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2,869,038,153,728 | arxiv | \section{Introduction}
\label{s1}
Let $V(n)$ be the set of all positive divisors of a positive integer $n$ as defined in~(\ref{e01}).
For instance, $V(20) = \{1, 2, 4, 5, 10, 20\}$.
The partial order called the {\em divides} relation, $a$ divides $b$ denoted $a|b$, is applied to $V(n)$ and
yields two types of directed acyclic graphs (henceforth referred simply as graphs) as shown in Figure~\ref{f01}.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\resizebox{!}{1.2in}{\includegraphics{f01b.eps}}
&
\resizebox{!}{1.2in}{\includegraphics{f01a.eps}}\\
(a) Transitive Closure $G^T(20)$ & (b) Hasse diagram $G^H(20)$
\end{tabular}
\end{center}
\caption{\label{f01} Two basic graphs derived from the divides relation.}
\end{figure}
The first graph is called the {\em transitive closure}, $G^T(n) = (V(n), E^T(n))$ where
\begin{equation}
V(n) = \{x \suchthat{ x \in Z^{+} \wedge x|n}\}
\label{e01}
\end{equation}
\begin{equation}
E^T(n) = \{(a,b) \suchthat{a,b \in V(n) \wedge a < b \wedge a|b}\}
\label{e02}
\end{equation}
Next, when all arcs in $G^T(n)$ with alternative transitive paths are excluded, the graph becomes a {\em Hasse diagram} denoted as $G^H(n) = (V(n), E^H(n))$ where $E^H(n)$ is defined in~(\ref{e03}).
\begin{equation}
E^H(n) = E^T(n) - \{(a,b) \in E^T(n) \suchthat{\exists c \in V(n) (a < c < b \wedge a|c \wedge c|b)}\}
\label{e03}
\end{equation}
Figures~\ref{f01} (a) and (b) show the {\em Transitive Closure} $G^T(20)$ and {\em Hasse diagram} $G^H(20)$ , respectively.
Note that $G^H(n) = G^T(n)$ if and only if $n$ is a prime.
Numerous integer sequences have been discovered from the divides relation from the number theory point of view (see ~\cite{oeis}).
In Section~\ref{s2}, this paper not only compiles
various existing integer sequences in~\cite{oeis}, but also discovers numerous integer sequences from the graph theory point of view, mainly from $G^H(n)$ and $G^T(n)$.
By the {\em Fundamental Theorem of Arithmetic}, every positive integer $n > 1$ can be represented by $\omega$ distinct prime numbers $p_1, p_2, \cdots, p_\omega$ and positive integers $m_1, m_2, \cdots, m_\omega$ as corresponding exponents such that $n = p_1^{m_1} p_2^{m_2} \cdots p_\omega^{m_\omega}$ where $p_1 < p_2 < \cdots < p_\omega$. Let $M(n)= (m_1, m_2, \cdots, m_\omega)$ be the sequence of the exponents.
In~\cite{HW1979}, Hardy and Wright used $\Omega(n)$ and $\omega(n)$ to denote the number of prime divisors of $n$ counted with multiplicity and the number of distinct prime factors of $n$, respectively.
For example, $20 = 2 \times 2 \times 5 = 2^2 \times 5^1$ has $\Omega(20) = 3$ and $\omega(20) = 2$.
Let $M'(n) = [m_1, m_2, \cdots, m_\omega]$ be the {\em multiset} known as the {\em prime signature} of $n$ where the order does not matter and repetitions are allowed.
For example, $M'(4500 = 2^2\times3^2\times5^3) = [2, 2, 3]$ has the same prime signature as $M'(33075 = 3^3\times5^2\times7^2) = [3,2,2]$.
The prime signature $M'(n)$ uniquely determines the structures of $G^H(n)$ and $G^T(n)$ and play a central role in this work as they partition the $G^H(n)$ and $G^T(n)$ into isomorphism classes and
are used as the labels of the nodes of $G^H(n)$ and $G^T(n)$ .
Any ordering of the prime signatures corresponds to an ordering of the isomorphism classes of $G^H(n)$ and $G^T(n)$ and consequently of their associated graph invariants, such as their order, size, and path counts.
Two kinds of orderings of prime signatures such as the {\em graded colexicographic} and
{\em canonical orderings} appear in the literature and the On-line Encyclopedia of Integer Sequences~\cite{oeis}.
Several integer sequences by prime signatures have been studied from the number theory point of view~\cite{AS1972,HW1979},
the earliest one of which dates from 1919~\cite{MacMahon1919}.
However, some sequences have interpretations different from the graph theory interpretations provided here.
Most importantly, over twenty new integer sequences of great interest are presented in Section~\ref{s3}.
\section{Graph Theoretic Properties and Invariants of the Divides Relation}
\label{s2}
In this section, fourteen graph invariants such as order, size, degree, etc. for the {\em Hasse Diagram} and/or {\em Transitive Closure} graphs are formally defined and investigated. Furthermore, various graph theoretic properties are also determined.
The first graph invariant of interest is the common {\em order} of $G^H(n)$ and $G^T(n)$, i.e., the number of nodes, $|V(n)|$. By definition, this is simply the number of divisors of $n$.
\begin{theorem}[Order of $G^H(n)$ and $G^T(n)$]
\label{Tmorder}
\begin{equation}
|V(n)| = |V(M(n))|= \prod_ {m_i \in M(n)}(m_i + 1)
\end{equation}
\end{theorem}
\begin{proof}
Each $p_i^{m_i}$ term contains $m_i + 1$ factors which can contribute to a divisor of $n$. Thus, the number of divisors of $n$ is $(m_1+1)\times(m_2+1)\times\cdots\times(m_\omega+1)$ by the {\em product rule of counting}.
\end{proof}
This classic and important integer sequence of $|V(n)|$ in natural order is given in Table~\ref{t01} and listed as A000005 in~\cite{oeis}.
Table~\ref{t01} lists 14 integer sequences of all forthcoming graph invariants with OEIS number if listed and blank in the OEIS column if not listed.
\begin{table}[hbtp]\vspace*{-3ex}
\caption[]{divides relation graph invariants in natural order}
\label{t01}
\centering
{\footnotesize
\begin{tabular}{cp{3.6in}c} \hline
\multicolumn{1}{c}{Invariant} &
\multicolumn{1}{c}{Integer sequence for $n = 1,\cdots,50$} &
\multicolumn{1}{c}{OEIS} \\ \hline \hline
$|V(n)|$ & 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, $\cdots$ & A000005\\ \hline
$|E^H(n)|$ &0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 12, 1, 7, 7, 4, 1, 13, $\cdots$ & A062799\\ \hline
$\Omega(n)$ &0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, $\cdots$ & A001222\\ \hline
$\omega(n)$ &0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, $\cdots$ & A001221 \\ \hline
$W_v(n)$ & 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, $\cdots$ & A096825\\ \hline
$W_e(n)$ & 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 4, 1, 2, 2, 3, 1, 6, 1, 3, 3, 2, 1, 3, 1, 3, $\cdots$ & -\\ \hline
$\Delta(n)$ &0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, $\cdots$ & -\\ \hline
$|P^H(n)|$ & 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, $\cdots$ & A008480\\ \hline
$|V_E(n)|$ & 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, $\cdots$ &A038548\\ \hline
$|V_O(n)|$ &0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 3, $\cdots$ & A056924\\ \hline
$|E_E(n)|$ & 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 2, 1, 5, 1, 2, 2, 4, 1, 6, 1, 3, 2, 2, 2, 6, 1, 2, 2, 5, 1, 6, 1, 4, 4, 2, 1, 7, 1, 4, $\cdots$ & - \\ \hline
$|E_O(n)|$ & 0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 2, 2, 0, 3, 0, 3, 2, 2, 0, 5, 1, 2, 1, 3, 0, 6, 0, 2, 2, 2, 2, 6, 0, 2, 2, 5, 0, 6, 0, 3, 3, 2, 0, 6, 1, 3, $\cdots$ & -\\ \hline
$|E^T(n)|$ & 0, 1, 1, 3, 1, 5, 1, 6, 3, 5, 1, 12, 1, 5, 5, 10, 1, 12, 1, 12, 5, 5, 1, 22, 3, 5, 6, 12, 1, 19, 1, 15, 5, 5, 5, 27, 1, 5, 5, 22, 1, 19, 1, 12, 12, 5, $\cdots$ & - \\ \hline
$|P^T(n)|$ & 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2, $\cdots$ & A002033 \\ \hline
\end{tabular} }
\end{table}
The next eleven graph invariants of interest are for $G^H(n)$ exclusively.
The second graph invariant of interest is the {\em size} of $G^H(n)$ which is the cardinality of the arc set $|E^H(n)| = |E^H(M(n))|$.
A recursive algorithm to compute $|E^H(n)|$ is given in Algorithm~\ref{Ahsize} which utilizes a size fact about the {\em Cartesian product} of two graphs.
\begin{algorithm}[Size of $G^H(n)$]
Let $m_i \in M'$ and the multiset, $M = M'(n)$ initially.
\label{Ahsize}
\begin{equation}
|E^H(M)|=
\left\{
\begin{array}{l l}
|E^H(M - \{m_i\})|\times(m_i+1) + m_i\times|V(M- \{m_i\})| & \textrm{if } |M| > 1 \\
m_1 & \textrm{if } |M| = 1
\end{array} \right.
\end{equation}
\end{algorithm}
\begin{theorem}[Algorithm~\ref{Ahsize} correctly computes $|E^H(n)|$]
\end{theorem}
\begin{proof}
In~\cite{Harary1972}, a theorem about the size of the {\em Cartesian product} of two graphs is given, i.e., the size of a {\em Cartesian product} of two graphs is the size of the first multiplied by the order of the second added to the size of the second multiplied by the order of the first.
Using this theorem and the fact that $G^H(n)$ is isomorphic to the Cartesian product of paths,
it is clear inductively that the recursive Algorithm~\ref{Ahsize} correctly computes the size of $G^H(n)$.
\end{proof}
The integer sequence of $|E^H(n)|$ is listed as A062799 with an alternative formula and described as the {\em inverse M\"{o}bius transform} of the number of distinct prime factors of $n$ in~\cite{oeis}.
For the purpose of illustrating the various concepts that are defined in what follows $G^H(540)$ is shown in Figure~\ref{f02}.
Note that $540 = 2^23^35$ and that the nodes of $G^H(540)$ are labeled with the sequence of exponents with respect of the order of $M(n)$. Each node $v \in V(n)$ is expressed as a sequence, $M_n(v) = (v_1,\cdots, v_{\omega(n)})$ where $0 \le v_i \le m_i$.
\begin{definition}[Node as a sequence] If $v \in V(n)$ and $n = p_1^{m_1}p_2^{m_2} \cdots p_\omega^{m_\omega}$, then
\begin{equation}
v = p_1^{v_1}p_2^{v_2} \cdots p_\omega^{v_\omega} \textrm{ and } M_n(v) = (v_1, v_2, \cdots, v_\omega)
\end{equation}
\end{definition}
To minimize clutter in Figure~\ref{f02} the sequences $(2,3,1), (2,3,0), \cdots, (0,0,0)$ are written \newline
2 3 1, 2 3 0, $\cdots$, 0 0 0.
\begin{figure}[htb]
\centering
\includegraphics[scale=.5]{f02.eps}
\caption{\label{f02} $G^H(540) = G^H(M(540)) = G^H((2,3,1))$.}
\end{figure}
Let $V_l(n)$ denote the set of nodes lying in the $l$ level of the decomposition of $G^H(n)$. For example in Figure~\ref{f02}, $V_5(540) = \{108, 180, 270\}$.
\begin{lemma}[The sum of the prime signature of a node equals its level]
\label{Lmlevel}
\begin{equation}
V_l(n) = \left\{v \in V(n) \suchthat{\sum_{v_i \in M_n(v)}v_i = l}\right\}
\end{equation}
\end{lemma}
\begin{proof}
If $v \in V(n)$, then $v = n/x$, where $x$ is the product of $\Omega(n) - l$ primes (multiplicities counted) contained in
$\{p_1, p_2, \cdots p_w\}$. Thus, the nodes in $V_l(n)$ are precisely the nodes with signature sum $\sum_{v_i \in M_n(v)}v_i = l$.
\end{proof}
\begin{observation} Nodes partitioned by their level.
\begin{equation}
V_{l_1}(n) \cap V_{l_2}(n) = {\O} \textrm{ if } l_1 \not= l_2 \wedge l_1, l_2 \in \{0,..,\Omega(n)\}
\end{equation}
\begin{equation}
V(n) = \bigcup_{l \in \{0,..,\Omega(n)\}} V_l(n)
\end{equation}
\begin{equation}
|V(n)| = \sum_{l \in \{0,..,\Omega(n)\}}|V_l(n)|
\end{equation}
\end{observation}
Let $P(x,y)$ be the set of paths from node $x$ to node $y$ in a directed acyclic graph where each path is a sequence of arcs from $x$ to $y$. For example in $G^H(20)$ as shown in Figure~\ref{f01} (b),
$P(1,20) = \{\langle(1,2),(2,4),(4,20)\rangle, \langle(1,2),(2,10),(10,20)\rangle, \langle(1,5),(5,10),(10,20)\rangle\}$.
Let $sp(x,y)$ and $lp(x,y)$ be the lengths of the shortest path and longest path from $x$ to $y$.
Let $G(n)$ be a directed acyclic graph with a single source node, $1$ and a single sink node, $n$.
Let $sp(G(n))$ and $lp(G(n))$ be the lengths of the shortest path and longest path from $1$ to $n$, respectively.
For simplicity sake, we shall denote $P^H(n)$ and $P^T(n)$ for $P(1,n)$ in $G^H(n)$ and $G^T(n)$, respectively.
The height of $G^H(n)$ is the maximum level in the level decomposition of $G^H(n)$, namely the number of prime factors.
\begin{theorem}[Height of $G^H(n)$]
\label{Tmdepth}
\begin{equation}
height(G^H(n)) = sp(G^H(n)) = \sum_ {m_i \in M(n)}m_i = \Omega(n)
\end{equation}
\end{theorem}
\begin{proof}
Follows directly from Lemma~\ref{Lmlevel}.
\end{proof}
\begin{corollary}[Length of Paths in $G^H(n)$ and $G^T(n)$]
\label{Cldepth}
\begin{equation}
sp(G^H(n)) = lp(G^H(n)) = lp(G^T(n)) = \Omega(n)
\end{equation}
\end{corollary}
\begin{proof}
Follows directly from Lemma~\ref{Lmlevel}.
\end{proof}
Note that $sp(G^T(n)) = 1$ since the arc with a single path, $(1,n) \in P^T(n)$.
\begin{theorem}[Symmetry of $V_l(n)$]
\label{Tmsymmv}
\begin{equation}
|V_l(n)| = |V_{\Omega(n)-l}(n)|
\end{equation}
\end{theorem}
\begin{proof}
A $1-1$ correspondence $f$ is defined between $V_l(n)$ and $V_{\Omega(n)-l}(n)$.
Let $v$ be a node in $V_l(n)$ and
$f$ the function from $V_l(n)$ to $V_{\Omega(n)-l}(n)$ defined by
\begin{equation}
f(v) = p_1^{m_1 - v_1} p_2^{m_2 - v_2}\cdots p_\omega^{m_\omega - v_\omega}
\label{ef}
\end{equation}
By Lemma~\ref{Lmlevel}, $f(v)$ is on level $\Omega(n) - l$ and $f$ is clearly $1-1$ into.
Similarly, the function $g$ from $V_{\Omega(n)-l}(n)$ to $V_l(n)$ defined by
\begin{equation}
g(u) = p_1^{m_1 - u_1} p_2^{m_2 - u_2}\cdots p_\omega^{m_\omega - n_\omega} \textrm{ where } u \in V_{\Omega(n)-l}(n)
\label{eg}
\end{equation}
is clearly $1-1$ into with $g(u)$ in $V_l(n)$. Thus, $g$ is $f^{-1}$ and $|V_l(n)| = |V_{\Omega(n)-l}(n)|$.
\end{proof}
Let $E^H_l(n)$ be the set of arcs from nodes in level $l$ to level $l+1$ and formally defined in Definition~\ref{dlev}.
\begin{definition}
\label{dlev}
\begin{equation}
E^H_l(n) = \{(a,b) \in E^H(n) | a \in V_l(n)\}
\end{equation}
\end{definition}
For example in Figure~\ref{f02}, $E^H_0(540) = \{(1,2), (1,3), (1,5)\}$ and \newline
$E^H_5(540) = \{(108,540), (180,540), (270,540)\}$.
The following is a symmetry property of $E^H(n)$.
\begin{theorem}[Symmetry of $E^H_l(n)$]
\label{Tmsymme}
\begin{equation}
|E^H_l(n)| = |E^H_{\Omega(n)-l-1}(n)|
\end{equation}
\end{theorem}
\begin{proof}
Let $a \in V_l(n)$ and $b \in V_{l+1}(n)$, and $(a,b)$ be an arc from $V_l(n)$ to $V_{l+1}(n)$.
Then, using $f$ in~(\ref{ef}), the function $F$ defined by $F(a,b) = (f(b),f(a))$ provides a $1-1$ into function from
$E^H_l(n)$ to $E^H_{\Omega(n)-l-1}(n)$.
This is seen by noting that
\begin{eqnarray}
f(b) & = & p_1^{m_1-b_1} p_2^{m_2-b_2} \cdots p_\omega^{m_\omega-b_\omega} \textrm{ is in } V_{\Omega(n) - l - 1}\\
f(a) & = & p_1^{m_1-a_1} p_2^{m_2-a_2} \cdots p_\omega^{m_\omega-a_\omega} \textrm{ is in } V_{\Omega(n) - l}\\
\frac{f(a)}{f(b)} & = & \frac{p_1^{m_1-a_1} p_2^{m_2-a_2} \cdots p_\omega^{m_\omega-a_\omega}}{p_1^{m_1-b_1} p_2^{m_2-b_2}\cdots p_\omega^{m_\omega-b_\omega}} \nonumber \\
& = & \frac{p_1^{m_1} p_2^{m_2} \cdots p_\omega^{m_\omega} p_1^{b_1} p_2^{b_2} \cdots p_\omega^{b_\omega}}
{p_1^{m_1} p_2^{m_2} \cdots p_\omega^{m_\omega}p_1^{a_1} p_2^{a_2} \cdots p_\omega^{a_\omega}} = \frac{b}{a} = p \label{epf1}
\end{eqnarray}
Thus, from~(\ref{epf1}), since $(a,b)$ is an arc, $(f(b),f(a))$ is an arc from $V_{\Omega(n) - l - 1}$ to $V_{\Omega(n) - l}$.
Therefore, $F$ provides a $1-1$ into function from $E^H_l(n)$ to $E^H_{\Omega(n)-l-1}(n)$.
Similarly, the function $G$ defined by $G(c,d) = (g(d),g(c))$ is a $1-1$ into function
from $E^H_{\Omega(n)-l-1}(n)$ to $E^H_l(n)$ .
Therefore, $|E^H_l(n)| = |E^H_{\Omega(n)-l-1}(n)|$.
\end{proof}
All $G^H(n)$ have a single source node, $1$ and a single sink node, $n$. Thus $|V_0(n)| = |V_{\Omega(n)}(n)| = 1$.
There are two other special levels with $\omega(n)$ as their cardinalities.
\begin{theorem}[Two special levels with $\omega(n)$ nodes]
\label{Tmuniqp}
\begin{equation}
|V_{\Omega(n)-1}(n)|= |V_1(n)| = \omega(n)
\end{equation}
\end{theorem}
\begin{proof}
$V_1(n)$ consists of the $\omega(n)$ distinct prime factors of $n$. By Theorem~\ref{Tmsymmv} $|V_1(n)| = |V_{\Omega(n)-1}(n)| = \omega(n) $.
\end{proof}
\begin{definition} Width of $G^H(n)$ in terms of nodes
\begin{equation}
W_v(n) = \max_{l \in \{0,..,\Omega(n)\}}|V_l(n)|
\end{equation}
\end{definition}
For example in Figure~\ref{f02}, $W_v(540) = 6$ at level $3$.
The $W_v(n)$ sequence is listed as A096825, the maximal size of an {\em antichain} in a divisor lattice in~\cite{oeis}.
