt      
+    B+B
+    O+O
+    Ca+Ca
+    Au+Au
+(f) 39GeV 0-10% 
+ q (MeV/c) 
+FIG. 5.
+Same as Fig. 4 but for nonidentical light (anti)nuclei. (a) and (b) for p-d momentum correlation functions (solid
+symbols) and the anti-one (open symbols). (c) and (d) for p-t momentum correlation functions (solid symbols), (e) and (f) for
+d-t momentum correlation functions (solid symbols).
+relative momentum q in Fig. 4 (a) and (b). Fig. 4 (c)
+shows results for proton−antiproton momentum correla-
+tion functions, which are different from the results for
+proton pairs in Fig. 2 (c), however, qualitatively agrees
+with the experimental results in Ref. [79, 80]. In addition,
+Fig. 4 (b) and (d) show system-size dependence of n-p
+(¯n-¯p) and p-¯p momentum correlation functions, which is
+almost unanimously with the identical (anti)particle one
+in Fig. 2 (b) and (d). We can also observe an enhanced
+strength of momentum correlation function for particle
+pairs in smaller systems. In the same way, we also inves-
+tigate the effect of different centralities and system-size
+on the momentum correlation functions of nonidentical
+light (anti)nuclei. The p-d (¯p- ¯d), p-t and d-t momentum
+correlation functions in Fig. 5 (a), (c) and (e) are all char-
+acterized by an anti-correlation feature. For the p-d (¯p- ¯d)
+momentum correlation functions in Fig. 5 (a), the anti-
+correlation shape is a little unlike to the proton−deuteron
+momentum correlation function in the intermediate en-
+ergy heavy-ion collision [36, 37], indicating that a compe-
+tition between the s-wave attraction and the Coulomb re-
+pulsion. The correlation functions of p-t and d-t in Fig. 4
+(c) and (e) also display the trend of below unity due to
+the dominant Coulomb repulsion, which is similar to the
+previous results in intermediate energy heavy-ion colli-
+sions [36, 37]. In Fig. 5 (b), the system-size dependence
+of p-d (¯p- ¯d) momentum correlation functions is shown, an
+enhancement of p-d (¯p- ¯d) momentum correlation function
+
+7
+-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
+3.0x10
+6
+6.0x10
+6
+0
+1
+2
+3
+4
+5
+ ∆v>0     ∆v<0     
+     
+    0-10%
+     
+  10-20%
+     
+  20-40%
+     
+  40-60%
+     
+  60-80%
+(a) ∆v = vn-vp 
+ Cnp(q)
+3.0x10
+6
+6.0x10
+6
+(b) ∆v = vn-vp 
+ counts (arb.unit)
+   0-10%
+ 10-20%
+ 20-40%
+ 40-60%
+ 60-80%
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+(c) ∆v = vp-vp 
+ Cpp(q)
+q (MeV/c) 
+(d) ∆v = vp-vp  
+  ∆v (c)
+FIG. 6.
+The velocity-gated momentum correlation functions
+(left) and velocity difference (∆v) spectra (right) for n-p and
+p-¯p as a function of five different centralities in mid-rapidity
+(|y| < 0.5) for 39 GeV 197
+79 Au +197
+79 Au collision. The velocity
+conditions are indicated in each panel: ∆v > 0 is remarked
+by solid symbol and the ∆v < 0 by open symbol.
+0
+1
+2
+3
+4
+5
+(a) ∆v = vn-vp 
+ ∆v>0     ∆v<0     
+     
+    B+B
+ 
+    
+    O+O
+     
+    Ca+Ca
+     
+    Au+Au
+     
+ 
+ Cnp(q)
+3.0x10
+6
+6.0x10
+6
+  counts (arb.unit)
+(b) ∆v = vn-vp 
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+(c) ∆v = vp-vp 
+ Cpp(q)
+q (MeV/c) 
+-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
+3.0x10
+6
+6.0x10
+6
+ 
+(d) ∆v = vp-vp  
+  ∆v (c)
+ B+B
+ O+O
+ Ca+Ca
+ Au+Au
+FIG. 7.
+Same as Fig. 6 but for 0 − 10 % central collisions
+of 10
+5 B +10
+5 B, 16
+8 O +16
+8 O, 40
+20Ca +40
+20 Ca as well as 197
+79 Au +197
+79
+Au systems at √sNN = 39 GeV. The velocity conditions are
+indicated in each panel: ∆v > 0 is remarked by solid symbol
+and the ∆v < 0 by open symbol.
+is observed in smaller systems. In Fig. 5 (d) and (f), the
+p-t and d-t momentum correlation functions appear more
+sensitive to system-size only in the large system such as
+Au and Ca.
+0.90
+0.95
+1.00
+1.05
+0
+50
+100 150 200 250 300
+0.90
+0.95
+1.00
+1.05
+0
+50
+100 150 200 250 300
+(a) 39GeV AuAu ∆v = vn-vp 
+C∆v>0/C∆v<0
+ 0-10%
+ 10-20%
+ 20-40%
+ 40-60%
+ 60-80%
+(b) 39GeV 0-10% ∆v = vn-vp 
+ B+B
+ O+O
+ Ca+Ca
+ Au+Au
+(c) 39GeV AuAu ∆v = vp-vp 
+q (MeV/c)
+(d) 39GeV 0-10% ∆v = vp-vp 
+q (MeV/c)
+FIG. 8.
+Ratios of the velocity-gated momentum correla-
+tion functions (left) of n-p (a) and p-¯p (c) pairs for 39 GeV
+197
+79 Au +197
+79 Au collision at mid-rapidity (|y| < 0.5) and five
+different centralities. Ratios of the velocity-gated momentum
+correlation functions (right) of n-p (b) and p-¯p (d) pairs for 0
+− 10 % central collisions of 10
+5 B+10
+5 B, 16
+8 O+16
+8 O, 40
+20Ca+40
+20 Ca
+as well as 197
+79 Au +197
+79 Au systems at √sNN = 39 GeV.
+D.
+Velocity selected nonidentical light nuclei
+momentum correlation functions
+The momentum correlation functions of unlike par-
+ticles can provide an independent constrain on their
+mean emission order by simply making velocity selec-
+tions [22, 34, 35, 81, 82]. The principle of comparing the
+velocity-gated momentum correlation functions for the
+nonidentical particle pair to infer their average emission
+order is as follows. Here the two nonidentical particles
+are named by “a”and “b”, respectively. If the ve-
+locity of “a”particle is lower than “b”particle, the
+(anti)correlation will be stronger when the “a”particle
+is emitted averagely early than the “b”particle, be-
+cause the space-size between them is reduced during the
+flight and the final-state interaction (FSI) is enhanced,
+and vice versa. In addition, the velocity difference (∆v)
+spectrum between the two nonidentical particles is also
+sensitive to the mean emission order. Fig. 6 presents the
+velocity-gated momentum correlation functions as well as
+velocity difference (∆v) spectra of unlike particles pairs
+n-p and p-¯p for 39 GeV 197
+79 Au +197
+79 Au collisions at dif-
+ferent centralities of 0 − 10 %, 10 − 20 %, 20 − 40 %,
+40 − 60 %, and 60 − 80 %. In Fig. 6 (a) and (c), the
+centrality dependence on the velocity-gated momentum
+correlation functions of n-p and p-¯p is similar to Fig. 4.
+In Fig. 6 (a), the momentum correlation function for n-p
+pair with vn > vp is similar to one with the reverse situ-
+ation. The symmetry of velocity difference (∆v) spectra
+for n-p pairs is shown in Fig. 6 (b). The results demon-
+strate that the average emission sequence of neutrons and
+protons is almost the same and is insensitive to the cen-
+trality. In Fig. 6 (c), the momentum correlation function
+for p-¯p pair with vp > v¯p is slightly higher than one with
+
+8
+0.0
+0.5
+1.0
+(a) ∆v = vp-vd 
+ Cpd(q)
+ ∆v>0     ∆v<0
+     
+     0-10%
+     
+   10-20%
+     
+   20-40%
+     
+   40-60%
+     
+   60-80%
+3.0x10
+6
+6.0x10
+6
+  counts (arb.unit)
+(b) ∆v = vp-vd 
+ 
+  0-10%
+ 10-20%
+ 20-40%
+ 40-60%
+ 60-80%
+0.0
+0.5
+1.0
+(c) ∆v = vp-vt 
+ Cpt(q)
+3.0x10
+6
+6.0x10
+6
+(d) ∆v = vp-vt 
+ 60-80%*10
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+(e) ∆v = vd-vt 
+ Cdt(q)
+q (MeV/c) 
+-0.8 -0.6 -0.4-0.2 0.0 0.2 0.4 0.6 0.8
+3.0x10
+6
+6.0x10
+6
+ 60-80%*10
+(f) ∆v = vd-vt 
+ 
+  ∆v (c)
+FIG. 9.
+Same as Fig. 6 but for p-d (a) and (b), p-t (c) and (d), and d-t (e) and (f) pairs.
+the reverse situation. The slight asymmetry of velocity
+difference (∆v) spectra for p-¯p pairs is shown in Fig. 6 (d),
+which indicates that the mean order of emission sequence
+between proton and antiproton may be a little different
+but is not sensitive to the centrality. In Fig. 7 (a), the
+momentum correlation functions for n-p pairs with vn >
+vp are always similar to one with the reverse situation
+with increasing system-size. The symmetry of velocity
+difference (∆v) spectra for n-p pairs in different systems
+is shown in Fig. 7 (b). The comparison of velocity-gated
+momentum correlation functions illustrates that the av-
+erage emission sequence between neutrons and protons is
+always identical for different centrality and system-size,
+which is also learned from their ratios in Fig. 8 (a) and
+(b). In Fig. 7 (c) and (d), the comparison of velocity-
+gated momentum correlation functions for p-¯p indicates
+that the mean order of emission sequence between pro-
+tons and antiprotons may be a little different but has no
+dependence of system-size, which is also learned by their
+ratios in Fig. 8 (c) and (d).
+Fig. 9 and Fig. 10 show centrality and system-size de-
+pendences of velocity-gated momentum correlation func-
+tions and velocity difference (∆v) spectra of p-d, p-t and
+d-t pairs, respectively.
+For p-d and p-t pairs, the mo-
+mentum correlation functions with vp < vd (vp < vt)
+are stronger than the ones with the reverse situation
+vp > vd (vp > vt) in Fig. 9.
+The comparison of two
+velocity-gated correlation strengths gives that the mean
+order of emission of protons are emitted averagely ear-
+lier than deuterons and tritons according to the above
+criteria. The similar trend for d-t pairs is not so obvi-
+ous overall, except for in peripheral collision the momen-
+tum correlation function with vd < vt is stronger and
+deuterons are emitted averagely earlier than tritons. In
+contrast with the emission order as shown in many pre-
+vious results of the intermediate energy heavy-ion colli-
+sions [12, 36, 37, 81], the average emission sequence of
+protons, deuterons, and tritons is opposite for 39 GeV
+
+9
+0.0
+0.5
+1.0
+(a) ∆v = vp-vd 
+ Cpd(q)
+  ∆v>0   ∆v<0
+ 
+   
+   B+B
+ 
+   
+   O+O
+ 
+   
+   Ca+Ca
+ 
+   
+   Au+Au
+3.0x10
+6
+6.0x10
+6
+  counts (arb.unit)
+(b) ∆v = vp-vd 
+ 
+ B+B
+ O+O
+ Ca+Ca
+ Au+Au
+0.0
+0.5
+1.0
+(c) ∆v = vp-vt 
+ Cpt(q)
+3.0x10
+6
+6.0x10
+6
+(d) ∆v = vp-vt 
+ B+B*10
+ O+O*10
+ 
+0
+50
+100
+150
+200
+250
+300
+0.0
+0.5
+1.0
+(e) ∆v = vd-vt 
+ Cdt(q)
+q (MeV/c) 
+-0.8 -0.6 -0.4-0.2 0.0 0.2 0.4 0.6 0.8
+0.0
+3.0x10
+6
+6.0x10
+6
+(f) ∆v = vd-vt 
+ B+B*10
+ O+O*10
+  ∆v (c)
+FIG. 10.
+Same as Fig. 7 but for p-d (a) (b), p-t (c) (d) and d-t (e) (f) pairs.
+heavy-ion collisions. Meanwile, Fig. 9 presents velocity
+difference spectra for p-d, p-t and d-t pairs, respectively.
+The velocity difference spectra are all asymmetric due to
+the mean emission order. In addition, an enhanced dif-
+ference between the momentum correlation functions for
+p-d (p-t or d-t) pairs with vp > vd (vp > vt or vd > vt) and
+ones on the reverse situation at larger centrality, which
+manifests the larger interval of the mean emission or-
+der for unlike light nuclei in peripheral collisions. Their
+ratios in Fig. 11 (a), (c) and (e) can also illustrate the
+above phenomenon. The system-size dependence for p-
+d, p-t and d-t pairs can be found by the fact that mo-
+mentum correlation functions with vp < vd (vp < vt or
+vd < vt) are stronger than the ones with the reverse sit-
+uation vp > vd (vp > vt or vd > vt) in Fig. 10. Cor-
+respondingly, the velocity difference spectra for p-d, p-t
+and d-t pairs are all asymmetric about ∆v = 0 caused by
+the average emission order in Fig. 10. Therefore, protons
+are emitted averagely earliest and deuterons are emitted
+averagely earlier than tritons in smaller system-size col-
+lision. The system-size dependence of the velocity-gated
+momentum correlation functions is also clearly seen by
+their ratios in Fig. 11. With decreasing system-size, we
+can also observe an enhanced difference between the mo-
+mentum correlation functions for p-d (p-t or d-t) pair with
+vp > vd (vp > vt or vd > vt) and the ones with the reverse
+situation in Fig. 11 (b), (d) and (f).
+IV.
+SUMMARY
+In summary, with the AMPT model complemented
+by the Lednick´y and Lyuboshitz analytical method, we
+have constructed and analyzed the momentum correla-
+tion functions of light (anti)nuclei formed by the coa-
+lescence mechanism of (anti)nucleons for heavy-ion colli-
+sions with different system sizes and centralities at √sNN
+= 39 GeV. We present a comparison of proton−proton
+
+10
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+0
+50
+100
+150
+200
+250
+300
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+0
+50
+100
+150
+200
+250
+300
+(a) 39GeV AuAu ∆v = vp-vd 
+ 0-10%
+ 10-20%
+ 20-40%
+ 40-60%
+ 60-80%
+(b) 39GeV 0-10%  ∆v = vp-vd 
+ B
+ O
+ Ca
+ Au
+ C∆v>0/C∆v<0
+(c) 39GeV AuAu ∆v = vp-vt 
+(d) 39GeV 0-10%  ∆v = vp-vt 
+(e) 39GeV AuAu ∆v = vd-vt 
+q (MeV/c)
+(f) 39GeV 0-10%  ∆v = vd-vt 
+q (MeV/c)
+FIG. 11.
+Same as Fig. 8 but for p-d (a) and (b), p-t (c) and (d) and d-t (e) and (f) pairs.
+and proton−antiproton momentum correlation functions
+with the experimental data from the RHIC-STAR col-
+laboration [79, 80].
+Taking the same transverse mo-
+mentum and rapidity phase space coverage correspond-
+ing to the experimental situation as well as the maxi-
+mum hadronic rescattering time of 700 fm/c in AMPT,
+it is found that the p-p and p-¯p momentum correla-
+tion functions simulated by the present model can match
+the experimental data. We further study centrality and
+system-size dependence of momentum correlation func-
+tions for identical and nonidentical light (anti)nuclei
+pairs, respectively, which is in the condition of the
+maximum hadronic rescattering time of 100 fm/c in
+AMPT. The shape of momentum correlation functions
+for light (anti)nuclei pairs is consistent with previous
+works [13, 30, 36, 37, 79, 80], which is caused by both
+QS and FSI. The similar structure between light nuclei
+momentum correlation functions and anti-ones indicates
+that the interaction between them are the same, which
+has been confirmed in Ref. [30] only about proton and
+antiproton.
+The centrality dependence of momentum
+correlation functions for light (anti)nuclei is investigated
+by 197
+79 Au +197
+79 Au collisions at different five centralities
+of 0 − 10 %, 10 − 20 %, 20 − 40 %, 40 − 60 %, and
+60 − 80 % at √sNN = 39 GeV. It is found that with
+increasing centralities from center to periphery, the mo-
+mentum correlation functions for light (anti)nuclei be-
+come stronger, which are probably emitted from smaller
+source.
+The momentum correlation functions of light
+(anti)nuclei are sensitive to system-size through studying
+10
+5 B +10
+5 B, 16
+8 O +16
+8 O, 40
+20Ca +40
+20 Ca and 197
+79 Au +197
+79 Au in
+central collisions, and used to obtain the emission source-
+size of light (anti)nuclei which is self-consistent with their
+system-size.
+Momentum correlation functions between
+
+11
+nonidentical light nuclei can provide important informa-
+tion about the average emission sequence of them. The
+average emission time scale between neutrons and pro-
+tons is almost identical. However, heavier light clusters
+(deuterons or tritons) are emitted later than protons in
+the small relative momentum region. In the future we
+can explore further the energy dependence of the aver-
+age emission sequence of light nuclei and understand the
+physical interpretation.
+ACKNOWLEDGMENTS
+T. T. Wang thanks for discussion with Ms.
+Yi-
+Ling Cheng for the AMPT data.
+This work was
+supported in part by the National Natural Science
+Foundation of China under contract Nos.
+11890710,
+11890714, 11875066, 11925502, 11961141003, 11935001,
+12147101 and 12047514, the Strategic Priority Research
+Program of CAS under Grant No.
+XDB34000000,
+National Key R&D Program of China under Grant
+No.
+2018YFE0104600 and 2016YFE0100900, Guang-
+dong Major Project of Basic and Applied Basic Research
+No. 2020B0301030008, and the China PostDoctoral Sci-
+ence Foundation under Grant No. 2020M681140.
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+De Haas - van Alphen Effect under Rotation
+Shu-Yun Yang,1, ∗ Ren-Da Dong,1 De-Fu Hou,1, † and Hai-Cang Ren2, 1, ‡
+1Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS),
+Central China Normal University, Wuhan 430079, China
+2Physics Department, The Rockefeller University,
+1230 York Avenue, New York, NY 10021-6399
+Abstract
+We explored the interplay between magnetic field and rotation in the de Hass - van Alphen oscillation.
+The effect is found to be reduced because of the re-weighting of different states within the same Landau
+level by rotation energy. The implications of our results on high energy physics and condensed matter
+physics are speculated.
+∗ [email protected]
+† Co-corresponding author: [email protected]
+‡ Co-corresponding author: [email protected]
+1
+arXiv:2301.02857v1  [hep-ph]  7 Jan 2023
+
+I.
+INTRODUCTION
+The experimental activities for recent years regarding the polarization [1–7] and chiral
+magnetic effects [8–10] in off-central relativistic heavy ion collisions promoted theoretical
+research interests in a rotating thermodynamic system in a magnetic field [11–16]. The same
+physical conditions are also present in a neutron star [17–20]. One of the inteplay between
+the magnetism and rotation, the Barnett effect (or Einstein-de Haas effect) [21–23] has been
+considered in hydrodynamic modeling of the collisions.
+In this work, we examine another
+interplay between magnetism and rotation, i.e. the de Haas - van Alphen effect [24, 25] in
+a strongly degenerate rotating Fermi gas.
+Though purely theoretical at present stage, the
+implications are expected to shed light on the magnetic properties of the quark matter core, if
+exists, in a neutron star and/or the QGP droplet of generated in the RHIC STAR fixed target
+experiment, where the quark density is towards the strong degeneracy. The conclusion may
+also be tested directly in condensaed matter physics.
+De Haas-van Alphen effect is the consequence of charged fermions filling discrete but highly
+degenerate Landau levels [26, 27] in a magnetic field. In the absence of rotation, all degenerate
+Landau levels are equally populated at thermal equilibrium and the disceteness of different
+Laudau level is refelected in the thermodynamic limit as the oscillatory terms with respect to
+the chemical potential and the magnetic field in the thermodynamic potential, magnetization
+and magnetic susceptibility as well as some transport coefficients.
