# Quantum enhancement of a single quantum battery by repeated interactions
with large spins

P. Chen  T. S. Yin  Physics Department of Zhejiang Sci-Tech University,
Hangzhou 310018, China  Z. Q. Jiang<EMAIL_ADDRESS>Physics Department
of Zhejiang Sci-Tech University, Hangzhou 310018, China  G. R. Jin
<EMAIL_ADDRESS>Physics Department of Zhejiang Sci-Tech University, Hangzhou
310018, China

###### Abstract

A generalized collision model is developed to investigate coherent charging a
single quantum battery by repeated interactions with many-atom large spins,
where collective atom operators are adopted and the battery is modeled by a
uniform energy ladder. For an initially empty battery, we derive analytical
results of the average number of excitations and hence the charging power in
the short-time limit. Our analytical results show that a faster charging and
an increased amount of the power in the coherent protocol uniquely arise from
the phase coherence of the atoms. Finally, we show that the charging power
defined by the so-called ergotropy almost follows our analytical result, due
to a nearly pure state of the battery in the short-time limit.

##  I Introduction

One of central goals of quantum thermodynamics is to improve the thermodynamic
processes via quantum resources and quantum operators Horodecki ; Niedenzu ;
Goold ; Vinjanampathy ; Campaioli . The simplest setup to achieve the quantum
advantages in the thermodynamics is the so-called quantum batteries (QBs)
Alicki ; Hovhannisyan ; Barra ; Bhattacharjee , i.e., a small quantum system
that stores and provides energy. Starting from seminal ideas developed in Ref.
Alicki , various quantum systems have been considered as the candidate of the
QBs, including collective spins Andolina99 ; Zhang ; Peng , interacting spin
chains Le ; Juli ; QZhao , and mechanical flywheels Levy ; SeahS . Different
to classical batteries, the QBs explore phase coherence Kamin  and quantum
entanglement Binder ; CampaioliF ; SantosA ; RossiniD ; Gyhm  as useful
resources to improve the performance of the QBs. A notable example is the
Dicke-model QBs Ferraro ; AndolinaD ; Andolina ; Monsel  based on collective
super-radiant coupling in cavity and waveguide QED setups, which has been
experimentally demonstrated by using fluorescent organic molecules in a
microcavity QuachK .

Recently, the QBs have been realized with a transmon qutrit Hu  and a solid-
state qubit Maillette ; Gemme , which clearly demonstrate the quantum
advantage at the level of a single battery Seah ; SalviaM ; Shaghaghi .
Especially, Seah et al. Seah  present a collision model to investigate the
repeated charging of a single battery by a sequence of identical qubits, where
a single two-level qubit is adopted as the charger in each interaction or
collision. The collision model describes a system that undergoes successive
interactions or collisions with the auxiliary systems Rau ; Caves ; Brun ;
Ciccarello954 . It has been used in various research areas, such as non-
Markovian quantum dynamics Ciccarello87 , quantum thermodynamics Scarani ;
Strasberg , quantum optics Ciccarello4 , and quantum gravity Altamirano .
Using the collision model, Seah et al. Seah  numerically show that the phase
coherence of the qubit can realize a faster charging and a larger amount of
the charging power in a comparison with that of the qubit state without any
coherence. To understand the role of the coherence, analytical results of the
charging energy and its power are necessary.

In this paper, we generalize the collision model to investigate coherent
charging of a single quantum battery by repeated interactions with many two-
level atoms (or equivalently, a large spin), where collective atomic operators
$\\{\hat{J}_{\pm},\hat{J}_{z}\\}$ are adopted and the battery is modeled by a
uniform energy ladder Mitchison ; Seah . Analytical results of the charging
energy and its power are derived by considering the initially empty battery
and the short-time limit of the interaction at each charging step. Following
Ref. Seah , we compare the charging processes with the atomic coherence
$\langle\hat{J}_{-}\rangle\neq 0$, corresponding to the coherent charging, and
$\langle\hat{J}_{-}\rangle=0$ for the incoherent charging. Our analytical
results show that the advantage of the coherent protocol uniquely comes from
the atomic coherence. Furthermore, we show that the amount of the charging
power reaches its maximum (proportional to the number of atoms $N_{A}$), when
all the atoms prepared in a superposition state, i.e., a coherent spin state
$|\theta_{0},\phi_{0}\rangle$ with $\theta_{0}=\pi/2$ and $\phi_{0}=0$.
Finally, we investigate the charging power defined by the so-called ergotropy
Allahverdyan . In the short-time limit, numerical results of the power show
good agreement with the analytical result, since the battery state almost
maintains in a pure state.

##  II Generalized collision model for the quantum battery

As illustrated schematically by Fig. 1, we consider a quantum battery
$\hat{\rho}_{B}$ modeled by a uniform energy ladder Mitchison ; Seah , which
undergoes successive interactions with identical two-level atoms confined in
several lattices. The interaction between the battery and the atoms at each
collision can be described by the Hamiltonian

$\hat{H}=\hat{H}_{0}+\hat{V}=\varepsilon\hat{J}_{z}+\varepsilon\hat{n}+\hbar
g(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}}),$ (1)

where collective atomic operators
$\hat{J}_{z}=\sum_{i=1}^{N_{A}}(|e\rangle_{ii}\langle e|-|g\rangle_{ii}\langle
g|)/2$ and $\hat{J}_{+}=\sum_{i=1}^{N_{A}}|e\rangle_{ii}\langle
g|=(\hat{J}_{-})^{{\dagger}}$ are introduced to describe finite $N_{A}$
identical atoms, with the ground state $|g\rangle$ and the excited state
$|e\rangle$. For the battery, one can define the number operator
$\hat{n}=\sum_{n=0}^{N_{B}}n|n\rangle\langle n|$, where $N_{B}$ denotes the
highest level of the battery, and also the ladder operators

$\hat{B}=\sum_{n=1}^{N_{B}}|n-1\rangle\langle n|,\text{ \ \
}\hat{B}^{{\dagger}}=\sum_{n=1}^{N_{B}}|n\rangle\langle n-1|,$ (2)

satisfying the commutation relation BrunnerN ; Erker :
$[\hat{B},\hat{B}^{{\dagger}}]=|0\rangle\langle 0|-|N_{B}\rangle\langle
N_{B}|$. When the occupations of $|0\rangle$ and $|N_{B}\rangle$ are
vanishing, it becomes $[\hat{B},\hat{B}^{{\dagger}}]\approx 0$, corresponding
to the no-boundary condition. If we only omit the upper boundary (i.e., the
occupation on $|N_{B}\rangle$ is vanishing), then the commutation relation is
simply given by $[\hat{B},\hat{B}^{{\dagger}}]\approx|0\rangle\langle 0|$.

Figure 1: Schematic picture of a single battery (modeled by a uniform energy
ladder with $N_{B}+1$ levels), which undergoes successive interactions with
finite $N_{A}$ two-level atoms confined in each lattice. The optimal atom
states $\hat{\rho}_{A}$ for the incoherent and the coherent charging protocols
correspond to the coherent spin states of a large spin
$\ket{\theta_{0},\phi_{0}}$, with $\theta_{0}=0$, $\pi/2$, as illustrated by
their quasi-probability distributions on the Bloch sphere.