A different width can be defined in terms of arc cardinality in each level as depicted in Figure~\ref{f03}.
\begin{definition} Width of $G^H(n)$ in terms of arcs
\begin{equation}
W_e(n) = \max_{l \in \{0,..,\Omega(n)-1\}}|E^H_l(n)|
\end{equation}
\end{definition}
\begin{figure}[htb]
\begin{center}
\resizebox{!}{2.4in}{\includegraphics{f03.eps}}
\end{center}
\caption{\label{f03} Anatomy of $(n)$.}
\end{figure}
For example in Figure~\ref{f02}, $W_e(540) = 12$ at levels $2$ and $3$.
The $W_e(n)$ sequence does not appear in~\cite{oeis}.
Since $G^H(n)$ is a digraph, each node, $v$ has an {\em in-degree}, $\Delta^-(v)$, number of incoming arcs and an {\em out-degree}, $\Delta^+(v)$, number of outgoing arcs and the degree of $v$ is defined $\Delta(v) = \Delta^+(v) + \Delta^-(v)$ .
\begin{lemma}[Upper bound for indegrees and outdegrees]
\label{Tmbinout}
For a node $v \in V(n)$, \newline
\[ \Delta^-(v) \le \omega(n), \Delta^+(v) \le \omega(n), \textrm{ and } \Delta(v) \le 2\omega(n)\]
\end{lemma}
\begin{proof}
For the outdegree, each node can add at most one more of each distinct prime to the product.
For the indegree, the product represented by the node was obtained by adding at most one prime to the product at the level just below.
\end{proof}
\begin{definition} The degree of the graph $G^H(n)$ denoted , $\Delta(G^H(n))$ is defined by
\begin{equation}
\Delta(G^H(n)) = \max_{v \in V(n)} \Delta(v)
\end{equation}
\end{definition}
For example from Figure~\ref{f02}, $\Delta(G^H(540)) = 5$ because the maximum node degree of $G^H(540)$ occurs at $90, 30, 18,$ and $6$.
The $\Delta(G^H(n))$ or simply $\Delta(n)$ sequence is not listed in~\cite{oeis}.
The $\Delta(G^H(n))$ can be computed very efficiently as stated in Theorem~\ref{Tdelta} using only $M'(n)$.
Let $G(n)$ be a sub-multiset of $M'(n)$.
\begin{eqnarray}
G(n) & =& [m_i \in M'(n) \suchthat m_i > 1] \\
|G(n)| & = & \sum_{m_i \in M(n)}gto(m_i) \textrm{ where } gto(m_i) =
\left\{
\begin{array}{l l}
1 & \textrm{if } m_i > 1 \\
0 & \textrm{otherwise }
\end{array} \right.
\end{eqnarray}
For example of $M'(540) = [2, 3, 1]$, $G(540) = [2, 3]$, and $|G(540)| = 2$.
\begin{theorem}[Degree of $G^H(n)$]
\label{Tdelta}
\begin{equation}
\Delta(G^H(n)) = \omega(n) + |G(n)|
\end{equation}
\end{theorem}
\begin{proof}
Consider $v \in V(n)$ with $M_n(v) = (v_1,\cdots, v_{\omega})$ where $0 \le v_i \le m_i$.
For a $v_i$ whose $m_i > 1$, $v$ has an incoming arc from a node $u$ whose $M_n(u) = (v_1,\cdots, (u_i = v_i - 1), \cdots, v_{\omega})$ provided $v_i > 0$ and $v$ has an outgoing arc to a node $w$ whose $M_n(w)= (v_1,\cdots, (w_i = v_i + 1), \cdots, v_{\omega})$ as long as $v_i < m_i$.
Every element in $G(n)$ contributes $2$ to $\Delta(v)$.
For a $v_i$ in the $M'(n) - G(n)$ multiset, whose $m_i = 1$, $v$ can have either only the incoming arc from a node $u$ whose $M_n(u) = (v_1,\cdots, (u_i = 0), \cdots, v_{\omega(n)})$ if $v_i = 1$ or the outgoing arc to a node $w$ whose $M_n(w) = [v_1,\cdots, (w_i = 1), \cdots, v_{\omega(n)}]$ if $v_i = 0$. There are $\omega(n) - |G(n)|$ number of such elements, $\le 1$.
Therefore, for every node $v \in V(n)$, $\Delta(v) \le 2 \times |G(n)|+ \omega(n) - |G(n)| = \omega(n) + |G(n)|$. There exists a node $v$ whose $\Delta(v) = \omega(n) + |G(n)|$. One such node is $v$ such that $M_n(v) = (m_1-1, m_2-1,\cdots,m_{\omega(n)}-1)$.
\end{proof}
For example in Figure~\ref{f02}, in $G^H((2,3,1))$, the node $18$ whose $M_n(18) = (1,2,0)$ has the maximum degree, $5$.
The next graph invariant of interest is the cardinality of paths, $|P(G^H(n))|$.
The first 200 integer sequence entries match with those labeled as A008480~\cite{oeis} which is the number of ordered prime factorizations of $n$ with its multinomial coefficient formula given in Theorem~\ref{Tmnumop}~\cite{AS1972,KKW1993}.
\begin{theorem}[the number of ordered prime factorizations of $n$~\cite{AS1972,KKW1993}]
\label{Tmnumop}
\begin{equation}
opf(n)= \frac{(\sum_{x \in M(n)} x)!}{ \prod_{x \in M(n)} x!}
\end{equation}
\end{theorem}
While a nice formula has been given in~\cite{AS1972,KKW1993}, a recursive definition is given here where the {\em dynamic programming} technique can be applied to quickly generate the integer sequence.
\begin{theorem}[Cardinality of $P(G^H(n))$]
\label{Tmpnumh}
\begin{equation}
|P(G^H(n))|=
\left\{
\begin{array}{l l}
\sum\limits_{v \in V_{\Omega(n)-1}(n)}|P(G^H(v))| & \textrm{if } \Omega(n) > 1 \\
1 & \textrm{if } \Omega(n) \le 1
\end{array} \right.
\end{equation}
\end{theorem}
\begin{proof}
All paths in $P(G^H(n))$ must contain exactly one node at level $\Omega(n) - 1$.
\end{proof}
The next four graph invariants involve the fact that $G^H(n)$ is {\em bipartite} as depicted in Figure~\ref{f04}.
\begin{theorem}[$G^H(n)$ is bipartite]
\label{Tmbipart}
\end{theorem}
\begin{proof}
Arcs join only even level nodes to odd level nodes and vice versa.
Thus, the nodes at even and odd levels form a bipartition of $V(n)$.
\end{proof}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{cc}
\resizebox{!}{1.8in}{\includegraphics{f04a.eps}}
&
\resizebox{!}{1.8in}{\includegraphics{f04b.eps}}\\
(a) Hasse diagram $G^H(60)$ & (b) $G^H(60)$ shown as a bipartite graph
\end{tabular}
\end{center}
\caption{\label{f04} $G^H(60)$}
\end{figure}
\begin{definition}
\label{Defoddv}
\begin{eqnarray}
V_E(n) & = & \{v \in V(n) \mid \sum_{m_i \in M_n(v)}m_i = even\} \\
V_O(n) & = & \{v \in V(n) \mid \sum_{m_i \in M_n(v)}m_i = odd\}
\end{eqnarray}
\end{definition}
The integer sequence of the cardinality of $V_E$ matches with A038548 which is the number of divisors of $n$ that are at most $\sqrt{n}$~\cite{oeis,Andrews2004}.
The integer sequence of $|V_O|$ also appears as A056924, described as the number of divisors of $n$ that are smaller than $\sqrt{n}$~\cite{oeis,Andrews2004}.
\begin{theorem}[Cardinality of $V_O(n)$]
\label{TmnumOv}
\begin{equation}
|V_O(n)| = \left \lfloor \frac{|V(n)|}{2}\right \rfloor
\end{equation}
\end{theorem}
\begin{proof}
The proof is by induction. For the base case $\omega = 1$, each divisor has a single exponent, i.e., $v_i \in \{p_1^0,p_1^1,\cdots,p_1^{m_1}\}$. Clearly, $|V_O| = \left \lfloor \frac{|V(n)|}{2}\right \rfloor$.
For the inductive step $\omega + 1$, let $M_{\omega + 1}$ be $M_{\omega}$ with $m_{\omega + 1}$ appended.
$V_O(M_{\omega+1})$ is the union of the cartesian product of $V_O(M_{\omega})$ and $V_E(m_{\omega + 1})$
together with the cartesian product of $V_E(M_{\omega})$ and $V_O(m_{\omega + 1})$, thus
\begin{equation}
|V_O(M_{\omega + 1})| = |V_O(M_{\omega})|\times|V_E(m_{\omega + 1})| + |V_E(M_{\omega})|\times|V_O(m_{\omega + 1})|
\end{equation}
There are four cases depending on the parities of $|V(M_{\omega})|$ and $m_{\omega + 1}$.
The following uses Theorem~\ref{Tmorder} and Definition~\ref{Defoddv}.
\newline
If $|V(M_{\omega})|$ is odd and $m_{\omega + 1}$ is odd,
\begin{eqnarray}
|V_O(M_{\omega + 1})| & = & \frac{|V(M_{\omega})|-1}{2} \times \frac{m_{\omega + 1}+1}{2} + \frac{|V(M_{\omega})|+1}{2} \times \frac{m_{\omega + 1}+1}{2} \nonumber \\
& = & \frac{|V(M_{\omega})|(m_{\omega + 1}+1)-(m_{\omega + 1}+1)}{4} + \frac{|V(M_{\omega})|(m_{\omega + 1}+1)+(m_{\omega + 1}+1)}{4} \nonumber \\
& = & \frac{|V(M_{\omega + 1})|-(m_{\omega + 1}+1) + |V(M_{\omega + 1})|+(m_{\omega + 1}+1)}{4}
= \left \lfloor \frac{|V(M_{\omega + 1})|}{2}\right \rfloor \nonumber
\end{eqnarray}
If $|V(M_{\omega})|$ is odd and $m_{\omega + 1}$ is even,
\begin{eqnarray}
|V_O(M_{\omega + 1})| & = & \frac{|V(M_{\omega})|-1}{2} \times \frac{m_{\omega + 1}+2}{2} + \frac{|V(M_{\omega})|+1}{2} \times \frac{m_{\omega + 1}}{2} \nonumber \\
& = & \frac{|V(M_{\omega})|(m_{\omega + 1})-(m_{\omega + 1} + 2)}{4} + \frac{|V(M_{\omega})|(m_{\omega + 1}+2)+m_{\omega + 1}}{4} \nonumber \\
& = & \frac{|V(M_{\omega})|(2m_{\omega + 1}+2)-(m_{\omega + 1} + 2)+m_{\omega + 1}}{4}
= \frac{|V(M_{\omega + 1})|-1}{2} = \left \lfloor \frac{|V(M_{\omega + 1})|}{2}\right \rfloor \nonumber
\end{eqnarray}
If $|V(M_{\omega})|$ is even and $m_{\omega + 1}$ is odd,
\begin{eqnarray}
|V_O(M_{\omega + 1})| & = & \frac{|V(M_{\omega})|}{2} \times \frac{m_{\omega + 1}+1}{2} + \frac{|V(M_{\omega})|}{2} \times \frac{m_{\omega + 1}+1}{2} \nonumber \\
& = & \frac{|V(M_{\omega})|(m_{\omega + 1}+1)}{4} + \frac{|V(M_{\omega})|(m_{\omega + 1}+1)}{4} = \frac{|V(M_{\omega + 1})|}{2} = \left \lfloor \frac{|V(M_{\omega + 1})|}{2}\right \rfloor \nonumber
\end{eqnarray}
If $|V(M_{\omega})|$ is even and $m_{\omega + 1}$ is even,
\begin{eqnarray}
|V_O(M_{\omega + 1})| & = & \frac{|V(M_{\omega})|}{2} \times \frac{m_{\omega + 1}+2}{2} + \frac{|V(M_{\omega})|}{2} \times \frac{m_{\omega + 1}}{2} \nonumber \\
& = & \frac{|V(M_{\omega})|(2m_{\omega + 1}+2)}{4} = \frac{|V(M_{\omega + 1})|}{2} = \left \lfloor \frac{|V(M_{\omega + 1})|}{2}\right \rfloor \nonumber
\end{eqnarray}
Therefore, $|V_O(M_{\omega + 1})| = \left \lfloor \frac{|V(M_{\omega + 1})|}{2}\right \rfloor$ in all four cases.
\end{proof}
\begin{corollary}[Cardinality of $V_E(n)$]
\label{TmnumEv}
\begin{equation}
|V_E(n)| =|V(n)| - |V_O(n)| =|V(n)| - \left \lfloor |V(n)|/2\right \rfloor
\end{equation}
\end{corollary}
\begin{proof}
Since $V_E(n)$ and $V_O(n)$ partition $V(n)$, $|V_E(n)| =|V(n)| - |V_O(n)|$.
\end{proof}
Similarly as with $V(n)$, $E(n)$ is bipartite as follows.
\begin{definition}
\begin{eqnarray}
E_E(n) & = & \{ (a,b) \in E^H(n) \mid \sum_{m_i \in M(a)}m_i = even\} \\
E_O(n) & = & \{ (a,b) \in E^H(n) \mid \sum_{m_i \in M(a)}m_i = odd\}
\end{eqnarray}
\end{definition}
Surprisingly, the integer sequences of $|E_E(n)|$ and $|E_O(n)|$ are not listed in~\cite{oeis}.
\begin{theorem}[Cardinality of $E_O(n)$]
\label{TmnumSe}
\begin{equation}
|E_O(n)| = \left \lfloor \frac{|E^H(n)|}{2}\right \rfloor
\end{equation}
\end{theorem}
\begin{proof}
An inductive proof, similar to the proof for the node parity decomposition in Theorem~\ref{TmnumOv}, can be applied using the cartesian product of two graphs~(\ref{eec}) and the arc parity decomposition~(\ref{eeo}).
\begin{equation}
|E^H(M_{\omega + 1})| = |E^H(M_{\omega})| \times |V(m_{\omega + 1})| + |V(M_{\omega})| \times |E^H(m_{\omega + 1})|
\label{eec}
\end{equation}
\begin{eqnarray}
|E_O(M_{\omega + 1})| & = & |E_O(M_{\omega})| \times |V_E(m_{\omega + 1})| + |E_E(M_{\omega})| \times |V_O(m_{\omega + 1})|
\label{eeo} \\
&& + |V_O(M_{\omega})| \times |E_E(m_{\omega + 1})| + |V_E(M_{\omega})| \times |E_O(m_{\omega + 1})| \nonumber
\end{eqnarray}
$|E_O(M_{\omega + 1})| = \left \lfloor |E^H(M_{\omega + 1})|/2 \right \rfloor$
in all eight cases bsed on parities of $|E^H(M_{\omega})|$, $|V(M_{\omega})|$, and $m_{\omega + 1}$.
\end{proof}
\begin{corollary}[Cardinality of $E_E(n)$]
\label{TmnumEe}
\begin{equation}
|E_E(n)| =|E^H(n)| - |E_O(n)| =|E^H(n)| - \left \lfloor \frac{|E^H(n)|}{2}\right \rfloor
\end{equation}
\end{corollary}
\begin{proof}
Since $E_E(n)$ and $E_O(n)$ partition $E^H(n)$, $|E_E(n)| =|E^H(n)| - |E_O(n)|$.
\end{proof}
The last two graph invariants of Table~\ref{t01} are exclusive to the transitive closure, $G^T(n)$, namely the size and the number of paths in $G^T(n)$.
Also surprisingly, the sequence for the size of $G^T(n)$ is not listed in~\cite{oeis}.
\begin{theorem}[Size of $G^T(n)$]
\label{TmsizeT}
\begin{equation}
|E^T(n)| = \sum_ {v \in V(n)}(|V(v)| - 1)
\end{equation}
\end{theorem}
\begin{proof}
The number of incoming arcs to node $v$ is the number of divisors of $v$ that are less than $v$ itself. Thus the indegree of $v$ is $|V(v)| - 1$ and the sum of the indegrees of all nodes in $G^T(n)$ is the size of $G^T(n)$.
\end{proof}
\begin{theorem}[Cardinality of $P(G^T(n))$]
\label{Tmpnumt}
\begin{equation}
|P(G^T(n))|=
\left\{
\begin{array}{l l}
\sum\limits_{v \in V(n)-\{n\}}|P(G^T(v))| & \textrm{if } \Omega(n) > 1 \\
1 & \textrm{if } \Omega(n) \le 1
\end{array} \right.
\end{equation}
\end{theorem}
\begin{proof}
Let $P(G^T(v))$ be the set of all paths from $1$ to $v$ where $v \ne n$. The addition of the arc $(v,n)$ to each path in $P(G^T(v))$ yields a path from $1$ to $n$. Thus, summing over all $v \in V(n) - \{n\}$ is equal to $|P(G^T(n))|$.
\end{proof}
The integer sequence of $|P(G^T(n))|$ matches with
A002033~\cite{oeis} and described as the number of {\em perfect partitions} of $n$~\cite{oeis,Comtet1974}.
Thus, the interpretation as the number of paths from $1$ to $n$ is part of the original contributions of this work.
\section{Graph Invariant Integer Sequences ordered by Prime Signature}
\label{s3}
The set of positive integers $> 1$ is partitioned by their prime signatures as exemplified in Table~\ref{t02}.
\begin{table}[b]\vspace*{-3ex}
\caption[]{Partitions of integers ($> 1$) by prime signature congruency.}
\label{t02}
\centering
{\footnotesize
\begin{tabular}{p{0.32in}p{3.4in}p{0.9in}} \hline
\multicolumn{1}{c}{$M$ / $S$} &
\multicolumn{1}{c}{Integer sequence for $n = 1,\cdots,20$} &
\multicolumn{1}{c}{OEIS} \\ \hline \hline
(1) & 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, $\cdots$ & A000040 \newline (Primes)\\ \hline
(2) & 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, $\cdots$ & A001248 \newline(Squared prime)\\ \hline
(1,1) & 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, $\cdots$ & A006881 \\ \hline
(3) & 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, $\cdots$ & A030078\newline (Cubed prime)\\ \hline
(2,1) & 12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, $\cdots$ & A054753\\ \hline
(1,1,1) &30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, $\cdots$ & A007304 \\ \hline
\multicolumn{1}{c}{$\vdots$} &
\multicolumn{1}{c}{$\vdots$} &
\multicolumn{1}{c}{$\vdots$} \\
\end{tabular} }
\end{table}
\begin{definition}
$n_x$ and $n_y$ are {\em prime signature congruent} iff $M(n_x) = M(n_y)$.
\end{definition}
Let $S(n)$ be a representative sequence of the {\em prime signature} $M(n)$ written in descending order.
More formally, $S(n) = (s_1, s_2, \cdots, s_\omega)$ is the permutation of the multiset, $M(n) = [m_1, m_2, \cdots, m_\omega]$ such that $s_1 \ge s_2 \ge \cdots \ge s_\omega$.
For example, $S(4500) = S(33075) = (3,2,2)$ because $M(4500 = 2^2\times3^2\times5^3) = [2, 2, 3]$ has the same prime signature as $M(33075 = 3^3\times5^2\times7^2) = [3,2,2]$.
Albeit there are numerous ways of ordering $S$, the set of all $S(n)$, two particular orderings such as the {\em graded colexicographic} and {\em canonical} orders of $S$ appear in the literature~\cite{HW1979,AS1972}. First in the {\em graded colexicographic order}, $S$ are first grouped by $\Omega(S)$ and then by $\omega(S)$ in ascending order. Finally, the reverse lexicographic order is applied to the sub-group. It is closely related to the {\em graded reflected colexicographic order} used and denoted as $\pi$ in~\cite{AS1972} .