+When the system is in
+rotation, the thermodynamic equilibrium is established under a nonzero macroscopc angular
+momentum. The equal distribution of different angular momentum states within a Laudau
+level is offset by the nonzero angular velocity with higher angular momenta more favored than
+lower ones, which amounts to lifts the degeneracy of the Landau level. The dHvA oscillation
+is thereby expected to be reduced by the rotation. Consider a cylindrical volume of radius R
+with a constant magnetic field parallel to its axis, the states of each fermion is characterized
+by the z-component of the momentum q, the z-component of the angular momentum M, and
+the radial quantum number of the wave function, n(≥ 0). A Laudau levels corresponds M > 0,
+the cyclotron motion in classical picture, and all M > 0 are degenerate up to M ∼ eBR2,
+when the cyclotron orbit reaches the boundary. While the energy of a Landau level depends
+only on q and n. The nonzero angular velocity ω weight different M differently through the
+Boltzmann factor eMω/T in the ensemble of a macroscopic angular momentum. On the other
+hand, the requirement of subluminal linear speed on the boundary limits the radius of the
+2
+
+cylinder R < 1/ω and the thermodynamic limit R → ∞ is unrealistic and the degeneracy of
+the Landau levels becomes finite. We shall take the thermodynamic approximation by retaining
+the leading term in power in 1/R in the thermodynamic potential, keeping in mind ωR = O(1)1,
+and a sharp cutoff in the summation over angular momentum states within a Landau level is
+introduced to tak care of the finite size effect of the spectrum. Consequently, the implication
+of the rotation in the dHvA oscillation dependes on the size of the size of the system and the
+angular velocity. As we shall see, the dHvA is completely suppressed for typical parameters
+appropriate in a neutron star but may lead to observaservable effect for a cold and dense QGP
+fire ball created in future RHIC project. For a strongly degenerate non-relativistic electron gas,
+the reduction of the dHvA may be detectable in a rotating metallic sample.
+This paper is organized as follows.
+In section II, the dHvA term of an rotating ultra-
+relativistic quark gas is calculated and its implications is discussed.
+The same effect for a
+non-relativistic electron is examined in section III. Section IV concludes the paper.
+II.
+ULTRA RELATIVISTIC FERMI GAS
+A.
+Solution of Dirac Equation in Cylindrical Cooredinate
+For a massless fermion of electric charge e in a constant magnetic field ⃗B = Bˆz reads, the
+Hamiltonian in chiral representation reads
+H = −i⃗α · (⃗∇ − ieA) =
+�
+� −i⃗σ · (⃗∇ − ieA)
+0
+0
+i⃗σ · (⃗∇ − ieA)
+�
+�
+(1)
+where the vector potential
+⃗A = 1
+2
+⃗B × ⃗r
+(2)
+We adapt the circular gauge instead of Landau gauge for the convenience of investigating a
+rotating Fermi gas. As the fermions of opposite chiralities have identical spectrum, we shall
+focus one of them in what follows with the Hamiltonian
+H = −i⃗σ · (⃗∇ − ieA)
+(3)
+1 In this case the kinetic energy of rotation grows with the volume, like other extensive thermodynamic
+quantities.
+3
+
+and the eigenvalue equation Hχ = Eχ. For the ansatz of the two-component wave function χ
+in cylindrical coordinates, i.e.
+χ(⃗r) =
+�
+� f(ρ)ei(M− 1
+2)φ
+g(ρ)ei(M+ 1
+2)φ
+�
+� eiqz
+(4)
+we have the equations for the radial functions f(ρ) and g(ρ)
+�
+�
+�
+�
+�
+qf(ρ) − i
+�
+d
+dρ +
+M+ 1
+2
+ρ
+− 1
+2eBρ
+�
+g(ρ) = Ef(ρ)
+−i
+�
+d
+dρ −
+M− 1
+2
+ρ
++ 1
+2eBρ
+�
+f(ρ) − qg(ρ) = Eg(ρ)
+(5)
+where, q and M are the eigenvalue of the momentum and total angular momentum in the
+direction of the magnetic field with M = ±1/2, ±3/2, .... The equation (5) can be solved in
+terms of the generaliized Laguerre polynomial Lµ
+n(z) and we end up with the normalized wave
+function [28],
+χnMqs(⃗r) = 1
+2π
+�
+n!
+(n + m)!e− ζ
+2
+�
+�
+�
+eB(E+q)
+2E
+ζ
+m
+2 Lm
+n (ζ)eimφ
+iseB
+√
+E(E−q)ζm+1Lm+1
+n−1 (ζ)ei(m+1)φ
+�
+� eiqz
+(6)
+for M > 0, and
+χnMqs(⃗r) = 1
+2π
+�
+n!
+(n + |m|)!e− ζ
+2
+�
+�
+�
+eB(E+q)
+2E
+ζ
+|m|
+2 L|m|
+n (ζ)eimφ
+− iseB(n+|m|)
+√
+E(E−q) ζ
+(|m|−1)
+2
+L|m|−1
+n
+(ζ)ei(m+1)φ
+�
+� eiqz
+(7)
+for M < 0, where ζ ≡ 1
+2eBρ2, m ≡ M − 1/2, n = 0, 1, 2, ... and s = ±. The corresponding
+eigenvalue of energy is E = sEnMq with
+EnMq =
+�
+�
+�
+�
+�
+�
+2neB + q2
+for M > 0
+�
+2(n + |m|)eB + q2
+for M < 0
+(8)
+Care must be exercised for the case n = 0 of the solution (6) because of the nonexistence of
+Lm+1
+−1
+and the sigularity at E = −q. For E = ±q, eq.(5) becomes
+�
+�
+�
+�
+�
+�
+d
+dρ + m+1
+ρ
+− 1
+2eBρ
+�
+g(ρ) = i(±q − q)f(ρ)
+�
+d
+dρ − m
+ρ + 1
+2eBρ
+�
+f(ρ) = i(±q + q)g(ρ)
+(9)
+A normalizable solution exists only if E = q and reads
+χ0Mqs(⃗r) = 2m+1
+√π (eB)
+m+1
+2 ρme− 1
+4 eBρ2+imφ+iqz
+�
+� 1
+0
+�
+�
+(10)
+4
+
+with s = sign(q), which implies up(down) mover for positive(negative) energy solution. The
+wave function (7) corresponds to the classical motion along the cyclotron orbit and the spectrum
+(8) constitues the entire set of Landau levels and is responsible to magnetic properties including
+de Haas - van Alphen effect to be discussed below in thermodynamic approximation. The wave
+function (7) and the spectrum (8) is specific to the cylindrical coordinates and is subleading in
+the thermodynamic approximation as we shall see below.
+B.
+Thermodynamic Pressure
+The Hamiltonian of massless fermion field in a magnetic filed is given by
+H =
+�
+d3⃗rψ†Hψ
+(11)
+where H the single particle Hamiltonian (3) and the field operator
+ψ(⃗r) =
+�
+nMq
+ηnM(q)(anMqχnMq+(⃗r) + b†
+nM−qχnMq−(⃗r))
+(12)
+where
+ηnM(q) =
+�
+�
+�
+�
+�
+θ(q)
+for M > 0 and n = 0
+1
+otherwise
+(13)
+We have
+H =
+�
+n,M,q
+ηnM(q)EnMq(a†
+nMqanMq + b†
+nMqbnMq)
+(14)
+Correspondingly, the fermion number operator
+Q =
+�
+d3⃗rψ†ψ
+=
+�
+n,M,q
+ηnM(q)(a†
+nMqanMq − b†
+nMqbnMq)
+(15)
+and the angular momemtum projection operator
+Jz =
+�
+d3⃗rψ†
+�
+−i ∂
+∂φ + 1
+2σz
+�
+ψ
+=
+�
+n,M,q
+ηnM(q)M(a†
+nMqanMq − b†
+nMqbnMq)
+(16)
+5
+
+Consequently, the thermodynamic pressure at temperature T and chemical potential µ of a
+system rotating about z-axis with an angular velocity ω is
+P =T
+Ω
+�
+n=0,M>0,q>0
+[ln
+�
+1 + e−β(|q|−Mω−µ)�
++ ln
+�
+1 + e−β(|q|+Mω+µ)�
+]
++ T
+Ω
+�
+n=0,M>0,q
+[ln
+�
+1 + e−β(√
+q2+2neB−Mω−µ)�
++ ln
+�
+1 + e−β(√
+q2+2neB+Mω+µ)�
+]
++ T
+Ω
+�
+n̸=0,M>0,q
+[ln
+�
+1 + e−β(√
+q2+2(n+M+ 1
+2 )eB+Mω−µ)�
++ ln
+�
+1 + e−β(√
+q2+2(n+M+ 1
+2 )eB−Mω+µ)�
+]
+where we have switched the sign of M of the lower branch of the spectrum (8) for clarity. For
+a cylinder of radius R and length L, Ω = πR2L,
+�
+n,M,q
+(...) =
+1
+πR2
+� ∞
+−∞
+dq
+2π
+�
+n,M
+(...)
+(17)
+To avoid superluminal linear speed on the boundary, we require v ≡ ωR < 1. So the true
+thermodynamic limit R → ∞ is not attainable but we may still take the thermodynamic
+approximation for sufficiently large R by sorting the terms according to its power keeping in
+mind that ωR = O(1). For a finite R summation over M is limited. If follows from eqs. (6) and
+(7) that the square of the wave function for large M and finite n is peaked at the maximum
+of ρ2|m| exp
+�
+− 1
+2eBρ2�
+, which gives rise to ρ2 = 2|m|/(eB). When this ρ becomes comparable
+with R the finite size effect will distore the spectrum (8). Therefore, we introduce a cutoff for
+the summation over M, i.e.
+M ≤ Mc = [1
+2eBR2] >> 1
+(18)
+with [...] tuncate the argument inside to its integer part. As will be shown below, this cutoff
+produces the dHvA effect obtained from the Landau gauge in the absence of rotation. Without
+solving the boundary value problem of the edge states, we assume the uncertainty δMc = O(1)
+of the cutoff.
+Assuming strong degeneracy, µ >> T, the antiparticle contributions may be ignored 2 and
+2 To be cautious, let us examine whether the combination E ≡
+�
+q2 + 2(n + M + 1
+2)eB − Mω in the last
+term of (17) can become negative and compete with µ for large M.
+For the maximum M(= Mc), E >
+√2MceB − Mcω ≃ eBR(1 − v/2) > 0. The approximation of dropping the antiparticle contribution appears
+safe.
+6
+
+we end up with
+P = T
+πR2
+� ∞
+0
+dq
+4π
+�
+M>0
+ln
+�
+1 + e−β(|q|−Mω−µ)�
++
+T
+πR2
+� ∞
+−∞
+dq
+2π
+�
+n>0,M>0
+ln
+�
+1 + e−β(√
+q2+2neB−Mω−µ)�
++
+T
+πR2
+� ∞
+−∞
+dq
+2π
+�
+n,M>0
+ln
+�
+1 + e−β(√
+q2+2(n+M+ 1
+2 )eB+Mω−µ)�
+(19)
+where the contribution of the lowest Landau level has been isolated from higher Landau levels
+because different integration domain of q. The summation over M in the third term of (19)
+converges in the limit Mc → ∞ and thereby does not contribute to the thermadynamic limit
+and we are left with the Landau level terms only, i.e.
+P = T
+πR2
+� ∞
+0
+dq
+4π
+�
+M>0
+ln
+�
+1 + e−β(|q|−Mω−µ)�
++
+T
+πR2
+� ∞
+−∞
+dq
+2π
+�
+n>0,M>0
+ln
+�
+1 + e−β(√
+q2+2neB−Mω−µ)�
+≡ 1
+πR2PM
+(20)
+where
+PM = T
+� ∞
+0
+dq
+4π ln
+�
+1 + e−β(|q|−µM)�
++ T
+� ∞
+−∞
+dq
+2π
+�
+n>0
+ln
+�
+1 + e−β(√
+q2+2neB−µM)�
+(21)
+with µM = µ + Mω.
+C.
+de Haas - van Alphen Oscillation
+As the standard derivation of the de Haas - van Alphen (dHvA) effect, the summation over
+the Landau level index n can be carried out with the aid of the Poisson formula
+∞
+�
+n=0
+f(n) =
+� ∞
+0
+f(n)dn + 2Re
+∞
+�
+l=1
+� ∞
+0
+f(n)e2iπlndx
+(22)
+We have
+FM = F0M + 2Re
+∞
+�
+l=1
+FlM
+(23)
+where
+FlM = T
+� ∞
+−∞
+dq
+2π
+� ∞
+0
+dnei2πln ln
+�
+1 + e−β(√
+q2+2neB−µM)�
+(24)
+The dHvA oscillation resides in the second term of (23) and we shall focus on it.
+Transforming the integration variables from q, n to q, ϵ with ϵ =
+�
+q2 + 2neB, we find, via
+twice integration by part with respect to ϵ, that
+FlM = IlM + IIlM + IIIlM
+(25)
+7
+
+for l > 0, where
+IlM = ieBT
+4π2l
+� ∞
+−∞
+dq ln
+�
+1 + e−β(q−µM�
+),
+(26)
+IIlM =
+eB
+4iπ2l
+�
+eB
+πl
+� ∞
+−∞
+dqe−i lπ
+eB q2 φ
+��
+lπ
+eB|q|
+�
+eβ(q−µM) + 1
+(27)
+and
+IIIlM = − eB
+4iπ2l
+�
+eB
+lπ
+� ∞
+0
+dϵφ
+��
+lπ
+eB ϵ
+�
+βeβ(ϵ−µM)
+[eβ(ϵ−µM) + 1]2
+� ϵ
+−ϵ
+dqe−i lπ
+eB q2
+(28)
+with
+φ(z) ≡
+� ∞
+z
+dxeix2
+(29)
+IlM is imaginary thereby does not contribute to (23). Assuming the condition
+T ≪
+√
+eB ≪ µ
+(30)
+the leading terms of IIlM and IIIlM can be worked out and we ontain that
+IIlM =
+eB
+4π3l2
+�
+ln
+��
+4lπ
+eB µM
+�
++ 1
+2γE − iπ
+4
+�
+(31)
+with γE = 0.5772... the Euler constant (See Appendix A for the derivation), and
+IIIlM = −(eB)
+1
+2T
+4π
+e
+i
+�
+lπ2
+eB µ2
+M− π
+4
+�
+l3/2 sinh 2lπ2T(µ+Mω)
+eB
+.
+(32)
+where the integration formula
+� ∞
+−∞
+dx
+ex+iα
+(ex + 1)2 =
+πα
+sinh πα
+(33)
+and the asymptotic form
+φ(z) = i
+2zeiz2 + ... for z → ∞
+(34)
+have been employed to reduce IIIM. The dHvA osillation stems from IIIM. Summing over M,
+we end up with the dHvA term of the thermodynamic pressure under rotation, i.e.
+PdHvA ≡
+1
+πR2
+�
+M>0
+�
+2Re
+∞
+�
+l=1
+IIIlM
+�
+= −(eB)
+1
+2
+2π2R2
+∞
+�
+l=1
+1
+l3/2
+�
+M>0
+cos
+� lπ
+eB(µ + Mω)2 − π
+4
+�
+sinh 2lπ2T(µ+Mω)
+eB
+(35)
+In the absence of rotation, ω = 0, eq.(35) becomes
+PdHvA = −T(eB)
+3
+2
+4π2
+∞
+�
+l=1
+1
+l3/2
+cos
+� lπ
+eBµ2 − π
+4
+�
+sinh 2lπ2Tµ
+eB
+→ (eB)
+5
+2
+8π4µ
+∞
+�
+l=1
+1
+l5/2 cos
+� lπ
+eB µ2 − π
+4
+�
+(36)
+8
+
+in agreement with the expression derived from the Landau gauge.
+Eq.(35) can be further simplified at zero temperature, i.e.
+PdHvA = −(eB)
+3
+2
+4π4R2
+�
+M>0
+1
+µ + Mω
+∞
+�
+l=1
+1
+l5/2 cos
+� lπ
+eB (µ + Mω)2 − π
+4
+�
+(37)
+The angular velocity and magnetic field considered throught this work satisfy the condition
+ω <<
+√
+eB and the summation over M can be approximated by an integral. Consequently
+PdHvA ≃ − (eB)
+3
+2
+4π4R2ω
+� µ+Mcω
+µ
+dx1
+x
+∞
+�
+l=1
+1
+l5/2 cos
+� lπ
+eB x2 − π
+4
+�
+= −
+(eB)
+3
+2
+8
+√
+2π4R2ω
+∞
+�
+l=1
+1
+l5/2
+�
+Ci
+� lπ
+eB (µ + Mcω)2
+�
+− Ci
+� lπ
+eB µ2
+�
++Si
+� lπ
+eB (µ + Mcω)2
+�
+− Si
+� lπ
+eB µ2
+��
+≃ (eB)
+5
+2
+8π5R2ω
+∞
+�
+l=1
+1
+l7/2
+�
+sin
+� lπ
+eBµ2 − π
+4
+�
+µ2
+− sin
+� lπ
+eB(µ + Mcω)2 − π
+4
+�
+(µ + Mcω)2
+�
+(38)
+where Ci(z) and Si(z) are cosine and sine integrals and the last step follows from their
+asymptotic forms for z ≫ 1, i.e.
+�
+�
+�
+�
+�
+Si(z) ≈ π
+2 − cos z
+z
+Ci(z) ≈ sin z
+z
+(39)
+are employed in the last step. If the maximum rotation energy Mcω dominates, i.e. Mcω >> µ,
+the second term of (38) can be dropped and we have
+PdHvA ≃
+(eB)
+5
+2
+8π5µ2R2ω
+∞
+�
+l=1
+1
+l7/2 sin
+� lπ
+eB µ2 − π
+4
+�
+(40)
+and the uncertainty of Mc does not contribute.
+D.
+Numerical Estimates
+As pointed out in the introduction, the rotation will lift the degeneracy of states within
+each Landau level and thereby reduce the de Haas - van Alphen oscillation. In this section,
+we shall estimate the amount of reduction using the parameters appropriate for two realistic
+rotating ultra-relativistic fermion system in a magnetic field, the quark matter core and a QGP
+droplet at high baryon density. Since the Fermi gas approximation of these two system tends
+to be poor and the condition of the latter syetem is highly transient, we are not attempting
+to model the two system. The signifinace of our result below is only in the sense of order of
+9
+
+magnitude. For the ultra-relativistic system, we shall use mπ = 130MeV as the scale of the
+chemical potential and temperature and m2
+π = 1014G as the scale of the magnetic field. The
+estimate of the impact of the de Haas - van Alphen effect in a non-relativistic fermion system
+is deferred to the next section.
+The quark matter core of a neutron star
+μ2=10mπ
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-1.5×10-12
+-1. ×10-12
+-5. ×10-13
+0
+5. ×10-13
+1. ×10-12
+1.5×10-12
+eB/mπ
+2
+PdHvA
+Neutron Star
+ωR = 0.06
+ωR = 0.045
+ωR = 0.03
+ωR = 0.015
+FIG. 1. The oscillatory term of pressure P1 as a function of magnetic field eB
+m2π . Here, mπ = 140MeV,
+R = 1km.
+The radius of a neutron star is of the order of 10km and we assume a quark matter core
+made of light flavors of smaller radius R with a chemical potential of several hundreds of MeV,
+i.e. few times of pion’s rest energy, mπ. The magnetic field inside a neutron star can reach as
+high as 1015G, i.e. 1.4×10−3m2
+π. For the fastest spinning neutron star, PSR J1748-2446ad, the
+frequency is 716Hz and the linear speed at the boundary of the core is v ≃ 0.015 (in the unit
+of the speed of light). Consequently
+µ
+Mcω = µ
+mπ
+· m2
+π
+eB ·
+10−16
+R(km)v << 1
+(41)
+PdHvA
+PdHvA∥ω=0
+∼
+2
+µRv ≃
+3.86 × 10−16
+µ(MeV)R(km)v
+(42)
+for a typical neutron star. The approximation (40) is valid and we estimate
+PdHvA
+PdHvA∥ω=0
+∼
+2
+µRv ≃
+3.86 × 10−16
+µ(MeV)R(km)v
+(43)
+leading to huge suppression of dHvA oscillation.
+The thermodynamic pressure at µ2 = 10m2
+π and zero temperature versus magnetic field
+0 < eB < 0.01m2
+π is plotted in Fig. 1 for several linear speeds at the boundary of the rotating
+quark matter core. As a benchmark, the thermodynamic pressure in the absence of rotation is
+10
+
+ω=0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-150000
+-100000
+-50000
+0
+50000
+100000
+150000
+eB/mπ
+2
+PdHvA
+μ2 = 20 mπ2
+μ2 = 15 mπ2
+μ2 = 10 mπ2
+μ2 = 5 mπ2
+FIG. 2. The oscillatory term of pressure P1 as a function of magnetic field eB
+m2π . Here, ω = 0 and T = 0.