In experiments, the identical atoms (considered here as the charger) can be
realized by an ensemble of large spins at the room temperature (e.g., the spin
$j=7/2$ of cesium atoms Fernholz ; Auccaise ), or ultracold bosonic gases in
optical lattice Widera ; Riedel ; Pedrozo . On the other hand, the energy
ladder can be realized by a cavity mode with a _finite_ number of energy
levels Shaghaghi ; SalviaM . In the interaction picture, the Hamiltonian
becomes
$\hat{V}_{\mathrm{int}}=\exp(i\hat{H}_{0}t/\hbar)\hat{V}\exp(-i\hat{H}_{0}t/\hbar)=\hat{V}$,
due to $[\hat{H}_{0},\hat{V}]=0$, which in turn gives the time evolution
operator
$\hat{U}_{\tau}=\exp[-i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})]$,
where $\tau=gt$ and $t$ denotes the interaction time at each charging step.
Starting from an initial state $\hat{\rho}_{B}(0)$, the battery state at the
$k$th collision becomes

$\hat{\rho}_{B}(k)=\mathrm{Tr}_{A}[\hat{U}_{\tau}\hat{\rho}_{B}(k-1)\otimes\hat{\rho}_{A}\hat{U}_{\tau}^{{\dagger}}],$
(3)

where $\mathrm{Tr}_{A}(\cdots)$ denotes the trace over the atom states.
Similar to Ref. Seah , we assume that the atom states are identical for all
the collisions (i.e., $\hat{\rho}_{A}$ is independent on $k$). In the limit
$\tau\rightarrow 0$, the time evolution operator can be approximated as
$\hat{U}_{\tau}\approx
1-i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})$, and therefore

$\hat{\rho}_{B}(k)\approx\hat{D}^{{\dagger}}(k\alpha)\hat{\rho}_{B}(0)\hat{D}(k\alpha),$
(4)

where $\hat{D}(\alpha)=\exp(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})$
and $\alpha=i\tau\langle\hat{J}_{-}\rangle$, with
$\langle(\cdots)\rangle=\mathrm{Tr}_{A}[\hat{\rho}_{A}(\cdots)]$. Similar
result of Eq. (4) has been obtained by Ref. SalviaM  (see also the Appendix
A).

Using the probability distribution $P_{n}(k)=\langle
n|\hat{\rho}_{B}(k)|n\rangle$, one can define the mean number of excitations
and the mean energy Seah :

$\bar{n}_{k}=\sum_{n=0}^{N_{B}}nP_{n}(k)\text{, \ \ \
}E(k)=\varepsilon\bar{n}_{k},$ (5)

where $\varepsilon$ denotes the energy spacing of the battery. Following Ref.
Seah , we first consider the no-boundary condition (i.e., the occupations on
$|0\rangle$ and $|N_{B}\rangle$ are vanishing), which allows us to obtain a
recursion relation (see the Appendix A),

$\bar{n}_{k}=\bar{n}_{k-1}+v+\mathrm{Im}\left(\Omega\beta_{k-1}^{\ast}\right),$
(6)

where $v=2\sin^{2}(\tau)\langle\hat{J}_{z}\rangle$,
$\Omega=\sin(2\tau)\langle\hat{J}_{-}\rangle$, and
$\beta_{k}=\mathrm{Tr}_{B}[\hat{\rho}_{B}(k)\hat{B}]$. Without any boundary,
we have $\beta_{k}=\beta_{0}$, and therefore
$\bar{n}_{k}=\bar{n}_{0}+k[v+\mathrm{Im}(\Omega\beta_{0}^{\ast})]$, where Eq.
(6) has iterated for $k$ times. One can easily find that the mean number of
excitations $\bar{n}_{k}$ and hence the mean energy of the battery $E(k)$ grow
linearly with respect to the number of charging steps $k$ Seah . When
$\bar{n}_{k}\approx N_{B}$, the battery can be regarded as being fully
charged.

Next we focus on the short-time limit (i.e., $\tau\rightarrow 0$) to obtain
$v\approx 2\tau^{2}\langle\hat{J}_{z}\rangle\sim 0$ and $\Omega\approx
2\tau\langle\hat{J}_{-}\rangle=-2i\alpha$, which yield

$\bar{n}_{k}\approx\bar{n}_{k-1}-(\alpha\beta_{k-1}^{\ast}+\alpha^{\ast}\beta_{k-1}),$
(7)

where the lower boundary $|0\rangle$ has been taken into account (see the
Appendix B), as

$\beta_{k}=\beta_{0}-\alpha\sum_{k^{\prime}=0}^{k-1}\langle
0|\hat{\rho}_{B}(k^{\prime})|0\rangle.$ (8)

Note that the above recursion relations of the average number of excitations,
i.e., Eqs. (6) and (7) are independent on any specific form of
$\hat{\rho}_{A}$, and even free from the initial state of $\hat{\rho}_{B}(0)$.
Next, we consider a specific form of $\hat{\rho}_{A}$ and extend the single-
particle case (i.e., $N_{A}=1$) Seah  into the many-particle case, for which
the total charging process becomes more faster and the amount of charging
power can be increased, dependent on $N_{A}$.

##  III Collectively Coherent charging

In Ref. Seah , Seah et al. consider the battery charged by a sequence of
single atom (i.e., $N_{A}=1$), with the atom state Radcliffe ; Arecchi ;
ZhangD ; Kitagawa ; Jin ; Ji

$\displaystyle\hat{\rho}_{A}$ $\displaystyle=$
$\displaystyle\cos^{2}\left(\frac{\theta_{0}}{2}\right)|e\rangle\langle
e|+\sin^{2}\left(\frac{\theta_{0}}{2}\right)|g\rangle\langle g|$ (9)
$\displaystyle+c\left(\frac{\sin(\theta_{0})}{2}e^{-i\phi_{0}}|e\rangle\langle
g|+\mathrm{H.c.}\right),$

where $\theta_{0}$ and $\phi_{0}$ determine the population imbalance and the
relative phase between the two states Kitagawa ; Jin ; Ji , as shown in Fig.
1. The parameter $c\in[0,1]$ is added artificially to distinguish the two
opposing protocols: the incoherent charging ($c=0$) and the coherent charging
($c=1$). The spin-$1/2$ can be mapped into a large spin system with
$j=N_{A}/2$, for which the atom state becomes

$\hat{\rho}_{A}=\sum_{m}\rho_{m,m}|j,m\rangle\langle j,m|+c\sum_{m\neq
m^{\prime}}\rho_{m,m^{\prime}}|j,m\rangle\langle j,m^{\prime}|,$ (10)

where $\\{|j,m\rangle\\}$ denote eigenstates of $\hat{J}_{z}$ and
$\rho_{m,m^{\prime}}=d_{m}d_{m^{\prime}}^{\ast}$, with

$d_{m}=\binom{2j}{j+m}^{1/2}\cos^{j+m}\left(\frac{\theta_{0}}{2}\right)\sin^{j-m}\left(\frac{\theta_{0}}{2}\right)e^{i(j-m)\phi_{0}},$

and $\binom{n}{m}=\frac{n!}{m!(n-m)!}$. For the coherent charging scheme
(i.e., $c=1$), $\hat{\rho}_{A}$ becomes a coherent spin state
$|\theta_{0},\phi_{0}\rangle$ Radcliffe ; Arecchi ; ZhangD ; Kitagawa ; Jin ,
which gives the population imbalance and the phase coherence determined by
$\langle\hat{J}_{z}\rangle=j\cos(\theta_{0})$ and
$\langle\hat{J}_{-}\rangle=cj\sin(\theta_{0})\exp(-i\phi_{0})$ Jin ; Ji ,
respectively. Hereafter, we choose the azimuthal angle $\phi_{0}=0$ and
therefore,

$v=2j\cos(\theta_{0})\sin^{2}(\tau)\text{, \ \ \ \ \
}\Omega=cj\sin(\theta_{0})\sin(2\tau),$ (11)

where $j=N_{A}/2$, and $c=\Omega=0$ for the incoherent protocol.

