††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Entanglement and work extraction in the central-spin quantum battery

Jia-Xuan Liu  School of Physics, Northwest University, Xi’an 710127, China
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, Universerity of Science and Technology of China, Hefei
Anhui 230026,China  Hai-Long Shi  State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
APM, Chinese Academy of Sciences, Wuhan 430071, China  University of Chinese
Academy of Sciences, Beijing 100049, China  Yun-Hao Shi  Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  University of Chinese
Academy of Sciences, Beijing 100049, China  Xiao-Hui Wang<EMAIL_ADDRESS>School of Physics, Northwest University, Xi’an 710127, China  Shaanxi Key
Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China  Peng Huanwu
Center for Fundamental Theory, Xi’an 710127, China  Wen-Li Yang  Institute of
Modern Physics, Northwest University, Xi’an 710127, China  Shaanxi Key
Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China  Peng Huanwu
Center for Fundamental Theory, Xi’an 710127, China

(August 29, 2024)

###### Abstract

We consider a central-spin battery where $N_{b}$ central spins serve as
battery cells and $N_{c}$ bath spins serve as charging units. It is shown that
the energy stored in the battery that can be extractable is quantified by the
ergotropy, and that battery-charger entanglement is quantified via the Von
Neumann entropy. By using an exact approach to a one-cell and two-cell
battery, our analytical results suggest that, during the charging process, the
extractable work slowly increases before the battery-charger entanglement
reaches its maximum and then it will rapidly increase when the entanglement
begins to decrease. In particular, we rigorously show that there is an inverse
relationship between the extractable work and the entanglement at the end of
the charging process. Moreover, we investigate different approaches to realize
optimal work extraction without wasted energy. Among them a central-spin
battery with an unpolarized Dicke state as the charger possesses a universal
charging time $\propto 1/N_{c}$, large extractable work, and
$\sqrt{N_{c}}$-improvement of charging power compared with the battery in the
Tavis-Cummings limit. The above-mentioned results have also been numerically
verified in multi-cell batteries. Our results pave the way to improve
extractable work storage in the central-spin battery and highlight a
competitive relation between the extractable work and the battery-charger
entanglement.

###### pacs:

##  I Introduction

The state-of-the-art technology of qubit manipulation enables us to exploit
quantum resource, such as entanglement or coherence, for technological
purposes [1, 2, 3, 4, 5, 6, 7]. A recent development of this direction is
studying “quantum batteries”, which harness the unique property of quantum
thermodynamics to speed up the charging process and extract more work compared
to their classical counterparts. The ideal of quantum batteries was first put
forward by Alicki and Fannes in 2013 [8]. They demonstrated that entangling
unitary controls (i.e., collective controls) perform better than individual
controls (i.e., parallel controls) in work extraction. Further research
uncovered that entanglement generation benefits the speedup of work extraction
[9]. Subsequently, the authors of Refs. [10, 11] argued that, in the charging
process, the collective charging scheme results in $k$ time faster than the
parallel charging scheme, where $k$ denotes a $k$-body interaction among
battery cells. Therefore, we now say quantum batteries have quantum advantage.

The above-mentioned works mainly focused on abstract time-evolution operators.
To realize such operators in practice, especially the collective controls,
various theoretical models have been considered [12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. For instances, the spin-
chain battery uses an intrinsic spin-spin interaction to realize the
collective controls [21] while the Dicke battery relies on the cavity photons
to generate an effective interaction among its battery cells [12, 29]. Based
on superconducting qubits, a quantum battery has just recently been realized
in experiment [32].

The central spin model, as an exactly solvable model, has played a vital role
in quantitatively understanding decoherence problem and entanglement dynamics
[33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. This model can be experimentally
realized by superconductors [43], quantum dots [44, 45, 46], and nitrogen-
vacancy centers in diamond [47]. Quantum battery problem can be naturally
described in this model and we call it the “central-spin (quantum) battery”,
where $N_{b}$ central spins serve as “battery cells” and $N_{c}$ bath spins
serve as “charging units”, see Fig. 1. However, research on the central-spin
battery remains limited [19]. Assuming that only few photons in the view of
the Holstein-Primakoff transformation, the central-spin battery will reduce to
the Tavis-Cummings (TC) battery, which is a simplification of the Dicke
battery. The charging performance of the central-spin battery in this limit
has been confirmed to have quantum advantage [19]. A recent Ref. [17] pointed
out that battery-charger entanglement is a major obstacle in work extraction
by comparing the performance of a classical charger (coherent state) with that
of nonclassical chargers (Fock and squeezed states) in the TC battery.
However, the correspondence between the central-spin battery and the TC
battery breaks when the number of photons is far greater than the number of
charging units.

To bridge this gap, we fix our attention on the central-spin battery,
especially the non-TC case, and discuss the following questions. Is there a
quantitative relationship between battery-charger entanglement and extractable
work? Under what conditions “optimal” work extraction can be achieved? The
“optimal” refers to that all energy stored in the battery can be extracted.