+displayed in Fig. 2. The parameters underlying both figures satisfy the approximation condition
+(30) for the analytic expressions. The effect is suppressed by 17 order of magnitude.
+A cold and dense QGP droplet
+μ2=10mπ
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-150000
+-100000
+-50000
+0
+50000
+100000
+150000
+eB/mπ
+2
+PdHvA
+R = 10 fm
+ωR = 0.03
+ωR = 0.02
+ωR = 0.01
+ωR = 0
+FIG. 3. The oscillatory term of pressure P1 as a function of magnetic field
+eB
+m2π . Here, we fix the
+chemical potential µ2 = 10m2
+π and the radius is R = 10fm.
+The suppression of dHvA in a neutron star may be attributed to its large size.
+Let us
+switch to a cold and dense QGP droplet where the suppression of dHvA oscillation with the
+angular velocity becomes modest. The dHvA term of the thermodynamic pressure of eq.(38) for
+R = 10fm versus the magnetic field at fixed chemical potential and temperature and is plotted
+for several angular velocity including ω = 0 in Fig. 3. The same equation at fixed chemical
+potential and a nonzero angular velocity is plotted for several temperatures in Fig. 4. The dHvA
+without rotation, eq.(36) at the same chemical potential and the same set of tempertatures is
+plotted in Fig. 5 for reference. Notice that the suppression of dHvA with temperature becomes
+milder with ω ̸= 0. The selection of the size, chemical potential and the magnetic field is
+11
+
+μ2=20mπ
+2
+0.30 0.32 0.34 0.36 0.38 0.40
+-1.5×10-11
+-1. ×10-11
+-5. ×10-120
+5. ×10-12
+1. ×10-11
+1.5× 10-11
+2. ×10-11
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+eB/mπ
+2
+PdHvA
+R = 10 fm
+T = 56 MeV
+T = 54 MeV
+T = 52 MeV
+T = 50 MeV
+FIG. 4. The oscillatory term of pressure
+P1
+(eB/m2π)30 as a function of magnetic field eB
+m2π . Here, we fix the
+chemical potential µ2 = 10m2
+π, v = 0.01 and the radius is R = 10fm.
+μ2=20 mπ
+2
+0.80
+0.85
+0.90
+0.95
+1.00
+-2. ×10-9
+-1. ×10-9
+0
+1. ×10-9
+2. ×10-9
+eB/mπ
+2
+PdHvA
+ω = 0
+T = 56 MeV
+T = 54 MeV
+T = 52 MeV
+T = 50 MeV
+FIG. 5. The oscillatory term of pressure
+P1
+(eB/m2π)30 as a function of magnetic field eB
+m2π . Here, we fix the
+chemical potential µ2 = 10m2
+π, and ω = 0.
+motivated by the conditions of the current heavy ion collisions in RHIC and LHC.
+While the RHIC STAR fixed target experiment is expected to generate QGP of lower energy
+and higher bayon density, i.e., closer to the density axis of the QCD phase diagram, there may
+still be a gap to meet the condition of the cold and dense QGP described above. Even it did,
+the rapid expansion would hinder the observability of the effect because of non-equilibrium. So
+our discussions here are highly speculative.
+III.
+NON RELATIVISTIC FERMI GAS
+The Hamiltonin of a non-relativistic electron reads
+H = − 1
+2me
+(⃗∇ − ie ⃗A)2 + 1
+2σzωB
+(44)
+12
+
+with the vector potential
+⃗A = 1
+2Bˆz × ⃗r,
+(45)
+where ωB = eB/me is the cyclotron frequency and σz = diag.(1, −1).
+The spectrum in
+cylindrical coordinates can be found in many textbook of quantum mechnics and are given
+by
+Enmqσ =
+q2
+2me
++
+�
+n + m − |m|
+2
++ 1
+2
+�
+ωB + 1
+2σωB
+(46)
+where q is the momentum along z-direction, n = 0, 1, 2, ... are radial quantum number and
+m=0,±1, ±2, ...,±Mc are the z-component of the orbital angular momentum and σ = ±
+labels spin projections. The Landau levels correspond to m ≥ 0 and are labeled by n. The
+corresponding wave function reads
+ψnmqσ(⃗r) =
+�
+n!eB
+2π(n + |m|)!Lζ
+|m|
+2 e− ζ
+2L|m|
+n (ζ)ei(mφ+qz)
+(47)
+In a cylinder of finite radius, the thermodynamic approximation limits the azimuthal quantum
+number as (18), i.e.
+|m| < mc = [1
+2eBR2] >> 1.
+(48)
+with an uncertainty δmc = O(1) as in the ultra-relativistic case.
+A.
+Thermodynamic Pressure and dHvA
+For a free non-relativistic electron gas, the dHvA can be extracted using the same Poisson
+formula (22) as in most of the textbooks in solid state physics. Here we adapt a more elegant
+approach via Mellin transformation [29].
+The thermodynamic pressure of the electron gas in a rotating cylindrical volume of radius
+R and length Lz reads
+P =
+1
+πR2
+�
+m
+Pm(ζm)
+(49)
+where
+Pm(ζm) = T
+Lz
+�
+n,q,σ
+ln
+�
+1 + 1
+ζm
+e−βEqnmσ
+�
+(50)
+with ω the angular velocity and
+ζm = e−β(µ+mω)
+(51)
+13
+
+The case of strong degeneracy corresponds to ζm << 1. The Mellin transformation of the
+function Pm(ζ) with respect to ζ is given by
+Q(s) =
+� ∞
+0
+dζζs−1Pm(ζ)
+=
+πT
+Lzs sin πs
+�
+n,q,σ
+e−sβ(Enmqσ− 1
+2 σω)
+(52)
+for 0 < Res < 1. The last equality follows from an integration by part and the formula
+� ∞
+0
+dx xs−1
+x + 1 =
+π
+sin πs
+(53)
+For the same reason as in the relativistic case, the contribution from m < 0 is subleading in
+the thermodynamic approximation and we focus only on the branch m ≥ 0 of the spectrum.
+We have for m ≥ 0
+FIG. 6. Contour integration [29].
+Q(s) =
+πT
+Lzs sin πs
+�
+q
+e− sβq2
+2me
+�
+n,σ
+e−(n+ 1
+2)sβωB− 1
+2 σsβ(ωB−ω)
+=
+πT
+λs3/2 sin πs
+cosh 1
+2sβ(ωB − ω)
+sinh 1
+2sβωB
+(54)
+where λ =
+�
+2π/(mT) is the thermal wavelength. It follows from the Mellin inversion formula
+that
+Pm(ζ) =
+� c+i∞
+c−i∞
+ds
+2πiζ−sQ(s)
+(55)
+with 0 < c < 1. The integrand on the complex s-plane consists of a branch cut running along
+the negative real axis, poles along both real and imaginary axes, i.e.
+s = l
+s = 2lπT
+ωB
+i
+(56)
+14
+
+Im s
+21元T
+wB
+Re swith l = 0, ±1, ±2, .... Closing the contour from the left as shown in Fig.6 for ζ < 1, we find
+Pm(ζ) = Im(ζ) + IIm(ζ)
+(57)
+where Im is the integral around the branch cut and IIm stems from the poles along the imaginary
+axis. The former contributes to the Landau diamagnetism and Pauli paramagnetism along with
+the Barnett effect and the latter gives rise to dHvA oscillation. Summing up the residues of
+the poles within the contour, we end up with
+IIm(ζm) = 2T
+λ
+� ωB
+2πT
+∞
+�
+l=1
+1
+l3/2csch2lπ2T
+ωB
+cos lπω
+ωB
+cos
+�2lπ(µ + mω)
+ωB
+− π
+4
+�
+(58)
+Summing up the orbital angular momentum, we obtain that
+PdHvA =
+1
+πR2
+mc
+�
+m=0
+IIm
+= −T(meωB)1/2
+π2R2
+∞
+�
+l=1
+cos lπω
+ωB
+sin
+�
+2lπµ
+ωB − lπω
+ωB − π
+4
+�
+− sin
+�
+2lπµ
+ωB + lπω
+ωB − π
+4 + 2lπmcω
+ωB
+�
+l3/2 sinh 2lπ2T
+ωB sin lπω
+ωB
+(59)
+Without rotation, ω = 0, the well-known dHvA formula
+PdHvA|ω=0 = −T(meωB)3/2
+2π2
+∞
+�
+l=1
+1
+l3/2csch2lπ2T
+ωB
+cos
+�2lπµ
+ωB
+− π
+4
+�
+(60)
+emerges. At zero temperature, eq. (59) becomes
+PdHvA|T=0 = −(meωB)3/2
+4π4meR2
+∞
+�
+l=1
+cos lπω
+ωB
+sin
+�
+2lπµ
+ωB + lπω
+ωB − π
+4 + lπmeωR2�
+− sin
+�
+2lπµ
+ωB − lπω
+ωB − π
+4
+�
+l5/2 sin lπω
+ωB
+≃ − (meωB)5/2
+4π5m2
+eωR2
+∞
+�
+l=1
+1
+l7/2
+�
+sin
+�2lπµ
+ωB
+− π
+4 + 2lπmcω
+ωB
+�
+− sin
+�2lπµ
+ωB
+− π
+4
+��
+(61)
+where the approximation ω << ωB is made for the typical parameters in condensed matter
+physics. This expression is to be compared with the zero temperature limit of (62), i.e.
+PdHvA|ω=0 = −(meωB)5/2
+4π4
+∞
+�
+l=1
+1
+l5/2 cos
+�2lπµ
+ωB
+− π
+4
+�
+.
+(62)
+At this point, it is interesting to compare the non-relativistic dHvA and the ultra-relativistic
+dHvA. As shown in eq.(46), given q and σ, the non-relativistic Landau levels (m>0) are
+equally spaced while the spacing between successive ultra-relativistic Landau levels in the upper
+equation of (8) decreases with the label n. Since the dHvA is sensitive to the energy levels
+around the chemical potential µ, the amplitude of the oscillation is expected to be independent
+15
+
+of µ in the non-relativistic case but decreases with µ in the ultra-relativistic case as reflected
+in the large µ suppression by sinh 2lπ2Tµ
+eB
+of (36) in the latter case. When rotation is turned
+on, the effective chemical potential increases with the angular momentum quantum number.
+Consequently, the non-relativistic dHvA appears less vulnerable than the ultra-relativistic one.
+B.
+Numerical Estimates
+The electron gas in a good metal at room temperature, T ∼ 1/40eV can be well approximated
+by a free Fermi in the strong degeneracy limit. The chemical potential is of 1 ∼ 10eV, which
+makes µ/T ∼ 40 ∼ 400 >> 1 and the zero temperature approximation works well. For a
+magnetic field up to few Tesla’s and an angular velocity is Hz, we have
+ω/ωB ≃ 5.57 × 10−12 ω(Hz)
+B(Tesla)
+(63)
+justifying the approximation made in the (61) for mechanical rotation achievable in laboratory.
+The same condition also makes the contribution of the uncertainty in the angular momentum
+cutoff mc to the phase of the oscillation in (59) and (61) negligible. The dHvA oscillation is
+expected to be significantly reduced when the largest rotation energy mcω within a Landau
+level exceeds the spacing between successive levels, ωB. With R in cm, the linear velocity of
+the corcumference v = ωR in terms of cm/s, it follows from (48) that
+mcω
+ωB
+≃ 0.43Rv,
+(64)
+independent of the magnetic field.
+ω = 0
+1.00000
+1.00002
+1.00004
+1.00006
+1.00008
+1.00010
+-0.0004
+-0.0003
+-0.0002
+-0.0001
+0.0000
+0.0001
+0.0002
+B(T)
+PdHvA
+μ = 7 eV
+μ = 5 eV
+μ = 3 eV
+μ = 1 eV
+FIG. 7. The oscillatory term of non-relativistic pressure P1 as a function of magnetic field B when
+T = 0 and ω = 0.
+16
+
+ωR = 2 cm/s
+1.00000
+1.00002
+1.00004
+1.00006
+1.00008
+1.00010
+-0.00010
+-0.00005
+0.00000
+0.00005
+B(T)
+PdHvA
+R = 1 cm
+μ = 7 eV
+μ = 5 eV
+μ = 3 eV
+μ = 1 eV
+FIG. 8. The oscillatory term of non-relativistic pressure P1 as a function of magnetic field B when
+T = 0. Here, we fix ωR = 2cm/s and R = 1 cm .
+μ = 5 eV
+1.00000
+1.00002
+1.00004
+1.00006
+1.00008
+1.00010
+-0.0003
+-0.0002
+-0.0001
+0.0000
+0.0001
+0.0002
+B(T)
+PdHvA
+R = 1 cm
+ωR = 6 cm/s
+ωR= 4 cm/s
+ωR = 2 cm/s
+ωR = 0
+FIG. 9. The oscillatory term of non-relativistic pressure P1 as a function of magnetic field B when
+T = 0. Here, we fix the chemical potential µ = 5eV and the radius is R = 1cm.
+The dHvA term of the thermodynamic pressure of a strongly degenerate electron gas versus
+magnetic field for a long cylinder of radius R = 1cm at T = 0 is plotted in Fig. 7, Fig. 8 and
+Fig. 9. The magnetic field varies in a small neighborhood of 1T and the angular velocity is taken
+such that RHS of (64) is of order one. The dHvA effect without rotation, eq.(62), for different
+chemical potentials is shown in Fig. 7 sas benchmark. The parallel setup for ωR = 2cm/s,
+eq.(61), is shown in Fig. 8 with similar profiles. More important is Fig. 9 where dHvA at
+different ωR is displayed and the suppression of the oscillation by rotation is evident.
+IV.
+CONCLUDING REMARKS
+Let us recaptulate what we presented in preceding sections. We examined the robustness
+of the de Haas-van Alphen effect in a strongly degenerate Fermi gas under rotation.
+We
+17
+
+derived the formula for dHvA oscillation in an long cylinder rotating about its axis in the
+ultra-relativistic limit and non-relativistic limit. As the macroscopic degeneracy of Landau
+levels is offset by rotation energy of states of different angular momentum within each Landau
+level. The amplitude of the scillation is reduced. The amount of reduction depends on the
+angular velocity ω and the radius of the cylinder R and the oscillation is expected to become
+insignificant for sufficiently large ω and R. The ultra-relativistic dHvA appear more vulnerable
+than the non-relativistic one because of decreasing Landau level spacing with energy.
+Applying the ultra-relativistic formula to estimate dHvA with typical parameters of a
+neutron star, and with typical parameters of a cold and dense QGP droplet, we noted that
+the dHvA oscillation is completely suppressed in the former case and remains in the latter.
+The non-relativistic formula, on the other hand showed that for a typical electron gas in a
+good metal, the variation of dHvA oscillation with angular velocity appears detectable, via
+magnetization and/or magnetic susceptibility.
+As self-criticism, our approximation of the finite size effect by introducing the maximum
+angular momentum within a Landau level in (18) and (48) may be crude. Limited by the
+analytical tractability, the cylindrical shape of the system is not suitable to model a neutron
+star or a QGP droplet. Though the effect is expected to remain for a Fermi liquid, the strong
+correlation in quark matter may modify significantly the quantitative prediction. In this sense,
+our result is very preliminary.
+ACKNOWLEDGMENTS
+We thank Ren-Hong Fang for fruitful discussions. This work is supported by the National
+Key Research and Development Program of China (No. 2022YFA1604900). This work also
+is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos.
+11735007, 11890711, 11890710, 12275104.
+Appendix A: Appendix
+For µ >> T, eq.(55) can be approximated as
+IIlM ≃
+1
+2iπ2l
+�
+eB
+lπ
+� µM
+0
+dqe−i lπ
+eB q2φ
+��
+lπ
+eB q
+�
+=
+eB
+2iπ3l2J
+(A1)
+18
+
+where
+J =
+� K
+0
+dxe−ix2φ(x) =
+� K
+0
+dxe−ix2 � ∞
+x
+dξeiξ2
+(A2)
+with K =
+�
+lπ
+eBµM. Introducing ξ = xt, we find
+J =
+� K
+0
+dxe−ix2x
+� ∞
+1
+dteit2x2 = 1
+2i
+� ∞
+1
+dteiK2(t2−1) − 1
+t2 − 1
+= −1
+2K2
+� ∞
+1
+dteiK2(t2−1)t ln t − 1
+t + 1
+(A3)
+where the last equality follows from an integration by part. Introducing z = t2 − 1, we have
+J = −1
+4K2
+� ∞
+0
+dzeiK2z ln
+√z + 1 − 1
+√z + 1 + 1
+(A4)
+If follows from the Jordan lemma that integration path can be rotated to the imaginary axis
+on the z− plane and we end up with
+J = − i
+4K2
+� ∞
+0
+dye−K2y ln
+√1 + iy − 1
+√1 + iy + 1
+(A5)
+For K >> 1, we have
+J ≃ − i
+4K2
+� ∞
+0
+dye−K2y ln iy
+4 = i
+2
+�
+ln(2K) + 1
+2γE
+�
++ π
+8
+(A6)
+This gives rise to RHS of (A1).
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+

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+Information content of note transitions in the music of J. S. Bach
+Suman Kulkarni,1 Sophia U. David,2, 3 Christopher W. Lynn,4, 5 and Dani S. Bassett1, 2, 6, 7, 8, 9, ∗
+1Department of Physics & Astronomy, College of Arts & Sciences,
+University of Pennsylvania, Philadelphia, PA 19104, USA
+2Department of Bioengineering, School of Engineering & Applied Science,
+University of Pennsylvania, Philadelphia, PA 19104, USA
+3Department of Psychology, Yale University, New Haven, CT 06520, USA
+4Initiative for the Theoretical Sciences, Graduate Center,
+City University of New York, New York, NY 10016, USA
+5Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544, USA
+6Department of Electrical & Systems Engineering, School of Engineering & Applied Science,
+University of Pennsylvania, Philadelphia, PA 19104, USA
+7Department of Neurology, Perelman School of Medicine,
+University of Pennsylvania, Philadelphia, PA 19104, USA
+8Department of Psychiatry, Perelman School of Medicine,
+University of Pennsylvania, Philadelphia, PA 19104, USA
+9Santa Fe Institute, Santa Fe, NM 87501, USA
+(Dated: January 3, 2023)
+Music has a complex structure that expresses emotion and conveys information. Humans process
+that information through imperfect cognitive instruments that produce a gestalt, smeared version of
+reality. What is the information that humans see? And how does their perception relate to (and dif-
+fer from) reality? To address these questions quantitatively, we analyze J. S. Bach’s music through
+the lens of network science and information theory. Regarded as one of the greatest composers in
+the Western music tradition, Bach’s work is highly mathematically structured and spans a wide
+range of compositional forms, such as fugues and choral pieces. Conceptualizing each composition
+as a network of note transitions, we quantify the information contained in each piece and find that
+different kinds of compositions can be grouped together according to their information content.
+Moreover, we find that Bach’s music is structured for efficient communication; that is, it commu-
+nicates large amounts of information while maintaining small deviations of the inferred network
+from reality. We probe the network structures that enable this rapid and efficient communication of
+information—namely, high heterogeneity and strong clustering. Taken together, our findings shed
+new light on the information and network properties of Bach’s compositions. More generally, we
+gain insight into features that make networks of information effective for communication.
+I.
+INTRODUCTION
+From Tibetan throat singing to Scottish piobaireachd
+to modern hip hop, music is a universal aspect of human
+culture, enjoyed by people of all ages from all around the
+world. It has even been proposed that music is a funda-
+mental part of being human [1]. The earliest confirmed
+musical instruments are nearly 40,000 years old, and evi-
+dence suggests that vocal music began much earlier [2, 3].
+While it is a point of controversy, some scientists believe
+that communication through music arose even before lan-
+guage [1, 4, 5]. Though styles, sounds, and instruments
+vary drastically from one culture and time period to an-
+other, it is indisputable that music has had a substantial
+impact on the development of humans and society [6, 7].
+Making and listening to music is more than just a recre-
+ational activity. Music is a medium of communication.
+Through music we can tell stories [8], convey messages
+[9], and imbue the strongest of emotions [10–12]. It is
+∗ To
+whom
+correspondence
+should
+be
+addressed.;
+[email protected]
+a common human experience to feel pensive or despon-
+dent after hearing a slow song in a minor key or to feel
+carefree or energized after hearing an upbeat song in a
+major key. But how does something as abstract as mu-
+sic communicate so much? Past literature has discussed
+music in terms of expectation and surprise [13, 14]. In or-
+der to be evolutionarily successful, our brains are adept
+at forming expectations based on prior events.
