# High-capacity and high-power collective charging with spin chargers

Yong Huangfu  Department of Physics, Zhejiang University, Hangzhou 310027,
Zhejiang, China  Jun Jing  Email address<EMAIL_ADDRESS>Department of
Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China

###### Abstract

Quantum battery works as a micro- or nano-device to store and redistribute
energy at the quantum level. Here we propose a spin-charger protocol, in which
the battery cells are charged by a finite number of spins through a general
Heisenberg XY interaction. Under the isotropic interaction, the spin-charger
protocol is endowed with a higher capacity in terms of the maximum stored
energy than the conventional protocols, where the battery is charged by a
continuous-variable system, e.g., a cavity mode. By tuning the charger size, a
trade-off between the maximum stored energy and the average charging power is
found in comparison to the cavity-charger protocol in the Tavis-Cummings
model. Quantum advantage of our protocol is manifested by the scaling behavior
of the optimal average power with respect to the battery size, in comparing
the collective charging scheme to its parallel counterpart. We also discuss
the detrimental effect on the charging performance from the anisotropic
interaction between the battery and the charger, the non-ideal initial states
for both of them, and the crosstalk among the charger spins. A strong charger-
charger interaction can be used to decouple the battery and the charger. Our
findings about the advantages of the spin-charger protocol over the
conventional cavity-charger protocols, including the high capacity of energy
storage and the superior power-law in the collective charging, provide an
insight to exploit an efficient quantum battery based on the spin-spin-
environment model.

##  I Introduction

Substantial effort in recent years has been devoted to promote the performance
of the microscopic battery in the quantum regime, which plays an analog role
of storing and extracting useful energy as its classical counterpart. Quantum
battery is not only a scientifically interesting subject, but is a practical
need. Devising a quantum battery in various forms has constituted an emerging
field of nano-devices for quantum thermodynamics, such as solid-state
batteries Aykol  _et al._ (2019); Swift and Qi (2019), topological batteries
Liu  _et al._ (2018); Qie  _et al._ (2018); Tserkovnyak and Xiao (2018); Patil
_et al._ (2020), and spin batteries Brataas  _et al._ (2002); Xie  _et al._
(2018); Caravelli  _et al._ (2020). The contemporary development of quantum
battery also inspires the relevant investigations in quantum information about
the work extraction Frenzel  _et al._ (2014) and the optimal finite-time
operation Gallego  _et al._ (2016); Anders and Esposito (2017); Julià-Farré
_et al._ (2020).

The working procedure in quantum battery can be divided into three ordered
steps: charging Ferraro  _et al._ (2018); Andolina  _et al._ (2018); Crescente
_et al._ (2020); Rossini  _et al._ (2020); Le  _et al._ (2018); Rossini  _et
al._ (2019); Andolina  _et al._ (2019a); Chetcuti  _et al._ (2020); Binder
_et al._ (2015); Kamin  _et al._ (2020); Campaioli  _et al._ (2017); Julià-
Farré  _et al._ (2020); Tabesh  _et al._ (2020), energy storage Gherardini
_et al._ (2020); Santos  _et al._ (2019, 2020); Liu  _et al._ (2019); Bai and
An (2020), and discharging Allahverdyan  _et al._ (2004); Francica  _et al._
(2017); Alicki and Fannes (2013); Andolina  _et al._ (2019b); Çakmak (2020).
The charger in convention is equipped by a continuous-variable system. The
conventional charging protocol starts from the Jaynes-Cummings model Andolina
_et al._ (2018), where the two-level system and the cavity mode serve as the
battery cell and the charger system, respectively. The exact solution about
the charging performance, including the maximum stored energy and optimal
average charging power, provides a benchmark for the subsequent studies of the
charging protocols assisted by a cavity mode. When the battery is extended to
adopt a collection of two-level systems, one can have quantum batteries based
on the Tavis-Cummings (TC) model Andolina  _et al._ (2019b) and the Dicke
model Ferraro  _et al._ (2018); Andolina  _et al._ (2018); Crescente  _et al._
(2020); Rossini  _et al._ (2020). When the charger becomes semi-classical, one
can have many-spin batteries charged by a driving field Le  _et al._ (2018);
Rossini  _et al._ (2019); Andolina  _et al._ (2019a); Chetcuti  _et al._
(2020). Powerful charging protocols were reported by virtue of the two-photon
process Crescente  _et al._ (2020), the global operations Binder  _et al._
(2015), and the non-Markovian effects Kamin  _et al._ (2020). Many
technologies were applied to a capable quantum battery for energy storage,
such as the quantum measurement on Zeno protection Gherardini  _et al._
(2020), the stimulated Raman adiabatic passage technique Santos  _et al._
(2019, 2020), the symmetry protected dark state living in a decoherence-free
subspace Liu  _et al._ (2019), and the Floquet engineering with two bound
states in the quasienergy spectrum Bai and An (2020). To promote the work
extraction, the amount of available energy from the active state to the
passive state is studied in the discharging process Francica  _et al._ (2017).
An observation is that the entangling unitary controls extract in general more
work than the separable controls Alicki and Fannes (2013).