We first consider the incoherent charging process to an initially empty
battery $\hat{\rho}_{B}(0)=|0\rangle\langle 0|$, using $\hat{\rho}_{A}$ with
$c=0$ and $\theta_{0}=\pi/3$. The red dashed lines of Fig. 2(a) show the
probability distributions $P_{n}(k)$ against $n$ for different charging times
$k\tau$, where $\tau=\pi/4$ is fixed Seah . One can find that the probability
distribution tends to a Gaussian as $k$ increases. Therefore, from Eq. (6),
the peak of $P_{n}(k)$ appears at

$n=\bar{n}_{k}\approx
vk+\Omega\sum_{k^{\prime}=0}^{k-1}\mathrm{Im}(\beta_{k^{\prime}}^{\ast})\sim(v+\Omega)k,$
(12)

where the last result holds when $\mathrm{Im}(\beta_{k})\sim 1$ (see the
Appendix B). From Fig. 2(a), one can see the locations of the peaks
$n\approx(v+\Omega)k$, as indicated by the vertical lines. When the peak
approaches to the highest level (i.e., $n\approx N_{B}$), the battery can be
regarded as being fully charged and the number of charging steps needed is
given by Seah :

$k_{\mathrm{est}}\approx\mathrm{Ceiling}\left[\frac{N_{B}}{v+\Omega}\right],$
(13)

where $\mathrm{Ceiling}[x]$ gives the smallest integer greater than or equal
to $x$. In Fig. 2(b), we show the mean value of the excitations $\bar{n}_{k}$
against the total charging time $k\tau$. As shown by the red dashed line,
$\bar{n}_{k}$ monotonically increases from $0$ to its maximal value $N_{B}$.
Similar result has been observed by considering the single-qubit case of
$\hat{\rho}_{A}$ Seah .

For the single-qubit case Seah , it has been shown that the number of
collisions $k_{\mathrm{est}}\approx 800$, corresponding to the total charging
time $k_{\mathrm{est}}\tau\approx 628$. For the coherent charging process
(i.e., $c=1$), it is about $k_{\mathrm{est}}=292$ Seah  and hence the total
charging time $k_{\mathrm{est}}\tau$ $\approx 229$, which is shorter than that
of the incoherent case by $2.74$ times. To understand the advantage of the
coherent protocol, one can note that the number of collisions
$k_{\mathrm{est}}$ can be reduced due to $\Omega\neq 0$ (it is maximized for
$\tau=\pi/4$ and is vanishing for the incoherent case), and therefore leads to
a faster charging. However, when $k>k_{\mathrm{est}}$, the coherent protocol
losses its advantage, due to a decay of $\bar{n}_{k}$. This is because the
probability distribution $P_{n}(k)$ is reflected by the upper boundary
($n=N_{B}$) Seah , which reduces $\bar{n}_{k}$ and hence the energy of the
battery $E(k)$. Such a phenomenon also occurs for the many-particle case.

For the many-particle case, e.g., $N_{A}=10$, the charging time can be further
reduced for both the incoherent and the coherent protocols. As shown by Fig.
2(b), the incoherent scheme requires $k_{\mathrm{est}}\tau\approx 63$ (the red
dashed line) to fully charging the battery. For the coherent case (the solid
line), the total charging time is about $k_{\mathrm{est}}\tau$ $\approx 23$.
Indeed, the number of charging steps $k_{\mathrm{est}}$ is reduced by $N_{A}$
times since both $v$ and $\Omega$ are proportional to $j$ ($=N_{A}/2$), as
Eqs. (11) and (13). Therefore, a more faster charging can be realized by using
the many-particle $\hat{\rho}_{A}$.

In Fig. 2(c) and (d), we further consider the above two charging processes in
the short-time limit (e.g., $\tau=10^{-2}$). For this case, the coherent
protocol always outperforms the incoherent one within the total charging time
$k\tau\in[0$, $70]$. For the incoherent case, both $P_{n}(k)$ and
$\bar{n}_{k}$ increase very slowly (the red dashed lines). Indeed, the battery
is fully charged when $k_{\mathrm{est}}\tau\approx 4\times 10^{3}$. In
contrast, the coherent scheme significantly reduces the total charging time.
One can see that $\bar{n}_{k}$ increases to its maximum at
$k_{\mathrm{est}}\tau\approx 23$, with a very small value of interaction time
$\tau=10^{-2}$. This result is somewhat counter-intuitive.

Figure 2: Probability distributions of $\hat{\rho}_{B}(k)$ (left panel) and
average number of the excitations (right panel), for different values of the
charging time $\tau=\pi/4$ (in a and b) and $0.01$ (c and d). Solid (red
dashed) lines correspond to the coherent (incoherent) charging protocol. All
curves in the left panel are rescaled to their associated maxima. Vertical
lines: location of the peak $n\approx(v+\Omega)k$ (left panel) and
$k_{\mathrm{est}}\tau$ (right panel), determined by Eqs. (12) and (13),
respectively. Circles in (d): analytical result of $\bar{n}_{k}$, given by Eq.
(17). Parameters: $N_{A}=10$, $N_{B}=200$, and $\theta_{0}=\pi/3$.

To understand the above result, we derive analytical results of $\bar{n}_{k}$
and hence the charging power in the short-time limit (i.e., $\tau\ll 1$).
According to Eqs. (7) and (8), $\bar{n}_{k}$ depends on $\beta_{k}$ and also
$\langle 0|\hat{\rho}_{B}(k)|0\rangle\approx|\langle
0|\hat{D}(k\alpha)|0\rangle|^{2}$, where

$\langle
0|\hat{D}(k\alpha)|0\rangle=\sum_{l=0}^{\infty}\frac{k^{l}}{l!}\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{l}|0\rangle,$ (14)

with $\alpha=i\tau\langle\hat{J}_{-}\rangle$. Performing a series expansion
over $\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{l}|0\rangle$, one can see
that it is vanishing for odd $l$ (see the Appendix B). As inspired by Fig. 3
(a), for even $l=2n$, we obtain $\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{2n}|0\rangle=(-1)^{n}|\alpha|^{2n}C_{n}$,
where $C_{n}=\frac{1}{n+1}\binom{2n}{n}$ denotes the Catalan number Hilton ;
Stanley , and therefore

$\langle
0|\hat{D}(k\alpha)|0\rangle=\sum_{n=0}^{\infty}\frac{(-1)^{n}(k|\alpha|)^{2n}C_{n}}{(2n)!}=\frac{J_{1}(2k|\alpha|)}{k|\alpha|}.$
(15)

Here $J_{1}(x)$ denotes the first-order Bessel function of the first kind. Now
Eq. (8) becomes

$\beta_{k}\approx\beta_{0}-\alpha\sum_{k^{\prime}=0}^{k-1}\left(\frac{J_{1}(2k^{\prime}|\alpha|)}{k^{\prime}|\alpha|}\right)^{2},$
(16)

where $\beta_{0}=\mathrm{Tr}_{B}[\hat{\rho}_{B}(0)\hat{B}]=0$. From Eq. (7),
we further obtain (see the Appendix B)

$\bar{n}_{k}\approx
2\sum_{k^{\prime}=0}^{k-2}(k-1-k^{\prime})\left(\frac{J_{1}(2k^{\prime}|\alpha|)}{k^{\prime}}\right)^{2},$
(17)

where $\bar{n}_{0}=0$. For a large enough $k$ (so that $k-2\approx k$), the
sum over $k^{\prime}$ can be replaced by an integral

$\bar{n}_{k}\approx
2\int_{0}^{k}(k-k^{\prime})\left(\frac{J_{1}(2k^{\prime}|\alpha|)}{k^{\prime}}\right)^{2}dk^{\prime}=2xf(x),$
(18)

where $x=k|\alpha|=k\tau\langle\hat{J}_{-}\rangle$, with
$\langle\hat{J}_{-}\rangle=j\sin(\theta_{0})$, and

$f(x)=\frac{1}{x}\int_{0}^{x}\left(x-x^{\prime}\right)\left[\frac{J_{1}(2x^{\prime})}{x^{\prime}}\right]^{2}dx^{\prime}.$
(19)

From Fig. 3(b), one can see that $f(x)$ is a monotonic function, which
increases from $0$ to its asymptotic value $f(\infty)=8/(3\pi)\approx 0.85$.
When $x=30$, $f(x)$ approaches to $f(\infty)$.