+When
+these expectations are contradicted by an experience, we
+feel surprised. With surprise can come a host of other
+emotions. We may feel relief when the dissonant sound
+we expected was actually consonant, or we may feel dis-
+tress when the musical resolution we expected did not
+occur [15]. But how do we quantify these expectations
+and surprises? How do we mathematically formalize the
+communicative success of a piece of music? Fundamen-
+tally, music is comprised of fleeting and elusive sounds,
+and hence may appear hard to measure. Even written
+on a page, the jumble of notes, rests, dynamic markings,
+and multilingual commands is daunting to describe with
+mathematical rigor.
+Here, we seek to extract order from music’s complexity
+by examining music through the lens of network science.
+A network consists of nodes and edges—representing en-
+arXiv:2301.00783v1  [physics.soc-ph]  2 Jan 2023
+
+2
+tities and the connections between them, respectively.
+Conceptualizing each note as a node and each transition
+between two notes as an edge, we can build a network for
+any piece of music. Using music networks, we provide a
+comprehensive analysis of Bach’s compositions. Bach is
+a natural case study given his prolific career, the wide
+appreciation his compositions have garnered, and the in-
+fluence he had over contemporaneous and subsequent
+composers. His diverse compositions (from chorales to
+fugues) for a wide range of musicians (from singers to
+orchestra members) often share a fundamental underly-
+ing structure of repeated—and almost mathematical—
+musical themes and motifs.
+These features of Bach’s
+compositions make them particularly interesting to study
+using a mathematical framework.
+As we listen to music, we form expectations.
+Upon
+hearing a particular note, we anticipate which notes
+might come next based on past transitions.
+The less
+likely the outcome, the more surprised we are upon hear-
+ing it. This “suprisal” can be quantified by the Shan-
+non information entropy [16].
+Ideas from information
+theory have led to illuminating insights in a wide range
+of settings, including language [17, 18], social networks
+[19, 20], transportation patterns [21] and music [22, 23].
+We draw upon these ideas to shed light on the features of
+Bach’s music that make it successful in communicating
+information to the human mind. Prior research has at-
+tempted to quantitatively identify patterns and features
+that might be present across different kinds of music [24–
+27]. However, understanding how humans perceive these
+patterns is more nuanced and complex than simply eval-
+uating the structure of compositions because humans are
+not perfect learners. Rather, humans assimilate patterns
+of information presented to them through imperfect per-
+ceptual systems, sacrificing the accuracy of their internal
+representation to conserve computational energy [28–30].
+This trade-off between the accuracy of the inferred transi-
+tion structure and the computational cost involved in its
+formation results in a slightly distorted version of tran-
+sition networks. The “learned” version of a network can
+be calculated using previous models of human percep-
+tion [31, 32].
+Networks for which the inferred version
+maintains a low deviation from the true network can be
+considered efficient in communicating information. This
+framework thus provides insight into the communicative
+success of a network, from the point of view of how the
+network interacts with our imperfect perceptual systems.
+In this work, we apply information theory to note tran-
+sition networks constructed from Bach’s musical compo-
+sitions. We seek to quantify the amount of information in
+these networks and understand what patterns or features
+allow these networks to successfully hold and accurately
+convey information. We begin by studying the informa-
+tion entropy of each piece.
+Here, we find that Bach’s
+music contains more information than expected from typ-
+ical (or random) transition structures. Strikingly, certain
+composition forms are clustered together based on their
+information content. We hypothesize that the higher in-
+formation content in Bach’s music and the differences
+observed across musical pieces can be explained by the
+heterogeneity in node degrees (or the number of distinct
+pitches that follow a given note). Next, to determine how
+accurately the transition structure of a composition can
+be inferred by a human observer, we use a free energy
+model of how humans perceive networks of information.
+We hypothesize that Bach’s music networks maintain a
+low deviation between the learned and original network,
+and this property is driven by tight clustering in the net-
+work. Additionally, we find that certain compositional
+forms can be distinguished based on the discrepancies
+between the original and the inferred network. Our find-
+ings illuminate how these music networks are structured
+to convey large amounts of information rapidly and accu-
+rately, thereby supporting successful communication. By
+performing this systematic study of how the information
+in a complex system, like music, is structured and per-
+ceived by humans, our work provides a new perspective
+on how humans experience the world around them.
+II.
+MUSIC AS A NETWORK OF NOTE
+TRANSITIONS
+We study a wide range of Bach’s compositions in-
+cluding: preludes, fugues, inventions, cantatas, English
+suites, French suites, chorales, Brandenburg concertos,
+toccatas, and concertos (see Materials and Methods sec-
+tion A 1 for further details).
+The audio files for these
+pieces were collected and read in MIDI format, from
+which the sequence of notes was extracted. Each note
+present in a piece is represented as a node in the network,
+with notes from different octaves represented as distinct
+nodes. The transitions between notes are calculated sep-
+arately for different instruments. If there is a transition
+from note i to note j, then we draw a directed edge from
+node i to node j (see Fig. 1A). For chords, where multiple
+notes occur at the same time, edges are drawn between
+all notes in the first chord to all notes in the second chord.
+To simplify our analysis, we remove any self loops in the
+network, thereby restricting ourselves to understanding
+the structure of transitions to the next different note in
+the piece. We begin by examining unweighted networks
+of note transitions to focus on how the network structure
+alone impacts the information content and perception of
+a musical piece. After understanding the skeleton of the
+transitions, we then add weights to the edges based on
+how frequently various transitions occur. This procedure
+allows us to disentangle the effects of the network struc-
+ture (comprising the set of possible note transitions) and
+edge weights (comprising the note transition probabili-
+ties).
+III.
+QUANTIFYING THE INFORMATION IN
+NETWORKS
+We seek to measure the amount of information pro-
+duced by a sequence of notes. Although note sequences
+
+3
+F
+D
+G
+D
+B
+C
+A
+B
+G
+E
+G
+E
+G
+E
+C
+G
+A
+B
+E
+G'
+D
+F
+E'
+C
+G
+A
+B
+E
+G'
+D
+F
+E'
+B
+A
+E
+D
+G'
+E'
+Model information production using random walks
+(iii) Network Entropy:
+(ii) Node-level Entropy:
+Model human perception using free energy principle
+C
+G
+A
+B
+E
+G'
+D
+F
+E'
+B.
+C.
+A
+High 
+Low
+Original Network
+Inferred Network
+Low
+High 
+(i)
+FIG. 1. By treating music as a network of note transitions, we build a model for how information is produced
+and the network is perceived by humans. (A) An example of a network constructed from a musical piece using the
+method described in our paper. At the top, we show a toy musical piece. Below, we show the network in which notes are nodes
+and transitions between notes, whether isolated or played simultaneously as part of a chord, are directed edges. The direction
+of the edge matches the temporal direction of the transition. (B) The model of information production using random walks. (i)
+An example of a random walk on the network of note transitions is shown using the blue dotted line. At each node, the walker
+chooses an outgoing edge to traverse, each weighted with equal probability. This walk generates a sequence of notes as shown
+below. (ii) The amount of information, or the entropy, generated when a walker traverses an edge from a node depends on the
+number of edges emanating from the node (called the degree of the node). When traversing nodes with a high versus low degree,
+the walker has more choices for which edge to pick and hence, such a transition generates more information. Thus, nodes with
+a higher degree (right) are said to have higher entropy than nodes with a low degree (left). (iii) To calculate the entropy of the
+entire network, one needs to weigh the contribution of each node by the probability that a walker will occupy it. For networks
+with the same average degree, those with a wider range of degrees (right) have a higher entropy than those with a narrower
+range of degrees (left). (C) The model for how humans form internal estimates of the network. Humans perceive sequences of
+information presented to them through imperfect perceptual systems, which results in an imperfect internal representation of
+the network (left). This inexact inferred version of the network contains extra edges due to biases that stem from imperfect
+perception. The color bar indicates the weight assigned to an edge. Based on models for this fuzzy perception, humans are
+most likely to jumble up transitive relationships, as shown on the right. Therefore, networks with a large number of these
+triangular clusters are resilient to the inaccuracies in human perception and are easier to learn.
+can have long-range temporal dependencies [33, 34], as a
+first analytical step, we focus on the Markov transition
+structure. That is, we study the information contained
+in individual note transitions. This information is quan-
+tified by the Shannon entropy of a random walk on the
+network [16, 35] (Fig.
+1B; see also the Materials and
+Methods section A 2 for further details). Given a net-
+work of transitions, the contribution of the ith node to
+the entropy can be written in terms of the entries of the
+transition probability matrix P as:
+Si = −
+�
+j
+Pij log Pij.
+(1)
+In the case of directed unweighted networks, Pij
+=
+1/kout
+i
+, where kout
+i
+is the out-degree of the node. Hence,
+for unweighted networks, the node-level entropy is Si =
+log (kout
+i
+), which is solely determined by the out-degree.
+To calculate the entropy of the entire network, the con-
+tributions of the nodes are weighted by their stationary
+distribution—the probability that a walker ends up at
+node i after infinite time—which we denote by πi. The
+entropy of the network is then [35]:
+S =
+�
+i
+πiSi = −
+�
+i
+πi
+�
+j
+Pij log Pij.
+(2)
+For undirected and unweighted networks, the stationary
+distribution has a simple analytical form πi = ki/2E,
+where ki is the degree of node i, and E is the total number
+of edges. The network entropy is then:
+S = 1
+2E
+�
+i
+ki log ki.
+(3)
+By contrast, for directed networks the stationary dis-
+tribution depends on the detailed structure of the net-
+work and cannot be written in closed form. Hence, for
+our directed music networks, we calculate the stationary
+
+Model information production using random walks
+Node-level Entropy:
+G
+F
+Si=
+>Pii log Pit
+D
+D
+C
+B)
+(B
+G
+Network Entropy:
+E
+TiPii log Pi
+ Model human perception using free energy principle4
+A
+B
+C
+FIG. 2. Quantifying the information of Bach’s music using the entropy of random walks on networks of note
+transitions. (A) Entropy of Bach’s music networks (Sreal) compared with random networks of the same size (Srand). We report
+the entropy of the corresponding random networks after averaging over 100 independent realizations. The error bars for Srand
+indicate the standard error of the sample. (B) The entropy of Bach’s music networks (Sreal) compared with random networks
+that preserve the in- and out-degree of each node (Sdeg). We report the entropy of the corresponding degree-preserving random
+networks after averaging over 100 independent realizations. The error bars for Srand indicate the standard error of the sample.
+(C) The entropy of the chorales as a function of the average in-degree heterogeneity Hin = Var(kin)/⟨kin⟩ (top) and out-degree
+heterogeneity Hout = Var(kout)/⟨kout⟩ (bottom) of the networks. In panels (A) and (B), each data point represents a single
+piece. Color and marker indicate the type of piece, as shown in the legend. The dashed line represents the line y = x. In panel
+(C), the dotted line indicates the best linear fit, and the reported rs value is the Spearman correlation coefficient.
+distribution numerically and use Eq. 2 to compute the
+entropy of each piece.
+To understand the amount of information produced
+by the music networks, we compare them to random-
+ized (or “null”) networks with equal number of nodes
+and edges (see the Materials and Methods section A 5 for
+details on generating null networks). If the note transi-
+tions in Bach’s music do have distinct properties that al-
+low them to communicate a large amount of information,
+then we would expect Bach’s networks to contain more
+information than random transition structures. By aver-
+aging over 100 random networks for each piece, we find
+that the real networks have consistently higher entropy—
+thereby containing more information—than their ran-
+dom counterparts (Fig.
+2A). Moreover, by comparing
+across pieces, we observe that the different kinds of com-
+positions cluster together based on their entropy. The
+chorales, typically meant to be sung by groups in eccle-
+siastical settings, have a markedly lower entropy than
+the rest of the compositions studied. By contrast, the
+toccatas and preludes have a much higher entropy. It is
+possible that the chorales’ functions of meditation, adora-
+tion, and supplication are best supported by predictabil-
+ity and hence low entropy, whereas the entertainment
+functions of the toccatas and preludes are best supported
+by unpredictability and hence high entropy.
+We know that the node-level entropy is defined only
+by the out-degrees of the nodes. Accordingly, it is use-
+ful to assess differences between the true networks and
+others wherein the node-level entropies have been fixed
+by preserving the true degree distribution. To perform
+this assessment, we compare the entropy of the real net-
+works with another set of null models: randomized net-
+works which preserve both the in- and out-degree of each
+node (see the Methods and Materials section A 5 for de-
+tails on generating these networks).
+We observe that
+the entropies of the networks are more or less preserved
+(see Fig. 2B). Although this preservation is expected for
+undirected networks (where the entropy is determined
+only by the degree distribution), it need not exist for di-
+rected networks (where the different stationary distribu-
+tions contribute to the entropy). We therefore find that
+the entropy of music networks is primarily determined
+by their degree distributions rather than their stationary
+distributions.
+Heterogeneity in degrees favors higher entropy
+To gain intuition for how the entropy of note tran-
+sitions depends on network structure, consider the case
+of unweighted and undirected networks. The network en-
+tropy takes a particularly simple form, as shown in Eq. 3.
+Following a Taylor expansion around the average degree
+of the network (see the Materials and Methods section
+A 2), one obtains:
+S = log⟨k⟩ + Var(k)
+2 ⟨k⟩2 + ...
+(4)
+where ⟨k⟩ is the average degree of the network and Var(k)
+is the variance of the degrees. To first order, we see that
+
+5
+the entropy increases logarithmically with the average
+degree of the network. To second order, the entropy in-
+creases with the variance or the heterogeneity of the de-
+grees, such that more information will be produced by
+networks with heterogeneous (or broader) degree distri-
+butions. We define the degree heterogeneity as:
+H = Var(k)
+⟨k⟩2 .
+(5)
+Many networks that we encounter in our daily lives are
+characterized by heterogeneous degree distributions, typ-
+ically with few high degree “hub” nodes and many low de-
+gree nodes [36–38]. By contrast, regular graphs—which
+have homogeneous degrees—produce random walks with
+the least entropy.
+Where does Bach’s music fall along this spectrum? We
+found in Fig. 2A that Bach’s music networks have consis-
+tently higher entropy than null networks with the same
+number of nodes and edges (in other words, randomized
+networks with the same average degree). In the Supple-
+mentary Information Sec. B 4, we show that this higher
+information content of Bach’s music networks is due to
+higher heterogeneity in their in- and out-degree distri-
+bution; that is, Bach’s music networks are more hetero-
+geneous in their degrees than expected from transition
+structures of their size, enabling them to pack more in-
+formation into their structure. In (Fig. 2A), we also ob-
+served that various pieces belonging to certain composi-
+tions were clustered together in their entropy. Consistent
+with this observation, we find that the pieces which are
+clustered together in their entropy have very similar de-
+grees (see Supplementary Information Sec. B 3). Exam-
+ples include English suites, French suites, and chorales.
+In contrast, fugues did not cluster together in their en-
+tropy as much as other composition types and displayed
+diverse average degrees. For the compositions that are
+grouped together in their entropy, we find that the dif-
+ferences observed among the pieces in the group can be
+explained by their degree heterogeneity (see Supplemen-
+tary Information Sec. B 4). We can, for example, see
+this relation in the chorales where the pieces which have
+a higher in- and out-degree heterogeneity tend to have a
+higher entropy, despite having similar degrees (Fig. 2C).
+We note that this relationship between the entropy and
+degree heterogeneity holds even in our data set of di-
+rected networks, likely because the in- and out-degrees
+tend to be correlated.
+IV.
+HOW HUMANS PERCEIVE NETWORKS
+OF INFORMATION
+Communication systems, such as music or language,
+convey information in sequences of discrete items. Hu-
+mans then assimilate this information and build repre-
+sentations of the underlying structure of inter-item rela-
+tionships. The information that is perceived by a human
+is the sum of the information present in the system and
+Internal Estimates
+0
+η
+0
+1
+1/4
+Learned probability
+Maximal accuracy
+Minimal accuracy
+Maximal complexity
+Minimal complexity
+Trade-off between accuracy and computational cost
+Balanced between accuracy
+and complexity
+(i)
+(ii)
+(iii)
+FIG. 3. How humans process networks of information.
+Humans strike a balance between accuracy and complexity
+when forming internal network models of the world. The pa-
+rameter η quantifies this trade-off between accuracy and cost.
+In panel (i), we see the example network built when solely
+maximizing the accuracy (η → 0), which forms a perfect rep-
+resentation of reality. However, building this network requires
+perfect memory and is computationally expensive. In panel
+(iii), we see the network built when solely minimizing the
+computational cost (η → 1), in which all nodes are connected
+to all other nodes, unlike the original network. Constructing
+this network does not require significant cost, but it provides
+no accuracy in representing the original information.
+Hu-
+mans tend to display intermediate values of η = 0.80 [31],
+thereby constructing networks that preserve some but not all
+of the true transition structure, as shown in panel (ii). Figure
+adapted with permission from Ref. [32].
+the inaccuracies that stem from the imperfect cognitive
+processes involved in perception [31]. In the previous sec-
+tion, we focused on quantifying the actual information
+present in the system (see Fig. 1B). We will now account
+for the second piece: the inaccuracies that arise due to
+the imperfect cognitive process of perceiving information
+(see Fig. 1C).
+When forming an internal network representation of
+the information presented to them, humans seek to max-
+imize the accuracy of their internal representation while
+simultaneously minimizing the computational cost in-
+volved in building it [30–32, 39]. One the one hand, a
+human could learn the structure with no errors, forming
+a perfectly accurate network of the transitions (Fig. 3(i))
+but that formation process would be computationally ex-
+pensive. On the other hand, one could disregard accuracy
+and have the least expensive representation (Fig. 3(iii)).
+Most humans do something in between by recalling the
+sequence of transitions sometimes accurately and some-
+times inaccurately, thereby forming a fuzzy perception
+of the true network (Fig. 3(ii)). Formally, the competi-
+tion between computational complexity and accuracy can
+be captured by a free energy model of people’s internal
+representation [30]. The learned transition probabilities
+
+6
+under this model can be written as follows:
+ˆP = (1 − η)P(I − ηP)−1,
+(6)
+where η ∈ [0, 1] captures the errors in representation.
+Using this model, we can compute the learned network
+for each musical piece.
+Prior work indicates that, on
+average, humans display an η = 0.80 in large-scale online
+laboratory experiments [31].
+Given a network of note
+transitions with transition probabilities P, we use this
+empirically measured value of η = 0.8 to calculate the
+average network that a human infers ˆP using Eq. 6.
+V.
+QUANTIFYING THE LEARNABILITY OF
+NOTE TRANSITIONS
+We are now prepared to investigate how a given music
+network differs from the internal representation formed
+by a human listener.
+The closer the learned network
+is to the original network, the more resilient the network
+structure is to human errors in learning, and the network
+is said to be more learnable. Mathematically, one can
+quantify the deviations between the inferred network ( ˆP)
+and the original network (P) using the Kullback-Leiber
+(KL) divergence:
+DKL(P|| ˆP) = −
+�
+i
+πi
+�
+j
+Pij log
+ˆPij
+Pij
+,
+(7)
+where πi is the stationary distribution of the original
+network.
+The lower the KL-divergence, the closer the
+learned transition structure is to the original transition
+structure, and hence the more learnable the network.
+Do Bach’s musical compositions possess distinct features
+that facilitate human learning? How do pieces differ in
+their learnability?
+What are the structural differences
+between the musical pieces that lead to such differences?
+To answer these questions, for each musical piece, we
+compute the KL-divergence between the true transition
+probabilities P and the learned transition probabilities
+ˆP. Then, to understand whether Bach’s music networks
+are structured in a manner that improves their learnabil-
+ity, we compare them against random networks with the
+same number of nodes and edges.
+The data confirms
+our intuition (Fig. 4A): Bach’s music networks have a
+lower KL-divergence than random networks of the same
+size. Even if we compare against null networks with the
+same in- and out-degree distributions, we still see that
+Bach’s music networks have a lower KL-divergence (Fig.
+4B). This finding suggests that the lower KL-divergence
+of these networks cannot be explained by their degree
+distributions alone. Additionally, we observe variations
+in the KL-divergence among the different compositions
+(Fig. 4). The chorales, at one extreme, seem to have the
+highest KL-divergence, while the preludes have the lowest
+KL-divergence. Our findings indicate that the note tran-
+sitions in Bach’s music are structured in a manner that is
+resilient to errors that humans make when learning infor-
+mation. Further, learnability differs across composition
+forms, with some being easier to learn than others.