Let $LI(S)$ denote the least integer of a prime signature in the graded (reflected or not) colexicographic order. This sequence is listed as A036035 in~\cite{oeis}.
\begin{figure}[htb]
\begin{center}
{\small
\begin{tabular}{ llllll }
1 (0) & 2 (1) & 3 (2) & 5 (3) & 8 (4) & 13 (5) \\
& & 4 (1,1) & 6 (2,1) & 9 (3,1) & 14 (4,1) \\
& & & 7 (1,1,1) & 10 (2,2) & 15 (3,2) \\
& & & & 11 (2,1,1) & 16 (3,1,1) \\
& & & & 12 (1,1,1,1) & 17 (2,2,1) \\
& & & & & 18 (2,1,1,1) \\
& & & & & 19 (1,1,1,1,1) \\
\end{tabular}}\\
{\small
\begin{tabular}{ l l l l l l ll }
Index & Graded Colexicographic & Canonical\\
20 & (6) & (6) \\
21 & (5,1) & (5,1) \\
22 & (4,2) & (4,2) \\
23 & (3,3) & (4,1,1) \\
24 & (4,1,1) & (3,3) \\
25 & (3,2,1) & (3,2,1) \\
26 & (2,2,2) & (3,1,1,1) \\
27 & (3,1,1,1) & (2,2,2) \\
28 & (2,2,1,1) & (2,2,1,1) \\
29 & (2,1,1,1,1) & (2,1,1,1,1) \\
30 & (1,1,1,1,1,1) & (1,1,1,1,1,1)
\end{tabular}}
\end{center}
\caption{\label{f05} First 30 prime signatures in colexicographic and canonical orders.}
\end{figure}
Next, the {\em canonical order}, also known as the {\em graded reverse lexicographic order}, is often used to order the partitions~\cite{HW1979}. It first groups prime signatures by $\Omega(S)$ and then uses the reverse lexicographic order. Although this order is identical to the {\em graded colexicographic} order for the first 22 prime signatures, they clearly differ at 23, 24, 26, 27, etc., as seen in Figure~\ref{f05}.
The integer sequence of the least integer, $LI(S)$ in canonical order is listed as the Canonical partition sequence encoded by prime factorization (A063008) in~\cite{oeis}.
\begin{figure}[htb]
\begin{center}
\resizebox{5.1in}{!}{\includegraphics{f05.eps}}\\
\end{center}
\caption{\label{f06} First seven Hasse diagrams ordered by prime signatures.}
\end{figure}
The $S(n)$ determine the structure of $G^H(S(n))$ and $G^T(S(n))$ as shown in Figure~\ref{f06} with the first few simple {\em Hasse diagrams}.
All integer sequences of graph invariants in natural order in Table~\ref{t01} can be ordered in the graded colexicographic order (Table~\ref{t04}) and the canonical order (Table~\ref{t05}).
However, very little has been investigated concerning these sequences since most of them are in fact new.
In~\cite{AS1972}, Abramowitz and Stegun labeled $\Omega(S)$, $\omega(S)$, and $|P^H(S)|$ in the graded colexicographic order as $n$, $m$, and $M_1$, respectively. Only these three graph invariants and the number of divisors, $|V(S)|$ are found in~\cite{oeis} for the graded colexicographic order. Only $|P^H(S)|$ is found in~\cite{oeis} for the canonical order.
\section{Conclusion}
\label{s4}
In this article, fourteen graph invariants were investigated for two classic graphs, the {\em Hasse diagram}, $G^H(n)$ and its {\em transitive closure}, $G^T(n)$.
Integer sequences with their first two hundred entries in natural order by $n$ are computed and compared to existing sequences in the On-Line Encyclopedia of Integer Sequences.
Five new integer sequences in natural order, shown in Table~\ref{t01} were discovered, i.e., not found in~\cite{oeis}.
New interpretations based on graph theory are provided for sequences found in ~\cite{oeis}.
Ten (Table~\ref{t04}) and thirteen (Table~\ref{t05}) new integer sequences were discovered for the graded colexicographic and canonical orders, respectively.
Here are some intriguing conjectures stated as open problems.
\begin{conjecture}[Cardinality of disjoint paths] Let $P'(G^H(n))$ be the set of {\em disjoint paths}. $|P'(G^H(n))|=\omega(n)$?
\label{Tmdpnum}
\end{conjecture}
\begin{conjecture}[Node width at middle level] $W_v(n) = |V_{\lceil \Omega(n)/2\rceil}(n)|$?
\label{Conj5}
\end{conjecture}
\begin{conjecture}[Relationship between widths by nodes and arcs] There always exists a level $l$ such that
if $|V_l(n)| = W_v(n)$, then $|E^H_l(n)| = W_e(n)$.
\begin{equation}
\argmax_{l \in \{0,..,\Omega(n)-1\}}|E^H_l(n)| = \argmax_{l \in \{0,..,\Omega(n)-1\}}|V_l(n)|?
\end{equation}
\label{Conj6}
\end{conjecture}
Other future work includes finding either a closed and/or a simpler recursive formula for the cardinality of $P(G^T(n))$ in Theorem~\ref{Tmpnumt}.
Note that entries for $|P^T(S)|$ in Table~\ref{t04} and~\ref{t05} are less than 50 as computing $|P^T(S)|$ by Theorem~\ref{Tmpnumt} took too long time.
\oneappendix
\section{Integer Sequences by Prime Signatures}
\begin{table}[btp]\vspace*{-3ex}
\caption[]{divides relation graph invariants in graded colexicographic order}
\label{t04}
\centering
{\footnotesize
\begin{tabular}{cp{3.7in}p{0.4in}} \hline
\multicolumn{1}{c}{Invariant} &
\multicolumn{1}{c}{Integer sequence for $S = [0],\cdots,[4,4]$} &
\multicolumn{1}{c}{OEIS} \\ \hline \hline
$LI(S)$ &
1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 1296, $\cdots$ & A036035 \\ \hline
$|V(S)|$ &1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64, 8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128, 9, 16, 21, 24, 25, $\cdots$ & A074139\\ \hline
$|E^H(S)|$ & 0, 1, 2, 4, 3, 7, 12, 4, 10, 12, 20, 32, 5, 13, 17, 28, 33, 52, 80, 6, 16, 22, 24, 36, 46, 54, 72, 84, 128, 192, 7, 19, 27, 31, 44, 59, 64, 75, 92, 116, 135, 176, 204, 304, 448, 8, 22, 32, 38, 40, $\cdots$ & -\\ \hline
$\Omega(S)$ &0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, $\cdots$ &A036042\\ \hline
$\omega(S)$ &0, 1, 1, 2, 1, 2, 3, 1, 2, 2, 3, 4, 1, 2, 2, 3, 3, 4, 5, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 1, 2, 2, 2, 2, $\cdots$ & A036043 \\ \hline
$W_v(S)$ & 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 2, 3, 4, 5, 7, 10, 1, 2, 3, 4, 4, 6, 7, 8, 10, 14, 20, 1, 2, 3, 4, 4, 6, 7, 8, 8, 11, 13, 15, 18, 25, 35, 1, 2, 3, 4, 5, $\cdots$ & -\\ \hline
$W_e(S)$ & 0, 1, 1, 2, 1, 3, 6, 1, 3, 4, 7, 12, 1, 3, 5, 8, 11, 18, 30, 1, 3, 5, 6, 8, 12, 15, 19, 24, 38, 60, 1, 3, 5, 7, 8, 13, 16, 19, 20, 30, 37, 46, 58, 90, 140, 1, 3, 5, 7, 8, $\cdots$ & -\\ \hline
$\Delta(S)$ & 0, 1, 2, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 4, 5, 5, 5, 2, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6, 2, 3, 4, 4, 4, 5, 5, 6, 5, 6, 7, 6, 7, 7, 7, 2, 3, 4, 4, 4, $\cdots$ & -\\ \hline
$|P^H(S)|$ & 1, 1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720, 1, 7, 21, 35, 42, 105, 140, 210, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, $\cdots$ & A036038\\ \hline
$|V_E(S)|$ &1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 3, 5, 6, 8, 9, 12, 16, 4, 6, 8, 8, 10, 12, 14, 16, 18, 24, 32, 4, 7, 9, 10, 12, 15, 16, 18, 20, 24, 27, 32, 36, 48, 64, 5, 8, 11, 12, 13, $\cdots$ & - \\ \hline
$|V_O(S)|$ & 0, 1, 1, 2, 2, 3, 4, 2, 4, 4, 6, 8, 3, 5, 6, 8, 9, 12, 16, 3, 6, 7, 8, 10, 12, 13, 16, 18, 24, 32, 4, 7, 9, 10, 12, 15, 16, 18, 20, 24, 27, 32, 36, 48, 64, 4, 8, 10, 12, 12, $\cdots$ & -\\ \hline
$|E_E(S)|$ & 0, 1, 1, 2, 2, 4, 6, 2, 5, 6, 10, 16, 3, 7, 9, 14, 17, 26, 40, 3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96, 4, 10, 14, 16, 22, 30, 32, 38, 46, 58, 68, 88, 102, 152, 224, 4, 11, 16, 19, 20, $\cdots$ & -\\ \hline
$|E_O(S)|$ &0, 0, 1, 2, 1, 3, 6, 2, 5, 6, 10, 16, 2, 6, 8, 14, 16, 26, 40, 3, 8, 11, 12, 18, 23, 27, 36, 42, 64, 96, 3, 9, 13, 15, 22, 29, 32, 37, 46, 58, 67, 88, 102, 152, 224, 4, 11, 16, 19, 20, $\cdots$ & -\\ \hline
$|E^T(S)|$ & 0, 1, 3, 5, 6, 12, 19, 10, 22, 27, 42, 65, 15, 35, 48, 74, 90, 138, 211, 21, 51, 75, 84, 115, 156, 189, 238, 288, 438, 665, 28, 70, 108, 130, 165, 240, 268, 324, 365, 492, 594, 746, 900, 1362, 2059, 36, 92, 147, 186, 200, $\cdots$ & - \\ \hline
$|P^T(S)|$ & 1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683, 64, 256, 544, 768, 976, 1888, 2316, 3172, 3408, 5740, 7880, 10404, 14300, 25988, $\cdots$ & - \\ \hline
\end{tabular} }
\end{table}
\begin{table}[btp]\vspace*{-3ex}
\caption[]{divides relation graph invariants in canonical order}
\label{t05}
\centering
{\footnotesize
\begin{tabular}{cp{3.7in}p{0.4in}} \hline
\multicolumn{1}{c}{Invariant} &
\multicolumn{1}{c}{Integer sequence for $S = [0],\cdots, [5,3]$} &
\multicolumn{1}{c}{OEIS} \\ \hline \hline
$LI(S)$ &1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 240, 216, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 480, 432, 720, 1680, 1080, 1800, 2520, 9240, 6300, 13860, 60060, 510510, 256, 384, 576, 960, 864, $\cdots$ & A063008 \\ \hline
$|V(S)|$ & 1, 2, 3, 4, 4, 6, 8, 5, 8, 9, 12, 16, 6, 10, 12, 16, 18, 24, 32, 7, 12, 15, 20, 16, 24, 32, 27, 36, 48, 64, 8, 14, 18, 24, 20, 30, 40, 32, 36, 48, 64, 54, 72, 96, 128, 9, 16, 21, 28, 24, $\cdots$ & - \\ \hline
$|E^H(S)|$ & 0, 1, 2, 4, 3, 7, 12, 4, 10, 12, 20, 32, 5, 13, 17, 28, 33, 52, 80, 6, 16, 22, 36, 24, 46, 72, 54, 84, 128, 192, 7, 19, 27, 44, 31, 59, 92, 64, 75, 116, 176, 135, 204, 304, 448, 8, 22, 32, 52, 38, $\cdots$ & -\\ \hline
$\Omega(S)$ & 0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, $\cdots$ & - \\ \hline
$\omega(S)$ & 0, 1, 1, 2, 1, 2, 3, 1, 2, 2, 3, 4, 1, 2, 2, 3, 3, 4, 5, 1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 5, 6, 7, 1, 2, 2, 3, 2, $\cdots$ & - \\ \hline
$W_v(S)$ & 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 2, 3, 4, 5, 7, 10, 1, 2, 3, 4, 4, 6, 8, 7, 10, 14, 20, 1, 2, 3, 4, 4, 6, 8, 7, 8, 11, 15, 13, 18, 25, 35, 1, 2, 3, 4, 4, $\cdots$ & -\\ \hline
$W_e(S)$ & 0, 1, 1, 2, 1, 3, 6, 1, 3, 4, 7, 12, 1, 3, 5, 8, 11, 18, 30, 1, 3, 5, 8, 6, 12, 19, 15, 24, 38, 60, 1, 3, 5, 8, 7, 13, 20, 16, 19, 30, 46, 37, 58, 90, 140, 1, 3, 5, 8, 7, $\cdots$ & -\\ \hline
$\Delta(S)$ & 0, 1, 2, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 4, 5, 5, 5, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 2, 3, 4, 4, 4, $\cdots$ & -\\ \hline
$|P^H(S)|$ & 1, 1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720, 1, 7, 21, 42, 35, 105, 210, 140, 210, 420, 840, 630, 1260, 2520, 5040, 1, 8, 28, 56, 56, $\cdots$ & A078760\\ \hline
$|V_E(S)|$ & 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 3, 5, 6, 8, 9, 12, 16, 4, 6, 8, 10, 8, 12, 16, 14, 18, 24, 32, 4, 7, 9, 12, 10, 15, 20, 16, 18, 24, 32, 27, 36, 48, 64, 5, 8, 11, 14, 12, $\cdots$ & - \\ \hline
$|V_O(S)|$ & 0, 1, 1, 2, 2, 3, 4, 2, 4, 4, 6, 8, 3, 5, 6, 8, 9, 12, 16, 3, 6, 7, 10, 8, 12, 16, 13, 18, 24, 32, 4, 7, 9, 12, 10, 15, 20, 16, 18, 24, 32, 27, 36, 48, 64, 4, 8, 10, 14, 12, $\cdots$ & -\\ \hline
$|E_E(S)|$ & 0, 1, 1, 2, 2, 4, 6, 2, 5, 6, 10, 16, 3, 7, 9, 14, 17, 26, 40, 3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96, 4, 10, 14, 22, 16, 30, 46, 32, 38, 58, 88, 68, 102, 152, 224, 4, 11, 16, 26, 19, $\cdots$ & -\\ \hline
$|E_O(S)|$ & 0, 0, 1, 2, 1, 3, 6, 2, 5, 6, 10, 16, 2, 6, 8, 14, 16, 26, 40, 3, 8, 11, 18, 12, 23, 36, 27, 42, 64, 96, 3, 9, 13, 22, 15, 29, 46, 32, 37, 58, 88, 67, 102, 152, 224, 4, 11, 16, 26, 19, $\cdots$ & -\\ \hline
$|E^T(S)|$ & 0, 1, 3, 5, 6, 12, 19, 10, 22, 27, 42, 65, 15, 35, 48, 74, 90, 138, 211, 21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665, 28, 70, 108, 165, 130, 240, 365, 268, 324, 492, 746, 594, 900, 1362, 2059, 36, 92, 147, 224, 186, $\cdots$ & - \\ \hline
$|P^T(S)|$ & 1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 368, 252, 604, 1076, 818, 1460, 2612, 4683, 64, 256, 544, 976, 768, 1888, 3408, 2316, 3172, 5740, 10404, 7880, $\cdots$ & - \\ \hline
\end{tabular} }
\end{table}
|
2,869,038,153,729 | arxiv | \section{Introduction}
\label{intro}
Einstein's equivalence principle (EEP) is at the core of our understanding of gravitation and is among the most important postulates of modern physics. It is under constant scrutiny since a violation of any of its pillars would lead to new physics beyond general relativity (GR) and would mark an important milestone in the search for a theory of everything (TOE). The EEP is comprised of three separate postulates: the Universality of Free Fall (UFF), Local Lorentz Invariance (LLI) and Local Position Invariance (LPI). Free fall experiments, as the one described in this letter, test the UFF by comparing the accelerations of two bodies of different internal structure and mass in a gravitational field. This inertial and gravitational mass equality is also known as the weak equivalence principle (WEP). To quantify a possible violation of the UFF it is common to normalise the acceleration difference between two test masses to the average local gravitational acceleration. This parametrization leads to the E\"otv\"os ratio defined by
$$\eta_{A,B}=2 \frac{g_{A}-g_{B}}{g_{A}+g_{B}},$$
with $g_{A,B}$ being the gravitational acceleration of test masses $A$ and $B$ respectively. The most straightforward way to do such a test is to directly measure the acceleration of two bodies in the same gravitational field. This class of tests is called Galilean and the most accurate to date was performed by comparing uranium and copper at a level of $10^{-10}$~\cite{Niebauer1987PRL}. The most accurate tests of the UFF were performed by the lunar laser ranging project (LLR), measuring the free fall of the moon and the earth in the gravitational field of the solar system. Since the UFF is a statement about the acting forces, not only Galilean type free fall experiments are performed to test it, but also force balance experiments with torsion balances. Torsion balances and LLR constrain possible violations of UFF to less than $10^{-13}$ in E\"otv\"os ratio~\cite{Williams2004PRL,Schlamminger2008PRL}. No violation was found so far. Future experiments with classical bodies are striving towards spaceborne platforms, to reduce the influence of external error source and allow measurements far beyond current state of the art~\cite{Nobili2012CQG,Touboul2012CQG}.\\
The use of atom interferometry broadens the field of test masses and allows an operation in the quantum regime. As such it is a complementary method to experiments with macroscopic bodies and will test aspects formerly inaccessible, such as violations linked to the coherence length of the test mass~\cite{Goklu2008CQG}, the possibility to employ cold atoms as accelerometers and clocks, and the possibility of spin-polarisation~\cite{Tarallo2014PRL}. A first measurement was performed by a device measuring gravity with a fountain of cold caesium atoms and comparing their fall rates to a commercial falling corner cube gravimeter at a level of $7\cdot 10^{-9}$~\cite{Peters1999Nat}. More recent experiments demonstrate tests of the UFF by using atom interferometry with two different quantum objects within the same device but do not yet reach the same precision. They are in part relying on two isotopes of the same species~\cite{Fray04PRL,Bonnin2013PRA,Tarallo2014PRL} but also on isotopes of two different elements~\cite{Schlippert2014PRL}. Especially tests with two isotopes want to take benefit from similarities for large noise suppression factors intrinsically arising
from the measurement’s arrangement. New experiments of both types are proposed to exceed the limits of current sensitivities, either on ground~\cite{Dickerson2013PRL,Dimopoulos2007PRL} or in micro-gravity environments~\cite{Rudolph2011MST,Geiger2011natcomm}, including the STE-QUEST space mission~\cite{Aguilera2014CQG}.\\
To employ this variety of test candidates in a precision experiment, a crucial point is the ability to trap both of the species not only simultaneously but rather in the same trap to have a well defined overlap of their initial positions and velocities. In this respect we propose quantum degenerate mixtures of rubidium and ytterbium for testing the UFF in a large scale device on ground.\\
In this paper we discuss the unique features of these mixtures that make them an ideal choice as test masses by calculating their violation parameters and comparing them to the ones used in other experiments and recent proposals. Focusing on the miscibility of different isotopes of these two elements, we will give a description on the source setup we aim for. Besides this description, we present possible scenarios for performing a UFF test with Bragg-type beam splitters. Along this we analyze noise contributions to the measured signal and estimate the performance of a test of the UFF to be $7 \cdot 10^{-13}$ in the E\"otv\"os ratio.
\subsection{Species miscibility and dynamical evolution}
\label{mixtures}
This ability to cool non-magnetic ytterbium isotopes to quantum degeneracy inside the 2\,$\mu$m dipole trap via evaporation without additional effort is a key motivation for our choice. Fermionic isotopes are not considered in this study since degenerate Fermi gases are large and expand with higher rates than BECs, which is an important parameter for long baseline interferometry. They might nevertheless be interesting for future tests and the device is designed to keep this option open. As table 2 shows, we are left with five bosonic isotopes where two of them, $^{172}$Yb and $^{176}$Yb, have negative intra-species scattering length. They would require a more complex experimental design including the manipulation of an optical Feshbach resonance to reach degeneracy. $^{174}$Yb is the most abundant isotope which was already condensed~\cite{Yamazaki2010PRL}. Nevertheless, due to the repulsive collisions to $^{87}$Rb (inter-species scattering lengths of $(880\pm 120)\,\textit{a}_0$), a binary mixture will not be stable due to three-body losses.
For all the reasons stated above, we focus our investigations on $^{168}$Yb, $^{170}$Yb and possible mixtures with $^{87}$Rb. Unfortunately, $^{168}$Yb and $^{170}$Yb are the least abundant isotopes making loading rates significantly low which constrains the cycling rate in the order of tens of seconds unless they are enriched. The $^{168}$Yb -$^{87}$Rb mixture features an inter-species positive scattering length of $39.2\pm 1.6\,\textit{a}_0$ meaning that this Yb isotope can be sympathetically cooled by $^{87}$Rb atoms.