The paper is organized as follows. In Sec. II, we introduce the central-spin
battery and discuss its correspondence with the TC battery. Time evolutions
for one-cell battery and two-cell battery are analytically solved in the same
section. Multi-cell battery cases are treated by a numerical method. With the
above preparation, we obtain analytical and numerical results about the
evolutions of battery-charger entanglement and battery’s extractable work in
Sec. III. We rigorously show that the battery-charger entanglement first
increases and then decreases during the charging process. At the end of the
charging process, the final extractable work is significantly enslaved to the
battery-charger entanglement. To be more exact, the extractable work increases
with the decrease of the battery-charger entanglement when the initial
chargers are restrained to the Dicke states. In Sec. IV, we show that optimal
work extraction can be realized when the central-spin battery lies in the non-
TC limit region (e.g. the initial charger is an unpolarized Dicke state).
Moreover, in this region, the charging time is proportional to $1/N_{c}$,
which is independent of the number of battery cells. On the other hand, if the
central-spin battery is in the TC limit, the optimal work extraction is not
always possible unless the number of photons is large but does not break the
TC limit. At this case, the charging time is proportional to $1/\sqrt{N_{c}}$.
Finally, a conclusion is given in Sec. V. Our analytical results shed light on
entanglement and extractable work in the central-spin battery and provide
different proposals able to optimally store extractable work.

##  II Central-spin battery

Figure 1: Schematic illustration of the central-spin battery. Central spins
serve as $N_{b}$ battery cells whereas bath spins serve as $N_{c}$ charging
units. In the limit of $N_{c}\to\infty$, if the number of spin-up charging
units is far less than $N_{c}$ then the central-spin battery reduces to the TC
battery and we call this case the TC limit, see the left panel. On the other
hand, if the number of spin-up charging units is proportional to $N_{c}$ but
not zero or one then the correspondence between the central-spin battery and
the TC battery breaks and we call this case the non-TC limit, see the right
panel.

The central-spin battery, just as its name implies, is governed by the
Hamiltonian of the central spin model,

$\displaystyle H=H_{b}+H_{c}+H_{I},$ (1) $\displaystyle H_{b}=BS^{z},\quad
H_{c}=hJ^{z},$ $\displaystyle H_{I}=A(S^{+}J^{-}+S^{-}J^{+})+2\Delta
S^{z}J^{z},$

where $S^{\alpha}=\sum_{j=1}^{N_{b}}\sigma_{j}^{\alpha}/2$ $(\alpha=x,y,z)$
are the total spin operators for $N_{b}$ battery cells and
$J^{\alpha}=\sum_{k=1}^{N_{c}}\sigma_{k}^{\alpha}/2$ for $N_{c}$ charging
units. $J^{\pm}=J^{x}\pm iJ^{y}$ and $S^{\pm}=S^{x}\pm iS^{y}$ are spin ladder
operators. The Hamiltonians of the battery and the charger are denoted by
$H_{b}$ and $H_{c}$, respectively. The parameter $A$ and $\Delta$ characterize
the flip-flop interaction and the Ising interaction, respectively.

At time $t<0$, the battery is prepared in the ground state of $H_{b}$, i.e.,
all spins are down
$|0\rangle_{b}\equiv|\downarrow_{1},\downarrow_{2},\ldots,\downarrow_{N_{b}}\rangle$.
The charger is prepared in a Dicke state with $m$ spins up

$\displaystyle|m\rangle_{c}\equiv\frac{1}{\sqrt{\binom{N_{c}}{m}}}\sum_{k}P_{k}(|\uparrow_{1},\cdots,\uparrow_{m},\downarrow_{m+1},\ldots,\downarrow_{N_{c}}\rangle)$
(2)

where $\binom{N_{c}}{m}=N_{c}!/[m!(N_{c}-m)!]$ and $P_{k}$ denotes the
complete set of all possible distinct permutations of the qubits.

At time $t=0$, by suddenly turning on the interaction Hamiltonian $H_{I}$ for
a finite time $T$, we aim to inject as much as possible energy into the
battery. Such a time $T$ is called the charging time. To evaluate the
performance of a quantum battery, we need obtain the reduced density matrix
for battery cells, which is given by
$\rho_{b}(t)=\mathrm{Tr}_{c}(e^{-iHt}\rho_{0}e^{iHt})$, where
$\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$ is the initial state and
$|\psi_{0}\rangle=|0\rangle_{b}\otimes|m\rangle_{c}$. To ensure that
$|0\rangle_{b}$ is the ground state of $H_{b}=BS^{z}$, we assume $B>0$ in the
following discussion.