+A.
+Transitive clustering
+In the previous section, we saw that the differences
+between the KL-divergences of the music networks and
+the null networks could not be explained by the distri-
+butions of degrees. Here, we seek to understand what
+network property leads to the observed differences. Pre-
+vious work has shown that in the case of undirected net-
+works, the KL-divergence decreases with the density of
+triangles in the network [31]. One can show this ana-
+lytically by substituting the expression for the averaged
+learned version of a network (Eq. 6) into the equation
+for the KL-divergence (Eq. 7). This substitution gives
+us an expression for the KL-divergence in terms of the
+adjacency matrix of the original network:
+DKL(P|| ˆP) = − log(1 − η) −
+η
+ln 2
+�
+i
+πi×
+�
+�
+�
+�
+j
+Aij
+�
+l
+1
+kout
+i
+Ail
+1
+kout
+l
+Alj
+�
+�
+� + O(η2).
+(8)
+Here we see that the KL-divergence depends on a prod-
+uct of the form AijAilAlj, which measures the transitive
+relationships present in the network. More explicitly, it
+depends on the number of directed triangles of the form
+i → j → k and i → k. Musically, the presence of a larger
+density of such triangles suggests that if there is a tran-
+sition between notes i and j, and notes i and k, there is
+likely also a transition between notes j and k.
+To quantify the extent to which a network has clusters
+of this form, we calculate the transitive clustering coef-
+ficient of the network. For each node, this quantity is
+measured by dividing the number of transitive triangles
+that node i is a part of (∆T
+i ) by the number of possible
+directed triangles:
+CT
+i =
+∆T
+i
+ktot
+i
+(ktot
+i
+− 1).
+(9)
+Here ktot
+i
+is the total degree (in + out) of the node. We
+average this quantity over all nodes in the network to re-
+port a single value for each piece. Eq. 8 indicates that the
+KL-divergence will be smaller for networks with a large
+number of transitive triangles. This intuition arises from
+the fact that humans can easily make swap errors among
+transitive relations. If node i is connected to node j and
+node j links to node k, a human learner may erroneously
+draw an edge between node i and node k. However, if
+the network had an edge connecting node i to node k to
+begin with, such an edge would not be an error. Hence,
+we expect networks that have more transitive relations
+to be more robust to errors made in learning. Indeed, we
+
+7
+B
+D
+C
+A
+FIG. 4. Quantifying the difference between the actual information and the perceived information in Bach’s
+music networks by calculating the KL-divergence between the actual and perceived network. (A) KL-divergence
+of the real music networks (Dreal
+KL ) compared with random networks of the same size (Drand
+KL ). We report the KL-divergence
+of the corresponding random networks after averaging over 100 independent realizations. The error bars for Drand
+KL
+indicate
+the standard error of the sample. (B) KL-divergence of the real music networks (Dreal
+KL ) compared with random networks that
+preserve the in- and out-degree of each node (Ddeg
+KL ). We report the KL-divergence of the corresponding degree-preserving
+random networks after averaging over 100 independent realizations. The error bars for Ddeg
+KL indicate the standard error of
+the sample. (C) KL-divergence of the real music networks as a function of the transitive clustering coefficient of the network
+C = ⟨∆T
+i /ktot
+i
+(ktot
+i
+− 1)⟩. (D) The transitive clustering coefficient of the real music networks compared with random networks
+that preserve the in- and out-degree of each node. The dotted line indicates the line y = x. For the degree-preserving random
+networks, we report the transitive clustering coefficient after averaging over 100 independent realizations, with error bars
+denoting the standard error of the sample. In all the panels, each data point represents a single piece. Color and marker
+indicate the type of piece, as shown in the legend. The dotted line in panels (A), (B), and (D) represents the line y = x.
+observe that the KL-divergence of the music networks is
+lower for networks that have a higher transitive cluster-
+ing coefficient (Fig.
+4C).
+In fact, the real music net-
+works have a higher transitive clustering coefficient than
+degree-preserving random networks (Fig. 4D), suggest-
+ing that this feature is not due to mere coincidence. From
+Fig 4D, we make an interesting observation: the chorale
+pieces generally have a higher transitive clustering coef-
+ficient than expected from null networks that preserve
+their size and degree distribution, while the preludes ap-
+pear to have a lower transitive clustering coefficient than
+the corresponding null networks. We probe this further
+in the Supporting Information and identify meso-scale
+structures that could lead to the observed differences be-
+tween the compositional forms.
+VI.
+ACCOUNTING FOR NOTE TRANSITION
+FREQUENCIES
+So far, we have focused our attention on the infor-
+mation content and perception of unweighted (or bi-
+nary) note transition networks created from Bach’s mu-
+sic. These networks only captured whether or not a tran-
+sition exists between two notes and were not sensitive to
+how frequently each transition occurs. The binary net-
+works enabled us to probe how the structure of the tran-
+
+8
+A
+B
+C
+FIG. 5. Accounting for the frequencies of the note transitions in our analysis. (A) Entropy of the weighted versions of
+Bach’s music networks (Sweighted) compared with the corresponding unweighted versions (Sunweighted). (B) The KL-divergence
+of the weighted versions of Bach’s music networks (Dreal,w
+KL
+) compared with the corresponding unweighted versions (Dreal
+KL ). (C)
+Top: Entropy of the weighted note transition networks (Sreal,w) compared with degree-preserving edge-rewired null networks
+(Sdeg, w). Bottom: The KL-divergence of the weighted note transition networks (Dreal,w
+KL
+) compared with degree-preserving
+edge-rewired null networks (Ddeg, w
+KL
+). In all panels, each data point represents a single piece. Color and marker indicate the
+type of piece, as shown in the legend. The dashed line represents the line y = x.
+sitions supports effective communication.
+However, in
+many real networks, not all transitions occur with the
+same frequency. To reflect the different frequencies with
+which transitions may occur, we construct networks in
+which transitions are weighted according to this. For ex-
+ample, if note i follows note j 90% of the time and note
+k follows note j 10% of the time, the edge from node j to
+node i will be more heavily weighted than the edge from
+node j to node k (see the Materials and Methods section
+A 1 for further details on network construction). Adding
+this piece of information to the networks leads us to new
+questions about the role that transition weights play in
+communicating information to listeners.
+For example,
+how is the information generated by a random walk on
+the network altered by differences in the frequencies of
+transitions? In Bach’s music, do these differences in fre-
+quencies make it easier for humans to learn the transition
+networks?
+A.
+Weights reduce the surprisal of transitions
+For unweighted networks, the node-level entropy of
+a random walk is determined solely by the out-degree
+(kout
+i
+), since each outgoing edge is traversed with prob-
+ability Pij = 1/kout
+i
+. If the edges are weighted by their
+transition frequencies, the Pij’s will no longer be uni-
+formly distributed, and each outgoing edge will not have
+an equal probability of being traversed. Hence, incorpo-
+rating the edge weights reduces the node-level entropy.
+This observation is intuitive since non-uniformities in any
+distribution lead to decreases in entropy. However, ex-
+tending this intuition to the entropy produced by the
+entire network is not as straightforward, since one must
+weigh the contribution of each node by the stationary
+distribution of the random walkers, which cannot be ex-
+pressed in closed form for directed networks. Generally,
+we find that the entropy of weighted networks is still
+lower than the corresponding unweighted networks (Fig.
+5A). This finding suggests that the different weights re-
+duce the overall surprisal generated by the networks.
+B.
+Weights reduce the deviations between the
+learned network and the original network
+Incorporating the transition frequencies also helps us
+to understand the role that the weights play in the hu-
+man inference of note transitions. We observe that the
+weighted networks of note transitions have lower KL-
+divergence than the binary networks (Fig. 5B). This ob-
+servation suggests that the weights aid in forming more
+accurate internal representations of the transition struc-
+tures, thereby improving their learnability.
+In light of these data, we next verify the role that the
+network structure plays in the communicative success of
+weighted networks by comparing the entropy and KL-
+divergence of the weighted music networks with edge-
+rewired null networks.
+In the analysis on unweighted
+networks, we observed that the entropy was primarily
+driven by the degree distribution of the network and not
+sensitive to the precise connectivity pattern. To make
+this observation, we had compared the entropy of the
+real music networks to randomized networks that pre-
+
+9
+served the exact degree distribution of each node and
+hence, held the node-level entropies fixed. Along simi-
+lar lines, here we make use of null models that keep the
+node-level entropies fixed by preserving the in- and out-
+degree of each node and the out-weights at each node
+(see the Materials and Methods section for details on the
+null models). By comparing the entropy of the weighted
+music networks to the degree-preserving weighted null
+models, we see that the entropies of real networks are
+still more or less unchanged, although the real networks
+have marginally higher entropies than the null networks
+(Fig.
+5C, top).
+These results support our conclusion
+that the entropy in the real networks is still primarily
+driven by their degree distribution. When we compare
+the KL-divergence of the real weighted networks with the
+degree-preserving weighted null models, we find that the
+real networks have a lower KL-divergence than the cor-
+responding null networks (Fig. 5C, bottom). Together,
+these results suggest that incorporating the weights into
+our network analysis does not alter the effects of network
+structure qualitatively.
+Accounting for the note transition frequencies in our
+network model leads to several interesting lines of inquiry.
+For instance, is it the specific distribution of weights
+that improves the learnability of music networks? Fu-
+ture work could evaluate this possibility by comparing
+the KL-divergence of the weighted networks with a class
+of null models that preserve the skeleton of the network,
+but permute the edge weights. It would also be interest-
+ing to test whether higher edge weights are concentrated
+in triangular clusters of the network, offering a potential
+explanation for the lower KL-divergence of the weighted
+networks compared to the binary networks.
+VII.
+DISCUSSION
+In this article, we study music composed by J. S. Bach
+through the lens of network science and information the-
+ory. Viewing Bach’s musical compositions as networks of
+note transitions, we quantify the information generated
+by the note transitions and study how this information is
+perceived by humans. We analyzed a total of 327 Bach
+compositions spread over a wide range of compositional
+forms, including preludes, fugues, inventions, cantatas,
+English suites, French suites, chorales, Brandenburg con-
+certos, toccatas, and concertos. For each musical piece,
+we construct a network of note transitions by drawing di-
+rected edges between notes that are played consecutively.
+We then quantify the amount of information generated
+by the network structure and find that different composi-
+tional forms are grouped together based on their entropy.
+Further, we find that the note transitions in Bach’s music
+contain more information than expected from transition
+structures of their size, which can be attributed to higher
+heterogeneity in their degree distribution.
+To quantify how the transition structure of Bach’s mu-
+sic is perceived by a human, we use a mathematical model
+for how humans infer networks of information [30, 31],
+which allows us to estimate the average “learned” net-
+work given any network of information. Using this model,
+we compute the inferred version for each music network,
+and quantify the information that arises due to discrep-
+ancies between the original and inferred networks. We
+find here that the discrepancies differ among the compo-
+sitional forms. Moreover, Bach’s music networks main-
+tain a consistently lower deviation between the original
+and inferred version compared to randomized null net-
+works of the same size and degree distribution. Probing
+the structural features that enable these music networks
+to be more resilient to biases in perception, we find that
+this property is driven by a high density of transitive
+triangular clusters in the network.
+Finally, we study how the frequencies of transitions
+influence the information content and perception of the
+musical pieces, by weighing the transitions by the number
+of times they occur. We find that the weights reduce the
+overall entropy or surprisal of the transitions, and also
+reduce the deviations between the inferred and actual
+network, suggesting that the weights aid the learnability
+of these transition structures. On comparing the infor-
+mation content and learnability of the weighted networks
+with degree-preserving null models, we find that qualita-
+tively, our results relating the information content and
+learnability to the network structure are still valid for
+the weighted networks.
+More generally, our findings here along with the re-
+sults in Ref. [31] provide insight into features that make
+a wide range of complex systems around us effective at
+communicating information. To communicate informa-
+tion successfully, networks of information in complex sys-
+tems tend to be structured in a manner that allows them
+to carry large amounts of information, while also being
+robust to inaccuracies that humans make when infer-
+ring relationships between items.
+Networks which are
+denser (have a higher average degree) produce more un-
+predictable random walk sequences, and hence produce
+more information (have a higher entropy). Further, for
+networks of comparable average degree, more heteroge-
+neous (higher variance in degree distribution) structures
+produce more information than those more regular or ho-
+mogeneous in their degree (Fig. 6A(i)). Additionally, we
+find that networks which contain a large number of tri-
+angular clusters can be inferred more accurately when
+viewed through an observer’s imperfect cognitive appa-
+ratus (Fig. 6A(ii)). Together, these findings suggest that
+for networks of a given size, rapid and accurate commu-
+nication of information is supported by structures that
+are simultaneously heterogeneous and clustered (Fig. 6).
+Future directions
+Our study has focused on analyzing the note transi-
+tions present in Bach’s music. It is important to note that
+music is a multifaceted art form that encompasses a range
+of structural and expressive elements. Future work could
+
+10
+Supports efficient communication
+1
+Low KL-divergence
+Easy to learn
+High KL-divergence
+Hard to learn
+High Entropy
+Contains more information
+Low Entropy
+Contains lesser information
+Does not support effective communication
+A.
+B.
+i.
+ii.
+i.
+ii.
+FIG. 6.
+Network structures that support effective communication of information.
+(A) Networks with a larger
+variance or heterogeneity in their node degrees, as shown in panel (i), pack more information into their structure and have a
+higher entropy. Clustering in the network, as shown in panel (ii), makes the structure more resilient to errors made by humans
+when building an internal representation of the information, allowing the network to be inferred more accurately. Together,
+these structures convey a large amount of information that can be learned by humans more accurately, and are hence more
+efficient for communication. (B) Networks with lower variance in their node degrees, as shown in panel (i), carry relatively
+lower information in their structure compared to networks that are of similar size but more heterogeneous in their degrees. A
+lower tendency for nodes to form clusters, as shown in panel (ii), makes the network more susceptible to errors when humans
+infer its transition structure. Together, these structures convey information less efficiently, rapidly, and accurately compared
+to those shown in panel (A).
+build upon our study by exploring other aspects of music,
+for example, considering networks of transitions between
+rhythms or harmonies.
+Beyond music, our study can
+also be extended to a range of complex systems present
+around us—such as language and social networks. For
+example, one could analyze works of literature and ask:
+Does the entropy of noun transitions in various works of
+Shakespeare differ based on their genre?
+More specif-
+ically, does the information content and learnability of
+noun transitions or relationships between characters dif-
+fer between tragedies and comedies?
+By providing an
+example of a systematic and comprehensive analysis of
+the actual and perceived information in music, our study
+complements and adds to the rich study of language, mu-
+sic, and art as complex systems [25, 40, 41].
+Systematically analyzing the information that we ex-
+tract from complex systems can provide new insights into
+the human experience. A question that often arises in
+the context of how humans experience music is: What
+makes a musical composition appealing to the human
+ear?
+While individual preferences in music can vary
+widely and is highly subjectively, there is still a gen-
+eral agreement on certain composers being considered
+“influential” or “great”.
+This fact raises the possibil-
+ity that there may be some inherent qualities that are
+common to musical pieces which are widely considered
+appealing.
+Identifying such features might give us in-
+sight into the creative process of composing music and
+also complement existing work using AI to generate mu-
+sic [42, 43]. Several attempts have been made to identify
+such patterns. For example, Ref. [24] analyzed note tran-
+sition networks in certain compositions by Bach, Chopin,
+and Mozart as well as Chinese pop music, and sug-
+gested that “good” music is characterized by the small-
+world property [44] and heavy-tailed degree distributions.
+On the other hand, Ref. [25] studied selected composi-
+tions from Bach’s Well-Tempered Clavier and found non-
+heavy-tailed degree distributions, suggesting that such
+distributions are not necessary for music to be appeal-
+ing. It would be interesting to devise future experiments
+to determine whether our findings relate to the aesthetic
+or emotional appeal of a piece. In our study, we found
+that Bach’s music networks had a higher number of tran-
+sitive triangular clusters, enabling them to be learned
+more efficiently than arbitrary transition structures. Are
+pieces with a larger number of these triangles also more
+appealing to a listener? Future work assess this possi-
+bility by conducting experiments that ask people to rate
+Bach’s compositions and analyzing whether these ratings
+correlate with the presence of triangular clusters. More
+generally, our work focuses not solely on the informa-
+tion inherent in the transition structure of music, but
+also on how the information in this transition structure
+is perceived by a human listener. This framework might
+be useful in studying cognitive aspects of music and in
+bridging patterns observed in data with cognitive theo-
+ries of music.
+In future work, it would be interesting to extend our
+analysis to study how music networks evolve with time.
+There are three potentially interesting lines of inquiry
+here: First, how do the entropy and KL-divergence of
+a musical piece change as the piece progresses?
+Does
+
+11
+this temporal change differ among the various compo-
+sitional forms?
+Second, how has the music of a spe-
+cific composer (whether Bach or otherwise) changed over
+the course of their lifetime? Has it become more intri-
+cate and complex, holding more information? Perhaps as
+the composer gains experience, their compositions con-
+vey information more efficiently and accurately, as re-
+flected in a reduced KL-divergence? If the exact dates
+of when each piece was composed were known, then the
+framework used in our paper might provide answers to
+these questions. Third, how has music of a given genre,
+say classical music, changed over the years across com-
+posers? Ref. [27], for example, studied the fluctuation in
+pitch between adjacent notes in compositions by Bach,
+Mozart, Beethoven, Mendelsohn, and Chopin, and found
+that the largest pitch fluctuations of a composer gradu-
+ally increased over time from Bach to Chopin. It would
+be interesting to expand our analysis to different com-
+posers, and see how the information and expectations
+vary across composers and time.
+Further considering how a genre changes with time, it
+would be of interest to assess how various styles or gen-
+res of music differ [45–47]. What are the key features by
+which a listener distinguishes between music from two
+eras, say the Classical and the Romantic eras? How do
+the differences in structure then impact how the piece is
+perceived by a listener? An analysis of the information
+content and perception of various genres of music could
+complement existing work in musicology, and potentially
+aid in systematically classifying pieces into genres that
+may not be a priori obvious. Classifying genres of music
+could also be beneficial for audio streaming services, and
+our framework could potentially complement existing ap-
+proaches to musical genre classification [46, 48–51].
+Methodological considerations
+Here we highlight the assumptions made in our study
+and the resulting methodological constraints in our re-
+search.
+First, in constructing networks of note transi-
+tions, the self loops present in the networks were ignored
+to simplify our analysis. This choice restricted us to un-
+derstanding only the structure of transitions between dif-
+ferent notes in a musical piece. However, these self loops
+may have interesting effects on the discrepancies between
+the actual and perceived information content from the
+network. Future work could include self loops, studying
+their impact on the information content and learnability
+of the network. Second, the production of information
+from the underlying transition structure has been mod-
+elled using Markov random walks. While this is a stan-
+dard first step in understanding complex systems, in re-
+ality, the transitions present in music possess long range
+correlations and constraints to their structure. Including
+these correlations (perhaps in the form of a biased ran-
+dom walk with memory) would be a fruitful direction to
+pursue to gain a better and more realistic understand-
+ing of the information we encounter from real complex
+systems around us.
+VIII.
+CONCLUSION
+In this work, we analyze Bach’s musical compositions
+as networks of note transitions conveying information to
+humans. Recent studies have shown that the information
+humans perceive from complex systems around them con-
+sists of two parts: the information inherent in the system
+and the information arising due to errors in their per-
+ception [30, 31]. Analyzing the information from these
+two parts, we find that different compositional forms can
+be distinguished from one another.
+Further, we gain
+insight into structural features that enable these music
+networks to communicate effectively: they communicate
+more information by having more heterogeneous degrees,
+and they convey information more accurately (minimiz-
+ing the discrepancies with human inferences) by having
+a higher density of transitive clusters (Fig. 6). Through
+this quantitative analysis of Bach’s music, our findings
+provide new methods to understand how humans share
+and experience information around them.
+ACKNOWLEDGMENTS
+We thank Chris Macklin for an early conversation on
+this topic and audience members who have asked prob-
+ing questions about our earlier work in communication
+networks.
+These interactions motivated our continued
+investigation in this space.
+This particular research
+was primarily supported by the Army Research Office
+award number DCIST-W911NF-17-2-0181 and the Na-
+tional Institutes of Mental Health award number 1-R21-
+MH-124121-01.