As shown in our systematics study in section \ref{requirements_accuracy}, the separation between the two components of a binary mixture has a dramatic effect on the performance of the UFF test. Therefore, quantum miscibility cannot be neglected in this density regime. Indeed, if the interspecies repulsion exceeds the miscibility threshold~\cite{Papp2008PRL}, the two atomic clouds spatially separate to minimize the interaction energy. This immiscible state is a hindrance for optimising the overlap of the centre of mass of the two wave packets fed into the interferometer for comparison. This makes it necessary to carefully check for the proposed isotopes if they can be prepared in overlapping pairs of spherical symmetry. We therefore solve a system of 3D-coupled Gross-Pitaevskii equations describing the ground state of the mixture \cite{Ho1996PRL}. The results of these simulations are shown in figure \ref{miscibilityplot}.
\begin{figure}[t]
\centering
\begin{subfigure}[$^{168}$Yb-$^{87}$Rb]{
\includegraphics[width=0.4\textwidth]{miscibilitya.pdf}
\label{miscibilitya}}
\end{subfigure}
\hspace{1cm}
\begin{subfigure}[$^{170}$Yb-$^{87}$Rb]{
\includegraphics[width=0.4\textwidth]{miscibilityb.pdf}
\label{miscibilityb}}
\end{subfigure}
\caption{Density plots of the ground states of the $^{170}$Yb/$^{168}$Yb and $^{87}$Rb mixtures. For each pair mixture, the wave functions are computed solving the Gross-Pitaevkii equation in 3D including the intra-species interactions of the two isotopes and the inter-species one with $^{87}$Rb.
The magnitudes of these interactions are the same shown in table~\ref{yb_isotopes}. We assume that each mixture is confined by the same external trap with frequencies solely differing due to the mass difference. The trapping frequencies are $2\pi\cdot 88$\,Hz for Rb and $2\pi\cdot 67$\,Hz for Yb. In both cases, a symmetric mixture ground state is found illustrating the miscibility of the two pairs without further tuning of external optical or magnetic parameters (Feshbach for example).}
\label{miscibilityplot}
\end{figure}
The calculations confirm the miscibility of $^{87}$Rb with the two Yb isotopes considered making it a suitable candidate for an UFF test. In contrast, the combination of $^{168}$Yb with $^{170}$Yb builds up a symmetric shell structure. These binary states numerically found are susceptible to and deformable by external fields (magnetic forces, gravitational sag, etc.) present in the science chamber. Therefore, this mixture is not considered for dynamics and systematics. \\
In order to reduce systematic errors of the atom interferometric comparison and allow for an extended interrogation time, it is crucial to reduce the size of the atomic samples. In the proposed facility, few seconds of free fall or launch time are used to reach the target accuracy of the UFF test. It is clear that thermal ensembles would reach very large sizes at these time scales. This motivates the use of degenerate matter waves characterized by a slow expansion. The state-of-the art in slowing down the expansion of BECs improved dramatically with the use of delta-kick cooling (DKC) techniques \cite{Muntinga2013PRL,Dickerson2013PRL}. In recent experiments with a comparable baseline \cite{Kovachy2014Arx}, it was experimentally demonstrated that the expansion energy of a degenerate $^{87}$Rb ensemble could be restricted to only few tens of pK in 2D. We anticipated such records when proposing space missions with more than 10\,s of free evolution time~\cite{Aguilera2014CQG} of a mixture of a $^{87}$Rb / $^{85}$Rb condensates. \\
The DKC manipulation~\cite{Chu1986pro} consists in collimating matter waves by suddenly reducing the frequency of the initial trap holding the atoms and cutting it when all atoms reach the turning points of the trap walls (at t$_p$/4, where t$_p$ is the trap period). The same result is expected by re-pulsing the initial trap after switching it off for some free expansion time. A substantial part of the atoms kinetic energy is absorbed by this process leading to a slowed expansion. The analogy with light beams collimation often led to label this manipulation as an atomic lens. We anticipate the use of a double lens to match the expansion rates of $^{87}$Rb an $^{170}$Yb. This match is mandatory to mitigate errors related to residual wave front curvatures and relaxes the requirements on the initial collimation and retro reflection mirror planarity.
\section{Requirements and error budget}
\label{requirements_accuracy}
\begin{table}[tb]
\caption{Contributions of the different error sources to the uncertainty in $\eta$ in different configurations. 1) Requires back correction via knowledge of g, $T_{zz}$, $T_{zzz}$, and $\Omega_{y}$}
\begin{tabular}{p{0.3\textwidth}p{0.18\textwidth}p{0.18\textwidth}p{0.18\textwidth}}\hline
Error source & Initial & Intermediate & Advanced \\
u$_{\eta}$ & in $10^{-12}$ & $10^{-13}$ & $10^{-14}$ \\ \hline
Gravity gradient + position overlap & 0.3 & 0.3 & 0.3 \\
Gravity gradient + velocity overlap & 0.15 & 0.15 & 0.4 \\
Gravity gradient + g, v$_{0}$ & 0.15 & 0.15 & 0.15 \\
Coriolis x & 0.23 & 0.23 & 0.23 \\
Coriolis y & 0.2 & 0.2 & 0.2 \\
Other terms $^{1)}$ & 1 & 1 & 1 \\\hline
Magnetic fields & 0.3 & 1 & 1 \\ \hline
Wave fronts & 5.1 & 5.2 & 5.7 \\ \hline
Mean field & 1.3 & 3.6 & 3.9 \\ \hline
Sum & 5.7 & 6.7 & 7.4 \\ \hline
\end{tabular}
\label{tab:error_budget_table}
\end{table} This chapter summarizes the requirements on experimental and environmental parameters to restrict statistical and systematic errors. These requirements are partly relaxed compared to single species gravimetry measurements~\cite{Louchet2011NJP,LeGouet2008APB}, because the simultaneous operation of the dual atom interferometer and certain parameters choices allow to engineer suppression ratios for inertial phase shifts and inhomogeneities in the beam splitting wave fronts. A detailed derivation and discussion of error terms for an UFF-test with $^{87}$Rb / $^{85}$Rb in the 10\,m tower in Stanford was reported in~\cite{Hogan08arXiv} and the error budget for a satellite based test can be found in~\cite{Aguilera2014CQG,Schubert13arXiv}. This paper utilizes the same approaches for error assessment and thus focuses on the results. \\
We consider three different scenarios. In the near future, atoms will be dropped from the top chamber, and the scaling factors $k_{Rb}T_{Rb}^{2}=k_{Yb}T_{Yb}^{2}$ will be matched. In this case of matched scaling factors, correlation between the two atom interferometers will then allow to extract the differential phase corresponding to the differential acceleration via ellipse fitting~\cite{Varoquaux2009NJP,Foster2002OL}. The next intermediate step is to use the same free evolution time $T_{Rb}=T_{Yb}$ which mitigates bias terms $\sim kT^{3}$, $\sim kT^{4}$ but requires a more complex read out scheme. Since the scale factors differ now, the correlated signal will not form an ellipse. Restricting phase excursion to below 2$\pi$ still allows the extraction of the differential phase via fitting the Lissajous figure~\cite{Chen2014PRA}. However, the expected vibration noise level is above 2$\pi$. As mentioned earlier this ambiguity may be lifted via correlation with classical sensor mounted in close proximity to the retro reflection mirror as demonstrated for an atom interferometer on a plane~\cite{Geiger2011natcomm} or by adapting the phase extraction algorithms. Finally, the advanced scenario considers launched atoms from the bottom chamber and increased momentum transfers by the beam splitters. A lattice launching technique inside a 10\,m fountain~\cite{Dickerson2013PRL} and high momentum transfer beam splitters~\cite{Chiow2009PRL,Chiow2011PRL} which meet the requirements of this paper were already successfully implemented by other experiments. Requirements for systematics are summed up in table~\ref{tab:error_budget_reqs} and the resulting uncertainties in table~\ref{tab:error_budget_table}. Statistical fluctuations in these parameters are allowed up to the levels reported in table~\ref{tab:statistical_errors_reqs} which implies the errors in table~\ref{tab:statistical_errors_table}. \\
\begin{table}[tb]
\begin{center}
\caption{Requirements on noise sources for the dual species atom interferometers in different configurations. All contributions are expected to be uncorrelated. The requirements were set to reach the shot noise limit. Where appropriate values are given as a requirement for a single measurement cycle. (1) Assuming correlation with an additional classical seismometer or advanced data fitting eliminating the $2\pi$ ambiguity.}
\begin{tabular}{p{0.25\textwidth}p{0.325\textwidth}p{0.325\textwidth}}\hline
Noise source & Near / intermediate & Advanced \\ \hline
Shot noise & \multicolumn{2}{c}{\textit{See tab.~\ref{tab:error_budget_reqs} for N, k, and T.}} \\
Beam splitter & \multicolumn{2}{c}{1\,kHz Lorentzian linewidth} \\
Linear vibrations & $10^{-6}\,\mathrm{m\,s}^{-2}\,\mathrm{Hz}^{-1/2}$ & $10^{-6}\,\mathrm{m\,s}^{-2}\,\mathrm{Hz}^{-1/2}$ $^{(1)}$ \\
Starting velocity & $\sigma_{v}<0.3$\,mm/s & $\sigma_{v}<3.8\,\mu$m/s \\
Overlap & $\sigma_{\Delta r}<10\,\mu$m, & $\sigma_{\Delta r}<0.3\,\mu$m, \\
& $\sigma_{\Delta v}<10\,\mu$m/s & $\sigma_{\Delta v}<0.3\,\mu$m/s \\
Magnetic fields & $\sigma_{\delta B}<0.5$\,mG/m & $\sigma_{\delta B}<45\,\mu$G/m \\
Wave fronts & \multicolumn{2}{c}{$\sigma_{df}=\sigma_{\Delta z}=100\,\mu$m, jitter telescope \& mirror position 1\,mm} \\
& \multicolumn{2}{c}{in z-direction (g)} \\
Mean field & 5\,\% jitter in beam splitting ratio, 20\,\% in atom numbers & 1\,\% jitter in beam splitting ratio, 20\,\% in atom numbers \\
Cycle times & 11\,s & 12.6\,s \\ \hline
\end{tabular}
\label{tab:statistical_errors_reqs}
\end{center}
\end{table}
\begin{table}[tb]
\begin{center}
\caption{Resulting noise contributions following tab.~\ref{tab:statistical_errors_reqs}. All contributions are expected to be uncorrelated. The requirements were set to reach the shot noise limit. All values are given as the noise of a single measurement.}
\begin{tabular}{p{0.25\textwidth}p{0.325\textwidth}p{0.325\textwidth}}\hline
Noise source & Near / intermediate & Advanced \\
& in $10^{-10}$\,m/s$^{2}$ & in $10^{-11}$\,m/s$^{2}$ \\ \hline
Shot noise & 4.8 & 1.8 \\
Beam splitter & 2.8 & 1 \\
Linear vibrations & 2.8 & 1.8 \\
Overlap & 1 & 0.3 \\
Starting velocity & 0.1 & 0.03 \\
Magnetic fields & 0.3 & 0.3 \\
Wave fronts & 0.12 / $<0.01$ & $<0.01$ \\
Mean field & 0.6 & 0.4 \\ \hline
Sum & 6.3 & 2.8 \\
- after 24 h & - 7.1$\cdot$10$^{-2}$ & - 3.4$\cdot$10$^{-2}$ \\ \hline
\end{tabular}
\label{tab:statistical_errors_table}
\end{center}
\end{table} To engineer a high common mode rejection ratio, the center of mass positions, center of mass velocities, size and expansion ratios of the two atomic species have to be matched. Coupled to gravity gradients and rotations, position and velocity differences in the center of mass positions cause spurious phase shifts in the differential signal. Using trapping frequencies of $2\pi\cdot 500\,$Hz implies a gravitational sag of 1\,$\mu$m which will need to be characterized to $1\,\%$ in the advanced scenario. Due to the lattice launch, we expect a differential velocity of 31\,$\mu$m/s. The corresponding biases will be subtracted from the signal which imposes the requirement of knowing the gravity gradient to 0.1\,\%. This will be measured with the apparatus itself in a gradiometer operation mode. Existing gradiometer experiments reached noise floor of down to $3\cdot10^{-8}\,$s$^{-2}$\,Hz$^{-1/2}$~\cite{McGuirk2002,Rosi2014Nat}. Furthermore, a counter rotation of the retro reflection mirror will reduce the bias due to the earth's rotation~\cite{Dickerson2013PRL}. Additional errors occur if the atoms map different parts of the beam splitter wave fronts to which imperfect collimation or the finite quality of the retro reflection mirror cause inhomogeneities. Commercially available mirrors are rated up to $\lambda/20$ (peak to valley) ~\cite{Fichou} which puts requirements onto the maximum allowable expansion rates. Demonstrated perfomances of lensing $^{87}$Rb atoms to 1\,nK in 3D~\cite{Muntinga2013PRL}, and to 50\,pK in 2D~\cite{Kovachy2014Arx} are sufficient for the experiment.\\
Additional sources for errors are magnetic fields inducing a second order Zeeman shift in the $^{87}$Rb interferometer and the scattering properties of the individual ensembles and the mixture. Suppression of magnetic stray fields with residual rms deviations of $\sim$0.8\,mG inside a three layer 8.8\,m $\mu$-metal shield were demonstrated~\cite{Dickerson2012RSI}. Therefore, additional calibration might be necessary to characterize the magnetic fields to the required level.
\section{Conclusion and outlook}
\label{outlook}
We presented a novel experimental scheme to test the EEP with two different atomic species, namely ytterbium and rubidium which is in the progress of being set up in Hanover in the new infrastructure of the Hanover Institute of Technology. Using this particular test pair for precision inertial sensing with atom interferometry imposes some challenges which are discussed in this letter together with appropriate specific solutions. Based on the knowledge of this kind of measurement we provide an assessment of the expected performances of the experiment and of the major systematic effects. They should allow to test the E\"otv\"os parameter at a level of $7 \cdot 10^{-13}$ in the next few years. The work described in this letter is the first step in a complete investigation of inertial sensing with an alkaline earth like element as ytterbium. In the framework of the collaborative research center geo-Q we will investigate possible applications of this technology for geodesy and further ways to improve ground based EEP tests beyond the level of tests with devices employing classical test masses. We expect this work to have a major influence on to the field of fundamental sciences by giving new limits to possible violation scenarios. Moreover, the possibility to investigate interferometric techniques on long time scales with a high repetition rate will benefit atom interferometry experiments in micro-gravity environment or space platforms.
\section{Choice of test pairs}
\label{parameters}
As already mentioned the common way of quantifying an experiment testing the UFF is the E\"otv\"os ratio, which scales a measured differential acceleration to the strength of the local gravitational field comparing any abnormal composition based forces to the composition independent force. While this is a reasonable way to quantify the result of the performed measurement it does not take into account the specific kind of composition dependence in question. By just using the E\"otv\"os parameter as a tool for comparing two tests, an experiment with two spin polarized samples of the same isotope would not be treated different than a comparison between hydrogen and anti-hydrogen as proposed in~\cite{Hamilton2014PRL} while being fundamentally different. Taking the specific composition difference into account is part of the interpretation of the data and is strongly dependent on the model used to assess a possible violation theory. The use of extended wave functions for testing UFF opens the path to formerly unexplored theoretical models which are probing the quantum nature of matter and its interaction with space time~\cite{Goklu2008CQG}. While this is a vast field of study, we will focus on models which allow us a comparison to classical experiments. Specifically we asses the dilaton scenario~\cite{Damour2012CQG} and a scenario-independent scaling approach based on the standard model extension (SME)~\cite{Hohensee13PRL}. Atom interferometry can provide several new aspects different with respect to classical test masses as the test masses are of high isotopic purity and the choices of test masses can be extended beyond non-magnetic, conducting solids which are typically used in torsion balances. \\
According to the dilation model~\cite{Damour2012CQG} a violation may be caused by forces acting differently on neutron and proton number. With the introduced effective charges \Q{A,B} and \QQ{A,B} calculated from the composition of a test particle a measurement of the E\"otv\"os ratio set bounds to the parameters $D_1$ and $D_2$ according to the formula
\begin{equation}
\eta_{\text{A,B}}~\widetilde =~ D_1(\Delta Q^{'1}_{\text{A,B}})+D_2(\Delta Q^{'2}_{\text{A,B}})\text{.}
\end{equation}\\
A similar kind of parametrization can be given for the standard model extension~\cite{Hohensee13PRL}
\begin{equation}
\eta_{\text{A,B}}~\widetilde = ~\Delta f_{-n}+\Delta f_{+n}+\bar{\Delta f_{-n}}+\bar{\Delta f_{+n}}
\end{equation}
with the defined violation parameters for matter and anti-matter linked to neutron excess and total baryon number
\begin{equation}
\begin{aligned}
\Delta f_{-n} = f_{\beta^{e+p-n}_{\text{A}}}\beta^{e+p-n} - f_{\beta^{e+p-n}_{\text{B}}}\beta^{e+p-n}\\
\Delta f_{+n} = f_{\beta^{e+p+n}_{\text{A}}}\beta^{e+p+n} - f_{\beta^{e+p+n}_{\text{B}}}\beta^{e+p+n}\\
\bar{\Delta f_{-n}} = f_{\beta^{\bar{e}+\bar{p}-\bar{n}}_{\text{A}}}\beta^{\bar{e}+\bar{p}-\bar{n}} - f_{\beta^{\bar{e}+\bar{p}-\bar{n}}_{\text{B}}}\beta^{\bar{e}+\bar{p}-\bar{n}}\\
\bar{\Delta f_{+n}} = f_{\beta^{\bar{e}+\bar{p}+\bar{n}}_{\text{A}}}\beta^{\bar{e}+\bar{p}+\bar{n}} -f_{\beta^{\bar{e}+\bar{p}+\bar{n}}_{\text{B}}}\beta^{\bar{e}+\bar{p}+\bar{n}}\text{.}
\end{aligned}
\end{equation}
In both models larger absolute differences in the sensitivity factors of the employed test mass pair give rise to a larger signal in case of a violation of the UFF. Vice versa, an experimental determination of the E\"otv\"os ratio for such a test mass choice better constrains the existence of violations than tests performed with lower sensitivity factors for the same accuracy. Moreover, different test mass pairs probe different linear combinations of suspected violations linked to the neutron excess and the total baryon number of the test masses. In order to unambiguously determine the origin of a violation, a minimum of two test mass pairs needs to be employed. Interestingly, as shown in Ref.~\cite{Mueller2013proc}, even a test performed at a lower accuracy as compared to state of the art tests can further constrain possible violations, when the used test masses are significantly different to previously utilized ones. The sensitivity factors for different choices of test pairs are presented in table~\ref{violation}. For example, in comparison to Be-Ti the combination of ytterbium and rubidium isotopes is a factor of 2 more sensitive to baryon number related violations and even three orders of magnitude more sensitive in the parameter $\bar{\Delta f_{-n}}$.
\begin{table*}[h!]