###  II.1 Two limits

Before calculating $\rho_{b}(t)$ we first discuss the relationship between our
battery and the TC battery. The Holstein-Primakoff (HP) transformation
establishes an exact map from the total spin operators $J^{\alpha}$ to a
bosonic mode $a$,

$\displaystyle
J^{+}\to\sqrt{N_{c}}a^{\dagger}\sqrt{1-\frac{a^{\dagger}a}{N_{c}}},$ (3)
$\displaystyle J^{-}\to\sqrt{N_{c}}\sqrt{1-\frac{a^{\dagger}a}{N_{c}}}a,$ (4)
$\displaystyle J^{z}\to-\frac{N_{c}}{2}+a^{\dagger}a.$ (5)

Note that the Dicke state $|m\rangle_{c}$ is an eigenstates of $J^{z}$ with
the eigenvalue $(-N_{c}/2+m)$. Thus, the number of up bath spins is exactly
the number of photons in the view of the HP transformation. For convenience we
will identify up spins with photons. If the number of photons is far less than
$N_{b}$ then the HP transformation simplifies to
$J^{+}\to\sqrt{N_{c}}a^{\dagger},J^{-}\to\sqrt{N_{c}}a$ and the central-spin
battery (1) reduces to the TC battery, i.e.,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
BS^{z}+h\left(-\frac{N_{c}}{2}+a^{\dagger}a\right)$
$\displaystyle+A\sqrt{N_{c}}(S^{+}a+S^{-}a^{\dagger})+2\Delta
S^{z}\left(-\frac{N_{c}}{2}+a^{\dagger}a\right).$

We call this limit

$\displaystyle\lim_{N_{c}\to\infty}\frac{\langle
a^{\dagger}a\rangle}{N_{c}}=\lim_{N_{c}\to\infty}\frac{m}{N_{c}}=0$ (7)

the TC limit, where $m$ is the number of up bath spins or the number of
photons.

Another case is that the number of photons is proportional to $N_{c}$ but not
zero or one, i.e.,

$\displaystyle\lim_{N_{c}\to\infty}\frac{\langle
a^{\dagger}a\rangle}{N_{c}}=\lim_{N_{c}\to\infty}\frac{m}{N_{c}}\equiv k\neq
0,1.$ (8)

We call it the non-TC limit. In this limit, the correspondence between the
central-spin battery and the TC battery breaks due to the non-negligible term
$\sqrt{1-a^{\dagger}a/N_{c}}$ in the HP transformation (3). Theses two limits
(7) and (8) are illustrated in Fig. (1).

###  II.2 Exact solutions

Now we evaluate the reduced density matrix of the battery. Without loss of
generality, we assume that the number of up-spin charging units is not less
than the number of battery cells, i.e., $m\geq N_{b}$. Due to the $u(1)$
symmetry, namely, $[H,J^{z}+S^{z}]=0$, the invariance subspace of $H$
containing the initial state
$|\psi_{0}\rangle=|0\rangle_{b}\otimes|m\rangle_{c}$ is given by

$\displaystyle\mathcal{H}_{m}=$$\displaystyle\\{|0\rangle_{b}|m\rangle_{c},|1\rangle_{b}|m-1\rangle_{c},\cdots,|N_{b}\rangle_{b}|m-N_{b}\rangle_{c}\\},$

where both of the states of battery and charger are expressed in term of the
Dicke state, see (2). For example, $|2\rangle_{b}|3\rangle_{c}$ refers to the
quantum state whose battery part is a Dicke state with 2 particles and whose
charger part is a Dicke state with 3 particles.

In terms of the basis (II.2), the Hamiltonian (1) can be represented as a
$(N_{b}+1)\times(N_{b}+1)$ matrix:

$\displaystyle\bm{H}=\begin{pmatrix}b_{0}&u_{1}&\\\ u_{1}&b_{1}&u_{2}&\\\
&\ddots&\ddots&\ddots&\\\ &&u_{N_{b}-1}&b_{N_{b}-1}&u_{N_{b}}\\\
&&&u_{N_{b}}&b_{N_{b}}\end{pmatrix},$ (10)

where $u_{j}=A\sqrt{j(N_{b}-j+1)(N_{c}-m+j)(m-j+1)}$ and
$b_{j}=B(j-N_{b}/2)+h(m-j-N_{c}/2)+2\Delta(j-N_{b}/2)(m-j-N_{c}/2)$. Suppose
that $\bm{H}$ is diagonalized by a unitary matrix $\bm{U}$, that is
$\bm{H}=\bm{U}\bm{D}\bm{U}^{\dagger}$ where $\bm{D}$ is a diagonal matrix.
Then, the matrix representation of the wavefunction for the whole system at
time $t$ is given by

$\displaystyle\bm{\psi}(t)=\bm{U}e^{-i\bm{D}t}\bm{U}^{\dagger}(1\ 0\ \ldots\
0)^{T},$ (11)

and

$\displaystyle|\psi(t)\rangle=\bm{\psi}_{1}(t)|0\rangle_{b}|m\rangle_{c}+\cdots+\bm{\psi}_{N_{b}+1}(t)|N_{b}\rangle_{b}|m-N_{b}\rangle_{c}.$

It thus follows that the reduced density matrix of the battery is given by

$\displaystyle\rho_{b}(t)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{c}(|\psi(t)\rangle\langle\psi(t)|)$ (13)
$\displaystyle=$ $\displaystyle|\bm{\psi}_{1}(t)|^{2}|0\rangle\langle
0|+\cdots+|\bm{\psi}_{N_{b}+1}|^{2}|N_{b}\rangle\langle N_{b}|.$

Eqs. (10, 11, 13) give a numerical approach to evaluate $\rho_{b}(t)$ in
multi-cell battery cases.