+D.S.B. would also like to acknowledge
+additional support from the John D. and Catherine T.
+MacArthur Foundation, the Alfred P. Sloan Foundation,
+the Institute for Scientific Interchange Foundation, and
+the Army Research Office (Grafton-W911NF-16-1-0474).
+The content is solely the responsibility of the authors and
+does not necessarily represent the official views of any of
+the funding agencies.
+CITATION DIVERSITY STATEMENT
+Recent work in several fields of science has identi-
+fied a bias in citation practices such that papers from
+women and other minority scholars are under-cited rel-
+ative to the number of such papers in the field [52–60].
+Here we sought to proactively consider choosing refer-
+ences that reflect the diversity of the field in thought,
+form of contribution, gender, race, ethnicity, and other
+factors. First, we obtained the predicted gender of the
+first and last author of each reference by using databases
+that store the probability of a first name being carried by
+
+12
+a woman [56, 61]. By this measure (and excluding self-
+citations to the first and last authors of our current pa-
+per), our references contain 9.37% woman (first)/woman
+(last), 18.67% man/woman, 19.29% woman/man, and
+52.67% man/man.
+This method is limited in that a)
+names, pronouns, and social media profiles used to con-
+struct the databases may not, in every case, be indica-
+tive of gender identity and b) it cannot account for in-
+tersex, non-binary, or transgender people.
+Second, we
+obtained predicted racial/ethnic category of the first and
+last author of each reference by databases that store the
+probability of a first and last name being carried by
+an author of color [62, 63].
+By this measure (and ex-
+cluding self-citations), our references contain 11.79% au-
+thor of color (first)/author of color (last), 11.60% white
+author/author of color, 16.05% author of color/white
+author, and 60.56% white author/white author.
+This
+method is limited in that a) names and Florida Voter
+Data to make the predictions may not be indicative of
+racial/ethnic identity, and b) it cannot account for In-
+digenous and mixed-race authors, or those who may face
+differential biases due to the ambiguous racialization or
+ethnicization of their names. We look forward to future
+work that could help us to better understand how to sup-
+port equitable practices in science.
+Appendix A: Materials and Methods
+1.
+Data Collection and Network Construction
+The music files were collected in the MIDI for-
+mat from various sources.
+The sources for the com-
+positions analyzed are as follows:
+preludes [64, 65],
+fugues [64, 65], inventions[64, 65], cantatas[66], English
+suites[67], French suites[67], chorales[65], Brandenburg
+concertos[65], toccatas[67], and concertos[67]. The pre-
+ludes and fugues are split based on whether they belong
+to the first or second part of The Well-Tempered Clavier,
+and are labelled ‘1’ or ‘2’. Certain compositions consist of
+different movements and our data set has separate MIDI
+files for each movement. We analyze each movement sep-
+arately and average our measurements over them to yield
+a single measured quantity for each piece, as indexed by
+a unique BWV number.
+The MIDI files were read in MATLAB using the
+readmidi function in MATLAB [68] to obtain informa-
+tion about the notes being played. Different instruments
+in a piece are stored in separate channels within each
+data file. The transitions between notes are calculated
+separately for each instrument or track. We assign each
+note present in a piece a node in the network, and notes
+from different octaves are assigned distinct nodes. We
+then draw an edge from note i to note j if there is a
+transition between them. If there are multiple notes be-
+ing played at a single time t (as is the case with chords),
+edges are drawn from the previously played note to all
+notes at time t, and from all the notes being played at
+time t to the subsequent note(s). This procedure gives
+us a directed binary network of note transitions. We also
+construct weighted versions of these networks, where each
+edge is weighted by the number of times the correspond-
+ing transition occurs.
+2.
+Entropy of random walks on networks
+We use random walks to model how a sequence of in-
+formation is generated from an underlying network of
+information. Under this model, a walker traverses the
+network by picking an outgoing edge to traverse at each
+node. Given a network with adjacency matrix A and ma-
+trix element Aij, the probability that a walker transitions
+from node i to node j in a standard Markov random walk
+is Pij = Aij/kout
+i
+, where kout
+i
+= �
+j Gij is the out-degree
+of a node. We are interested in quantifying how much
+information is contained in the resulting sequence, which
+is captured by the entropy of the random walk:
+S = −
+�
+i
+πi
+�
+j
+Pij log Pij,
+where π is the stationary distribution of the walkers,
+which satisfies the condition Pπ = π. For the simplest
+possible case of an undirected and unweighted network,
+Pij = 1/ki and πi = ki/2E, where ki is the degree of
+the ith node and E = �
+i,j Aij/2 is the total number of
+edges. The entropy in this case simplifies to:
+S = 1
+2E
+�
+i
+ki log ki = ⟨k log k⟩
+⟨k⟩
+.
+(A1)
+We can apply a Taylor expansion to this expression
+around the average degree of the network, and thereby
+obtain:
+S = log⟨k⟩ + Var(k)
+2 ⟨k⟩2 + ...
+(A2)
+Hence we find that the entropy of random walks increase
+logarithmically with the average degree of the network.
+Additionally, it grows as the variance of the degrees in-
+creases. This formalization enables us to relate the in-
+formation content of various music networks to their net-
+work structure.
+3.
+Model for how humans learn networks
+As discussed in the main text, when forming internal
+representations of information around them, each human
+arbitrates a trade-off between accuracy and cost [30, 31].
+In striking this balance, evidence suggests that humans
+perform a fuzzy temporal integration of transition struc-
+tures over time [29, 30, 69–71]. This process results in
+humans connecting items in the sequence that are not
+directly adjacent to each other. Mathematically, we can
+
+13
+express the inferred transition structure ˆP in terms of
+the true transition structure P under this model of fuzzy
+temporal integration as:
+ˆP =
+∞
+�
+∆t=0
+f(∆t)P ∆t+1,
+(A3)
+where f(∆t) is the weight given to the higher powers of
+P and is a decreasing function of ∆t.
+The functional form of f(∆t) is obtained using a
+free energy model that captures the accuracy-complexity
+trade-off described in Ref. [30]. Under this theory, the
+optimal distribution for f(∆t) is a Boltzmann distribu-
+tion with a parameter β that quantifies the trade-off be-
+tween cost and accuracy in forming an internal represen-
+tation of the information:
+f(∆t) = e−β∆t/Z,
+(A4)
+where Z = � e−β∆t = (1 − e−β)−1 is a normalization
+constant.
+Substituting this expression to simplify Eq.
+A3, we obtain an equation that relates the inferred tran-
+sition probabilities ˆP to the true transition probabilities
+P:
+ˆP =(1 − e−β)−1
+∞
+�
+∆t=0
+e−β∆tP ∆t+1
+=(1 − η)P(I − ηP)−1,
+(A5)
+where η = e−β. Prior work has estimated the value of
+η to be 0.8 from large-scale online experiments in hu-
+mans [31]. Using this measured value of η, we use Eq.
+A5 to calculate the learned network for any given music
+network.
+4.
+KL-divergence
+To quantify how much the distorted learned transition
+structure ˆP differs from the original transition structure
+P, we calculate the Kullback-Leiber (KL) divergence be-
+tween the two transition structures. The Kullback-Leiber
+divergence is a measure of how different a probability dis-
+tribution is from a reference distribution, and is given by:
+DKL(P|| ˆP) = −
+�
+i
+πi
+�
+j
+Pij log
+ˆPij
+Pij
+,
+(A6)
+where ⃗π is the stationary probability distribution of the
+transition matrix P, obtained by solving Pπ = π. The
+KL-divergence between two quantities is always non-
+negative and attains the value zero if and only if P = ˆP.
+The larger the KL-divergence, the more the inferred net-
+work ˆP differs from the original network.
+Hence, this
+quantity acts as a measure of the extent to which a net-
+work gets scrambled by the inaccuracies of human of
+learning—or in other words, how learnable a network
+structure is.
+5.
+Null Models
+We aim to identify distinct features in the music net-
+works that enable them to convey information effectively.
+To assess whether our observations are merely due to ran-
+dom chance or are instead a unique feature of our dataset,
+we compare our measurements on the real music networks
+with the following null network models [72, 73].
+1. Null networks with the same number of nodes and
+edges. These are obtained by generating random
+networks with the same number of nodes and edges,
+and enable us to assess whether the quantity we
+have measured is to be expected merely based on
+network size.
+2. Degree-preserving null networks.
+These are ran-
+domized networks of the same size, with the ad-
+ditional constraint that the in- and out-degrees of
+each node in the network are preserved. Such net-
+works are constructed by swapping edges between
+pairs of nodes in the network iteratively, such that
+the in- and out-degrees of each node are preserved
+but the connectivity (or topology) of the network
+is randomized. This class of null models enable us
+to evaluate the role that connectivity or topology
+plays in the quantity we are measuring.
+We can generalize the degree-preserving null networks
+to weighted networks.
+We are interested in degree-
+preserving randomized networks since these keep the
+node-level entropies fixed and allow us to study the im-
+pact of topology on the quantities we are measuring. In
+the case of weighted networks, the node-level entropies
+are determined by the out-weights and out-degrees of the
+nodes. Hence, our procedure of swapping edges between
+pairs of nodes in the network still works since it pre-
+served the out-weights of each node in addition to the in-
+and out-degrees. With these null models, we can bench-
+mark the presence of the quantities we are interested in,
+and identify the role that the connectivity pattern or size
+plays.
+6.
+Transitive Clustering Coefficient
+Along the lines of the clustering coefficient of a node
+[44, 74], we define the transitive clustering coefficient as
+a measure of the degree to which nodes in a directed net-
+work tend to form transitive relationships. The transitive
+clustering coefficient of a node i (for an unweighted graph
+with no self loops) is given by:
+CT
+i =
+∆T
+i
+ktot
+i
+(ktot
+i
+− 1),
+(A7)
+where ∆T
+i denotes the number of transitive triangles that
+node i is a part of and ktot
+i
+is the total degree (in + out)
+of the node. The denominator simply counts the number
+
+14
+of triangles that could exist within the neighborhood of
+node i.
+FIG. 7. The 8 different possible triangles with node i as a
+vertex in a directed graph.
+The triangles which represent
+transitive relationships are marked using the letter ’T’.
+The possible directed triangles involving node i can
+be divided into two categories—those representing cyclic
+relationships and those representing transitive relation-
+ships (Fig. 7). The number of transitive triangles involv-
+ing node i that actually exist can be expressed in terms
+of the adjacency matrix of the graph A,
+CT
+i = (A + AT )3
+ii − A3
+ii − (AT )3
+ii
+2 ktot
+i
+(ktot
+i
+− 1)
+.
+(A8)
+This expression counts a subset of the total number of
+triangles, and is a special case of the expression derived
+in Ref. [75]. We will use this expression to measure the
+transitive clustering coefficient of each music networks.
+Appendix B: Supplementary Information
+1.
+Introduction
+In this Supplementary Information, we provide ex-
+tended analysis and discussion to support the results pre-
+sented in the main text. In Sec. B 2, we expand upon
+our analysis of the information content of Bach’s music
+networks and how it relates to network structure. In Sec.
+B 5, we examine the transitive clustering coefficient more
+closely and study meso-scale features that might explain
+the differences observed across compositional forms.
+2.
+Information content
+To better visualize the variation in information content
+among the musical compositions, we assign each piece
+an index number and plot the information entropy for
+each piece as a function of its index number (Fig. 8A).
+We observe here more clearly how different compositional
+forms tend to have pieces clustered together in their en-
+tropies. As reported in the main text, we find that the
+chorales have a markedly lower entropy than the rest
+of the compositions studied.
+In contrast, the toccatas
+and the second set of preludes have a much higher en-
+tropy.
+To relate the information entropy of the music
+networks to their structure, we compare their entropy to
+corresponding null networks (Fig. 2A and B in the main
+text), where we conclude that the information entropy is
+primarily determined by the degree distributions. In the
+case of undirected and unweighted networks, the network
+entropy depends upon the logarithm of the average de-
+gree of the network and the heterogeneity in the degree
+distribution (Eq.
+4) to first and second order, respec-
+tively [31, 35]. We now provide supplementary results
+that relate the information entropy of the music networks
+to their structure.
+3.
+Understanding the information entropy to first
+order: average degree
+On plotting the information entropy of the music net-
+works as a function of their average degree (Fig. 8B), we
+see that the differences in the information entropy of the
+compositional forms to first order arise due to differences
+in their average degrees. Although we observed in Fig.
+8A that the compositional forms are clustered together
+in their entropy, it is clear that some pieces—such as the
+chorales, French suites, English suites, and cantatas—
+are more tightly clustered than the fugues and first set of
+preludes. These differences can be explained by the how
+much the average degrees vary across pieces. In Fig. 9,
+we plot the entropy of the music networks as a function
+of the average network degree, separately for each com-
+position type. Additionally, we also report the standard
+deviation in the average degree of the pieces for each com-
+position type. Studying these plots, we observe that the
+English suites, French suites, and chorales (which clus-
+tered more tightly in their entropies) have tighter degree
+distributions, while the fugues (which are more spread
+out in their entropy) display more diverse average de-
+grees.
+4.
+Understanding the information entropy to
+second order: degree heterogeneity
+In Fig.
+2A of the main text, we observed that the
+entropy of the real music networks is larger than corre-
+sponding randomized null networks with the same num-
+ber of nodes and edges. Since the average degree is the
+same for the two networks, we hypothesize that the differ-
+ences arise due to higher in- and out-degree heterogene-
+ity as per Eq. 4. To test our hypothesis, we compare the
+in- and out-degree heterogeneity of the music networks
+(calculated using Eq. 5) with their corresponding null
+networks in Fig. 10. In general, we observe that Bach’s
+music networks are indeed more heterogeneous than ex-
+pected from the random networks of the same size. This
+organization allows them to pack more information into
+their structure.
+The heterogeneity in degrees can also explain the dif-
+ferences in entropies observed between pieces that are
+
+1
+2
+0
+2
+215
+A
+B
+FIG. 8. The entropy of Bach’s music networks and its relation to the average degree of the network. (A) The
+entropy of Bach’s music networks (Sreal) indexed by the pieces. (B) The entropy of Bach’s music networks (Sreal) as a function
+of the average degree of the network ⟨k⟩. Each data point in panels (A) and (B) represents a single piece. Colors and markers
+indicate the type of pieces, as shown in the legend.
+tightly clustered together in their entropy. As observed
+earlier, compositions such as the chorales, French suites,
+English suites, and cantatas have pieces that are clus-
+tered together in their average degree and consequen-
+tially, in their entropy. We expect that the differences
+observed among the pieces in each group can be explained
+by differences in their degree heterogeneity. In Fig. 11
+and Fig. 2C, we plot the entropies of the pieces that clus-
+tered together as a function of their in- and out-degree
+heterogeneity, and in general observe that the pieces with
+higher heterogeneity have a higher information entropy.
+However, we note that our sample size for most com-
+positional forms is small and hence, we only report the
+chorales in the main text.
+5.
+Further analysis of the transitive clustering
+coefficient
+In our analysis of the discrepancies between the ac-
+tual and perceived information content of note transi-
+tions in Bach’s musical compositions, we found that these
+discrepancies were primarily driven by the presence of
+transitive triangular clusters. These transitive triangular
+clusters tend to bring the inferred network closer to the
+actual network, making the network more learnable. As
+shown in Fig. 12A, the real (unweighted) music networks
+tend to have a higher transitive clustering coefficient than
+random networks that preserve the degree of each node,
+indicating that this is a distinct feature of the music net-
+works that is not merely due to coincidence. The data
+in Fig. 12A has a striking shape, which we elaborate on
+and analyze in this section. First we observe that the
+chorale pieces tend to have a higher transitive clustering
+coefficient than expected from networks of their same
+size and degree distribution. Second, although the pre-
+ludes have a higher transitive clustering coefficient than
+other compositional forms, the value was still lower than
+expected from networks of their same size and degree
+distribution. Indeed, by examining only the x-axis, we
+notice that the null networks corresponding to the pre-
+ludes have a higher transitive clustering coefficient than
+the null networks corresponding to chorales. However,
+by examining the y-axis, we see that the deviation be-
+tween the real chorales and the prelude networks are not
+that pronounced. We hypothesize that these differences
+might be due to the presence of mesoscale features in the
+networks, such as core-periphery structure.
+a.
+Core-periphery structure
+Core-periphery structure in a network refers to the
+presence of two components: a tightly connected “core”
+and a sparsely connected “periphery. The core consists of
+nodes which are well-connected to each other and to the
+periphery, while the nodes in the periphery are sparsely
+connected to one another and to the nodes in the core
+[76, 77]. We hypothesize that the presence of a relatively
+larger core might explain why the chorales have a higher
+clustering coefficient than expected given their size and
+degree. Similarly, a smaller than expected core for the
+preludes might be explain why their clustering coefficient
+was lower than expected from networks of the same size
+and degree distribution. Since the core consists of nodes
+that are well-connected to themselves and the periphery,
+
+16
+A
+B
+C
+D
+E
+F
+G
+H
+I
+J
+K
+L
+FIG. 9. The relation between the information entropy and the average degree of the music networks plotted
+separately for each compositional form. The entropy of Bach’s music networks (Sreal) plotted against the average degree
+of the network ⟨k⟩. Each data point represents a single piece. Colors and markers indicate the type of pieces, as shown in the
+legend.
+if there are a larger number of edges occurring within
+the core and between the core and periphery than be-
+tween the periphery nodes, it is likely that these edges
+will form the clusters that we are interested in. We de-
+
+17
+A
+B
+FIG. 10. Comparing the heterogeneity of Bach’s music networks to randomized null networks of the same size.
+(A) The in-degree heterogeneity of the music networks compared with random networks of the same size. (B) The out-degree
+heterogeneity of the music networks compared with random networks of the same size. Each data point in panels (A) and (B)
+represents a single piece. Colors and markers indicate the type of pieces, as shown in the legend. For each random network,
+we report the in- and out- degree heterogeneity after averaging over 100 independent realizations. Error bars on the x-axis
+represent the standard error of the sample.
+A
+B
+C
+D
+FIG. 11. The relation between the information entropy of Bach’s music networks and its degree heterogeneity.
+The entropy of Bach’s music networks (Sreal) plotted against the network in- and out-degree heterogeneity. Each data point
+represents a single piece. Colors and markers indicate the type of pieces, as shown in the legend. The dotted line in each
+panel indicates the best linear fit, and the reported rs value is the Spearman correlation coefficient between the x- and y-axis
+variables.
+note the edges between two nodes that belong to the
+core by core-core (CC), those between nodes that belong
+to the periphery by periphery-periphery (PP), and those
+between the nodes in the core and the nodes in the pe-
+riphery by core-periphery (CP).
+To test our hypothesis, we compute the core-periphery
+
+18
+A
+B
+FIG. 12. Core-periphery analysis of the music networks. (A) The transitive clustering coefficient of the real music
+networks compared to null networks that preserve the in- and out-degree of each node. For the degree-preserving null networks,
+we report the average over 100 independent realizations, with error bars denoting the standard error of the sample. (B) The
+ratio of the number of core-core (CC) edges and core-periphery (CP) edges to the number of periphery-periphery (PP) edges in
+the real music networks compared to degree-preserving null networks. For the degree-preserving null networks, we report the
+average value computed over 100 independent random graphs. In both panels, the dotted line indicates the line y = x. Colors
+and markers indicate the type of piece, as shown in the legend.
+structure for each music network using the method de-
+scribed by Borgatti and Everett [77]. We then compute
+the ratio of the sum of the number of core-core (CC)
+edges and core-periphery (CP) edges to the number of
+periphery-periphery (PP) edges for each network.
+To
+understand this ratio, we compare it to corresponding
+degree-preserving null networks (Fig. 12B). Strikingly,
+we observe that the chorales have a higher fraction of
+edges that are within or emanating from the core than
+expected from their corresponding null networks.
+The
+preludes are at the other end, and have a lower frac-
+tion of edges that are within or emanating from the core
+than expected from their corresponding null networks.
+This pattern of findings suggests that the chorales have a
+more pronounced core-periphery structure than expected
+by chance, while the preludes have a less pronounced
+core-periphery structure than expected.
+Although the
+preludes still have a slightly higher transitive clustering
+coefficient than the other pieces, the differences are not
+as pronounced as one would expect because of these dif-
+ferences in their core-periphery structure.
+By performing this additional analysis, we provide an
+example of how the music networks display interesting
+meso-scale structures that differ from one compositional
+form to another, resulting in differences in how their net-
+work structure is perceived.