\caption{Comparison of choices for test masses A and B employed in existing and planned tests of the UFF parametrized for violation scenarios with respect to their effective charges \Q{A,B}, \QQ{A,B}~and \fbplus{A,B}, \fbminus{A,B}, \fbbarminus{A,B}, \fbbarplus{A,B} calculated according to \cite{Damour2012CQG} and \cite{Hohensee13PRL}. Nuclide data is used from~\cite{Audi03} and for Ti a natural occurrence of isotopes is assumed~\cite{Laeter09}.}
\begin{tabular}{ c c c | c c c c c c } \hline
\multirow{2}{*}{A}& \multirow{2}{*}{B}&\multirow{2}{*}{Ref.} &\multicolumn{1}{c}{$\Delta$\Q{A,B}}&\multicolumn{1}{c}{$\Delta$\QQ{A,B}}& \multicolumn{1}{c}{$\Delta f_{-n}$} & \multicolumn{1}{c}{$\Delta f_{+n}$} & \multicolumn{1}{c}{$\bar{\Delta f_{-n}}$} & \multicolumn{1}{c}{$\bar{\Delta f_{+n}}$} \\
&&&\multicolumn{1}{c}{$\cdot 10^4$}&\multicolumn{1}{c}{$\cdot 10^4$}&\multicolumn{1}{c}{$\cdot 10^2$}&\multicolumn{1}{c}{$\cdot 10^4$}&\multicolumn{1}{c}{$\cdot 10^5$}&\multicolumn{1}{c}{$\cdot 10^4$}\\
\hline
\textsuperscript{9}Be& Ti&\cite{Schlamminger2008PRL} &-15.46 &-71.20& 1.48 &-4.16 & -0.24 &-16.24\\
Cu &\textsuperscript{238}U&\cite{Niebauer1987PRL} &-19.09 & -28.62 &-7.08& -8.31 &-89.89& -2.38\\
\textsuperscript{6}Li&\textsuperscript{7}Li &\cite{Hohensee2011JMO} &0.79& -10.07 &-7.26& 7.79 &-72.05& 5.82\\
\textsuperscript{85}Rb&\textsuperscript{87}Rb&\cite{Fray04PRL,Fray2009SSR,Bonnin13PRA} &0.84& -0.79 &-1.01& 1.81 &1.04& 1.67\\
\textsuperscript{87}Sr&\textsuperscript{88}Sr &\cite{Tarallo2014PRL} &0.42& -0.39 &-0.49& 2.04 &10.81& 1.85\\
\textsuperscript{39}K&\textsuperscript{87}Rb&\cite{Schlippert2014PRL}& -6.69& -23.69& -6.31& 1.90& -62.30& 0.64\\
\textsuperscript{87}Rb&\textsuperscript{170}Yb&[This work]& -12.87 &-13.92 &-1.36& -8.64 &86.00 &-5.46\\ \hline
\end{tabular}
\label{violation}
\end{table*}
\section{Atom interferometry in a 10~m atomic fountain}
\label{vlbai}
The inertial sensitive interferometry with cold rubidium clouds is well covered by state-of-the-art experiments for measuring gravity~\cite{Hauth2013APB,Gillot2014Met}, gravity gradients \cite{Rosi2014Nat} and rotations~\cite{Tackmann2012NJP} as well as for measuring fundamental constants \cite{Bouchendira2011PRL}. Similarly laser-cooled ytterbium is by now very successfully utilized in optical clocks, especially optical lattice clocks~\cite{Hinkley2013Sci}. A key prerequisite to perform interferometry over long baselines is the preparation of a very narrow velocity distribution even beyond the ones of typical Bose-Einstein condensates which was already demonstrated for both species~\cite{Anderson1995Sci,Cornish2000PRL,Takasu2003PRL,Yamazaki2010PRL}. This can be reached by delta-kick cooling~\cite{Muntinga2013PRL,Kovachy2014Arx}. The facility we want to employ for a test of the UFF is the {\it VLBAI-Teststand} located at the new founded Hanover Institute for Technology (HITec)~\cite{HITEC_WEBPAGE}. This device will provide two experimental chambers for the preparation of atomic ensembles with two independent source chambers for a maximum flexibility in the choice of atomic species. A 10\,m ultra-high vacuum-tube with a magnetically shielded region of approximately 9\,m forms the baseline for an extended free fall. Since operation of the equivalence principle test only occurs in the magnetically shielded region we anticipate a free fall time of 1\,s and up to 2.6\,s if the atoms are launched. Assuming a measurement with $1\cdot 10^{5}$ ytterbium atoms and $2\cdot 10^{5}$ rubidium atoms produced in 10\,s, this leads to a shot noise limited performance of $1.6\cdot 10^{-10}\,\mathrm{Hz}^{-1/2}$ and $6.5\cdot 10^{-12}\,\mathrm{Hz}^{-1/2}$ in the E\"otv\"os ratio respectively. The second value relies on higher order beam splitters, as explained in chapter \ref{requirements_accuracy}.\\
\begin{figure}[t]
\centering
\begin{subfigure}[Mach Zehnder geometry]{
\includegraphics[width=7cm]{machzehnder.png}
\label{machzehnderscheme}}
\end{subfigure}
\hspace{1 cm}
\begin{subfigure}[Setup]{
\includegraphics[width=2cm]{scheme.png}
\label{experimentalscheme}}
\end{subfigure}
\caption{Mode of operation in Mach-Zehnder configuration and sketch of the experimental setup. Shown in \ref{experimentalscheme} is an operation in drop configuration.}
\end{figure}
\subsection{Interferometer sequence}
\label{sequence}
As described earlier, performing an UFF-test is equivalent to a simultaneous measurement of the gravitational acceleration $g_{A,B}$ acting on the two test masses. To perform this measurement with atoms a sequence of light pulses has to be applied to interrogate them with respect to a common reference mirror which acts as a phase front reference. The most prominent configuration for inertial sensitive atom interferometry is the Mach-Zehnder-type $\pi/2-\pi-\pi/2$ sequence with a time $T$ of free evolution in between each of the pulses. Two different modes of operation can be distinguished: (i) dropping atoms from a source on the top of the device and (ii) launching atoms onto a parabolic trajectory from a source at the bottom of the device. While the first mode is characterized by a good control over the initial conditions at free evolution times of $2T=1-1.3$\,s at a baseline of roughly 9\,m, the second one offers the perspective to increase the overall length of the interferometer up to $2T=2.6$\,s. Launching over approximately 10~m was already demonstrated for rubidium in an accelerated optical lattice by coherently transferring a large number of photons at a decent efficiency \cite{Dickerson2013PRL} and appears also realizable for ytterbium with similar parameters. Nevertheless, this fountain mode requires a well controlled launching velocity of both test masses.
\subsection{Beam splitting and match of scaling factor.}
A major limitation for inertial measurements with atom interferometers is seismic noise which scales similar to the acceleration signal with $T^2$ and thus limits the maximum time of interferometry where the signal to noise ratio is still improving. When using a common mirror for a differential measurement, as planned for this experiment, the seismic noise for both interferometers is common and thus suppressed in the difference signal~\cite{Varoquaux2009NJP,Chen2014PRA}.
To fully benefit from the non magnetic properties of the ytterbium $^1S_0$ state and allow for higher order beam splitting we plan to use Bragg type beam splitters, coupling momentum states of the respective ground states. The used off resonant transitions are the $^1S_0$-$^1P_1$ transition for ytterbium at 399\,nm and the $5^2S_{1/2}$-$5^2P_{3/2}$ transition for rubidium at 780\,nm.
The suppression factor depends on the match of the scaling factor $kT^2$, with the effective wave vectors $k$, and of the sensitivity function which is itself dependent on the timing of the interferometer pulse sequence. The basic approach is to match the scaling factors by tuning the interferometry time $T$ for each species individually~\cite{Varoquaux2009NJP}. This will lead to a small difference in the frequency response of the two interferometers and will not properly suppress contributions scaling differently with $T$ but allows for a simple data analysis scheme.\\
In the case of mismatched effective wave vectors and same pulse timing, the phase frequency response is similar between the two species but rescaled according to the appropriate wave vector. As long as the resulting phase noise is smaller than 1\,rad the phase information can still be fully recovered by weighting the results with the wave vector ratio. An analysis of this case can be found in~\cite{Chen2014PRA}. Even in the case of noise above $\pi$ most of the information can be recovered at the cost of signal to noise ratio. In the case of higher common noise contributions the resulting 2$\pi$ ambiguity can be fully resolved by operating an additional classical sensor \cite{Geiger2011natcomm}. Another option is to adapt the model used for data interpretation and recover at least some level of suppression by fitting an appropriate probability distribution.
\section{Concept for a dual species source of rubidium and ytterbium}
\label{source}
Mixtures of rubidium and ytterbium have been studied before in various experiments \cite{Munchow2011PCC,Baumer2011PRA} but were not yet used for precision interferometry. The construction of a dual species source capable of supporting an EEP test experiment faces a variety of challenges which are studied in the first phase of the experiment described in this work. A source has to fulfill the following characteristics:
\begin{itemize}
\item The clouds have to be able to be cooled down to quantum degeneracy to fully exploit the long time of free fall achievable in the used infrastructure. Although this is relaxed by employing so called delta kick cooling, the efficiency of this process is strongly dependent on the initial temperature.
\item The initial collocation has to be very well known and controlled. To a certain degree this excludes isotope combination which are immiscible as discussed in chapter~\ref{mixtures}.
\item The initial velocity distribution of the two species has to be matched to a high degree to allow for differential suppression of systematic effects, like wave front curvature or residual rotations.
\item To achieve the target performance, $1\cdot 10^{5}$ ytterbium atoms and $2\cdot 10^{5}$ rubidium atoms have to be brought to degeneracy in less than 10\,s. If this performance is not reached, it will increase the time needed for integration, but is not prohibitive to the overall experiment.
\end{itemize}
\subsection{MOT Operation} Rubidium has two stable isotopes with mass numbers 87 and 85, both are bosonic and can be brought to degeneracy with common methods \cite{Anderson1995Sci,Cornish2000PRL}. Since both are also natural abundant and can be cooled similar well by standard laser cooling techniques, the specific decision for a rubidium species will be taken based on the miscibility with the ytterbium isotopes. The widely spread method for the preparation of rubidium ensembles is laser-cooling on the $5^2S_{1/2}$-$5^2P_{3/2}$ transition with a subsequent optical molasses step for achieving sub-Doppler temperatures down to approximately $2\,\mu$K. With a combination of a multi-layer atom chip allowing for an efficient transfer of laser cooled atoms to a magnetic trap and a 2D$^+$-MOT, quantum degenerated ensembles with $4\cdot10^{5}$ rubidium atoms were produced in 1.6\,s~\cite{Rudolph2015arXiv}.\\
With in total five bosonic and two fermionic stable isotopes that have all been brought to quantum degeneracy before \cite{Takasu2003PRL,Yamazaki2010PRL}, ytterbium offers a variety of choices for test masses as seen in table~\ref{yb_isotopes}. The bosonic isotopes have no hyperfine splitting and therefore a very low magnetic sensitivity compared to rubidium for example \cite{Taichenachev2006PRL}. While this is beneficial to counteract systematic effects, the missing possibility to drive Raman-transitions between the hyperfine states is limiting the implementation scenarios. Ytterbium, an alkaline earth like element, offers the possibility to perform narrow-line cooling on the inter-combination transition $^1S_0$-$^3P_1$ with a Doppler-temperature of $T_D=4.4\,\mu$K. Due to a low vapor pressure one has to face the challenge to pre-cool the hot source for efficient MOT operation. The common method is the use of a Zeeman-slower with a transversal cooling stage at the singlet transition $^1S_0$-$^1P_1$ \cite{Miranda2012PRA}. Another comparably new option is the use of 2D-MOT at same transition \cite{Dorscher2013RSI}. Experimentally loading rates of $6\cdot 10^7$ $^{174}$Yb atoms per second have been achieved by both methods. The 2D-MOT seems preferable over the Zeeman-slower setup in terms of vacuum quality in the main chamber due to the use of differential pumping stages and offers higher scalability with available laser power at 398.9\,nm.
\begin{table*}[h!]
\caption{Stable isotopes of ytterbium and their relative natural abundance~\cite{Lide2008CRC} in $\%$, character of spin-statistic, intra-species scattering length~\cite{Kitagawa2008}, inter-species scattering length with $^{87}$Rb in $a_0$~\cite{Borkowski2013arXiv}, as well as isotope-shift relative to $^{174}$Yb of the relevant cooling transitions in MHz.}
\begin{tabular}{ c | c c c c c c c } \hline
Isotope & Abund. & Spin st. &$a_{Yb/Yb}$ & $a_{Yb/Rb}$ & $J$ & $^1S_0$-$^3P_1$ & $^1S_0$-$^1P_1$\\
\hline
$^{168}$Yb & 0.13 & boson & $252 \pm 3$ & $39.2 \pm 1.6$ & & 3655 & 1887.4\\
$^{170}$Yb & 3.05 & boson & $64 \pm 2$ & $-11.5 \pm 2.5$ & & 2287 & 1192.4\\
$^{171}$Yb & 14.3 & fermion & $-2.8 \pm 3.6$& $58.9 \pm 4.4$ &(1/2-1/2)& -2132& 1153.7\\
& & & & &(1/2-3/2)& 3805 & 832.4\\
$^{172}$Yb & 21.9 & boson & $-599 \pm 64$ & $-161 \pm 11$ && 1000 & 1887.4\\
$^{173}$Yb & 16.1 & fermion & $199 \pm 2$ & $626 \pm 88$ &(5/2-5/2)& 2312 & -253.4 \\
& & & & &(5/2-7/2)& -2386 & 588\\
& & & & &(5/2-3/2)& 3806 & 516\\
$^{174}$Yb & 31.8 & boson & $105 \pm 2$ & $880 \pm 120$ & & 0 & 0\\
$^{176}$Yb & 12.7 & boson & $-24 \pm 4$ & $216.8 \pm 4.7$ & & -955 & -509.3\\ \hline
\end{tabular}
\label{yb_isotopes}
\end{table*}
\subsection{Trapping and evaporation} Since we aim for a combined trap of both species, magnetic traps are not an option for the magnetically not trappable ytterbium. As a result a far detuned optical dipole-trap in the mid-infrared will be used as a common trap. Figure~\ref{polarsim} shows the scalar polarisability at a certain wavelength with respect to the inter-combination MOT for ytterbium. The differential polarisability shows mainly two remarkable results: Ytterbium is not trapped at $1\,\mu$m and there is a zero-crossing close to $1.5\,\mu$m, that would potentially allow for AC-Stark shift compensated dipole-trap. A more conservative and less demanding solution would be the use of a dipole-trap beyond the zero-crossing for example at 1960\,nm. To compensate AC-Stark shift dispersion over the cloud, which would be large due to the narrow linewidth of the transition a low-intensity blue detuned compensation beam can be used \cite{Kaplan2002PRA} with a detuning of $\Delta_{\text{comp.}} = 2\pi\cdot1$\,GHz and a power of $I_{\text{comp.}} = 8.84$\,mW. The Bose-Einstein condensation in a single beam dipole-trap at this wavelength for $^{87}$Rb was already shown in a weak hybrid trap configuration in~\cite{Zaiser11PRA}. Therefore, a 1960\,nm trap appears to be an ideal solution and lasers with output powers up to 100\,W are available.
\begin{figure}
\centering
\begin{subfigure}[Scalar polarizability $^1S_0$]{
\includegraphics[width=0.45\textwidth]{polarizability_Yb.pdf}
\label{scPol-S}}
\end{subfigure}
\begin{subfigure}[Scalar polarizability $^3P_1$]{
\includegraphics[width=0.45\textwidth]{polarizability_YbP.pdf}
\label{scPol-P}}
\end{subfigure}
\begin{subfigure}[Differential scalar polarizability $^1S_0$-$^3P_1$]{
\includegraphics[width=0.45\textwidth]{diffpolarizability_Yb.pdf}
\label{diffPo-SP}}
\end{subfigure}
\begin{subfigure}[Differential AC-Stark shift]{
\includegraphics[width=0.45\textwidth]{ODT.pdf}
\label{AC-Stark}}
\end{subfigure}
\caption{Scalar polarisability and effective AC-Stark shift. The upper curves \ref{scPol-S} and \ref{scPol-P} show the laser wavelength dependent scalar polarisability of the states in the transition used for the intercombination line cooling. The lower curves show in \ref{diffPo-SP} the differential polarisability and in \ref{AC-Stark} the resulting differential AC Stark shift imposed on the intercombination line by a 1960\,nm ODT with 100\,W, a 50\,$\mu$m waist and using an additional 8.84\,mW dressing beam with 1\,GHz blue detuned to the transition.}
\label{polarsim}
\end{figure}
\subsection{Dual species loading sequence} The cycle time of the experiment will be limited by smaller loading rates of the ytterbium, even with the use of a 2D$^{+}$-MOT and the expected increase in flux, due to the use of higher laser power. In addition the $^1S_0$-$^1P_1$ transition cannot be driven together with the rubidium cooling transition $5^2S_{1/2}$-$5^2P_{3/2}$, since the ionization energy of the upper state of rubidium is 2.59\,eV that corresponds to 478.7\,nm. Therefore the dual species sequence will first completely undergo the loading steps for cooling and trapping ytterbium into the dipole trap before we start the fast loading of the rubidium MOT. To avoid losses due to collisions at this stage of the experiment it is possible to shift the center of the rubidium MOT against the dipole trap via adjusting the magnetic field gradient before both isotopes are co-located inside the dipole trap.
\section*{Acknowledgements}
This work is supported by the DFG in the scope of the SFB geo-Q and will facilitate the major research instrumentation { \it VLBAI-Teststand } applied for at the DFG. The authors would like to also acknowledge the support of the German Space Agency (DLR) with funds provided by the Federal
Ministry of Economic affairs and Energy (BMWi) due to an enactment of the German Bundestag under Grant No. DLR 50WM1131-1137 (project QUANTUS-III). We would like to thank M. Kasevich, J. Hogan and A. Wanner for their help during the planning of the { \it VLBAI-Teststand}. We thank H. Mueller, M. Hohensee, W. Schleich and A. Roura for support concerning the calculation and interpretation of the violation parameters. We thank C. Klempt for fruitful discussions and L. Richardson, P. Berg and E. Wodey for proof reading this document. \\
\section*{References}
|
2,869,038,153,730 | arxiv | \section{Introduction}
Black holes are believed to play a key role in a number of highly energetic astrophysical phenomena, from active galactic nuclei to gamma-ray bursts to ultraluminous X-ray binaries.
The extraordinary amounts of energy released during such events may have two different origins. It can be the gravitational potential energy of the matter falling toward an existing or forming black hole during accretion or a gravitational collapse. Or it can also be the energy of the black hole itself. Indeed, a remarkable prediction of general relativity is that a spinning black hole has free energy available to be tapped. How this occurs has fundamental implications for our understanding of high energy astrophysical phenomena powered by black holes.
It was shown by Christodoulou \cite{christodoulou70} that for a spinning (Kerr) black hole having mass $M$ and dimensionless spin parameter $a$, a portion of the black hole mass is ``irreducible'',
\begin{equation}
M_{\rm irr} = M \sqrt{\frac{1}{2} \left( {1+\sqrt{1-a^2}} \right)} \, .
\end{equation}
The irreducible mass has a one-to-one connection with the surface area of the event horizon, $A_H =4\pi(r_H^2+a^2) = 16 \pi M_{\rm irr}^2$, which is proportional to the black hole enthropy $S_{\rm BH} = ({k_B c^3}/{4 G \hbar}) A_H$ \cite{bekenstein72,bekenstein73,hawking74,hawking75}, where $k_B$, $G$, $\hbar$, and $c$ denote, respectively, the Boltzmann constant, the gravitational constant, the reduced Planck constant, and the speed of light in vacuum. Thus, the maximum amount of energy that can be extracted from a black hole without violating the second law of thermodynamics is the rotational energy
\begin{equation}
E_{\rm rot} = \left[ {1-\sqrt{\frac{1}{2} \left( {1+\sqrt{1-a^2}} \right)}} \right] M c^2 \, .
\end{equation}
For a maximally rotating black hole ($a =1$), this gives $E_{\rm rot} = (1-1/\sqrt{2}) M c^2 \simeq 0.29 M c^2$. Therefore, a substantial fraction of black hole energy can, in principle, be extracted \cite{note1}.
The possibility of extracting black hole rotational energy was first realized by Penrose \cite{penrose69}, who envisioned a thought experiment in which particle fission ($0 \rightarrow 1 + 2$) occurs in the ergosphere surrounding a rotating black hole. If the angular momentum of particle $1$ is opposite to that of the black hole and is sufficiently high, then the energy of particle $1$, as viewed from infinity, may be negative. Hence, since the total energy at infinity is conserved, the energy of particle $2$ as measured from infinity will be larger than that of the initial particle $0$. When the particle with negative energy at infinity ($1$) falls into the black hole's event horizon, the total energy of the black hole decreases. Therefore, the energy of the escaping particle $2$, which is higher than that of the original particle $0$, is increased at the expense of the rotational energy of the black hole.
Although the Penrose process indicates that it is possible to extract energy from a black hole, it is believed to be impractical in astrophysical scenarios. Indeed, energy extraction by means of the Penrose process requires that the two newborn particles separate with a relative velocity that is greater than half of the speed of light \cite{Bardeen_1972,wald74apj}, and the expected rate of such events is too rare to extract a sizable amount of black hole's rotational energy. On the other hand, Penrose's suggestion sparked the interest to find alternative mechanisms for extracting black hole rotational energy, such as superradiant scattering \cite{TP74}, the collisional Penrose process \cite{Piran75}, the Blandford-Znajek process \cite{BZ77} and the magnetohydrodynamic (MHD) Penrose process \cite{Takahashi90}.