For the one-cell battery case $N_{b}=1$, we find that
$\bm{U}=\exp(-i\theta\sigma^{y}/2)$ and $\bm{D}={\rm diag}(d_{1},d_{2})$,
where $\theta$ is determined by

$\displaystyle\sin\theta=\frac{u_{1}}{\sqrt{u_{1}^{2}+\frac{1}{4}(b_{0}-b_{1})^{2}}},$
$\displaystyle\cos\theta=\frac{b_{0}-b_{1}}{2\sqrt{u_{1}^{2}+\frac{1}{4}(b_{0}-b_{1})^{2}}},$
(14)

and the energy levels are given by

$\displaystyle
d_{1}=\frac{b_{0}+b_{1}}{2}+\sqrt{u_{1}^{2}+\frac{1}{4}(b_{0}-b_{1})^{2}},$
(15) $\displaystyle
d_{2}=\frac{b_{0}+b_{1}}{2}-\sqrt{u_{1}^{2}+\frac{1}{4}(b_{0}-b_{1})^{2}}.$
(16)

The reduced density matrix of the battery is thus given by

$\displaystyle\rho_{b}(t)=\frac{1+r(t)}{2}|0\rangle\langle
0|+\frac{1-r(t)}{2}|1\rangle\langle 1|,$ (17)

where

$\displaystyle r(t)=\cos^{2}\theta+\cos((d_{1}-d_{2})t)\sin^{2}\theta.$ (18)

For the two-cell battery case $N_{b}=2$, we let $h=B$ and $\Delta=0$ to
simplify the calculation. Under this assumption, the matrices $\bm{U}$ and
$\bm{D}$ are given by

$\displaystyle\bm{U}=\frac{1}{\sqrt{2(u_{1}^{2}+u_{2}^{2})}}\begin{pmatrix}\sqrt{2}u_{2}&u_{1}&u_{1}\\\
0&\sqrt{u_{1}^{2}+u_{2}^{2}}&-\sqrt{u_{1}^{2}+u_{2}^{2}}\\\
-\sqrt{2}u_{1}&u_{2}&u_{2}\end{pmatrix},$

and $\bm{D}={\rm diag}(e_{1},e_{2},e_{3})$, where $u_{1}$ and $u_{2}$ are the
same as the ones in Eq. (10) and

$\displaystyle e_{1}=B(m-1-N_{c}/2),$ (20) $\displaystyle
e_{2}=B(m-1-N_{c}/2)+\sqrt{u_{1}^{2}+u_{2}^{2}},$ (21) $\displaystyle
e_{3}=B(m-1-N_{c}/2)-\sqrt{u_{1}^{2}+u_{2}^{2}}.$ (22)

The reduced density matrix of the battery is given by

$\displaystyle\rho_{b}(t)=\rho_{11}(t)|0\rangle\langle
0|+\rho_{22}(t)|1\rangle\langle 1|+\rho_{33}(t)|2\rangle\langle 2|,$ (23)

where

$\displaystyle\rho_{11}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{(u_{1}^{2}+u_{2}^{2})^{2}}(u_{2}^{2}+u_{1}^{2}\cos(\omega
t))^{2},$ $\displaystyle\rho_{22}(t)$ $\displaystyle=$
$\displaystyle\frac{u_{1}^{2}}{u_{1}^{2}+u_{2}^{2}}(1-\cos^{2}(\omega t)),$
$\displaystyle\rho_{33}(t)$ $\displaystyle=$
$\displaystyle\frac{u_{1}^{2}u_{2}^{2}}{(u_{1}^{2}+u_{2}^{2})^{2}}(1-\cos(\omega
t))^{2},$ (24)

and $\omega=e_{2}-e_{1}=\sqrt{u_{1}^{2}+u_{2}^{2}}$.

##  III Entanglement and extractable work

Figure 2:  Evolution of the occupation numbers for the battery part are
plotted in panel (a) for $N_{b}=1$ and panel (c) for $N_{b}=2$. Evolutions of
battery-charger entanglement $S(t)$ and extractable work $\mathcal{E}(t)$ are
plotted in panel (b) for $N_{b}=1$, panel (d) for $N_{b}=2$ and panel (e) for
$N_{b}=10$. $t^{*}$ denotes the time when the battery-charger entanglement
reaches its maximum. $T$ denotes the time when the extractable work reaches
its maximum. The process of $0\sim T$ is called the charging process. The
cartoon (f) illustrates the changes of battery-charger entanglement during the
charging process. The other parameters are set to be
$B=1,h=4,A=1,\Delta=0,N_{b}=1,N_{c}=20,m=20$ in panels (a,b);
$B=h=1,,A=1,\Delta=0,N_{b}=2,N_{c}=200,m=100$ in panels (c,d);
$B=h=1,,A=1,\Delta=0,N_{b}=10,N_{c}=20,m=20$ in panel (e).