Figure 3: (a) All possible paths for $\langle
0|(\alpha^{\ast}\hat{B}-\alpha\hat{B}^{{\dagger}})^{2n}|0\rangle\neq 0$, with
the number of paths given by the Catalan number $C_{n}$. (b) The monotonic
function $f(x)$ (solid), defined by Eq. (19), and $f(x)+xf^{\prime}(x)$ (red
dashed). In (a), the number of paths $C_{n}=5$ for $n=3$. Horizontal line in
(b): $8/(3\pi)$.

Note that Eqs. (17) and (18) are valid for $\hat{\rho}_{B}(0)=|0\rangle\langle
0|$ and $k\tau\leq k_{\mathrm{est}}\tau$, where the charging time per
collision $\tau\ll 1$. In the short-time limit, from Eq. (13), we obtain
$k_{\mathrm{est}}\tau\approx\tau\mathrm{Ceiling}[N_{B}/\Omega]\sim
N_{B}/N_{A}$, provided $\theta_{0}=\pi/2$. As depicted by Fig. 2(d), our
analytical results (the circles) shows a good agreement with the numerical
result of $\bar{n}_{k}$, as long as $k\tau\leq k_{\mathrm{est}}\tau\approx
23$. Furthermore, Eq. (17) at $k=k_{\mathrm{est}}$ gives $\bar{n}_{k}\approx
0.82N_{B}$, coincident quite well with the numerical result $0.87N_{B}$.
Numerical results of $\bar{n}_{k}$ decreases after $k\tau\gtrsim
k_{\mathrm{est}}\tau$, which can not be predicted by our analytical result.

##  IV Charging power

The performance of the battery can be quantified by the charging power

$P=\frac{E(k)-E(0)}{kt}=g\varepsilon\left(\frac{\bar{n}_{k}-\bar{n}_{0}}{k\tau}\right),$
(20)

where $E(0)=\varepsilon\bar{n}_{0}$ is the mean energy of the initially
uncharged battery. For the fully empty battery,
$\hat{\rho}_{B}(0)=|0\rangle\langle 0|$, we have $E(0)=\bar{n}_{0}=0$. Using
Eq. (12), one can obtain an approximate upper bound of the power,

$P\approx\frac{g\varepsilon}{\tau}(v+\Omega)=g\varepsilon
N_{A}\frac{\sin(\tau)}{\tau}\sin(\tau+\theta_{0})\leq g\varepsilon N_{A},$
(21)

where $v$ and $\Omega$ are given in Eq. (11). The first result comes from Eq.
(12) when $\mathrm{Im}(\beta_{k})\sim 1$, valid for the no-boundary condition.
The second inequality is a natural result of Eq. (11) with $c=1$. One can
easily find that maximum of the power can be reached at
$\theta_{0}=\pi/2-\tau$. Therefore, in the limit $\tau\rightarrow 0$ (i.e.,
$\theta_{0}\approx\pi/2$), the power is possible to reach its upper bound
$g\varepsilon N_{A}$.

To reach the upper bound, we now consider the coherent charging of the
initially empty battery (i.e., $c=1$ and $\hat{\rho}_{B}(0)=|0\rangle\langle
0|$), using an optimal atom state $|\pi/2,0\rangle$. In the short-time limit
$\tau\rightarrow 0$, Fig. 2(d) suggests a large enough number of the atom-
battery interactions (i.e., $k\rightarrow\infty$, but with a finite total
charging time $k\tau$). Using Eq. (18), we obtain the analytical result of the
power

$P_{\mathrm{coh}}\approx\frac{2g\varepsilon}{k\tau}xf(x)=2g\varepsilon\langle\hat{J}_{-}\rangle
f(x),$ (22)

where $x=k\tau\langle\hat{J}_{-}\rangle$, with
$\langle\hat{J}_{-}\rangle=j\sin(\theta_{0})$. The optimal atom state
$|\theta_{0},0\rangle$ with $\theta_{0}=\pi/2$ can be obtained from the
following equation

$0=\frac{\partial P_{\mathrm{coh}}}{\partial\theta_{0}}\propto\frac{\partial
x}{\partial\theta_{0}}\left[f(x)+x\frac{\partial f(x)}{\partial x}\right],$
(23)

or equivalently, $0=\partial
x/\partial\theta_{0}\propto\partial\langle\hat{J}_{-}\rangle/\partial\theta_{0}=j\cos(\theta_{0})$,
i.e., $\theta_{0}=\pi/2$. With the optimal state, the charging power
$P_{\mathrm{coh}}$ can reach its maximum at $k_{\mathrm{est}}\tau\sim
N_{B}/N_{A}$, with

$P_{\mathrm{coh,\max}}\approx N_{A}g\varepsilon
f(\infty)=\frac{8}{3\pi}N_{A}g\varepsilon,$ (24)

where $f(\infty)=8/(3\pi)\approx 0.85$, as shown by Fig. 3 (b). In the single-
particle picture, the optimal atom state can be rewritten as a direct product
$|\pi/2,0\rangle\varpropto(|e\rangle+|g\rangle)^{\otimes N_{A}}$,
corresponding to all the spins pointed to the $\hat{J}_{x}$ axis, as depicted
in Fig. 1. The coherent spin state with $\theta_{0}=\pi/2$ has been prepared
in the large spin system at the room temperature Fernholz ; Auccaise  and the
ultracold bosonic gases in optical lattice Widera ; Riedel ; Pedrozo .

The coherent charging scheme is robust to imperfections in preparing the
atomic states. As shown in Fig. 4(a), one can see that $P_{\mathrm{coh}}$
varies smoothly with $\theta_{0}$ for different values of $N_{A}$. No peak or
dip at $\theta_{0}\sim\pi/2$ means that the coherent scheme works well for a
sub-optimal atom state. When $\theta_{0}$ largely departures from $\pi/2$
(e.g., $\theta_{0}=\pi/8$), the squares of Fig. 4(b) show that the power
$P_{\mathrm{coh,\max}}\approx 0.85g\varepsilon N_{A}$ can almost maintain by
taking a relatively larger charging time $\tau$, which has not been
investigated in Ref. Seah .

To confirm the above results, we consider the coherent charging to the
initially empty battery (i.e., $c=1$ and $\hat{\rho}_{B}(0)=|0\rangle\langle
0|$). In Fig. 4, we choose a fixed charging time $k\tau=60/N_{A}$ to calculate
numerical results of $\bar{n}_{k}$ and hence the power $P$, which depend on
the choices of $\theta_{0}$ and $\tau$. For given $N_{A}=1$ (the open
squares), $2$ (the squares), $4$ (the open circles), in Fig. 4(a), we take
$\tau=0.01$ (i.e., from bottom to top, $k=6000$, $3000$, and $1500$) and show
the scaled power as a function of $\theta_{0}$. As $\tau\ll 1$, one can see
that our analytical results work well (the curves) and the maximum of the
power appears at $\theta_{0}=\pi/2$. When $\theta_{0}$ largely departures from
$\pi/2$, e.g., $\theta_{0}=\pi/8$ in Fig. 4(b), we show the power as a
function of $N_{A}$ for $\tau=0.01$ (the open squares), $0.3$ (the circles),
$0.88$ (the squares). From the squares, one can see that the optimal result
$P_{\mathrm{coh,\max}}$ that obtained for $\theta_{0}=\pi/2$ and $\tau\ll 1$
(the open circles) almost maintains by choosing a relatively larger charging
time $\tau$.