Among them, the Blandford-Znajek process, in which energy is extracted electromagnetically through the magnetic field supported by an accretion disk around the black hole, is thought to be the leading mechanism for powering the relativistic jets of active galactic nuclei (AGNs) \citep[e.g.][]{McKGamm04,Hawley06,komissarov07,Tchekho11} and gamma-ray bursts (GRBs) \citep[e.g.][]{HKLee2000,Tchekho08,komissarov09}.
While different mechanisms of energy extraction have been carefully analyzed over the years, the possibility of extracting black hole rotational energy as a result of rapid reconnection of magnetic field lines has been generally overlooked. An exploratory study conducted by Koide and Arai \cite{KA} analyzed the feasibility conditions for energy extraction by means of the outflow jets produced in a laminar reconnection configuration with a purely toroidal magnetic field. In this simplified scenario, they suggested that relativistic reconnection was required for energy extraction, but the extracted power and the efficiency of the reconnection process were not evaluated. This is necessary for determining whether magnetic reconnection can play a significant role in the extraction of black hole energy.
The recent advent of general-relativistic kinetic simulations of black hole magnetospheres \cite{parfrey} do indeed suggest that particles accelerated during magnetic reconnection may spread onto negative energy-at-infinity trajectories, and that the energy extraction via negative-energy particles could be comparable to the energy extracted through the Blandford-Znajek process.
In this paper we provide an analytical analysis of black hole energy extraction via fast magnetic reconnection as a function of the key parameters that regulate the process: black hole spin, reconnection location, orientation of the reconnecting magnetic field, and plasma magnetization.
Our main objective is to evaluate the viability, feasibility conditions, and efficiency of magnetic reconnection as a black hole energy extraction mechanism.
In Section \ref{section2} we delineate how we envision the extraction of black hole rotational energy by means of fast magnetic reconnection, and we derive the conditions under which such energy extraction occurs. In Section \ref{section3} we show that magnetic reconnection is a viable mechanism of energy extraction for a substantial region of the parameter space.
In Section \ref{section4} we quantify the rate of energy extraction and the reconnection efficiency in order to evaluate whether magnetic reconnection is an effective energy extraction mechanism for astrophysical purposes. We further compare the power extracted by fast magnetic reconnection with the power that can be extracted through the Blandford-Znajek mechanism. Finally, we summarize our results in Section \ref{section5}.
\section{Energy Extraction by Magnetic Reconnection} \label{section2}
The possibility of extracting black hole rotational energy via negative-energy particles requires magnetic reconnection to take place in the ergosphere of the spinning black hole since the static limit is the boundary of the region containing negative-energy orbits. Magnetic reconnection inside the ergosphere is expected to occur routinely for fast rotating black holes. Indeed, a configuration with antiparallel magnetic field lines that is prone to magnetic reconnection is caused naturally by the frame-dragging effect of a rapidly spinning black hole.
In this paper, we envision the situation illustrated in Fig. \ref{fig1}, where the fast rotation of the black hole leads to antiparallel magnetic field lines adjacent to the equatorial plane.
This scenario is also consistent with numerical simulations of rapidly spinning black holes \citep[e.g.][]{parfrey,komissarov05,East18,ripperda20}.
The change in magnetic field direction at the equatorial plane produces an equatorial current sheet.
This current sheet forms dynamically and is destroyed by the plasmoid instability (permitted by non-ideal magnetohydrodynamic effects such as thermal-inertial effects, pressure agyrotropy, or electric resistivity) when the current sheet exceeds a critical aspect ratio \cite{Comisso_2016,UzdLou_2016,Comisso_2017}. The formation of plasmoids/flux ropes (see circular sections in the zoomed-in region of Fig. \ref{fig1}) drives fast magnetic reconnection \citep[e.g.][]{daughton09,bhatta09}, which rapidly converts the available magnetic energy into plasma particle energy.
Eventually, the plasma is expelled out of the reconnection layer and the magnetic tension that drives the plasma outflow relaxes. The field lines are then stretched again by the frame-dragging effect and a current layer prone to fast plasmoid-mediated reconnection forms again. This leads to reconnecting current sheets that form rapidly and intermittently.
\begin{figure}[]
\begin{center}
\includegraphics[width=8.5cm]{Luca_BlackHole_100820.pdf}
\end{center}
\caption{Schematic illustration of the mechanism of energy extraction from a rotating black hole by magnetic reconnection in the black hole ergosphere.
A configuration with antiparallel magnetic field lines adjacent to the equatorial plane is favored by the frame-dragging effect of the rapidly spinning black hole (panels (a) and (b) portray meridional and equatorial views, respectively), and the resulting equatorial current sheet is prone to fast plasmoid-mediated magnetic reconnection (see circular structures in the zoomed-in region \cite{noteplasmoids3D}).
Magnetic reconnection in the plasma that rotates in the equatorial plane extracts black hole energy if the decelerated plasma that is swallowed by the black hole has negative energy as viewed from infinity, while the accelerated plasma with a component in the same direction of the black hole rotation escapes to infinity.
The outer boundary (static limit) of the ergosphere is indicated by the short-dashed lines in both panels. In panel (b), long-dashed and solid lines indicate magnetic field lines below and above of the equatorial plane, respectively. Finally, the dashed lines in the zoomed region indicate the two magnetic reconnection separatrices intersecting at the dominant magnetic reconnection $X$-point.}
\label{fig1}
\end{figure}
Magnetic reconnection in the plasma that rotates around the black hole has the effect of accelerating part of the plasma and decelerating another part. If the decelerated plasma has negative energy at infinity and the accelerated one has energy at infinity larger than its rest mass and thermal energies (see the example regions in orange in Fig. \ref{fig1}(b)), then the plasma that escapes to infinity acquires energy at the expense of the black hole rotational energy when the negative-energy particles are swallowed by the black hole as in the standard Penrose process \cite{penrose69}. Therefore, we want to examine when magnetic reconnection in the ergosphere of the black hole redistributes the angular momentum of the plasma in such a way to satisfy these conditions. Furthermore, we want to evaluate if the extraction of black hole rotational energy via fast plasmoid-mediated reconnection can constitute a major energy release channel.
We describe the spacetime around the rotating black hole by using the Kerr metric in Boyer-Lindquist coordinates $x^\mu=(t, r, \theta, \phi)$, where $r$ is the radial distance, $\theta$ is the polar angle, and $\phi$ is the azimuthal angle. The Kerr metric can be expressed in terms of the square of the line element $d{s^2} = g_{\mu \nu} d{x^\mu}d{x^\nu}$ as \citep[e.g.][]{MTW}
\begin{equation} \label{BL_coord}
d{s^2} = g_{tt} d{t^2} + 2 g_{t\phi} dt d\phi + g_{\phi\phi} d{\phi^2} + g_{rr} d{r^2} + g_{\theta\theta} d{\theta^2} \, ,
\end{equation}
where the non-zero components of the metric are given by
\begin{equation}
g_{tt} = \frac{2 Mr}{\Sigma} -1 \, , \; \; \; g_{t\phi} = - \frac{2 M^2 a r \sin^2 \theta}{\Sigma} \, ,
\end{equation}
\begin{equation}
g_{\phi\phi} = \frac{A}{\Sigma} \sin^2 \theta \, , \quad g_{rr} = \frac{\Sigma}{\Delta} \, , \quad g_{\theta\theta} = \Sigma \, ,
\end{equation}
with
\begin{equation}
\Sigma = {r^2} + {\left( {aM} \right)^2}{\cos ^2}\theta \, ,
\end{equation}
\begin{equation}
\Delta = {r^2} - 2Mr + {\left( {aM} \right)^2} \, ,
\end{equation}
\begin{equation}
A = \big[ {{r^2} + {{\left( {a M} \right)}^2}} \big]^2 - {\left( {aM} \right)^2} \Delta \, {\sin ^2}\theta \, .
\end{equation}
The only two parameters that appear in the metric are the black hole mass, $M$, and the black hole dimensionless spin, $0 \leq a \leq 1$. Here, and in all subsequent expressions, we use geometrized units with $G=c=1$.
The inner boundary of the ergosphere of the Kerr black hole, which coincides with the outer event horizon, is given by the radial distance
\begin{equation}\label{outerevent}
r_{H}=M+ M ({1 - a^2})^{1/2} \, ,
\end{equation}
while the outer boundary (static limit) is given by
\begin{equation}\label{outerergo}
r_{E} = M+ M ({1- a^2 \cos^2 \theta})^{1/2} \, ,
\end{equation}
which yields $r_{E} =2M $ at the equatorial plane $\theta=\pi/2$.
In this paper we make the simplifying assumption that magnetic reconnection happens in the bulk plasma that rotates circularly around the black hole at the equatorial plane.
This corresponds to a Keplerian angular velocity
\begin{equation}\label{keplerOmega}
\Omega_K= \pm \frac{M^{1/2}}{r^{3/2} \pm a M^{3/2}} \, ,
\end{equation}
as seen by an observer at infinity. The upper sign refers to corotating orbits, while the lower sign applies to counter-rotating orbits. Circular orbits can exist from $r \rightarrow \infty$ down to the limiting circular photon orbit, whose radius is given by
\begin{equation}\label{circularorbitphotonrad}
r_{\rm ph}=2M \left[ 1+\cos\left(\frac{2}{3} \arccos(\mp a) \right)\right] \, .
\end{equation}
For a maximally rotating black hole ($a =1$), one has $r_{\rm ph}=M$ (corotating orbit) or $r_{\rm ph}=4M$ (counter-rotating orbit).
However, for $r > r_{\rm ph}$ not all circular orbits are stable. Non-spinning test particles can stably orbit the black hole if they are at distances larger than or equal to the innermost stable circular orbit \cite{Bardeen_1972}
\begin{equation}\label{rmargbsc}
r_{\rm isco}=M\left[3+Z_2 \mp {\Big( {(3-Z_1)(3+Z_1+2Z_2)} \Big)^{1/2}} \right] \, ,
\end{equation}
where
\begin{equation}\label{}
Z_1=1+(1-a^2)^{1/3}[(1+a)^{1/3}+(1-a)^{1/3}] \, ,
\end{equation}
\begin{equation}\label{}
Z_2=(3a^2+Z_1^2)^{1/2} \, .
\end{equation}
For a maximally rotating black hole $r_{\rm isco}=M$ (corotating orbit) or $r_{\rm isco}=9M$ (counter-rotating orbit). Here we focus on corotating orbits since we are interested in magnetic reconnection occurring inside the ergosphere.
We also assume that the plasma acceleration through magnetic reconnection is localized in a small region (close to the dominant reconnection $X$-point) compared to the size of the black hole ergosphere.
In what follows, it is convenient to analyze the plasma energy density in a locally nonrotating frame, the so called ``zero-angular-momentum-observer'' (ZAMO) frame \cite{Bardeen_1972}. In the ZAMO frame, the square of the line element is given by $d{s^2} = - d{{\hat t}^2} + \sum\nolimits_{i=1}^3 {{{(d{{\hat x}^i})}^2}} = {\eta _{\mu \nu }}d{{\hat x}^\mu }d{{\hat x}^\nu }$, where
\begin{equation}
d\hat t = \alpha \, dt \, , \quad \; d{{\hat x}^i} = \sqrt{g_{ii}} \, d{x^i} - \alpha {\beta^i}dt \,
\end{equation}
(no implicit summation is assumed over $i$), with $\alpha$ indicating the lapse function
\begin{equation}
\alpha= \left( { -g_{tt} + \frac{g_{\phi t}^2}{g_{\phi\phi}} } \right)^{1/2} = \left(\frac{\Delta \Sigma}{A} \right)^{1/2} \,
\end{equation}
and $\beta^i$ indicating the shift vector $(0, 0, \beta^\phi)$, with
\begin{equation}
\beta^\phi = \frac{\sqrt{g_{\phi\phi}} \, \omega^\phi}{\alpha} = \frac{\omega^\phi}{\alpha} \left(\frac{A}{\Sigma} \right)^{1/2} \sin\theta \,
\end{equation}
and $\omega^\phi = - g_{\phi t}/g_{\phi\phi} = 2 M^2 a r/A$ being the angular velocity of the frame dragging. An advantage of this reference frame is that equations become intuitive since the spacetime is locally Minkowskian for observers in this frame. Hereinafter, quantities observed in the ZAMO frame are denoted with hats.
Vectors in the ZAMO frame are related to the vectors in the Boyer-Lindquist coordinates as $\hat b^{0}=\alpha b^{0}$ and $\hat b^{i}= \sqrt{g_{ii}} \, b^{i} - \alpha\beta^i b^{0}$ for the contravariant components, while $\hat b_{0}=b_{0}/\alpha + \sum\nolimits_{i=1}^3 {(\beta^i/\sqrt{g_{ii}}) \, b_i} $ and $\hat b_i= b_i/\sqrt{g_{ii}}$ for the covariant components.
We evaluate the capability of magnetic reconnection to extract black hole energy by examining the conditions for the formation of negative energy at infinity and escaping to infinity of the plasma accelerated/decelerated by the reconnection process in the ergosphere (in this work we do not address the origin of the plasma properties but rather assume a plasma with a given particle density and pressure). From the energy-momentum tensor in the one-fluid approximation,
\begin{equation}
T^{\mu \nu} = p g^{\mu \nu} + w U^{\mu} U^{\nu} + {F^\mu}_{\delta} F^{\nu \delta} - \frac{1}{4} g^{\mu \nu} F^{\rho \delta} F_{\rho \delta} \, ,
\end{equation}
where, $p$, $w$, $U^{\mu}$, and $F^{\mu \nu}$ are the proper plasma pressure, enthalpy density, four-velocity, and electromagnetic field tensor, respectively, one has the ``energy-at-infinity'' density $e^\infty = - \alpha g_{\mu 0} T^{\mu 0}$. Therefore, the energy-at-infinity density is given by
\begin{equation}
e^\infty = \alpha {\hat e} + {\alpha \beta^\phi {\hat P}^\phi} \, ,
\label{einfty}
\end{equation}
where
\begin{equation}
{\hat e} = w \hat\gamma^2 -p + \frac{{\hat B}^2 + {\hat E}^2}{2} \,
\end{equation}
is the total energy density and
\begin{equation}
{\hat P}^\phi = w \hat\gamma^2 {\hat v}^\phi + {\big({\bm{\hat{B}}} \times {\bm{\hat{E}}}\big)^\phi} \,
\end{equation}
is the azimuthal component of the momentum density, with $\hat\gamma = \hat U^0 = \big[ 1 - \sum\nolimits_{i=1}^3 {{{(d{{\hat v}^i})}^2}} \big]^{-1/2}$, $\hat B^i = \epsilon^{ijk} \hat F_{jk}/2$, and $\hat E^i = \eta^{ij} \hat F_{j0} = \hat F_{i0}$.
The energy-at-infinity density can be conveniently separated into hydrodynamic and electromagnetic components as $e^\infty = e^\infty_{\rm hyd} + e^\infty_{\rm em}$, where
\begin{equation}\label{enerhyd}
e^\infty_{\rm hyd} = \alpha {\hat e}_{\rm hyd} + {\alpha \beta^\phi w \hat\gamma^2 {\hat v}^\phi } \,
\end{equation}
is the hydrodynamic energy-at-infinity density and
\begin{equation}\label{enerem}
e^\infty_{\rm em} = \alpha {\hat e}_{\rm em} + {\alpha \beta^\phi {\big({\bm{\hat{B}}} \times {\bm{\hat{E}}}\big)_\phi} } \,
\end{equation}
is the electromagnetic energy-at-infinity density, with ${\hat e}_{\rm hyd} = w \hat\gamma^2 - p$ and ${\hat e}_{\rm em} = ({\hat B}^2 + {\hat E}^2)/{2} $ indicating the hydrodynamic and electromagnetic energy densities observed in the ZAMO frame.
In this paper we assume an efficient magnetic reconnection process that converts most of the magnetic energy into kinetic energy, so that the electromagnetic energy at infinity is negligible with respect to the hydrodynamic energy at infinity. Then, from Eq. \eqref{enerhyd}, we can evaluate the energy-at-infinity density of the expelled plasma using the approximation that the plasma element is incompressible and adiabatic, which leads to \cite{KA}
\begin{equation}\label{enerhydincompress}
e^\infty_{\rm hyd} = \alpha \Big[ (\hat\gamma + \beta^\phi \hat\gamma {\hat v}^\phi)w - \frac{p}{\hat\gamma} \Big] \, .
\end{equation}
To analyze the localized reconnection process, we introduce the local rest frame $x^{\mu \prime}=(x^{0 \prime}, x^{1 \prime}, x^{2 \prime}, x^{3 \prime})$ of the bulk plasma that rotates with Keplerian angular velocity $\Omega_K$ in the equatorial plane. We choose the frame $x^{\mu \prime}$ in such a way that the direction of $x^{1 \prime}$ is parallel to the radial direction $x^{1}=r$ and the direction of $x^{3 \prime}$ is parallel to the azimuthal direction $x^{3}=\phi$. The orientation of the reconnecting magnetic field lines is kept arbitrary as it ultimately depends on the large scale magnetic field configuration, the black hole spin, and is also time dependent. Indeed, the complex nonlinear dynamics around the spinning black hole induces magnetic field line stretching, with magnetic reconnection causing a topological change of the macroscopic magnetic field configuration on short time scales.
Therefore, here we introduce the orientation angle
\begin{equation}
\xi=\arctan \big({{v}_{\rm out}^{1 \prime}}/{{v}_{\rm out}^{3 \prime}} \big) \, ,
\label{anglexi}
\end{equation}
where ${{v}_{\rm out}^{1 \prime}}$ and ${{v}_{\rm out}^{3 \prime}}$ are the radial and azimuthal components of the outward-directed plasma in the frame $x^{\mu \prime}$. Accordingly, the plasma escaping from the reconnection layer has velocities ${\bm{v}}_{\pm}^{\prime}=v_{\rm out} (\pm \cos\xi\, {\bm{e}}_3^{\prime} \mp \sin\xi\, {\bm{e}}_1^{\prime})$, with $v_{\rm out}$ indicating the magnitude of the outflow velocity observed in the frame $x^{\mu \prime}$ and the subscripts $+$ and $-$ indicating the corotating and counterrotating outflow direction, respectively. In the plasmoid-mediated reconnection regime, a large fraction of the plasma is evacuated through plasmoid-like structures \cite{noteplasmoids}, which can also contain a significant component of nonthermal particles. Such particles gain most of their energy from the motional electric field \citep[e.g.][]{GuoPoP20} and are carried out by the plasmoids (where most of them are trapped) in the outflow direction \citep[e.g.][]{sironi16}.
The outflow Lorentz factor $\hat\gamma$ and the outflow velocity component ${\hat v}^\phi$ observed by the ZAMO can be conveniently expressed in terms of the Keplerian velocity in the ZAMO frame and the outflow velocities in the local frame $x^{\mu \prime}$. From Eq. \eqref{keplerOmega}, we can express the corotating Keplerian velocity observed in the ZAMO frame as
\begin{equation}\label{keplerv}
\hat v_K = \frac{A}{\Delta^{1/2}} {\left[ { \frac{ (M/r)^{1/2} -a (M/r)^2 }{r^3-a^2 M^3} } \right]} -\beta^\phi \, .
\end{equation}
Then, using ${\hat v}_{\pm}^\phi = ({\hat v_K} \pm v_{\rm out} \cos \xi)/(1 \pm {\hat v_K} v_{\rm out} \cos \xi)$ for the azimuthal components of the two outflow velocities and introducing the Lorentz factors $\hat\gamma_K=(1-\hat v_K^2)^{-1/2}$ and $\gamma_{\rm out} =(1-v_{\rm out}^2)^{-1/2}$, we can write the energy-at-infinity density of the reconnection outflows as
\begin{eqnarray}\label{energuis}
e^\infty_{{\rm hyd},\pm}& \!=\! &\alpha \hat\gamma_K \Bigg[ \left(1 \!+\! \hat v_K \beta^\phi \right) \gamma_{\rm out} w \nonumber \\
&& \pm \cos\xi \left(\hat v_K \!+\! \beta^\phi \right) \gamma_{\rm out} v_{\rm out} w \nonumber \\
&& -\frac{p}{\left(1 \!\pm\! \cos\xi \, \hat v_K v_{\rm out} \right) \gamma_{\rm out} \hat\gamma_K^2} \Bigg] \, ,
\end{eqnarray}
where the subscripts $+$ and $-$ indicate the energy-at-infinity density associated with corotating (${\bm{v}}_{+}^{\prime}$) and counterrotating (${\bm{v}}_{-}^{\prime}$) outflow directions as observed in the local frame $x^{\mu \prime}$.