Since the quantum state of battery-charger system remains a pure state during
the time evolution, the battery-charger entanglement can be well characterized
by the Von Neumann entropy of the battery part, which is given by [1]

$\displaystyle S(t)=-\mathrm{Tr}(\rho_{b}(t)\log_{2}\rho_{b}(t)).$ (25)

Another important quantity is the extractable work. During the charging
process, the energy injected from the charger to the battery is described by

$\displaystyle\Delta E(t)=E(t)-E(0),$ (26)

where $E(t)=\mathrm{Tr}(\rho(t)H_{b})$ denotes the mean energy of the battery
state $\rho_{b}(t)$. However, not all energy can be extracted from the
battery, which is known as the second law of thermodynamics. A proper measure
of the extractable work for the state $\rho_{b}(t)$ is provided by the
ergotropy [48]

$\displaystyle\mathcal{E}(t)=E(t)-E_{p}(t),$ (27)

where $E_{p}(t)=\mathrm{Tr}(\mkern
1.5mu\overline{\mkern-1.5mu\rho_{b}(t)\mkern-1.5mu}\mkern 1.5muH_{b})$ is the
energy of the passive counterpart $\mkern
1.5mu\overline{\mkern-1.5mu\rho_{b}(t)\mkern-1.5mu}\mkern 1.5mu$ of
$\rho_{b}(t)$. Passive states are incapable of giving out energy via any
cyclic Hamiltonian process. In our settings, the passive state $\mkern
1.5mu\overline{\mkern-1.5mu\rho_{b}(t)\mkern-1.5mu}\mkern 1.5mu$ is given by

$\displaystyle\mkern 1.5mu\overline{\mkern-1.5mu\rho_{b}(t)\mkern-1.5mu}\mkern
1.5mu=\sum_{j=1}^{N_{b}+1}|\bm{\psi}_{\eta_{j}}|^{2}|j-1\rangle\langle j-1|,$
(28)

where $\bm{\psi}$ is given in Eq. (13) and $\eta$ is a permutation of
$1,2,\ldots,N_{b}+1$ so that
$|\bm{\psi}_{\eta_{j}}|^{2}\geq|\bm{\psi}_{\eta_{j+1}}|^{2}$ for
$j=1,\ldots,N_{b}$.

Now we consider the single-cell battery case, i.e., $N_{b}=1$. The
entanglement for this case reduces to the binary entropy

$\displaystyle
S(t)\\!=\\!-\frac{1+r(t)}{2}\log_{2}\frac{1+r(t)}{2}-\frac{1-r(t)}{2}\log_{2}\frac{1-r(t)}{2},$

which shows an inverse relationship between $S(t)$ and $|r(t)|$. However, the
ergotropy is

$\displaystyle\mathcal{E}(t)=\left\\{\begin{array}[]{lrr}0,&&r(t)\geq 0\\\
-Br(t),&&r(t)<0\\\ \end{array}\right.$ (32)

which is positively related to the $|r(t)|$ when $\mathcal{E}(t)\neq 0$. Thus,
we have the following
_Theorem1_ : For $N_{b}=1$, if the ergotropy $\mathcal{E}(t)$ is not equal
zero then $\mathcal{E}(t)$ is in inversely related to the battery-charger
entanglement $S(t)$. In particular, at the end of the charging process,
$\mathcal{E}(T)$ is also in inversely related to $S(T)$.

As shown in Figs. 2(a) and 2(b), $r(t)$ keeps falling in the time interval
$0\sim T$. At time $t^{*}$, the occupation numbers of the ground state
$|0\rangle$ and the excited state $|1\rangle$ are balanced and thus the
battery state is maximally entangled with the charger. When time is far away
from $t^{*}$, the occupation numbers begin to reverse resulting in the
decrease of entanglement and the appearance of nonzero extractable work. At
time $T$, the charging process is finished with the most energy stored in the
battery and the battery-charger entanglement decreases to a minimum. According
to theorem 1, the extracted work $\mathcal{E}(T)$ essentially depend on the
battery-charger entanglement $S(T)$. The smaller $S(T)$ the larger
$\mathcal{E}(T)$.

For $N_{b}=2$ case, the complex expression of Eq. (II.2) impedes the further
discussion. We notice that $u_{1}=A\sqrt{2m(N_{c}-m+1)}$ can be considered as
the same as $u_{2}=A\sqrt{2(N_{c}-m+2)(m-1)}$ under some special limits, e.g.,
the Non-TC limit (8). Then Eq. (II.2) immediately reduces to a simple form:

$\displaystyle\rho_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1+\cos(\omega t))^{2},$ $\displaystyle\rho_{22}$
$\displaystyle=$ $\displaystyle\frac{1}{2}(1-\cos^{2}(\omega t)),$
$\displaystyle\rho_{33}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\cos(\omega t))^{2}.$ (33)