Figure 4: (a) Scaled power $P/(g\varepsilon)$ against $\theta_{0}$ for a fixed
$\tau=0.01$ and different numbers of the atoms, (b) the charging power against
$N_{A}$ for $\theta_{0}=\pi/8$ and different values of $\tau$, (c) and (d)
numerical and analytical results of $P/(g\varepsilon N_{A})$ as functions of
$\theta_{0}$ and $\tau$. All the numerical results are obtained at
$k\tau=60/N_{A}$. In (a): from top to bottom, $N_{A}=4$ (open circles), $2$
(squares), $1$ (open squares), coincident with the curves for Eq. (22). In
(b): from top to bottom, $\tau=0.88$ (squares), $0.3$ (circles), $0.01$ (open
squares), to comparing with the optimal result (the open circles for
$\theta_{0}=\pi/2$ and $\tau=0.01$), and the curves also from Eq. (22). The
white dashed line of (c): local maximum of the power at $k\tau=60/N_{A}$ for
$N_{A}=10$ and different values of $(\theta_{0},\tau)$, to comparing with its
analytical result $\tau=1.17(1-2\theta_{0}/\pi)$ (solid line), obtained from
the second result of Eq. (21). All for $N_{B}=200$.

In Fig. 4(c), we take $N_{A}=10$ and show the scaled power $P/(g\varepsilon
N_{A})$ against $\theta_{0}$ and $\tau$. When $\tau\ll 1$, the optimal atom
state corresponds to $\theta_{0}=\pi/2$, as expected. While for each value of
$\theta_{0}<\pi/2$, there exists an optimal value of the charging time $\tau$
(the white dashed line). In Fig. 4(d), we show the scaled power using the
second result of Eq. (21), where the white background region indicates
$P\propto\sin(\tau+\theta_{0})<0$. This is unphysical. Analytically, we show
that the maximum of the power can be obtained for
$\tau=1.17(1-2\theta_{0}/\pi)$ (the solid line), coincident quite well with
the white dashed line of Fig. 4(c). When $\theta_{0}=\pi/8$, we have
$\tau=0.88$, as depicted by the squares of Fig. 4(b). As
$\theta_{0}\rightarrow 0$, the maximum of the power appears at $\tau=1.17$,
which can be understood by considering the incoherent charging protocol.

For the incoherent charging case (i.e., $c=\Omega=0$), Eq. (6) becomes
$\bar{n}_{k}\approx kv$, and therefore

$P_{\mathrm{inc}}\approx
2g\varepsilon\langle\hat{J}_{z}\rangle\frac{\sin^{2}(\tau)}{\tau},$ (25)

where $\langle\hat{J}_{z}\rangle=j\cos(\theta_{0})$. One can easily find that
an optimal value of $P_{\mathrm{inc}}$ can be obtained for $\theta_{0}=0$,
corresponding to an optimal atom state $|e\rangle^{\otimes N_{A}}$, as shown
by Fig. 1. Furthermore, $P_{\mathrm{inc}}$ reaches its maximum at a finite
$\tau$, determined by

$0=\left.\frac{\partial
P_{\mathrm{inc}}}{\partial\tau}\right|_{\tau_{0}}\propto\frac{\sin^{2}\tau_{0}}{\tau_{0}^{2}}\left(2\tau_{0}\cot\tau_{0}-1\right),$
(26)

or equivalently, $\tau_{0}\cot\tau_{0}=1/2$. This is a transcendental equation
with one of the roots $\tau_{0}\approx 1.17$ Seah , which gives
$P_{\mathrm{inc,\max}}\approx 0.72g\varepsilon N_{A}$. Note that
$P_{\mathrm{inc,\max}}$ is the achievable power attained from the incoherent
charging protocol, as depicted by the top edge of the gray area in Fig. 5(a)
and (b). Using the optimal state with $\theta_{0}=0$, Eq. (13) indicates that
fully charging the battery with the incoherent protocol requires the time
$k_{\mathrm{est}}\tau_{0}\approx\tau_{0}\mathrm{Ceiling}[N_{B}/v]\sim
1.38N_{B}/N_{A}$, larger than that of the coherent charging protocol
($k_{\mathrm{est}}\tau\sim N_{B}/N_{A}$ for $\theta_{0}=\pi/2$).

Figure 5: With a given $\tau=0.01$, $0.1$, and $0.3$, the scaled power
$\mathcal{P}/(g\varepsilon N_{A})$ defined by the ergotropy and the purity of
$\hat{\rho}_{B}(k)$ for $N_{A}=1$ (left panel) and $4$ (right panel). The
curves in (a) and (b): the scaled power $P_{\mathrm{coh}}/(g\varepsilon
N_{A})$ for $N_{A}=1$ (dot-dashed), $2$ (dashed), and $4$ (solid), obtained
from Eq. (22) with $\theta_{0}=\pi/2$. The horizontal lines in (a) and (b):
the power of the incoherent charging scheme $P_{\mathrm{inc}}/(g\varepsilon
N_{A})$ for different values of $\tau=0.01$, $0.1$, and $0.3$ (from bottom to
top). The top edge of the gray area: the maximum power of the incoherent
charging protocol for $\tau=1.17$, i.e., $P_{\mathrm{inc,\max}}/(g\varepsilon
N_{A})\simeq 0.72$. All for $N_{B}=200$.

Comparing Eqs. (22) and (25), one can easily find that the coherent protocol
depends on the atomic coherence $\langle\hat{J}_{-}\rangle$, and the power
varies with the time $x=k\tau\langle\hat{J}_{-}\rangle$. For the incoherent
one, however, the power is a function of $\tau$, independent on the atomic
coherence. Our analytical results show that the advantages of the coherent
protocol in the charging time and that of the charging power _uniquely_ arise
from the phase coherence of the atoms. For the coherent spin state with
$\theta_{0}=\pi/2$, the coherence becomes maximum and the charging power
$P_{\mathrm{coh,\max}}\approx 0.85g\varepsilon N_{A}$ can be obtained for the
charging time $\tau\rightarrow 0$, which significantly reduces the role of
noise during each charging step. Furthermore, with a finite total charging
time $k_{\mathrm{est}}\tau\sim N_{B}/N_{A}$, one can reduce the number of
charging steps $k_{\mathrm{est}}$ by using large $N_{A}$, which is
inaccessible from Ref. Seah .