The outflow velocity $v_{\rm out}$ can be evaluated by assuming that the local current sheet at the dominant $X$-point has a small inverse aspect ratio $\delta_X /L_X \ll 1$, where $\delta_X$ and $L_X$ are the half-thickness and half-length of this local current sheet.
If we consider that the rest frame rotating with Keplerian velocity is in a gravity-free state and neglect general relativistic corrections \cite{AsenjComisPRL,comiAsenjblackhole,AsenjComiPRD19}, then, the conservation of momentum along the reconnection neutral line gives
\begin{equation}
w \gamma_{\rm out}^2 v_{\rm out}^2/L_X + {{B}_{\rm up}^2} \delta_X^2/L_X^3 \simeq ({{B}_{\rm up}}/\delta_X) ({{B}_{\rm up}} \delta_X/L_X) \, ,
\label{mom_eq}
\end{equation}
where $B_{\rm up}$ is the local magnetic field strength immediately upstream of the local current sheet. Here we have used Maxwell's equations to estimate the current density at the neutral line in addition to the outflow magnetic field strength \cite{Lyubarsky,comiAsenjoPRLspecial}. We also assumed that the thermal pressure gradient force in the outflow direction is small compared to the magnetic tension force, as verified by numerical simulations of relativistic reconnection with antiparallel magnetic fields \cite{Liu17}. Then, from Eq. \eqref{mom_eq} one gets
\begin{eqnarray}\label{velocityBup}
v_{\rm out} \simeq \left[ {\frac{ \left( 1-\delta_X^2/L_X^2 \right) \sigma_{\rm up}}{1 + \left( 1-\delta_X^2/L_X^2 \right) \sigma_{\rm up}}} \right]^{1/2} \, ,
\end{eqnarray}
where $\sigma_{\rm up} = B_{\rm up}^2/w_0$ is the plasma magnetization immediately upstream of the local current sheet at the dominant $X$-point. Consequently, for $\delta_X /L_X \ll 1$, the outflow velocity reduces to $v_{\rm out} \simeq \left[ {{\sigma_{\rm up}}/{(1 + \sigma_{\rm up})}} \right]^{1/2}$.
The local magnetic field $B_{\rm up}$ can be connected to the asymptotic macro-scale magnetic field $B_0$ by considering force balance along the inflow direction.
In the magnetically dominated regime, thermal pressure is negligible, and the inward-directed magnetic pressure gradient force must be balanced by the outward-directed magnetic tension (the inertia of the inflowing plasma is negligible if $\delta_X /L_X \ll 1$). Then, from geometrical considerations one gets \cite{Liu17}
\begin{equation}
B_{\rm up} = \frac{1- (\tan \varphi)^2}{1+ (\tan \varphi)^2} B_0 \, ,
\label{drop_B_eq}
\end{equation}
where $\varphi$ is the opening angle of the magnetic reconnection separatrix. Estimating $\tan \varphi \simeq \delta_X/L_X$, we have simply
\begin{eqnarray}\label{velocityB0}
v_{\rm out} \simeq {\left( {\frac{\sigma_0}{1 + \sigma_0}} \right)^{1/2}} \, ,\quad \gamma_{\rm out} \simeq {\left( {1+\sigma_0} \right)^{1/2}} \, ,
\end{eqnarray}
where we have defined $\sigma_0 = B_0^2/w_0$ as the plasma magnetization upstream of the reconnection layer. Accordingly, in the magnetically dominated regime $\sigma_0 \gg 1$, the reconnection outflow velocity approaches the speed of light. We finally note that in the presence of significant embedding of the local current sheet, the scaling of the outflow velocity could be weakened with respect to $B_0$, while Eq. \eqref{velocityBup} remains accurate \cite{Liu17,sironi16}.
We must point out that in the plasmoid-mediated reconnection regime considered here, the continuous formation of plasmoids/flux ropes prevents the formation of extremely elongated ``laminar'' reconnection layers, thereby permitting a high reconnection rate \citep[e.g.][]{daughton09,bhatta09}. Depending on the plasma collisionality regime, plasmoid-mediated reconnection yields an inflow velocity (as observed in the frame $x^{\mu \prime}$)
\begin{equation} \label{recvelocity}
v_{\rm in} =
\begin{cases}
\mathcal{O}(10^{-2}) & {\rm for} \quad \delta_X > \ell_k \; [44\!-\!47] \\
\mathcal{O}(10^{-1}) & {\rm for} \quad \delta_X \lesssim \ell_k \; [42, 43] \, ,
\end{cases}
\end{equation}
where $\ell_k$ is the relevant kinetic scale that determines the transition between the collisional and collisionless regimes. The collisional regime is characterized by $\delta_X > \ell_k$, while the collisionless regime occurs if $\delta_X \lesssim \ell_k$. For a pair (${e^-} {e^+}$) dominated plasma, we have \cite{comiAsenjoPRLspecial} $\ell_k = \sqrt{\gamma_{{\rm th},e}} \, \lambda_e$, where $\lambda_e$ is the nonrelativistic plasma skin depth and ${\gamma_{{\rm th},e}}$ is the electron/positron thermal Lorentz factor.
If there is also a significant ion component, then \cite{daughton09} $\ell_k = \sqrt{\gamma_{{\rm th},i}} \, \lambda_i$, where $\lambda_i$ is the nonrelativistic ion inertial length and ${\gamma_{{\rm th},i}}$ is the ion thermal Lorentz factor.
We emphasize that the reconnection rate is independent of the microscopic plasma parameters when magnetic reconnection proceeds in the plasmoid-mediated regime. In particular, plasmoid-mediated reconnection in the collisionless regime has an inflow velocity $v_{\rm in}$ that is a significant fraction of the speed of light, which potentially allows for a high energy extraction rate from the black hole (see Sec. \ref{section4}).
The expression for the energy at infinity associated with the accelerated/decelerated plasma as a function of the critical parameters ($a$, $r/M$, $\sigma_0$, $\xi$) can be finally obtained by substituting the magnetization dependence of the outflow velocity into Eq. \eqref{energuis}. Then, the hydrodynamic energy at infinity per enthalpy $\epsilon^\infty_\pm = e^\infty_{{\rm hyd},\pm}/w$ becomes
\begin{eqnarray}\label{energuisMagnet}
\epsilon^\infty_\pm& \!=\! &\alpha \hat\gamma_K \Bigg[ \left(1 \!+\! \beta^\phi \hat v_K\right) {\left( {1 \!+\! \sigma_0} \right)^{1/2}} \pm \cos{\xi} \left(\beta^\phi \!+\! \hat v_K \right) \sigma_0^{1/2} \nonumber\\
&&\qquad\qquad - \frac{1}{4} \frac{{\left( {1 \!+\! \sigma_0} \right)^{1/2}} \mp \cos{\xi} \, \hat v_K \sigma_0^{1/2}}{\hat\gamma_K^2 (1+\sigma_0 \!-\! \cos^2{\xi} \, \hat v_K^2 \sigma_0)}\, \Bigg]\, ,
\end{eqnarray}
where we have assumed a relativistically hot plasma with polytropic index $\Gamma=4/3$.
Similarly to the original Penrose process \cite{penrose69}, energy extraction from the black hole through magnetic reconnection occurs when
\begin{equation}\label{conditionsenergy}
\epsilon^\infty_-<0\, \quad {\rm and} \quad \Delta \epsilon^\infty_+ >0 \, ,
\end{equation}
where
\begin{equation}\label{conditionsenergy2}
\Delta \epsilon^\infty_+ = \epsilon^\infty_+ - \left( {1-\frac{\Gamma}{\Gamma-1} \frac{p}{w} } \right) = \epsilon^\infty_+
\end{equation}
for a relativistically hot plasma.
Therefore, black hole rotational energy is extracted if the decelerated plasma acquires negative energy as measured at infinity, while the plasma that is accelerated acquires energy at infinity larger than its rest mass and thermal energies.
\begin{figure}[]
\begin{center}
\vspace{0.20cm}
\includegraphics[width=8.4cm]{Fig2.pdf}
\vspace{-0.30cm}
\end{center}
\caption{Energy at infinity per enthalpy $\epsilon^\infty_+$ (gray line) and $\epsilon^\infty_-$ (orange line) for optimal energy extraction conditions ($a, r/M \rightarrow 1$ and $\xi \rightarrow 0$). Energy extraction requires $\sigma_0 > 1/3$. For $\sigma_0 \gg 1$, $\epsilon^\infty_+ \simeq \sqrt{3 \sigma_0}$ (dash-dotted black line) and $\epsilon^\infty_- \simeq - \sqrt{\sigma_0/3}$ (dashed black line).}
\label{fig2}
\end{figure}
The energy at infinity per enthalpy $\epsilon^\infty_\pm$ given by Eq. \eqref{energuisMagnet} depends on the black hole spin $a$ and the $X$-point distance $r/M$, as well as the plasma magnetization $\sigma_0$ and the orientation angle $\xi$, which encodes the information of the magnetic field configuration surrounding the black hole. Eqs. \eqref{energuisMagnet}-\eqref{conditionsenergy2} indicate that energy extraction is favored by lower values of the orientation angle $\xi$ and higher values of the magnetization $\sigma_0$. It is instructive to consider the limit $a \rightarrow 1$, $\xi \rightarrow 0$, and $r \rightarrow M$ (the metric \eqref{BL_coord} has a coordinate singularity at the event horizon that can be removed by a coordinate transformation). In this case, from Eq. \eqref{energuisMagnet} we obtain $\epsilon^\infty_+>0$ and $\epsilon^\infty_-<0$ when
\begin{equation}\label{}
\sigma_0 > {1}/{3} \, .
\end{equation}
Therefore, in principle, it is possible to extract rotational energy via magnetic reconnection for values of $\sigma_0$ below unity. However, higher $\sigma_0$ values are required to extract sizable amounts of energy. If, in addition to $a, r/M \rightarrow 1$ and $\xi \rightarrow 0$, we also consider $\sigma_0 \gg 1$, from Eq. \eqref{energuisMagnet} we obtain
\begin{equation}\label{energ_mas_simple}
\epsilon^\infty_+ \simeq \sqrt{3 g_{\phi\phi}} \, {\omega^\phi} \gamma_{\rm out} v_{\rm out} \simeq \sqrt{3 \sigma_0} \, ,
\end{equation}
\begin{equation}\label{energ_minus_simple}
\epsilon^\infty_- \simeq - {\sqrt{\frac{g_{\phi\phi}}{3}} \, \omega^\phi} \gamma_{\rm out} v_{\rm out} \simeq - \sqrt{\frac{\sigma_0}{3}} \, .
\end{equation}
These relations give us the energy at infinity per enthalpy of the accelerated ($+$) and decelerated ($-$) plasma in the maximal energy extraction regime (as can be seen from Fig. \ref{fig2}, they provide a fairly accurate estimate already at values of $\sigma_0$ moderately larger then unity).
In the next sections, we will show that magnetic reconnection is a viable mechanism for extracting energy from rotating black holes for a significant region of the parameter space, we will evaluate the rate of black hole energy extraction, and we will determine the efficiency of the reconnection process.
\section{Energy Extraction Assessment in Phase Space} \label{section3}
We analyze the viability of energy extraction via magnetic reconnection by considering solutions of Eq. \eqref{energuisMagnet}. In particular, in Figs. \ref{fig3} and \ref{fig4} we display the regions of the phase-space $\{a,r/M\}$ where $\epsilon^\infty_- <0$ and $ \Delta \epsilon^\infty_+ >0$, which correspond to the conditions for energy extraction. This is done for a reconnecting magnetic field with orientation angle $\xi = \pi/12$ and different values of the magnetization parameter $\sigma_0 \in \left\{ {1,3,10,30,100} \right\}$ (Fig. \ref{fig3}), and for a plasma magnetization $\sigma_0 = 100$ and different values of the orientation angle $\xi \in \left\{ {\pi/20,\pi/12,\pi/6,\pi/4} \right\}$ (Fig. \ref{fig4}).
\begin{figure}[]
\begin{center}
\includegraphics[width=8.4cm]{Fig3.pdf}
\vspace{-0.30cm}
\end{center}
\caption{Regions of the phase-space $\{a,r/M\}$ where the energies at infinity per enthalpy from Eq. \eqref{energuisMagnet} are such that $\Delta \epsilon^\infty_+ >0$ (gray area) and $\epsilon^\infty_- <0$ (orange to red areas), for a reconnecting magnetic field having orientation angle $\xi = \pi/12$ and different values of the magnetization parameter $\sigma_0 \in \left\{ {1,3,10,30,100} \right\}$. The area with $\epsilon^\infty_- <0$ increases monotonically as $\sigma_0$ increases.
The solid black line indicates the limit of the outer event horizon, Eq. \eqref{outerevent}, the dashed black line represents the limiting corotating circular photon orbit, Eq. \eqref{circularorbitphotonrad}, while the dash-dotted black line corresponds to the innermost stable circular orbit, Eq. \eqref{rmargbsc}. The limit $r/M = 2$ corresponds to the outer boundary of the ergosphere at $\theta = \pi/2$.}
\label{fig3}
\end{figure}
\begin{figure}[]
\begin{center}
\includegraphics[width=8.4cm]{Fig4.pdf}
\vspace{-0.30cm}
\end{center}
\caption{Regions of the phase-space $\{a,r/M\}$ where the energies at infinity per enthalpy from Eq. \eqref{energuisMagnet} are such that $\Delta \epsilon^\infty_+ >0$ (gray area) and $\epsilon^\infty_- <0$ (green areas), for plasma magnetization $\sigma_0 = 100$ and different values of the orientation angle $\xi \in \left\{ {\pi/20,\pi/12,\pi/6,\pi/4} \right\}$. Other lines are the same as in Figure \ref{fig3}. The area with $\epsilon^\infty_- <0$ increases monotonically as $\xi$ decreases.}
\label{fig4}
\end{figure}
As the magnetization of the plasma increases, the region of the phase-space $\{a,r/M\}$ where magnetic reconnection extracts black hole rotational energy extends to larger $r/M$ values and lower values of the dimensionless spin $a$ (Fig. \ref{fig3}). From Eq. \eqref{energuisMagnet} we can see that $\epsilon^\infty_-$ is a monotonically decreasing function of $\sigma_0$, while $\epsilon^\infty_+$ monotonically increases with $\sigma_0$. $\epsilon^\infty_+ > 0$ is easily satisfied for $r_{\rm ph} < r < r_E$, $a>0$, and $\xi < \pi/2$. On the other hand, $\epsilon^\infty_- < 0$ requires $\sigma_0 \gg 1$ in order for reconnection to extract black hole energy in a significant region of the phase-space $\{a,r/M\}$. High values of the plasma magnetization can extend the energy extraction region up to the outer boundary of the ergosphere, while energy extraction for moderate values of the spin parameter $a$ is subject to the occurrence of particle orbits inside the ergosphere.
Energy extraction via magnetic reconnection is also favored by reconnection outflows whose orientation is close to the azimuthal direction. The region of the phase-space $\{a,r/M\}$ where energy extraction occurs increases to larger $r/M$ values and lower $a$ values as the orientation angle $\xi$ decreases. Notwithstanding, even an angle as large as $\xi = \pi/4$ admits a modest region of the phase-space where magnetic reconnection extracts rotational energy. The increase of the energy extraction region for decreasing angle $\xi$ is due to the fact that only the azimuthal component of the outflow velocity contributes to the extraction of rotational energy. For an angle $\xi = \pi/20$, the extraction of black hole energy happens for $X$-points up to $r/M \approx 1.96$ (for $\sigma_0 =100$), while $\xi \rightarrow 0$ can extend this margin up to the outer boundary of the ergosphere.
The ergosphere of spinning black holes ($r_{H} <r < r_{E}$) can reach very high plasma magnetizations (e.g, $\sigma_0 \gg 100$ close to the event horizon of the black hole M87* \cite{EHT_5_2019}). Furthermore, for rapidly spinning ($a$ close to unity) black holes, we expect a reconnecting magnetic field with small orientation angle, $\xi \lesssim \pi/6$, as the strong frame-dragging effect inside the ergosphere stretches the magnetic field lines along the azimuthal direction \citep[e.g.][]{Koide02,Semenov04}. Therefore, the plots shown in Figs. \ref{fig3} and \ref{fig4} indicate that magnetic reconnection is a viable mechanism for extracting energy from rotating black holes with dimensionless spin $a$ close to unity. On the other hand, energy extraction via magnetic reconnection becomes negligible for spin values $a \lesssim 0.8$. The availability of reconnection regions inside the ergosphere decreases as the spin parameter decreases, with no circular orbits inside the ergosphere for spin $a \leq 1/\sqrt{2}$. Magnetic reconnection could still be capable of extracting energy in such cases if a circular orbit is sustained thanks to the help of the magnetic field or if one considers non-circular orbits.
\section{Energy Extraction Rate and Reconnection Efficiency} \label{section4}
We now evaluate the rate of black hole energy extraction. This depends on the amount of plasma with negative energy at infinity that is swallowed by the black hole in the unit time. Therefore, a high reconnection rate can potentially induce a high energy extraction rate. The power $P_{\rm extr}$ extracted from the black hole by the escaping plasma can be estimated as
\begin{equation} \label{Pextr}
P_{\rm extr} = - \epsilon_-^\infty w_0 A_{\rm in} U_{\rm in} \, ,
\end{equation}
where $U_{\rm in} = \mathcal{O}(10^{-1})$ for the collisionless regime, while $U_{\rm in} = \mathcal{O}(10^{-2})$ for the collisional one. $A_{\rm in}$ is the cross-sectional area of the inflowing plasma, which can be estimated as ${A}_{\rm in} \sim (r_E^2 - r_{{\rm ph}}^2)$ for rapidly spinning black holes. In particular, for $a \rightarrow 1$ one has $(r_E^2 - r_{{\rm ph}}^2) = (r_{E}^2 - r_{H}^2) = 3M^2$.
We show in Fig. \ref{fig5} the ratio $P_{\rm extr}/w_0$ as a function of the dominant $X$-point location $r/M$ for a rapidly spinning black hole with $a=0.99$ and magnetic reconnection in the collisionless regime.
This is done for a typical reconnecting magnetic field with orientation angle $\xi = \pi/12$ and different values of the magnetization parameter $\sigma_0 \in \left\{ {10,10^2,10^3,10^4,10^5} \right\}$ (top panel), and for a typical magnetization $\sigma_0 = 10^4$ and different values of the orientation angle $\xi \in \left\{ {0,\pi/20,\pi/12,\pi/8,\pi/6} \right\}$ (bottom panel).