The corresponding extractable work and entanglement are thus given by

$\displaystyle\frac{\mathcal{E}(t)}{B}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lrr}0,&&0\leq t<t_{1}\\\
-\frac{3}{4}\left(\cos(\omega
t)+\frac{1}{3}\right)^{2}+\frac{1}{3},&&t_{1}\leq t<t^{*}\\\
-\frac{3}{4}\left(\cos(\omega t)+1\right)^{2}+1,&&t^{*}\leq t<t_{2}\\\
-2\cos(\omega t),&&t_{2}\leq t\leq T\\\ \end{array}\right.$ (38)

and

$\displaystyle S(t)$ $\displaystyle=$
$\displaystyle\frac{3}{2}+\frac{1}{2}\cos^{2}(\omega
t)-\frac{1}{2}(\cos(\omega t)+1)^{2}\log_{2}(\cos(\omega t)+1)$ (39)
$\displaystyle-\frac{1}{2}(\cos(\omega t)-1)^{2}\log_{2}(1-\cos(\omega t))$
$\displaystyle-\frac{1}{2}(1-\cos^{2}(\omega t))\log_{2}(1-\cos^{2}(\omega
t)),$

where
$t_{1}=\arccos(1/3)/\omega,t_{2}=\arccos(-1/3)/\omega,t^{*}=\pi/(2\omega),$
and $T=\pi/\omega$. The analytical results are consistent with the numerical
results obtained in the settings of $N_{c}=200,m=100$, see Figs. 2(c) and
2(d). Unlike the $N_{b}=1$ case, the extractable work $\mathcal{E}(t)$ is not
always zero but still small before the entanglement $E(t)$ reaches its maximum
$E(t^{*})$ because at time $t_{1}$ the occupation numbers of the ground state
and the first excited state are starting to inverse. Another difference lies
in that $S(T)\neq 0$ for $N_{b}=1$ while $S(T)=0$ for $N_{b}=2$. Note that the
choice of parameters $N_{b},N_{c},m$ largely determines whether $S(T)$ is
zero. In the next section, we will discuss this question in detail and further
highlight a strong link between the entanglement and the extractable work. Now
we claim that
_Theorem2_ : For $N_{b}=2$, the ergotropy $\mathcal{E}(T)$ is in inversely
related to the battery-charger entanglement $S(T)$. Here, we need not make the
simplification by requiring the condition $u_{1}=u_{2}$.
_Proof:_ The charging time $T$ is determined by maximizing the energy injected
to the battery, i.e., $\Delta E(t)$. According to the general expressions of
the occupation numbers (II.2) and the definition (26), we have

$\displaystyle\Delta E(t)$ $\displaystyle=$ $\displaystyle
B+\frac{B}{(u_{1}^{2}+u_{2}^{2})^{2}}[u_{1}^{2}(u_{2}^{2}-u_{1}^{2})x^{2}-4u_{1}^{2}u_{2}^{2}x$
(40) $\displaystyle+u_{2}^{2}(u_{1}^{2}-u_{2}^{2})],$

where $x=\cos(\omega t)$. If $u_{2}^{2}-u_{1}^{2}\geq 0$ then $\Delta E(t)$
takes the maximum value when $x=-1$. Otherwise, the maximum value of the
quadratic function $\Delta E(t)$ is taken at
$x_{*}=-2u_{2}^{2}/(u_{1}^{2}-u_{2}^{2})$ by ignoring the constraint $-1\leq
x\leq 1$. It is not difficult to show that $x_{*}<-1$ by using the condition
$m\geq N_{b}=2$. Thus, we deduce that the charging time $T$ is determined by
$\cos(\omega T)=-1$. It immediately follows that the occupation number of the
first excited state $\rho_{22}(T)=0$. Therefore, this theorem can be proven by
using the same argument in proving theorem 1.

Except for the cases of $N_{b}=1$ and $N_{b}=2$, it is difficult to carry out
analytical calculations in multi-cell battery cases. Thus, we perform a
numerical verification of the results obtained above for the case of
$N_{b}=10$, see Fig. 2(e). We see that the minimal value of entanglement
$E(T)$ marks the maximal extractable work $\mathcal{E}(T)$. The inverse
relationship between $S(T)$ and $\mathcal{E}(T)$ is checked in Fig. 3.

The cartoon 2(f) shows a variation of the battery-charger entanglement during
the charging process. Initially, the spin-down battery cells are interacted
with a well prepared charger that is a Dicke state with $m$ up spins. Thus,
there is no entanglement between the battery and the charger. When the
charging process begins, the spin down battery cells will be excited due to
the flip-flop interaction, resulting in increasing battery-charger
entanglement. At time $t^{*}$, the entanglement reaches its maximum and the
occupation numbers for the battery display a nearly balanced distribution. As
the charging process continues, the battery cells will continue to be excited
while the battery-charger entanglement begins to decrease. At the end of the
charging process, namely time $T$, the energy of battery reaches its maximum
while the battery-charger entanglement reaches its minimum. However, the
extractable work $\mathcal{E}(T)$ is less than the energy stored in the
battery and is inversely related to the battery-charger entanglement $S(T)$.