Finally, it should be mentioned that the useful energy of $\hat{\rho}_{B}(k)$
that can be extracted is given by the so-called ergotropy Allahverdyan :

$\mathcal{E}_{B}(k)=E(k)-\mathrm{Tr}_{B}[\hat{\sigma}_{B}(k)\hat{H}_{B}],$
(27)

where $\hat{H}_{B}=\varepsilon\hat{n}$ is the free Hamiltonian of the battery,
and $\hat{\sigma}_{B}(k)=\sum_{n}r_{n}|n\rangle\langle n|$, known as the
passive state Pusz ; Lenard , dependent on the eigenvalues of
$\hat{\rho}_{B}(k)$ that arranged in descending order (i.e., $r_{n}\leq
r_{n+1}$). In terms of the ergotropy, one can define another kind of the power

$\mathcal{P}=\frac{\mathcal{E}_{B}(k)}{kt}=g\varepsilon\left(\frac{\bar{n}_{k}-\mathrm{Tr}_{B}[\hat{\sigma}_{B}(k)\hat{n}]}{k\tau}\right).$
(28)

Note that $\mathcal{E}_{B}(k)\leq E(k)$ and hence $\mathcal{P}\leq P$, where
the equality holds for a pure state of $\hat{\rho}_{B}(k)$. As shown in Fig.
5(a) and (b), we show numerical results of $\mathcal{P}$ for $N_{A}=1$ and $4$
by taking fixed charging times $\tau=0.3$ (the open squares), $0.1$ (the
crosses), and $0.01$ (the circles). With the short-time case (i.e.,
$\tau=0.01$), one can see that the circles show a good agreement with the
analytical results of Eq. (22). To understand it, we calculate the purity of
the battery state $\varrho=\mathrm{Tr}_{B}[\hat{\rho}_{B}^{2}(k)]$, where
$\varrho=1$ for a pure state and $\varrho<1$ for a mixed state. The purity
$\varrho$ has also been investigated in the ultra-strong atom-field
interaction Shaghaghi  to show the pure state of the battery. As depicted in
Fig. 5(c) and (d), one can see $\varrho\approx 1$ for $\tau=0.01$, indicating
a nearly pure state of $\hat{\rho}_{B}(k)$, which in turn gives
$\mathcal{E}_{B}(k)\approx E(k)$ and hence $\mathcal{P}\approx P$. For a
larger charging time $\tau=0.3$, the battery state becomes more and more
mixed, leading to a departure of $\mathcal{P}$ from $P$.

##  V Discussion and Conclusion

We have generalized the repeated atom-battery interaction model (i.e., the so-
called collisional battery Seah ) from the spin-$1/2$ charger to the case of a
large spin $j=N_{A}/2$, where the battery is modeled by the energy ladder with
a finite number of levels $N_{B}+1$. Assuming little population over the
battery states $|0\rangle$ and $|N_{B}\rangle$ (corresponding to the no-
boundary problem), we first derive a recursion relation of the averaged
excitation that stored in the battery (see Eq. (6) and the Appendix A), which
is independent from any specific form of the atom state, and also free from
the initial state of $\hat{\rho}_{B}(0)$. Similar to the single-atom case
(i.e., the number of two-level atoms $N_{A}=1$) Seah , the incoherent and the
coherent charging protocols have been investigated by considering the atoms
prepared in a mixed state and a coherent spin state
$|\theta_{0},\phi_{0}\rangle$, respectively. For the coherent protocol, the
atomic coherence $\langle\hat{J}_{-}\rangle=j\sin(\theta_{0})\neq 0$, leading
to a reduced charging time $k_{\mathrm{est}}\tau\sim N_{B}/N_{A}$, where
$k_{\mathrm{est}}$ is the number of collisions that for the battery being
fully charged and $\tau$ is the charging time per collision.

Next, we focus on the coherent charging process over the initially empty
battery in the short-time limit (i.e., $\tau\rightarrow 0$). Analytical
results of the average number of excitations [see Eqs. (17) and (18)] have
been derived, which are related to the first-order Bessel function of the
first kind. Our results show that maximum of the charging power
$P_{\mathrm{coh,\max}}\approx 0.85g\varepsilon N_{A}$ can be obtained by using
the optimal coherent state $|\theta_{0},0\rangle$ with $\theta_{0}=\pi/2$.
With a fixed charging time $k\tau=60/N_{A}$, we calculate the power against
$\theta_{0}$ and $\tau$. For $\tau\ll 1$, the optimal state corresponds to
$\theta_{0}=\pi/2$. When the atom state is imperfect and $\theta_{0}<\pi/2$,
the power almost follows $P_{\mathrm{coh,\max}}$, provided that a relatively
larger value of $\tau\sim 1.17(1-2\theta_{0}/\pi)$ is adopted. As
$\theta_{0}\rightarrow 0$, the maximum power $P_{\mathrm{inc,\max}}\approx
0.72g\varepsilon N_{A}$ appears at $\tau=1.17$, coincident with the single-
atom case Seah . Finally, another kind of the charging power has been
investigated in terms of the so-called ergotropy. In the short-time limit, we
find that numerical results of the power show good agreement with its
analytical result, due to a nearly pure state of the battery in the early
charging steps (i.e., $k\lesssim 0.1k_{\mathrm{est}}$).

In summary, we have generalized the collision model to investigate coherent
charging of a single quantum battery by repeated interactions with finite
$N_{A}$ two-level atoms. Analytical results of the average number of
excitations and hence the charging power have been derived in the short-time
limit. Using an optimal coherent spin state with $\theta_{0}=\pi/2$, we obtain
the total charging time $k_{\mathrm{est}}\tau\sim N_{B}/N_{A}$ and the
achievable charging power $0.85g\varepsilon N_{A}$, where $N_{B}$ is the
number of the levels of the battery and $g\varepsilon N_{A}$ is the upper
bound of the charging power. The faster charging time and the increased amount
of the power in comparison with the incoherent charging protocol uniquely
arise from the phase coherence of the atoms. With a fixed charging time
$k\tau$, we investigate the optimal choices of the initial atom state
$\theta_{0}$ and the charging time per collision $\tau$. When $\theta_{0}$
largely departures from its optimal value $\pi/2$, the achievable charging
power can almost maintain by choosing a relative large value of $\tau\sim
1.17(1-2\theta_{0}/\pi)$. Finally, we show that the charging power defined by
the ergotropy almost follows its analytical result, since the purity of the
battery is almost equal to $1$, indicating a nearly pure state of the battery
in the short-time limit. The above results rely on the assumption that the
atom states are identical for all the collisions. Indeed, it is interesting to
investigate the dependence of the atom states on the charging steps (e.g., a
defect state randomly appeared at one of the steps).

###### Acknowledgements.

This work has been supported by the National Natural Science Foundation of
China (Nos. 12075209, 12005189, 11975205), and the Science Foundation of
Zhejiang Sci-Tech University (No. 18062145-Y).

##  Appendix A Details of Eqs. (4) and (6)

In the short-time limit, we have $\hat{U}_{\tau}\approx
1-i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})$ and therefore Eq.
(3) becomes

$\displaystyle\hat{\rho}_{B}(k)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{A}[\hat{U}_{\tau}\hat{\rho}_{B}(k-1)\otimes\rho_{A}\hat{U}_{\tau}^{{\dagger}}]$
$\displaystyle\approx$
$\displaystyle\hat{\rho}_{B}(k-1)+\left[(\alpha^{\ast}\hat{B}-\alpha\hat{B}^{{\dagger}})\hat{\rho}_{B}(k-1)+H.c.\right]$
$\displaystyle\approx$
$\displaystyle(1+\alpha^{\ast}\hat{B}-\alpha\hat{B}^{{\dagger}})\hat{\rho}_{B}(k-1)(1-\alpha^{\ast}\hat{B}+\alpha\hat{B}^{{\dagger}}),$

with $\alpha=i\tau\langle\hat{J}_{-}\rangle$ and
$\langle(\cdots)\rangle=\mathrm{Tr}_{A}[\hat{\rho}_{A}(\cdots)]$. Iterating
the above equation for $k$ times, we obtain

$\displaystyle\hat{\rho}_{B}(k)$ $\displaystyle\approx$
$\displaystyle(1+\alpha^{\ast}\hat{B}-\alpha\hat{B}^{{\dagger}})^{k}\rho_{B}(0)(1-\alpha^{\ast}\hat{B}+\alpha\hat{B}^{{\dagger}})^{k}$
(29) $\displaystyle\approx$
$\displaystyle\hat{D}^{{\dagger}}(k\alpha)\hat{\rho}_{B}(0)\hat{D}(k\alpha),$

as Eq. (4) in main text.