+
+19
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+XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 
+Quantum Encryption in Phase Space using 
+Displacement Operator for QPSK Data Modulation 
+ 
+ 
+ 
+ 
+Randy Kuang 
+Quantropi Inc. 
+Ottawa, Canada 
+[email protected] 
+ORCID: 000-0002-5567-2192 
+ 
+ 
+Adrian Chan 
+Quantropi Inc. 
+Ottawa, Canada 
+[email protected]  
+ 
+ 
+ 
+ 
+ 
+Abstract—Quantum Public Key Distribution or QPKE with 
+the randomized phase shift gate was proposed by Kuang and 
+Bettenburg in 2020. It has been implemented theoretically with 
+simulations and experimentally over existing fiber optical 
+networks since then. QPKE can be considered as an RSA-type 
+scheme in optical analogue domain. QPKE was renamed as 
+Quantum Encryption in Phase Space or QEPS to reflect the 
+encryption of coherent states in phase space. QEPS with the phase 
+shift gate can only be applied to data modulation scheme with 
+phase shift keying such as quadrature phase shift keying or QPSK. 
+It would leak data information in amplitude once it is applied to 
+quadrature amplitude modulation or QAM schemes. Kuang and 
+Chan recently proposed a new version of QEPS called Quantum 
+Encryption in Phase Space with the displacement gate or QEPS-d. 
+It demonstrated to overcome the limitation of QEPS with the 
+phase shift gate. We introduced a reduced displacement operator 
+by ignoring the global phase factor then the reduced displacement 
+operators are commutable. This commutability helps our 
+implementation at both transmission and receiving. An arbitrary 
+displacement operator can be decoupled into a standard QAM 
+modulation with a phase shift modulation to ease our encryption 
+and decryption. This paper simulates the QEPS-d encryption for 
+QPSK data modulation to demonstrate how QEPS-d works.    
+Keywords—quantum cryptography, post-quantum cryptography, 
+PQC, quantum encryption, coherent state, phase shift gate, 
+displacement gate, quadrature amplitude modulation, QAM, 
+quadrature phase shift keying, QPSK 
+I. INTRODUCTION  
+ After Shor proposed his algorithm with quantum bit or qubit 
+for integer factorization in 1994 [1], it has been well-understood 
+that classical public key algorithms such as RSA based on the 
+factorization problem, Diffie-Hellman or elliptic Diffie-
+Hellman based on the discrete logarithm are breakable once fault 
+tolerate quantum computers are available. However, breaking 
+RSA-2048 requires a fault tolerate quantum computer to have 
+more than 4000 logic qubits or 4 million physical qubits. The 
+latest released IBM quantum computer Osprey offers 433 
+physical qubits [2]. The IBM roadmap shows that they will 
+release their next quantum computer Condor with 1121 qubits 
+in 2023 and qubits will raise over 100,000 in 2026. Very 
+recently, Yan, et al. proposed a new algorithm by combining 
+classical lattice reduction with quantum optimization called 
+Sublinear-resource Quantum Integer Factorization (SQIF) [3]. 
+SQIF works in a noise quantum computer with a quantum 
+resource reduction or qubits of 4 magnitudes from 4 million of 
+physical qubits to less than 400 physical qubits. They have 
+demonstrated it for a 48-bit integer factorization with as little as 
+a 10-qubit quantum processor.  
+National Institute of Standards and Technology or NIST 
+started the standardization process in the late of 2017 and 
+completed its three rounds in 2021 [4] and announced its final 
+standardized algorithms for key encapsulation mechanism or 
+KEM and digital signature algorithms [5]. The lattice-based 
+Kyber [6] becomes the standardized winner for KEM and the 
+lattice-based Dillithium [7] and Falcon [8], as well as hash-
+based SPHINCS+ [9] become the standardized algorithms for 
+digital signature. NIST continues its standardization for KEM in 
+its round 4 and reopens its standardization of digital signature 
+for submissions in the early 2023. 
+Some major cryptanalyses have made NIST finalists 
+vulnerable in 2022.  Beullens broke Rainbow signature with a 
+laptop over a weekend [10], Robert broke SIDH [11] and 
+Castryck and Decru made its more efficient to break SIDH level 
+I in one hour with a single core computer [12]. Wenger, et al.  
+reported their secret recovery of lattice-based PQC with 
+machine learning by training the transformer with 300,000 
+samples and achieved the complete secret recovery for up to a 
+mid-size lattice dimension.  
+Some recent developments in PQC KEM and digital 
+signature were proposed by Kuang’s team, called Multivariate 
+Polynomial Public Key or MPPK by leveraging the NP-
+complete problem of the Modular Diophantine Equation 
+Problem [14, 15, 16, 17]. MPPK offers relatively small public 
+key size, cipher size, and signature size, comparable to the 
+classical public key schemes. They also outperform NIST 
+finalists in performances of key generation, encryption, 
+decryption, signing and verification. MPPK could become good 
+alternatives to NIST finalists for generic use cases. MPPK 
+digital signature scheme is planned to participate in the NIST 
+reopening submission for digital signature. 
+On the other hand, Quantum Key Distribution or QKD was 
+developed over three decades since it was proposed in 1984. 
+Shor and Preskill proved that QKD offers the information 
+theoretical security in 2000 [18]. It has become commercial 
+ready for a distance at around 100km. To break the distance 
+
+boundary, Lucamarini, et. Al. proposed Twin-Field QKD or TF-
+QKD in 2018 [19]. TF-QKD has been widely explored since 
+then and the longest distance of 830km was reported by Wang, 
+et al. in 2022 [20]. QKD generally offers a key rate at kbps level 
+and TF-QKD [20] achieved a key rate at 0.014 bps at 830km, 
+requiring more than 5 hours to establish a 256 bits of AES key.  
+ Considering the pre-shared secret for QKD authentication, 
+Kuang and Bettenburg in 2020 proposed a new mechanism 
+using Quantum Permutation Pad or QPP to digitally distribute 
+quantum random [21]. The pre-shared secret is not only used for 
+authentication but also used to map to a QPP pad for encoding 
+at the sender and decoding at the receiver. QPP is implemented 
+into matrices operating on data column vector or Dirac ket. 
+Permutation matrix is unitary and reversable, so the decoding 
+side uses the reversed QPP. Kuang and Barbeau proposed a 
+universal quantum safe cryptography using QPP in 2022 [22]. 
+QPP has been developed as a platform for digital QKD and 
+benchmarked by Deutsche Telekom in 2022 [23]. Leveraging 
+the quantum gate property of QPP, quantum encryption with 
+QPP implemented inside quantum computers was reported by 
+Kuang and Perepechaenko in 2022 [24], Perepechaenko and 
+Kuang in 2022 [25, 26].  
+To eliminate the pre-shared key in quantum key distribution 
+in coherent optical domain, Kuang and Bettenburg in 2020 
+proposed Quantum Public Key Envelope or QPKE using 
+randomized phase shift gate in a round-trip scheme [27], 
+leveraging the self-shared random secret to drive the phase shift 
+encoding without the specific requirement of the pre-shared 
+secret. QPKE was designed to operate in the existing coherent 
+optical networks with the same coherent detection module. It has 
+been simulated and experimentally implemented through the 
+collaborations with McGill University [28, 29, 30, 31]. QPKE 
+mimics the RSA-type public key scheme in coherent optical 
+domain. The experiment implementation with off-shelf optical 
+modules demonstrated the speed at 200 gbps for a distance 80km 
+between two communication peers. To mimicking its 
+implementation in a symmetric fashion with a pre-shared secret, 
+QPKE was renamed as Quantum Encryption in Phase Space or 
+QEPS with the randomized phase shift gate, reflecting to its 
+possible implementation in photonic quantum computer with 
+phase shift gate. There is one limitation of QEPS with phase shift 
+gate, or only applicable for data modulation schemes with phase 
+shift keying such as QPSK or M-PSK. Once the data modulation 
+is quadrature amplitude modulation or QAM, the amplitude bits 
+would be leaked out.  
+To overcome this limitation, Kuang and Chan recently 
+proposed to use coherent displacement operator ����� where � 
+denotes a coherent state [32].  This paper will report its 
+simulation results with QPSK data modulation.  Section 2 will 
+briefly summarize the QEPS with the displacement operator and 
+section 3 will present the simulation result and the conclusion is 
+at the end.   
+II. QEPS WITH DISPLACEMENT OPERATOR 
+A. Coherent State and Displacement Operator 
+ 
+A coherent state is the specific quantum state of quantum 
+harmonic oscillator denoted by a Dirac ket |�⟩ where � is a 
+complex variable in the phase space. � can be expressed either 
+in terms of in-phase and quadrature as � = �� + � ��   or 
+amplitude and phase � = |�|���. Then a coherent state can be 
+written as 
+ 
+|�⟩ = ��� + � ��� = | |�|��� ⟩   
+                        (1)  
+And the displacement operator is defined with creation and 
+annihilation operators ��� and �� through following equation 
+ 
+ 
+|�⟩ = �� �� ���∗ �� |0⟩ = ����� |0⟩    
+         (2) 
+So  
+ 
+ 
+����� = �� �� ���∗ �� 
+                                       (3)  
+which indicates the displacement operator is unitary and 
+reversable: 
+ 
+ 
+������ = ��� �� ���∗ ���
+�
+=  ���−�� = ��� ���       (4) 
+Let’s apply the displacement operator ����� to a coherent state 
+|�⟩ 
+     ����� |�⟩ = ����� ����� |0⟩ = ��!∗��∗!���� + ��|0⟩   (5) 
+And in the same way 
+����� |�⟩ = ����� ����� |0⟩ = �!�∗�!∗����� + ��|0⟩    (6) 
+So, it is clear that ����� and ����� are not commutable due to the 
+global phase factor ��!∗��∗!  but that does not impact our 
+physical measurements on the amplitude and phase of a coherent 
+state. Therefore, we can ignore the global phase factor and 
+introduce 
+a 
+reduced 
+displacement 
+operator 
+"#��� =
+ ���!∗$�∗!�����. Then the reduced displacement operator "#��� 
+and "#��� are commutable.  
+B. QEPS with Reduced Displacement Operator 
+From Eq. (5), QEPS encryption with a reduced displacement 
+operator "#��� can be expressed as follows 
+"#��� |�⟩ =  "#�� + ��|0⟩ = |� + �⟩ = |%⟩                    (7) 
+with |�⟩ to be a plain coherent state, "#��� to be an encryption 
+operator and |%⟩ to be the encrypted cipher coherent state. Eq. 
+(7) indicates that QEPS encryption with the reduced 
+displacement operator or QEPS-d essentially performs an 
+addition of two coherent states |�⟩ and |�⟩ as shown in Fig. 1. 
+A general displacement operator would change both the 
+amplitude and phase of a plain coherent state. But it can also 
+only change the phase of the plain coherent as shown in Fig. 1. 
+In this special case, the displacement operator behaves like a 
+phase shift operator. 
+The encryptor "#��� can be controlled by a pre-shared secret 
+in a symmetric encryption or a self-shared secret in an 
+asymmetric encryption as shown in QPKE [27]. In the ideal 
+communication case, the receiver would decrypt the cipher 
+coherent state |%⟩  with "#� ��� = "#�−�� : "#� ���|%⟩ =
+"#�−��|%⟩ = |−� + %⟩ = |�⟩. 
+In coherent optical communications, optical line path would 
+impact a coherent state during transmission from the sender to 
+the receiver such as dispersion, attenuation, polarization, noise, 
+
+environment factors, etc. Thanks to the digital signal processing 
+or DSP, all those impacts could be compensated and corrected 
+in the electrical digital domain. Based on that, we only consider 
+the encryption and decryption in the ideal transmission situation. 
+A displacement operator can be decomposed into two or 
+more displacement operators as follows 
+"#��� = "#�� � "#��&� … "#��(� 
+And  
+"#���|�⟩ = "#�� � "#��&� … "#��(�|�⟩ 
+                            = |� + �& + ⋯ �( + �⟩  
+This decomposition feature helps us to ease the implementation 
+of a general displacement operator with two operators: "#�� � 
+implemented with a standard modulation such as QAM and 
+"#��&�  with a phase shift operator. By doing that, we can 
+overcome the weakness of original QPKE scheme [27]. 
+III. QEPS-D SIMULATION 
+The simulation is performed with OptiSystem and the 
+simulation layout is illustrated in Fig. 2. The major modules are 
+explained in the figure caption. The only extra components are 
+needed to discuss here are QEPS and RNG. All others are 
+common for typical coherent optical communications. The 
+random number generator or RNG should be a cryptographic 
+PRNG or pseudo–Quantum Random Number Generator or 
+pQRNG [33] with generated random number meeting 
+cryptographic requirement. pQRNG is capable to take upto 16 
+KB of the pre-shared secret and produces pseudo random 
+number with excellent randomness [33]. QEPS consists of two 
+operators: "#�� � implemented with standard data modulation 
+such as 16-QAM or QPSK and "#��&� implemented with a 
+random phase shift operator. These two operators together offer 
+a coherent encryption with a generic displacement operator 
+"#���.   QEPS produces a complex modulation form based on the 
+rand number generated from RNG module. The complex 
+modulation form dictates the signal generator to produce 
+voltages for IQ modulator. In Fig. 2, we omitted the data input 
+which is combined with QEPS. Once the coherent states are 
+generated from CW and pass IQ Modulator, their amplitude and 
+phase would be modulated by IQ modulator then the encrypted 
+cipher coherent states are transmitted over 80 km fiber to 
+coherent detector at the receiver side. Typical coherent detection 
+is applied to produce electrical digital signal and QEPS-d 
+decryption is done before DSP processing. The simulation 
+parameters are given in Table 1.  
+We simulated QEPS encryption with the reduced 
+displacement operator for QPSK data modulations and plot 
+constellation diagrams in 3 cases: 
+1. Constellation right after coherent detection as shown in Fig. 
+3. This constellation diagram displays the detections of 
+cipher coherent states together with fiber path impacts. 
+2. Constellation diagram after applying the digital signal 
+processing as shown in Fig. 4. 
+TABLE 1. SIMULATION PARAMETERS ARE TABULATED. 
+ 
+Layout 
+Parameter 
+Sequence length  
+Baudrate  
+PM period  
+65,536 bits 
+28 Gbaud 
+1024 
+CW Laser and 
+LO Laser 
+Center wavelength 
+Power 
+Linewidth 
+Azimuth 
+1550 nm 
+5 dBm 
+0.1 MHz 
+0.45 degree 
+IQ Modulator 
+Extinction ratio 
+Switching bias 
+Insertion loss 
+20 dB 
+3 V 
+5 dB 
+EDFA 
+Forward pump power 
+Forward pump wavelength 
+Loss at 1550 nm 
+Loss at 980 nm 
+13-14 mW 
+980 nm 
+0.1dB/m 
+0.15 dB/m 
+Optical Fiber 
+Length (1 spool) 
+Attenuation 
+Dispersion 
+Dispersion slope 
+Differential group delay 
+Effective area 
+80 km 
+0.2 dB/km 
+0.3 16.75 ps/nm/km 
+0.4 0.075 ps/nm2/km 
+0.5 0.2ps/km 
+80 μm2 
+Figure 1. Illustration of QEPS-d is plotted in the phase space. A 
+special case of QEPS with phase shift operator is also plotted for 
+demonstration purpose of a general displacement operator "#���. 
+Figure 2. Simulation layout is illustrated. CW: continuous wave 
+source, IQ Modulator: in-phase and quadrature modulator, +,, .,and 
++/ , .0: in-phase and quadrature components for IQ modulator, QEPS: 
+coherent encryption module driven by a random number generator or 
+RNG seeded with a pre-shared secret, EDFA: Erbium-Doped Fiber 
+Amplifier, Coherent Receiver: coherent detection, LO: local oscillator, 
+QEPS and DSP: digital QEPS decryption and DSP.   
+QEPS 
+and 
+DSP 
+
+Qy
+LO
+DSP
+Signal Generator
+Quantum
+QEPS
+Encoding :
+RNGAliceTransmitter
+Bob Receiver
+BPF
+Ix
+Digital
+cW
+IQModulator
+Qx
+Phase De-
+80 km
+Coherent
+EDFA
+EDFA
+randomiz
+Ix
+Qx
+Receiver
+1y
+ationandy) =a(α)Iβ)= [α +β)
+a(a)
+lα)
+Iβ)
+p3. Constellation after applying digital QEPS decryption and 
+DSP compensations as shown in Fig. 5. 
+ 
+Fig. 3 is used to mimic the attacker’s coherent detection by 
+assuming the attacker taped good portion of the transmitted 
+cipher coherent signals. Then he/she would obtain a coherent 
+constellation diagram as shown in Fig. 3, which is randomly 
+scattered points. Then we also assume that the attacker knows 
+the data modulation scheme to be QPSK so he/she can apply 
+DSP to compensate and correct the impacts from the fiber path. 
+After applying DSP processing, he/she obtains a constellation 
+diagram as shown in Fig. 4 with a huge Bit-Error-Rate or BER 
+at 0.38.  That means, it is impossible to extract any meaningful 
+transmitted data. If we carefully look at Fig. 4, we will notice 
+that there is a square-typed band with 2-unit amplitude, 
+indicating two QPSK modulations through QEPS-d encryption 
+"#�� � on a QPSK data modulation. The square band reflects the 
+phase shift operator "#��&�  driving by the random number 
+generated from RNG. The central disk reflects the QPSK data 
+modulations have the opposite phases of "#�� � so they cancel 
+out and give the “zero” amplitudes.  
+ 
+In QPSK data modulation scheme, data values are 
+modulated into phases not in amplitude, so Fig. 4 would not leak 
+transmitted data information. So, they transmission is totally 
+secure.  
+Coherent detection turns coherent optical domain into 
+coherent electrical domain so digital signal processing can 
+compensate and correct the impacts from the optical path. That 
+is fantastic for QEPS encryption: encryption in coherent optical 
+domain or analogue encryption then decryption in electrical 
+digital domain before DSP processing. That means, QEPS 
+encryption is an analogue encryption which blocks attackers to 
+Figure 3. Constellation diagram of directly detected cipher coherent 
+states is displayed. 
+Figure 4. Constellation diagram of directly detected cipher coherent 
+states is displayed after applying the DSP processing.  The BER is 
+0.38. 
+Figure 5. Constellation diagram of QEPS decryption and DSP 
+processing. BER is 0. 
+
+Electrical Constellation Visualizer
+2
+-1
+0
+2
+Amplitude -I (a.u.)Electrical Constellation Visualizer
+Amplitude
+C
+2
+1
+0
+2
+Amplitude -I (a.u.)Electrical Constelation Visualizer_1
+山
+'n'e)
+Q :
+-10 m
+0
+10 m
+Amplitude -I (a.u.)extract transmitted digital data. Of course, one can apply AES 
+encryption in data then transmit with coherent optical 
+communications which would allow attackers to extract AES 
+ciphertexts. That is the major difference between QEPS and 
+other encryption schemes. 
+Leveraging the feature of coherent detection, we apply 
+QEPS-d decryption with "#�−�� driving by the synchronized 
+RNG seeded with the pre-shared secret. Fig. 5 illustrates the 
+constellation diagram with QEPS-d decryption then DSP 
+processing. It is clearly seen that a QPSK constellation with 
+BER to be zero.  
+The described technique in the above can be implemented in 
+a round trip as shown in QPKE [27, 30] where Alice becomes 
+Alice Transmission and Alice receiving with a self-shared 
+random secret for encryption and decryption then Bob only 
+performs data modulations, Alice would securely extract Bob’s 
+transmitted data without pre-share secret. Using this way, one 
+trick needs to be remembered: phase shift operator must be in a 
+reverse 
+order 
+of 
+transmission 
+side. 
+The 
+round-trip 
+implementation can be also used for true random number 
+distributions, as an alternative of traditional QKD but the key 
+rate can be dramatically increased to 100s gbps. For example, in 
+this simulation, we could achieve 56 gbps with a single 
+polarization and 112 gbps with dual polarizations. 
+The distance can be extended with EDFA amplification as 
+what we have used in today’s coherent optical communications.  