The power extracted from the black hole increases monotonically for increasing values of the plasma magnetization and for lower values of the orientation angle. It reaches a peak for $X$-point locations that are close to the limiting circular orbit until it drops off. The peak of the extracted power can continue to raise up to a maximum value that is achieved for $r/M \rightarrow 1$ if $a \rightarrow 1$. The theoretical limit of the maximum power is given by
\begin{equation} \label{PextrMAX}
P_{\rm extr}^{\rm max} \simeq \sqrt{\sigma_0/3} \, w_0 A_{\rm in} U_{\rm in} \sim 0.1 M^2 \sqrt{\sigma_0} \, w_0 \, ,
\end{equation}
which follows directly from Eqs. \eqref{energ_minus_simple} and \eqref{Pextr}. We can see from Fig. \ref{fig5} that the peak of the extracted power is already close to the maximum theoretical limit when $\xi \lesssim \pi/12$.
\begin{figure}[]
\begin{center}
\includegraphics[width=8.4cm]{Fig5a.pdf}
\bigskip $\,$
\hspace*{-0.05cm}\includegraphics[width=8.4cm]{Fig5b.pdf}
\vspace{-0.30cm}
\end{center}
\caption{${P_{\rm extr}}/w_0 = - \epsilon_-^\infty A_{\rm in} U_{\rm in}$ as a function of the dominant $X$-point location $r/M$ for a rapidly spinning black hole with $a = 0.99$ and reconnection inflow four-velocity $U_{\rm in} = 0.1$ (i.e., collisionless reconnection regime). $\epsilon_-^\infty$ is evaluated using Eq. \eqref{energuisMagnet}, while $A_{\rm in} = (r_{{\rm ph}}^2 - r_{H}^2)$. We have also set $M=1$. Different colors (from indigo to red) refer to different plasma magnetizations (from $\sigma_0 = 10$ to $\sigma_0 = 10^5$) and $\xi = \pi/12$ (top panel) or different orientation angles (from $\xi = \pi/6$ to $\xi = 0$) and $\sigma_0 = 10^4$ (bottom panel). The vertical dashed line indicates the limiting circular orbit $r_{\rm ph}(a=0.99)$.}
\label{fig5}
\end{figure}
\begin{figure}[]
\begin{center}
\includegraphics[width=8.4cm]{Fig6.pdf}
\vspace{-0.30cm}
\end{center}
\caption{Efficiency $\eta$ of the reconnection process as a function of the dominant $X$-point location $r/M$ for a reconnection layer with upstream plasma magnetization $\sigma_0 = 100$ and reconnecting magnetic field having orientation angle $\xi = \pi/20$. Different colors (from indigo to red) refer to different black hole spin values (from $a = 0.9$ to $a = 1$). }
\label{fig6}
\end{figure}
The proposed mechanism of energy extraction via magnetic reconnection generates energetic plasma outflows that steal energy from the black hole, but it also necessitates magnetic field energy to operate. Magnetic energy is indeed needed in order to redistribute the angular momentum of the particles in such a way to generate particles with negative energy at infinity and particles escaping to infinity. Therefore, it is convenient to define the efficiency of the plasma energization process via magnetic reconnection as
\begin{equation} \label{eff}
\eta = \frac{\epsilon^\infty_+}{\epsilon^\infty_+ + \epsilon^\infty_-} \, .
\end{equation}
Extraction of energy from the black hole takes place when $\eta > 1$. Figure \ref{fig6} shows the efficiency $\eta$ as a function of the dominant $X$-point location $r/M$ for a reconnection layer with magnetization parameter $\sigma_0=100$, orientation angle $\xi = \pi/20$, and different black hole spin values $a \in \left\{ {0.90,0.96,0.99,0.999,1} \right\}$. The efficiency $\eta$ significantly increases for reconnection $X$-points that are closer to the black hole event horizon and falls off below unity when the inner radius reaches $r_{\rm ph}$. The maximum efficiency can be evaluated by considering the optimal energy extraction conditions ($a, r/M \rightarrow 1$, $\xi \rightarrow 0$) and $\sigma_0 \gg 1$. In this case, Eq. \eqref{eff} gives
\begin{equation} \label{effmax}
\eta_{\rm max} \simeq \frac{\sqrt{3 \sigma_0}}{ \sqrt{3 \sigma_0} - \sqrt{\sigma_0/3}} = {3}/{2} \, .
\end{equation}
Therefore, the additional energy extracted from the black hole, while non-negligible, do not extensively modify the energetics of the escaping plasma.
We can also compare the power extracted from the black hole by fast magnetic reconnection with the one that can be extracted via the Blandford-Znajek mechanism, in which the rotational energy is extracted electromagnetically through a magnetic field that threads the black hole event horizon.
For maximum efficiency conditions \cite{MT82,Thorne86,komissarov01}, the rate of black hole energy extraction via the Blandford-Znajek mechanism is given by \cite{BZ77,Tchekhovskoy10}
\begin{equation} \label{P_BZ}
P_{\rm BZ} \simeq \kappa \Phi_{\rm BH}^2 \left( {\Omega_H^2 + \chi \Omega_H^4 + \zeta \Omega_H^6 } \right) \, ,
\end{equation}
where $\Phi_{\rm BH} = \frac{1}{2} \int_{\theta} \int_{\phi} |B^r| dA_{\theta \phi}$ is the magnetic flux threading one hemisphere of the black hole horizon (with $dA_{\theta \phi} = \sqrt{-g} \, d\theta d\phi$ indicating the area element in the $\theta$-$\phi$ plane), $\Omega_H = a /2 r_{H}$ is the angular frequency of the black hole horizon, while $\kappa$, $\chi$, and $\zeta$ are numerical constants. The numerical prefactor $\kappa$ depends on the magnetic field geometry near the black hole ($\kappa \approx 0.053$ for a split monopole geometry and $\kappa \approx 0.044$ for a parabolic geometry), while $\chi \approx 1.38$ and $\zeta \approx -9.2$ \cite{Tchekhovskoy10}.
Eq. \eqref{P_BZ} is a generalization of the original Blandford-Znajek scaling \cite{BZ77} $P_{\rm BZ} \simeq \kappa \Phi_{\rm BH}^2 (a/4M)^2$, which is recovered in the small spin limit $a \ll 1$.
\begin{figure}[]
\begin{center}
\includegraphics[width=8.4cm]{Fig7.pdf}
\vspace{-0.30cm}
\end{center}
\caption{Power ratio ${P_{\rm extr}}/{P_{\rm BZ}}$ as a function of the plasma magnetization $\sigma_0$ for a black hole with dimensionless spin $a = 0.99$ and a reconnecting magnetic field having orientation angle $\xi = \pi/12$. Different colors (from indigo to red) refer to different dominant $X$-point locations $r/M \in \left\{ {1.3,1.4,1.5,1.6,1.7} \right\}$. We considered $U_{\rm in} = 0.1$ (i.e., collisionless reconnection regime), $A_{\rm in} = (r_{{\rm ph}}^2 - r_{H}^2)$, and $\kappa = 0.05$.}
\label{Fig7}
\end{figure}
In order to provide a rough order of magnitude estimate of the power extracted during the occurrence of fast magnetic reconnection with respect to the approximately steady-state Blandford-Znajek process,
we assume $\Phi_{\rm BH} \sim |B^r| r_{H}^2 \sim B_0 {\sin \xi} \, r_{H}^2$ (we point out that a precise evaluation of $\Phi_{\rm BH}$ requires direct numerical simulations that reproduce the detailed magnetic field configuration at all latitudes, while the angle $\xi$ is a good estimate for the magnetic field configuration only at low latitudes \citep[e.g.][]{Koide02,Semenov04}). Then, we can evaluate the ratio ${P_{\rm extr}}/{P_{\rm BZ}}$ as
\begin{equation} \label{powerratiowithBZ1}
\frac{P_{\rm extr}}{P_{\rm BZ}} \sim\frac{ - \epsilon_-^\infty A_{\rm in} U_{\rm in}} {\kappa \, \Omega_H^2 r_{H}^4 \sigma_0 \sin^2 \xi \, (1+ \chi \Omega_H^2 + \zeta \Omega_H^4)}\, .
\end{equation}
Fig. \ref{Fig7} shows the ratio ${P_{\rm extr}}/{P_{\rm BZ}}$ given by the right-hand side of Eq. \eqref{powerratiowithBZ1} as a function of the plasma magnetization $\sigma_0$ for the fast collisionless reconnection regime. ${P_{\rm extr}}/{P_{\rm BZ}} \gg 1$ for an extended range of plasma magnetizations. For $\sigma_0 \sim 1$, the force-free approximation (the inertia of the plasma is ignored, i.e. $w_0 \rightarrow 0$) that is used to derive the extracted power in the Blandford-Znajek process becomes invalid. In this case, magnetic reconnection is an effective mechanism of energy extraction provided that the plasma magnetization is sufficient to satisfy the condition $ \epsilon_-^\infty < 0$ (as well as $\Delta \epsilon^\infty_+ >0$). On the other hand, for $\sigma_0 \rightarrow \infty$, energy extraction via fast magnetic reconnection is always subdominant to the Blandford-Znajek process since ${P_{\rm extr}}/{P_{\rm BZ}} \rightarrow 0$ in this limit.
If we neglect higher order corrections with respect to $\Omega_H^2$ (which leads to an overprediction of $P_{\rm BZ}$ by about 25\% as $a \rightarrow 1$ \cite{Tchekhovskoy10}), and recalling that $\Omega_H = 1/2M$ for $a \rightarrow 1$, we can estimate the ratio ${P_{\rm extr}}/{P_{\rm BZ}}$ for a rapidly spinning black hole as
\begin{equation} \label{powerratiowithBZ2}
\frac{P_{\rm extr}}{P_{\rm BZ}}\sim\frac{- \epsilon_-^\infty}{\kappa \, \sigma_0 \sin^2 \xi}\, ,
\end{equation}
where we considered plasmoid-mediated reconnection in the collisionless regime. Therefore, the power extracted via fast collisionless magnetic reconnection can exceed the one extracted through the Blandford-Znajek process for an extended range of plasma magnetizations if there is a significant toroidal component of the magnetic field in the black hole ergosphere. Note, however, that energy extraction by fast magnetic reconnection is localized in time, since it requires a certain time to build-up the magnetic field configuration storing the magnetic energy that is eventually dissipated via fast magnetic reconnection.
\section{Conclusions}
\label{section5}
In this paper, we envisioned the possibility of extracting black hole rotational energy via fast magnetic reconnection in the black hole ergosphere. We considered a configuration with antiparallel magnetic field lines near the equatorial plane, which is induced by the frame dragging of the spinning black hole. The change in magnetic field direction at the equatorial plane produces an equatorial current sheet that is disrupted by the plasmoid instability when its aspect ratio reaches a critical value (for a collisionless relativistic pair plasma, the critical aspect ratio condition is derived in Ref. \cite{Comisso2019}). The formation of plasmoids/flux ropes drives fast magnetic reconnection, which rapidly converts the available magnetic energy into plasma particle energy. When the plasma is expelled out of the reconnection layer, the magnetic tension that drives the plasma outflow relaxes. The field lines are then stretched again as a consequence of the frame dragging and a current layer prone to fast plasmoid-mediated reconnection forms again. This process leads to reconnecting current sheets that form rapidly and intermittently.
Magnetic reconnection accelerates part of the plasma in the direction of the black hole rotation, while another part of the plasma is accelerated in the opposite direction and falls into the black hole. Black hole energy extraction occurs if the plasma that is swallowed by the black hole has negative energy as viewed from infinity, while the accelerated plasma that gains energy from the black hole escapes to infinity. Therefore, differently from the Blandford-Znajek process, in which the extraction of rotational energy is obtained through a purely electromagnetic mechanism, the energy extraction mechanism described here requires non-zero particle inertia. This mechanism is also different from the original Penrose process, since dissipation of magnetic energy is required to produce the negative-energy particles. Clearly, all mechanisms extract black hole rotational energy by feeding the black hole with negative energy and angular momentum.
We showed analytically that energy extraction via magnetic reconnection is possible when the black hole spin is high (dimensionless spin $a \sim 1$) and the plasma is strongly magnetized (plasma magnetization $\sigma_0 > 1/3$).
Magnetic reconnection is assumed to occur in a circularly rotating plasma with a reconnecting field having both azimuthal and radial components. The region of the phase-space $\{a,r/M\}$ where magnetic reconnection is capable of extracting black hole energy depends on the plasma magnetization $\sigma_0$ and the orientation $\xi$ of the reconnecting magnetic field. We showed that high values of the plasma magnetization and mostly azimuthal reconnecting fields can expand the energy extraction region up to the outer boundary of the ergosphere. For a dimensionless spin parameter that approaches unity, the extraction of black hole energy is maximal when the dominant reconnection $X$-point (where the two magnetic reconnection separatrices intersect) is close to the event horizon. For $\sigma_0 \gg 1$, we showed that the asymptotic negative energy at infinity per enthalpy of the plasma that is swallowed by the black hole is $\epsilon^\infty_- \simeq - \gamma_{\rm out} v_{\rm out}/ {\sqrt{3}} \simeq - \sqrt{\sigma_0/3}$. On the other hand, the plasma that escapes to infinity and takes away black hole energy asymptotes the energy at infinity per enthalpy $\epsilon^\infty_+ \simeq \sqrt{3} \, \gamma_{\rm out} v_{\rm out} \simeq \sqrt{3 \sigma_0}$.
We calculated the power extracted from the black hole by the escaping plasma and evaluated its maximum when the dominant reconnection $X$-point is close to the event horizon. This corresponds to $P_{\rm extr}^{\rm max} \sim 0.1 M^2 \sqrt{\sigma_0} \, w_0$ for the collisionless plasma regime and one order of magnitude lower for the collisional regime. The overall efficiency of the plasma energization process via magnetic reconnection can reach a maximum of $\eta_{\rm max} \simeq 3/2$. Therefore, the additional energy extracted from the black hole, while important, do not extensively modify the energetics of the escaping plasma. On the other hand, the power extracted via fast magnetic reconnection can induce a significant reduction of the rotational energy of the black hole, ${d E_{\rm rot}}/{dt} = \epsilon_-^\infty w_0 A_{\rm in} U_{\rm in}$. This is effective when $a$ is close to unity. Therefore, if we consider a black hole with dimensionless spin parameter close to unity and define $\varpi = 1-a \ll 1$, we have ${d E_{\rm rot}}/{dt} \simeq - (M/4 \sqrt{\varpi}) d\varpi/dt$ and the spindown time can be obtained as
\begin{equation} \label{}
{t_{\rm sd}} = \frac{\mathcal{O}(10)}{2 \sqrt{\sigma_0} \, w_0 M} (\sqrt{\varpi_{\rm f}}-\sqrt{\varpi_{\rm i}}) \, ,
\end{equation}
where the subscripts ${\rm f}$ and ${\rm i}$ are used to label final and initial values, respectively. This indicates that magnetic reconnection can cause a significant spindown of the black hole when $a \sim 1$. For example, fast magnetic reconnection in the ergosphere can reduce the black hole dimensionless spin from $a=0.999$ to $a=0.99$ in ${t_{\rm sd}} \sim 1/(\sqrt{\sigma_0} \, w_0 M)$. On the other hand, at lower spin values, especially for $a <0.9$, magnetic reconnection loses its efficacy as the plasma available in the ergosphere diminishes.
Various systems hosting a black hole are expected to have magnetization $\sigma_0 \gtrsim 1$ in the ergosphere.
For the typical conditions around supermassive black holes in active galactic nuclei (AGNs), the energy density of the electromagnetic field far exceeds the enthalpy density of the plasma and $\sigma_0 \sim 10^{4}$ or larger \cite{DoddsEden2010,Ponti17,EHT_5_2019} is foreseeable. Likewise, long and short gamma-ray bursts (GRBs) may have $\sigma_0 \sim 1$ or larger \cite{MacFadyen99,vanPutten99,Kiuchi15,Ruiz19} in the ergosphere (a central black hole is assumed).
Under these magnetization conditions (in addition to $a \sim 1$), magnetic reconnection is capable of extracting energy from the black hole. For $\sigma_0 \sim 1 - 10^4$, we have shown that the bursty energy extraction rate occurring during fast magnetic reconnection can exceed the more steady energy extraction rate expected from the Blandford-Znajek mechanism. On the other hand, as the plasma magnetization increases, energy extraction via fast magnetic reconnection becomes always subdominant since it requires non-vanishing plasma inertia.
In the scenario proposed here, fast magnetic reconnection occurs rapidly and intermittently, so that the associated emission within a few gravitational radii from the black hole is expected to be bursty in nature. This bursty behavior of fast magnetic reconnection might be responsible for triggering flares in the vicinity of rotating black holes. Indeed, frequent X-ray and near-infrared flares are detected on a regular basis from the Galactic Center black hole Sgr A* \citep[e.g.][]{Baganoff01,Genzel03,Meyer08,Neilsen13}, and magnetic reconnection close to the black hole is often conjectured to induce these flares \citep[e.g.][]{DoddsEden2010,ripperda20,Dexter20}. Recent observations by the GRAVITY collaboration \cite{Gravity2018} have been able to pin down the motion of near-infrared flares originating near the last stable circular orbit of Sgr A*.
Reconnection layers originate naturally in the ergosphere of rotating black holes and produce plasmoids/flux ropes that are filled with energized plasma with an energy budget that can exceed the energy originally stored in the magnetic field.
In this paper we have assumed that the plasma rotates circularly around the black hole. This assumption may be relaxed in order to treat more complex scenarios in which reconnection occurs in non-circular orbits. In this case, the plasma could approach the event horizon even when the black hole spin is not particularly high, expanding the parameter space region where magnetic reconnection can extract black hole energy.
Another situation that could increase the efficacy of magnetic reconnection is the simultaneous presence of equatorial and non-equatorial current sheets \cite{ripperda20}, which may result in an increase of the extracted power to some degree.
Finally, for reconnecting magnetic fields that have a significant radial component, particle acceleration owing to the reconnection electric field can increase the rate of energy extraction and the overall efficiency of the reconnection process.
$\,$
\begin{acknowledgments}
We gratefully acknowledge discussions with Lorenzo Sironi, Daniel Gro\v{s}elj, Russell Kulsrud, Manasvi Lingam, Yi-Hsin Liu, Joonas N\"attil\"a, Kyle Parfrey, Bart Ripperda, Daniel Siegel, and Yajie Yuan. L.C. acknowledges support by the NASA ATP NNX17AG21G and NSF PHY-1903412 grants. F.A.A. acknowledges support by the Fondecyt-Chile Grant No. 1180139.
\end{acknowledgments}
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2,869,038,153,731 | arxiv | "\\section{Introduction}\nIn this article we show that the adele class space\\footnote{More specific(...TRUNCATED) |
2,869,038,153,732 | arxiv | "\n\\section{Introduction}\n\\label{sec:intro}\n\nFlow states that remain exactly the same under som(...TRUNCATED) |
2,869,038,153,733 | arxiv | "\\section{Introduction}\n\\noindent\nThe symmetric group $S_n$ may be viewed as the subgroup of the(...TRUNCATED) |
2,869,038,153,734 | arxiv | "\\section{Introduction} \\vspace{-\\parskip}\nBulges of early-type spirals\nand elliptical galaxies(...TRUNCATED) |
2,869,038,153,735 | arxiv | "\\section{Introduction}\\label{sec:intro}\n\nThe cycle 4 {\\it Spitzer Space Telescope} Legacy proj(...TRUNCATED) |
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2,869,038,153,737 | arxiv | "\\section*{Introduction}\n\n\nStarting with the seminal works \\cite{Beil} and \\cite{BGG} on deriv(...TRUNCATED) |
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Dataset Description
To facilitate researchers to use NanoLM for comparative analysis across different model designs, we build a curated pre-training dataset from those of existing large-scale models (i.e., Llama, Falcon, GPT-3). It covers diverse domains to improve the generalization capabilities of the resultant models.
Dataset Creation
The data is mainly post-processed and filtered from RedPajama and RedPajamaV2. We develop a series of cleaning steps to remove redundant formatting, garbled characters, formula errors, duplicated paragraphs, low-quality text, and other unwanted content. After interleaved deduplication on document level of each independent subset, we finally obtain a high-quality dataset.
Dataset Summary
| Dataset | Num Tokens (B) |
|---|---|
| CommonCrawl | 67.00 |
| C4 | 15.00 |
| Wikipedia (En) | 5.14 |
| Books | 4.48 |
| ArXiv | 2.50 |
| StackExchange | 2.00 |
| Total | 97.12 |
We release the data with approximate 100B tokens. Furthermore, we recommend users to add code dataset such as Starcode, The Stack V2 to enrich model's performance on code and reasoning.
Citation
To cite NanoLM, please use:
@misc{yao2024nanolm,
title={nanoLM: an Affordable LLM Pre-training Benchmark via Accurate Loss Prediction across Scales},
author={Yiqun Yao and Siqi fan and Xiusheng Huang and Xuezhi Fang and Xiang Li and Ziyi Ni and Xin Jiang and Xuying Meng and Peng Han and Shuo Shang and Kang Liu and Aixin Sun and Yequan Wang},
year={2024},
eprint={2304.06875},
archivePrefix={arXiv},
primaryClass={cs.CL}
}
Acknowledgement
The data is mainly curated and filtered from RedPajama and RedPajamaV2. We extend our gratitude to the original authors for their innovative work and for making it available to the community.
License
The code of NanoLM used to process the dataset and loss prediction is licensed under the Apache 2.0 license.
For curated data, please refer to the licenses of the original ones.
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