Figure 3:  Panel (a) plots the battery-charger entanglement $S(t)$ and the
extractable work $\mathcal{E}(t)$ for $N_{b}=2,N_{c}=10$. From panel (a) we
can determine $T$ at which $\mathcal{E}(t)$ reaches its maximum
$\mathcal{E}(T)$. Panels (b-d) plots $\mathcal{E}(T)$ and $S(T)$ versus $m$.
$m$ is the number of spin-up charging units. Here we set
$h=B=A=1,\Delta=0,N_{c}=10$ for panels (a-d), $N_{b}=2$ for panels (a,b),
$N_{b}=4$ for panel (c), and $N_{b}=6$ for panel (d).

##  IV Optimal work Extraction

In Sec. III we establish an inverse relationship between the battery-charger
entanglement $S(T)$ and the extractable work $\mathcal{E}(T)$, where $T$ is
the charging time. In this section, we will discuss how to realize the optimal
work extraction. Review the definitions given in Eqs. (26) and (27).
Generally, the extractable work $\mathcal{E}(T)$ is less than the energy
$\Delta E(T)$ stored in the battery since we can not extract any work from the
passive state. Thus, “optimal” in here refers to that $\mathcal{E}(T)=\Delta
E(T)$, or equivalently, $E_{p}(T)=E(0)$. For convenience, we can let the
initial energy $E(0)=0$ by subtracting a constant $E(0)$ from the Hamiltonian
(1). Then, the condition of optimal work extraction can be stated as that at
the end of the charging process, the energy of passive state is zero, i.e.,
$E_{p}(T)=0$. Another important observation is that vanishing battery-charger
entanglement $S(T)=0$ implies that the battery $\rho_{b}(T)$ is a pure state.
Then, the corresponding passive state is exactly the initial state
$|0\rangle_{b}$ and thus $E_{p}(T)=E(0)=0$. Therefore, the battery-charger
entanglement $S(T)$ also indicates the quality of extractable work. The zero
entanglement means that the battery possesses optimal extractable work.

For the $N_{b}=1$ case, we deduce from Eqs. (18, 32) that $T$ is determined by
$\cos((d_{1}-d_{2})T)=-1$. Thus, we have

$\displaystyle\mathcal{E}(T)=B\left[1-\frac{2(h-B)^{2}}{4A^{2}m(N_{c}-m+1)+(B-h)^{2}}\right],$
(41)

where $m$ is the number of spin-up charging units and we assume here
$\Delta=0$. Eq. (41) indicates that extractable work $\mathcal{E}(T)$
decreases when $m$ is far away $(N_{c}+1)/2$. According to Eqs. (II.2, 18,
III) the entanglement $S(T)=0$ for $m=(N_{c}+1)/2$ when $h=B$ and $\Delta=0$.

For the $N_{b}=2$ case, the charging time $T$ is determined by $\cos(\omega
T)=-1$. Substituting this condition into Eq. (II.2) and the definition of
ergotropy (27), we obtain

$\displaystyle\mathcal{E}(T)=2B\left[-1+\frac{8}{\frac{u_{1}^{2}}{u_{2}^{2}}+\frac{u_{2}^{2}}{u_{1}^{2}}+2}\right].$
(42)

It is clear that as $u_{1}/u_{2}=\sqrt{m(N_{c}-m+1)/[(m-1)(N_{c}-m+2)]}$
deviates from 1, or equivalently, $m$ deviates from $(N_{c}+2)/2$, the
extractable work $\mathcal{E}(T)$ will decrease, see Fig. 3(b). The expression
(39) of entanglement for the case of $m=(N_{c}+2)/2$ shows that $S(T)=0$ and
thus optimal work extraction can be realized.

Figure 4:  The energy $E_{p}(T)$ (a,c) of passive state at time $T$ versus the
number $m$ of charging units with initially spin up. $T$ is the charging time.
$T_{NTC}$ and $T_{TC}$ are obtained from Eqs. (44,45). Here, we set
$N_{b}=2,N_{c}=500,h=B=A=1,\Delta=0$ for panels (a,b) and
$N_{b}=10,N_{c}=500,h=10,B=A=1,\Delta=0$ for panels (c,d).

The above analytical analysis implies a conjecture that for a fixed $N_{b}$,
the extractable work $\mathcal{E}(T)$ decreases as $m$ deviates from
$(N_{c}+N_{b})/2$. Numerical calculations (Figs. 3(b-d)) support this
conjecture . However, such maximal extractable work may not be optimal. As
shown in Fig. 3(d), $S(T)$ is not zero for $m=(N_{b}+N_{c})/2=8$.