Next, we calculate the mean number of the excitations

$\displaystyle\bar{n}_{k}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{B}[\hat{\rho}_{B}(k)\hat{n}]$ (30) $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\hat{U}_{\tau}\hat{\rho}_{B}(k-1)\otimes\hat{\rho}_{A}\hat{U}_{\tau}^{{\dagger}}\hat{n}\right]$
$\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\hat{\rho}_{B}(k-1)\otimes\hat{\rho}_{A}\hat{U}_{\tau}^{{\dagger}}\hat{n}\hat{U}_{\tau}\right],$

where, in the second step, we have used Eq. (3) in main text, and
$\hat{n}=\sum_{n=0}^{N_{B}}n|n\rangle\langle n|$, satisfying

$[\hat{B},\hat{n}]=\hat{B},\text{\ \ \ \
}~{}[\hat{B}^{{\dagger}},\hat{n}]=-\hat{B^{{\dagger}}}.$ (31)

Therefore, one can expand the term
$\hat{U}_{\tau}^{{\dagger}}\hat{n}\hat{U}_{\tau}$ using the Baker-Campbell-
Hausdorff formula,

$\hat{U}_{\tau}^{{\dagger}}\hat{n}\hat{U}_{\tau}=e^{i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})}\hat{n}e^{-i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})}=\sum_{k=0}^{\infty}\frac{1}{k!}\hat{C}_{k},$
(32)

where
$\hat{C}_{k+1}=i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{C}_{k}]$.
Starting from $\hat{C}_{0}=\hat{n}$, we obtain

$\displaystyle\hat{C}_{1}$ $\displaystyle=$ $\displaystyle
i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{C}_{0}]$ (33)
$\displaystyle=$ $\displaystyle
i\tau(\hat{J}_{+}[\hat{B},\hat{n}]+\hat{J}_{-}[\hat{B}^{{\dagger}},\hat{n}])$
$\displaystyle=$ $\displaystyle
i\tau(\hat{J}_{+}\hat{B}-\hat{J}_{-}\hat{B}^{{\dagger}}),$

where we have used Eq. (31). Next, we obtain

$\displaystyle\hat{C}_{2}$ $\displaystyle=$ $\displaystyle
i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{C}_{1}]=2\tau^{2}[\hat{J}_{+}\hat{B},\hat{J}_{-}\hat{B}^{{\dagger}}]$
(34) $\displaystyle=$ $\displaystyle
2\tau^{2}(\hat{J}_{+}[\hat{B},\hat{J}_{-}\hat{B}^{{\dagger}}]+[\hat{J}_{+},\hat{J}_{-}\hat{B}^{{\dagger}}]\hat{B})\approx(2\tau)^{2}\hat{J}_{z},$

where we have used the commutation relation
$[\hat{J}_{+},\hat{J}_{-}]=2\hat{J}_{z}$, as well as
$[\hat{B},\hat{B}^{{\dagger}}]\approx 0$ and
$\hat{B}^{{\dagger}}\hat{B}=1-|0\rangle\langle 0|\approx 1$, valid for the no-
boundary condition (i.e., the occupations of $|0\rangle$ and $|N_{B}\rangle$
being vanishing). Similarly, we obtain

$\displaystyle\hat{C}_{3}$ $\displaystyle=$ $\displaystyle
i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{C}_{2}]=-4i\tau^{3}(\hat{J}_{+}\hat{B}-\hat{J}_{-}\hat{B}^{{\dagger}}),$
$\displaystyle\hat{C}_{4}$ $\displaystyle=$ $\displaystyle
i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{C}_{3}]\approx-\left(2\tau\right)^{4}\hat{J}_{z},$

and so on. Finally, one can easily obtain

$\displaystyle\hat{U}_{\tau}^{{\dagger}}\hat{n}\hat{U}_{\tau}$
$\displaystyle\approx$
$\displaystyle\hat{n}+\frac{\left(\hat{J}_{-}\hat{B}^{{\dagger}}-\hat{J}_{+}\hat{B}\right)}{2i}\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{\left(2k+1\right)!}\left(2\tau\right)^{2k+1}$
$\displaystyle+\hat{J}_{z}\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{\left(2k+2\right)!}\left(2\tau\right)^{2k+2}$
$\displaystyle=$
$\displaystyle\hat{n}+\sin\left(2\tau\right)\frac{\left(\hat{J}_{-}\hat{B}^{{\dagger}}-\hat{J}_{+}\hat{B}\right)}{2i}+2\sin^{2}\left(\tau\right)\hat{J}_{z}.$

Therefore, Eq. (30) becomes

$\bar{n}_{k}\approx\bar{n}_{k-1}+2\sin^{2}(\tau)\langle\hat{J}_{z}\rangle+\sin(2\tau)\mathrm{Im}\left(\langle\hat{J}_{-}\rangle\beta_{k-1}^{\ast}\right),$
(35)

as Eq. (6) in main text, where
$\beta_{k}=\mathrm{Tr}_{B}[\hat{\rho}_{B}(k)\hat{B}]$.

##  Appendix B Details of Eqs. (8) and (17)

First, we calculate $\beta_{k}$ for the battery state $\hat{\rho}_{B}(k)$
defined by Eq.(3) in main text,

$\displaystyle\beta_{k}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{B}[\hat{\rho}_{B}(k)\hat{B}]$ (36) $\displaystyle=$
$\displaystyle\mathrm{Tr}[\hat{U}_{\tau}\hat{\rho}_{B}(k-1)\otimes\hat{\rho}_{A}\hat{U}_{\tau}^{{\dagger}}\hat{B}]$
$\displaystyle=$
$\displaystyle\mathrm{Tr}[\hat{\rho}_{B}(k-1)\otimes\hat{\rho}_{A}\hat{U}_{\tau}^{{\dagger}}\hat{B}\hat{U}_{\tau}].$

Similar to Eq. (32), we deal with the term

$\hat{U}_{\tau}^{{\dagger}}\hat{B}\hat{U}_{\tau}=e^{i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})}\hat{B}e^{-i\tau(\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}})}=\sum_{k=0}^{\infty}\frac{1}{k!}\hat{D}_{k},$
(37)

where
$\hat{D}_{k+1}=i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{D}_{k}]$,
with $\hat{D}_{0}=\hat{B}$. In the short-time limit, the first-order expansion
is enough, i.e.,

$\hat{U}_{\tau}^{{\dagger}}\hat{B}\hat{U}_{\tau}\approx\hat{B}+\hat{D}_{1},$
(38)

where

$\displaystyle\hat{D}_{1}$ $\displaystyle=$ $\displaystyle
i\tau[\hat{J}_{+}\hat{B}+\hat{J}_{-}\hat{B}^{{\dagger}},\hat{D}_{0}]$ (39)
$\displaystyle=$ $\displaystyle
i\tau\hat{J}_{-}[\hat{B}^{{\dagger}},\hat{B}]=i\tau\hat{J}_{-}\left(|N_{B}\rangle\langle
N_{B}|-|0\rangle\langle 0|\right).$

Therefore, we obtain

$\displaystyle\beta_{k}$ $\displaystyle\approx$
$\displaystyle\mathrm{Tr}_{B}\left[\hat{\rho}_{B}(k-1)\hat{B}+\alpha\hat{\rho}_{B}(k-1)\hat{D}_{1}\right]$
(40) $\displaystyle=$
$\displaystyle\beta_{k-1}+\alpha\mathrm{Tr}_{B}\left[\hat{\rho}_{B}(k-1)\left(|N_{B}\rangle\langle
N_{B}|-|0\rangle\langle 0|\right)\right]$ $\displaystyle=$
$\displaystyle\beta_{k-1}+\alpha\left(\langle
N_{B}|\hat{\rho}_{B}(k-1)|N_{B}\rangle-\langle
0|\hat{\rho}_{B}(k-1)|0\rangle\right)$ $\displaystyle=$
$\displaystyle\beta_{0}+\sum_{k^{\prime}=0}^{k-1}\alpha\left(\langle
N_{B}|\hat{\rho}_{B}(k^{\prime})|N_{B}\rangle-\langle
0|\hat{\rho}_{B}(k^{\prime})|0\rangle\right),$

where the last second result has been iterated for $k$ times. With the no-
boundary condition, we simply obtain $\beta_{k}\approx\beta_{0}$; Neglecting
only the upper boundary $|N_{B}\rangle$, we obtain Eqs. (8) and (16) in main
text, where the lower boundary $|0\rangle$ has been taken into accounted.