+ 
+IV. CONCLUSION 
+We briefly introduced QEPS with the reduced displacement 
+operator proposed in [32] and applied it for QPSK data 
+modulation 
+with 
+QPSK 
+implementation 
+of 
+the 
+first 
+displacement operator "#�� �  and a randomized phase shift 
+operator of the second displacement operator "#��&� . The 
+simulation demonstrates QEPS-d offers security in analogue 
+domain encryption and the transmitted cipher coherent states 
+can not be extracted without knowing the pre-shared secret in 
+symmetric implementation mode. It can be also implemented in 
+a roundtrip scheme without the pre-shared secret which can be 
+used 
+for 
+key 
+distributions 
+over 
+coherent 
+optical 
+communications. The simulation shows that we can achieve 56 
+gbps distributions rate with a single polarization and 112 gbps 
+with dual polarizations. As what we have demonstrated in [32] 
+that the displacement operator can also be implemented with 
+QAM schemes such as 16-QAM or 32-QAM. That makes 
+QEPS-d be a generic encryption in coherent optical domain or 
+analogue encryption. In the future, we plan to implement it 
+experimentally. 
+  
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+ 
+

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+page_content='00 ©20XX IEEE  Quantum Encryption in Phase Space using  Displacement Operator for QPSK Data Modulation  Randy Kuang  Quantropi Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Ottawa, Canada  randy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='kuang@quantropi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='com  ORCID: 000-0002-5567-2192  Adrian Chan  Quantropi Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Ottawa, Canada  adrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='chan@quantropi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='com  Abstract—Quantum Public Key Distribution or QPKE with  the randomized phase shift gate was proposed by Kuang and  Bettenburg in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' It has been implemented theoretically with  simulations and experimentally over existing fiber optical  networks since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QPKE can be considered as an RSA-type  scheme in optical analogue domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QPKE was renamed as  Quantum Encryption in Phase Space or QEPS to reflect the  encryption of coherent states in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QEPS with the phase  shift gate can only be applied to data modulation scheme with  phase shift keying such as quadrature phase shift keying or QPSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  It would leak data information in amplitude once it is applied to  quadrature amplitude modulation or QAM schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Kuang and  Chan recently proposed a new version of QEPS called Quantum  Encryption in Phase Space with the displacement gate or QEPS-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  It demonstrated to overcome the limitation of QEPS with the  phase shift gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' We introduced a reduced displacement operator  by ignoring the global phase factor then the reduced displacement  operators are commutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' This commutability helps our  implementation at both transmission and receiving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' An arbitrary  displacement operator can be decoupled into a standard QAM  modulation with a phase shift modulation to ease our encryption  and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' This paper simulates the QEPS-d encryption for  QPSK data modulation to demonstrate how QEPS-d works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Keywords—quantum cryptography, post-quantum cryptography,  PQC, quantum encryption, coherent state, phase shift gate,  displacement gate, quadrature amplitude modulation, QAM,  quadrature phase shift keying, QPSK  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' INTRODUCTION  After Shor proposed his algorithm with quantum bit or qubit  for integer factorization in 1994 [1], it has been well-understood  that classical public key algorithms such as RSA based on the  factorization problem, Diffie-Hellman or elliptic Diffie-  Hellman based on the discrete logarithm are breakable once fault  tolerate quantum computers are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' However, breaking  RSA-2048 requires a fault tolerate quantum computer to have  more than 4000 logic qubits or 4 million physical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The  latest released IBM quantum computer Osprey offers 433  physical qubits [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The IBM roadmap shows that they will  release their next quantum computer Condor with 1121 qubits  in 2023 and qubits will raise over 100,000 in 2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Very  recently, Yan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' proposed a new algorithm by combining  classical lattice reduction with quantum optimization called  Sublinear-resource Quantum Integer Factorization (SQIF) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  SQIF works in a noise quantum computer with a quantum  resource reduction or qubits of 4 magnitudes from 4 million of  physical qubits to less than 400 physical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' They have  demonstrated it for a 48-bit integer factorization with as little as  a 10-qubit quantum processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  National Institute of Standards and Technology or NIST  started the standardization process in the late of 2017 and  completed its three rounds in 2021 [4] and announced its final  standardized algorithms for key encapsulation mechanism or  KEM and digital signature algorithms [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The lattice-based  Kyber [6] becomes the standardized winner for KEM and the  lattice-based Dillithium [7] and Falcon [8], as well as hash-  based SPHINCS+ [9] become the standardized algorithms for  digital signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' NIST continues its standardization for KEM in  its round 4 and reopens its standardization of digital signature  for submissions in the early 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Some major cryptanalyses have made NIST finalists  vulnerable in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Beullens broke Rainbow signature with a  laptop over a weekend [10], Robert broke SIDH [11] and  Castryck and Decru made its more efficient to break SIDH level  I in one hour with a single core computer [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Wenger, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  reported their secret recovery of lattice-based PQC with  machine learning by training the transformer with 300,000  samples and achieved the complete secret recovery for up to a  mid-size lattice dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Some recent developments in PQC KEM and digital  signature were proposed by Kuang’s team, called Multivariate  Polynomial Public Key or MPPK by leveraging the NP-  complete problem of the Modular Diophantine Equation  Problem [14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' MPPK offers relatively small public  key size, cipher size, and signature size, comparable to the  classical public key schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' They also outperform NIST  finalists in performances of key generation, encryption,  decryption, signing and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' MPPK could become good  alternatives to NIST finalists for generic use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' MPPK  digital signature scheme is planned to participate in the NIST  reopening submission for digital signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  On the other hand, Quantum Key Distribution or QKD was  developed over three decades since it was proposed in 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Shor and Preskill proved that QKD offers the information  theoretical security in 2000 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' It has become commercial  ready for a distance at around 100km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' To break the distance  boundary, Lucamarini, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' proposed Twin-Field QKD or TF-  QKD in 2018 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' TF-QKD has been widely explored since  then and the longest distance of 830km was reported by Wang,  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' in 2022 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QKD generally offers a key rate at kbps level  and TF-QKD [20] achieved a key rate at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='014 bps at 830km,  requiring more than 5 hours to establish a 256 bits of AES key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Considering the pre-shared secret for QKD authentication,  Kuang and Bettenburg in 2020 proposed a new mechanism  using Quantum Permutation Pad or QPP to digitally distribute  quantum random [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The pre-shared secret is not only used for  authentication but also used to map to a QPP pad for encoding  at the sender and decoding at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QPP is implemented  into matrices operating on data column vector or Dirac ket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Permutation matrix is unitary and reversable, so the decoding  side uses the reversed QPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Kuang and Barbeau proposed a  universal quantum safe cryptography using QPP in 2022 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  QPP has been developed as a platform for digital QKD and  benchmarked by Deutsche Telekom in 2022 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Leveraging  the quantum gate property of QPP, quantum encryption with  QPP implemented inside quantum computers was reported by  Kuang and Perepechaenko in 2022 [24], Perepechaenko and  Kuang in 2022 [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  To eliminate the pre-shared key in quantum key distribution  in coherent optical domain, Kuang and Bettenburg in 2020  proposed Quantum Public Key Envelope or QPKE using  randomized phase shift gate in a round-trip scheme [27],  leveraging the self-shared random secret to drive the phase shift  encoding without the specific requirement of the pre-shared  secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QPKE was designed to operate in the existing coherent  optical networks with the same coherent detection module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' It has  been simulated and experimentally implemented through the  collaborations with McGill University [28, 29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QPKE  mimics the RSA-type public key scheme in coherent optical  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The experiment implementation with off-shelf optical  modules demonstrated the speed at 200 gbps for a distance 80km  between two communication peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' To mimicking its  implementation in a symmetric fashion with a pre-shared secret,  QPKE was renamed as Quantum Encryption in Phase Space or  QEPS with the randomized phase shift gate, reflecting to its  possible implementation in photonic quantum computer with  phase shift gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' There is one limitation of QEPS with phase shift  gate, or only applicable for data modulation schemes with phase  shift keying such as QPSK or M-PSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Once the data modulation  is quadrature amplitude modulation or QAM, the amplitude bits  would be leaked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  To overcome this limitation, Kuang and Chan recently  proposed to use coherent displacement operator ����� where �  denotes a coherent state [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' This paper will report its  simulation results with QPSK data modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Section 2 will  briefly summarize the QEPS with the displacement operator and  section 3 will present the simulation result and the conclusion is  at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QEPS WITH DISPLACEMENT OPERATOR  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Coherent State and Displacement Operator  A coherent state is the specific quantum state of quantum  harmonic oscillator denoted by a Dirac ket |�⟩ where � is a  complex variable in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' � can be expressed either  in terms of in-phase and quadrature as � = �� + � ��   or  amplitude and phase � = |�|���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Then a coherent state can be  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='written as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='|�⟩ = ��� + � ��� = | |�|��� ⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='And the displacement operator is defined with creation and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='annihilation operators ��� and �� through following equation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='|�⟩ = �� �� ���∗ �� |0⟩ = ����� |0⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='So  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='����� = �� �� ���∗ ��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='which indicates the displacement operator is unitary and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='reversable:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='������ = ��� �� ���∗ ���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='���−�� = ��� ���       ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='Let’s apply the displacement operator ����� to a coherent state  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='|�⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='����� |�⟩ = ����� ����� |0⟩ = ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='∗��∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='���� + ��|0⟩   (5)  And in the same way  ����� |�⟩ = ����� ����� |0⟩ = �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='�∗�!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='∗����� + ��|0⟩    (6)  So, it is clear that ����� and ����� are not commutable due to the  global phase factor ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='∗��∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' but that does not impact our  physical measurements on the amplitude and phase of a coherent  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Therefore, we can ignore the global phase factor and  introduce  a  reduced  displacement  operator  "#��� =  ���!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='∗$�∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='�����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Then the reduced displacement operator "#���  and "#��� are commutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QEPS with Reduced Displacement Operator  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' (5), QEPS encryption with a reduced displacement  operator "#��� can be expressed as follows  "#��� |�⟩ =  "#�� + ��|0⟩ = |� + �⟩ = |%⟩                    (7)  with |�⟩ to be a plain coherent state, "#��� to be an encryption  operator and |%⟩ to be the encrypted cipher coherent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  (7) indicates that QEPS encryption with the reduced  displacement operator or QEPS-d essentially performs an  addition of two coherent states |�⟩ and |�⟩ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  A general displacement operator would change both the  amplitude and phase of a plain coherent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' But it can also  only change the phase of the plain coherent as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  In this special case, the displacement operator behaves like a  phase shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  The encryptor "#��� can be controlled by a pre-shared secret  in a symmetric encryption or a self-shared secret in an  asymmetric encryption as shown in QPKE [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' In the ideal  communication case, the receiver would decrypt the cipher  coherent state |%⟩  with "#� ��� = "#�−�� : "#� ���|%⟩ =  "#�−��|%⟩ = |−� + %⟩ = |�⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  In coherent optical communications, optical line path would  impact a coherent state during transmission from the sender to  the receiver such as dispersion, attenuation, polarization, noise,  environment factors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Thanks to the digital signal processing  or DSP, all those impacts could be compensated and corrected  in the electrical digital domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Based on that, we only consider  the encryption and decryption in the ideal transmission situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  A displacement operator can be decomposed into two or  more displacement operators as follows  "#��� = "#�� � "#��&� … "#��(�  And  "#���|�⟩ = "#�� � "#��&� … "#��(�|�⟩  = |� + �& + ⋯ �( + �⟩  This decomposition feature helps us to ease the implementation  of a general displacement operator with two operators: "#�� �  implemented with a standard modulation such as QAM and  "#��&�  with a phase shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' By doing that, we can  overcome the weakness of original QPKE scheme [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QEPS-D SIMULATION  The simulation is performed with OptiSystem and the  simulation layout is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The major modules are  explained in the figure caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The only extra components are  needed to discuss here are QEPS and RNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' All others are  common for typical coherent optical communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The  random number generator or RNG should be a cryptographic  PRNG or pseudo–Quantum Random Number Generator or  pQRNG [33] with generated random number meeting  cryptographic requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' pQRNG is capable to take upto 16  KB of the pre-shared secret and produces pseudo random  number with excellent randomness [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QEPS consists of two  operators: "#�� � implemented with standard data modulation  such as 16-QAM or QPSK and "#��&� implemented with a  random phase shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' These two operators together offer  a coherent encryption with a generic displacement operator  "#���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' QEPS produces a complex modulation form based on the  rand number generated from RNG module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The complex  modulation form dictates the signal generator to produce  voltages for IQ modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 2, we omitted the data input  which is combined with QEPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Once the coherent states are  generated from CW and pass IQ Modulator, their amplitude and  phase would be modulated by IQ modulator then the encrypted  cipher coherent states are transmitted over 80 km fiber to  coherent detector at the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Typical coherent detection  is applied to produce electrical digital signal and QEPS-d  decryption is done before DSP processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The simulation  parameters are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  We simulated QEPS encryption with the reduced  displacement operator for QPSK data modulations and plot  constellation diagrams in 3 cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Constellation right after coherent detection as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' This constellation diagram displays the detections of  cipher coherent states together with fiber path impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Constellation diagram after applying the digital signal  processing as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' SIMULATION PARAMETERS ARE TABULATED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Layout  Parameter  Sequence length  Baudrate  PM period  65,536 bits  28 Gbaud  1024  CW Laser and  LO Laser  Center wavelength  Power  Linewidth  Azimuth  1550 nm  5 dBm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='1 MHz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='45 degree  IQ Modulator  Extinction ratio  Switching bias  Insertion loss  20 dB  3 V  5 dB  EDFA  Forward pump power  Forward pump wavelength  Loss at 1550 nm  Loss at 980 nm  13-14 mW  980 nm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='1dB/m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='15 dB/m  Optical Fiber  Length (1 spool)  Attenuation  Dispersion  Dispersion slope  Differential group delay  Effective area  80 km  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='2 dB/km  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='75 ps/nm/km  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='075 ps/nm2/km  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='2ps/km  80 μm2  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Illustration of QEPS-d is plotted in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' A  special case of QEPS with phase shift operator is also plotted for  demonstration purpose of a general displacement operator "#���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Simulation layout is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' CW: continuous wave  source, IQ Modulator: in-phase and quadrature modulator, +,, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=',and  +/ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='0: in-phase and quadrature components for IQ modulator, QEPS:  coherent encryption module driven by a random number generator or  RNG seeded with a pre-shared secret, EDFA: Erbium-Doped Fiber  Amplifier, Coherent Receiver: coherent detection, LO: local oscillator,  QEPS and DSP: digital QEPS decryption and DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  QEPS  and  DSP  Qy  LO  DSP  Signal Generator  Quantum  QEPS  Encoding :  RNGAliceTransmitter  Bob Receiver  BPF  Ix  Digital  cW  IQModulator  Qx  Phase De-  80 km  Coherent  EDFA  EDFA  randomiz  Ix  Qx  Receiver  1y  ationandy) =a(α)Iβ)= [α +β)  a(a)  lα)  Iβ)  p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Constellation after applying digital QEPS decryption and  DSP compensations as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 3 is used to mimic the attacker’s coherent detection by  assuming the attacker taped good portion of the transmitted  cipher coherent signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Then he/she would obtain a coherent  constellation diagram as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 3, which is randomly  scattered points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Then we also assume that the attacker knows  the data modulation scheme to be QPSK so he/she can apply  DSP to compensate and correct the impacts from the fiber path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  After applying DSP processing, he/she obtains a constellation  diagram as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 4 with a huge Bit-Error-Rate or BER  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' That means, it is impossible to extract any meaningful  transmitted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' If we carefully look at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 4, we will notice  that there is a square-typed band with 2-unit amplitude,  indicating two QPSK modulations through QEPS-d encryption  "#�� � on a QPSK data modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The square band reflects the  phase shift operator "#��&�  driving by the random number  generated from RNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The central disk reflects the QPSK data  modulations have the opposite phases of "#�� � so they cancel  out and give the “zero” amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  In QPSK data modulation scheme, data values are  modulated into phases not in amplitude, so Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 4 would not leak  transmitted data information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' So, they transmission is totally  secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Coherent detection turns coherent optical domain into  coherent electrical domain so digital signal processing can  compensate and correct the impacts from the optical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' That  is fantastic for QEPS encryption: encryption in coherent optical  domain or analogue encryption then decryption in electrical  digital domain before DSP processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' That means, QEPS  encryption is an analogue encryption which blocks attackers to  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Constellation diagram of directly detected cipher coherent  states is displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Constellation diagram of directly detected cipher coherent  states is displayed after applying the DSP processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The BER is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Constellation diagram of QEPS decryption and DSP  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' BER is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Electrical Constellation Visualizer  2  1  0  2  Amplitude -I (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' )Electrical Constellation Visualizer  Amplitude  C  2  1  0  2  Amplitude -I (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=" )Electrical Constelation Visualizer_1  山  'n'e)  Q :  10 m  0  10 m  Amplitude -I (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' )extract transmitted digital data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Of course, one can apply AES  encryption in data then transmit with coherent optical  communications which would allow attackers to extract AES  ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' That is the major difference between QEPS and  other encryption schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Leveraging the feature of coherent detection, we apply  QEPS-d decryption with "#�−�� driving by the synchronized  RNG seeded with the pre-shared secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' 5 illustrates the  constellation diagram with QEPS-d decryption then DSP  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' It is clearly seen that a QPSK constellation with  BER to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  The described technique in the above can be implemented in  a round trip as shown in QPKE [27, 30] where Alice becomes  Alice Transmission and Alice receiving with a self-shared  random secret for encryption and decryption then Bob only  performs data modulations, Alice would securely extract Bob’s  transmitted data without pre-share secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Using this way, one  trick needs to be remembered: phase shift operator must be in a  reverse  order  of  transmission  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  The  round-trip  implementation can be also used for true random number  distributions, as an alternative of traditional QKD but the key  rate can be dramatically increased to 100s gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' For example, in  this simulation, we could achieve 56 gbps with a single  polarization and 112 gbps with dual polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  The distance can be extended with EDFA amplification as  what we have used in today’s coherent optical communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' CONCLUSION  We briefly introduced QEPS with the reduced displacement  operator proposed in [32] and applied it for QPSK data  modulation  with  QPSK  implementation  of  the  first  displacement operator "#�� �  and a randomized phase shift  operator of the second displacement operator "#��&� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The  simulation demonstrates QEPS-d offers security in analogue  domain encryption and the transmitted cipher coherent states  can not be extracted without knowing the pre-shared secret in  symmetric implementation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' It can be also implemented in  a roundtrip scheme without the pre-shared secret which can be  used  for  key  distributions  over  coherent  optical  communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' The simulation shows that we can achieve 56  gbps distributions rate with a single polarization and 112 gbps  with dual polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' As what we have demonstrated in [32]  that the displacement operator can also be implemented with  QAM schemes such as 16-QAM or 32-QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' That makes  QEPS-d be a generic encryption in coherent optical domain or  analogue encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' In the future, we plan to implement it  experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='gov/publications/  detail/nistir/8413/final (2022)  [6] Bos J, Ducas L, Kiltz E, Lepoint T, Lyubashevsky V, Schanck JM,  Schwabe P, Seiler G, Stehle D (2018) CRYSTALS - Kyber: A CCA-  secure module-lattice-based KEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='  [8] Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard lattices  and new cryptographic constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Proceedings of the Fortieth Annual  ACM Symposium on Theory of Computing STOC ’08 (Association for  Computing Machinery, New York, NY, USA), pp 197–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=' Advances in Cryptology – ASIACRYPT  2019, eds Galbraith SD, Moriai S (Springer International Publishing,  Cham), pp 33–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  [10] Beullens, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=': Breaking Rainbow Takes a Weekend on a Laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  Cryptology  ePrint  Archive,  Paper  2022/214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='org/2022/214 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=': Breaking SIDH in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Cryptology ePrint  Archive, Paper 2022/1038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='org/2022/1038 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  [12] Castryck, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=': An efficient key recovery attack on SIDH  (preliminary version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Cryptology ePrint Archive, Paper 2022/975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='org/2022/975 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  [13] Wenger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=': SALSA: Attacking Lattice  Cryptography with Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Cryptology ePrint Archive, Paper  2022/935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' https://eprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='iacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='org/2022/935 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content='  [14] Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=' & Barbeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' A new quantum-safe  multivariate polynomial public key digital signature algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='1038/s41598-022-15843-x  [15] Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=' A new post-quantum  multivariate polynomial public key encapsulation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
+page_content=' Quantum Inf  Process 21, 360 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=' PMID: 10991303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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+page_content=' Plant,  "Security Performance of Physical-Layer Encryption Based on  Randomized Phase Space in Optical Fiber Communication," 2022 IEEE  Photonics  Conference  (IPC),  2022,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfEwOD/content/2301.02894v1.pdf'}
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