Whether optimal work extraction can be achieved under $N_{c}\to\infty$? Taking
$N_{b}=2$ as an instructive example, the condition $u_{1}=u_{2}$ ensures that
the battery-charger entanglement $S(T)$ (39) becomes zero at the end of the
charging process. In Sec. III we have discussed that the condition
$u_{1}=u_{2}$ can be realized under the non-TC limit. In the TC-limit (7) an
additional condition $m\to\infty$ is required to ensure $u_{1}=u_{2}$.
Although these two limits ensure the realization of optimal work extraction,
the corresponding charging times are indeed different. In the non-TC limit,
the charging time is given by

$\displaystyle T_{NTC}=\frac{\pi}{2A\sqrt{k(1-k)}N_{c}},$ (43)

where $k=\lim_{N_{c}\to\infty}m/N_{c}$ is given in the non-TC limit (8). Since
$m=(N_{b}+N_{c})/2$ benefits the work extraction, we take $k=1/2$ in Eq. (8)
and thus

$\displaystyle T_{NTC}=\frac{\pi}{AN_{c}}.$ (44)

In the TC limit, the charging time is given by

$\displaystyle T_{TC}=\frac{\pi}{2A\sqrt{mN_{c}}},$ (45)

which indicates the charging power is $\propto\sqrt{N_{c}}$. Such
$\sqrt{N_{c}}$-speedup of charging has also been founded in Ref. [19].
Compared with the performance of the battery in the TC limit, however, the
$\sqrt{N_{c}}$-improvement of charing power in the non-TC limit manifests the
uniqueness of the central-spin battery distinguished from the TC battery.

Figs. 4(a,b) plot the energy of passive state and the charging time $T$ for
$N_{b}=2,N_{c}=500$. With the increase of $m$, the central-spin battery
firstly behaves like the TC battery since the charging time $T$ corresponds to
$T_{TC}$, see Fig. 4(b). Only when $m$ is large enough ($>10$ in Fig. 4(a))
can the energy of passive state becomes zero, so optimal work extraction can
be achieved. With the further increase of $m$, the central-spin battery enters
to the non-TC limit and the charging time $T$ conforms to $T_{NTC}$, see Fig.
4(b). It is worth noting that the central-spin battery behaves like under the
non-TC limit for a wide range of parameter $m$ (from 100 to 400 in Fig. 4(b)).
The case of $N_{b}=10,B\neq h$ is plotted in Figs. 4(c,d). In Fig. 4(d), the
inconsistency between $T_{TC}$ and $T$ arises since $T_{TC}$ (Eq. (45)) is
derived for $N_{b}=2$ case. However, $T_{NTC}$ is still consistent with the
$T$ when $m$ is around $(N_{c}+N_{b})/2$ even if $h\neq B$. In this sense,
$T_{NTC}$ is universal and independent of the number of quantum cells $N_{b}$
and the strength of magnetic fields $B$ and $h$.

##  V Conclusion

In this work, we have studied extractable work and entanglement in the
central-spin battery. It is shown that the extractable work is quantified by
the ergotropy and the battery-charger entanglement is quantified by the Von
Neumann entropy. Exact dynamics for them have been obtained in the cases of
single-battery cell and two-battery cells. Using these exact expressions, we
rigorously show that during the charging process the battery-charger
entanglement first increases to the maximum which marks a nearly balanced
distribution of occupation numbers, and then entanglement decreases with the
rapid increase of extractable work. At the end of charging process, the final
extractable work from battery is in inversely related to the battery-charger
entanglement. The optimal extractable work, namely, no wasted energy, can be
achieved if there is no battery-charger entanglement at the end of charging
process. We show that there are two conditions to obtain optimal extractable
work: one is the TC limit with $m\to\infty$ and another is the non-TC limit.
For the former, the central-spin battery reduces to the TC battery with the
charging time $T_{TC}\propto 1/\sqrt{N_{c}}$, while the latter, the non-TC
limit, is unique for the central-spin battery and does not correspond to the
TC battery. We can realize the non-TC limit by preparing the charger in an
unpolarized Dicke state. The advantage of this setting is short charging time
$T_{NTC}=\pi/(AN_{c})$, large extractable work, and realization of optimal
work extraction. Moreover, $T_{NTC}$ is independent of the number of battery
cells and thus universal. The above-mentioned analytical results have also
been verified by numerical calculations for a ten-cell battery. It is worth
noting that our present results are applicable to quantum batteries where no
coherence is involved during the charging process. Our work deepens the
understanding of entanglement and extractable work in such incoherent quantum
batteries and sheds light on how to realize optimal work extraction in the
central-spin battery. For a more general situation, other quantum correlations
will be involved and affect the performance of work extraction. The relations
between extractable work and other quantum correlations will becomes subtle
and complicated. Uncovering their relations in more general settings will
become important.

## ACKNOWLEDGMENTS

Qing-Kun Wan and Shu Ding are acknowledged for useful discussion. This work
was supported by NSFC (Grants Nos. 12047502, 11975183 and 11875220), the Key
Innovative Research Team of Quantum Many-body theory and Quantum Control in
Shaanxi Province (Grant No. 2017KCT-12), the Major Basic Research Program of
Natural Science of Shaanxi Province (Grant No. 2017ZDJC-32), and the Double
First-Class University Construction Project of Northwest University.

H.-L. Shi and J.-X. Liu contributed equally to the numerical and analytical
studies for this research.

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