Next, we analysis $\beta_{k}$ for an arbitrary state
$\hat{\rho}_{B}(k)=\sum_{i}p_{i}|\psi_{B}^{(i)}\rangle\langle\psi_{B}^{(i)}|$,
where $|\psi_{B}^{(i)}\rangle=\sum_{n=0}^{N_{B}}c_{n}^{(i)}|n\rangle$ and
$\sum_{i}p_{i}=1$, which gives the Cauchy-Schwartz inequality:

$\displaystyle|\beta_{k}|$ $\displaystyle=$
$\displaystyle\left|\sum_{n=0}^{N_{B}}\left\langle
n\right|\hat{\rho}_{B}(k)\hat{B}\left|n\right\rangle\right|\leq\sum_{i}p_{i}\left|\sum_{n=1}^{N_{B}}c_{n}^{(i)}c_{n-1}^{(i)\ast}\right|$
$\displaystyle\leq$
$\displaystyle\sum_{i}p_{i}\sqrt{\sum_{n=1}^{N_{B}}\left|c_{n}^{(i)}\right|^{2}\sum_{n^{\prime}=1}^{N_{B}}\left|c_{n^{\prime}-1}^{(i)}\right|^{2}}$
$\displaystyle=$
$\displaystyle\sum_{i}p_{i}\sqrt{\left(1-\left|c_{0}^{(i)}\right|^{2}\right)\left(1-\left|c_{N_{B}}^{(i)}\right|^{2}\right)}$
$\displaystyle\leq$ $\displaystyle 1.$

Note that the equality in the second step holds for
$c_{n}^{(i)}c_{n-1}^{(i)\ast}\in\mathbb{R}$. The following two equalities hold
for $c_{n}^{(i)}=c_{n-1}^{(i)\ast}$ and $c_{0}^{(i)}=c_{N_{B}}^{(i)}=0$,
respectively. Therefore, it is easy to obtain $|\beta_{k}|\in[0,1)$, which has
been used in Eq. (12).

Finally, we calculate Eq. (30) in the short-time limit to derive Eq. (17) in
main text. Using
$\hat{U}_{\tau}^{{\dagger}}\hat{n}\hat{U}_{\tau}\approx\hat{n}+\hat{C}_{1}=\hat{n}+i\tau(\hat{J}_{+}\hat{B}-\hat{J}_{-}\hat{B}^{{\dagger}})$,
we obtain

$\displaystyle\bar{n}_{k}$ $\displaystyle\approx$
$\displaystyle\mathrm{Tr}_{B}\left[\hat{\rho}_{B}(k-1)\hat{n}-\hat{\rho}_{B}(k-1)(\alpha\hat{B}^{{\dagger}}+\alpha^{\ast}\hat{B})\right]$
(41) $\displaystyle=$
$\displaystyle\bar{n}_{k-1}-(\alpha\beta_{k-1}^{\ast}+\alpha^{\ast}\beta_{k-1}),$

as Eq. (7) in main text. Iterating the above equation for $k$ times, we
further obtain

$\bar{n}_{k}\approx\bar{n}_{0}-\sum_{k^{\prime}=0}^{k-1}(\alpha\beta_{k^{\prime}}^{\ast}+\alpha^{\ast}\beta_{k^{\prime}}),$
(42)

where $\bar{n}_{0}=0$ for $\hat{\rho}_{B}(0)=|0\rangle\langle 0|$, and
$\beta_{k}$ depends on the terms $\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{l}|0\rangle$, as Eq. (14)
in main text. For odd $l$, e.g., $l=1$, it is easy to see $\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})|0\rangle=\alpha\langle
0|1\rangle=0$. As a result, we only consider even $l=2n$, e.g., $n=1$,

$\displaystyle\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{2}|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\left[(\alpha\hat{B}^{{\dagger}})^{2}+(-\alpha^{\ast}\hat{B})^{2}-|\alpha|^{2}\hat{B}^{{\dagger}}\hat{B}-|\alpha|^{2}\hat{B}\hat{B}^{{\dagger}}\right]|0\rangle$
$\displaystyle=$ $\displaystyle-|\alpha|^{2}\langle
0|\left[\hat{B}^{{\dagger}}\hat{B}+\hat{B}\hat{B}^{{\dagger}}\right]|0\rangle=-|\alpha|^{2}\langle
0|\hat{B}\hat{B}^{{\dagger}}|0\rangle$ $\displaystyle=$
$\displaystyle-|\alpha|^{2}.$

To obtain $\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{2n}|0\rangle\neq 0$, it
requires the numbers of $\hat{B}$ and $\hat{B}^{{\dagger}}$ are equal.
Furthermore, the ordering over the ladder operators $\hat{B}$ and
$\hat{B}^{{\dagger}}$ corresponds to the evolution paths from the “initial”
state $|0\rangle$ to the “final” state $|0\rangle$, as depicted by Fig. 3 (a),
where each path gives the same value $(-|\alpha|^{2})^{n}$ and the number of
all possible paths is given by the Catalan number
$C_{n}=\frac{1}{n+1}\binom{2n}{n}$. Therefore, we obtain

$\langle
0|(\alpha\hat{B}^{{\dagger}}-\alpha^{\ast}\hat{B})^{2n}|0\rangle=(-1)^{n}|\alpha|^{2n}C_{n}.$
(43)

Substituting it into Eq. (14), we obtain Eqs. (15) and (16) in main text, and
therefore

$\displaystyle\bar{n}_{k}$ $\displaystyle\approx$ $\displaystyle
2\sum_{k^{\prime}=0}^{k-1}\sum_{l=0}^{k^{\prime}-1}\left(\frac{J_{1}(2l|\alpha|)}{l}\right)^{2}=2\sum_{k^{\prime}=1}^{k-1}\sum_{l=0}^{k^{\prime}-1}\left(\frac{J_{1}(2l|\alpha|)}{l}\right)^{2}$
(44) $\displaystyle=$ $\displaystyle
2\sum_{l=0}^{k-2}\sum_{k^{\prime}=l+1}^{k-1}\left(\frac{J_{1}(2l|\alpha|)}{l}\right)^{2}$
$\displaystyle=$ $\displaystyle
2\sum_{l=0}^{k-2}(k-1-l)\left(\frac{J_{1}(2l|\alpha|)}{l}\right)^{2},$

where, in the last two steps, we have interchanged the order of summation
$\sum_{k^{\prime}=1}^{k-1}\sum_{l=0}^{k^{\prime}-1}(\cdots)=\sum_{l=0}^{k-2}\sum_{k^{\prime}=l+1}^{k-1}(\cdots)$,
and note the inner sum $\sum_{k^{\prime}=l+1}^{k-1}(\cdots)=(k-l-1)(\cdots)$.

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