Among the studies of quantum battery in the collective scheme, quantum
advantage is significantly reflected by the scaling behavior of the charging
performance, especially of the charging power. The quantum spin battery in the
Lipkin-Meshkov-Glick model Julià-Farré  _et al._ (2020) shows that quantum
entanglement enhances the charging power at the cost of the charging speed. In
the Sachdev-Ye-Kitaev model Rossini  _et al._ (2020), the power enhancement is
linked to the variance of the charging Hamiltonian. In contrast to the
parallel scheme in which each battery cell is individually charged, all cells
in the collective scheme are charged through the global operations on the
overall battery-charger system. The collective effect can boost the charging
power of the quantum battery in terms of the scaling law of the number of
battery cells $M$ Campaioli  _et al._ (2017). The Dicke quantum battery gains
a collective advantage with a $M^{1/2}$ scaling law Ferraro  _et al._ (2018).
It is soon followed by a strict comparison between the quantum battery and its
classical counterpart Andolina  _et al._ (2019a) to display the quantum
collective advantage, in which the Dicke quantum battery holds a much better
performance in the charging power.

In this work, we consider a closed quantum battery-charger system and focus
only on the charging process via unitary, deterministic, and reversible
evolutions. The discharging dynamics is then exactly the reverse process to
the charging dynamics, i.e., the energy stored in the battery can be
completely extracted in a reverse way. Rather than the conventional charging
protocols assisted by a cavity mode, we study the charging performance in a
protocol based on a spin-spin-environment model. In particular, the battery
cells are coupled collectively to a finite number of non-interacting spin
chargers through a general Heisenberg XY interaction. This model was
significant for the central spins would hold coherence in a comparatively long
time Kurucz  _et al._ (2009); Zadrozny  _et al._ (2015); Abobeih  _et al._
(2018) under a spin environment. It constitutes an important platform to
investigate the entanglement dynamics Xu  _et al._ (2011); Yang  _et al._
(2016); Jing and Wu (2018) and the quantum control strategy Wu (2010). In this
spin-charger protocol, it is interesting to find a trade-off between the
maximum stored energy and the optimal average charging power by tuning the
charger size and the collective advantage represented by the superior power-
law scaling with respect to the battery size.

The rest of this work is organized as following. In Sec. II, we present the
full Hamiltonian for the quantum spin batteries coupled to the charger spins.
The charging performance is analysed under various anisotropic coefficient
$\gamma$ and the ratios of the numbers of charger spins $N$ and battery cells
$M$. All the results are compared with the conventional cavity-charging
protocol based on TC model, that serves as a benchmark in the thermodynamical
limit of our model. The dynamics of two spin batteries under the isotropic
charging is provided in appendix A, where the maximum stored energy is found
to be consistent with the benchmark result. In Sec. III, we investigate the
collective advantage of our spin-charger protocol against its parallel
counterpart. In Sec. IV, we discuss the spin-charging protocol in the presence
of either the charger-spin crosstalk or the non-ideal initial states. In Sec.
V, we summarize the whole work.

##  II Model of quantum battery under spin-chargers

Figure 1: (Color online) (a) The quantum battery is consisted of $M$
spin-$1/2$ cells. (b) The charging protocol is based on the spin-spin-
environment model where $M$ quantum battery cells are coupled collectively to
$N$ non-interacting charger spins.

In the quantum battery shown in Fig. 1, we consider $M$ spin-$1/2$ cells
charged collectively by $N$ non-interacting charger spins-$1/2$. The overall
Hamiltonian driving the charging process can be written as ($\hbar\equiv 1$)

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{1},$ (1)
$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle H_{\rm B}+H_{\rm
C}=\frac{\omega_{0}}{2}\left(\sum^{M}_{k=1}\sigma^{z}_{k}+\sum^{N}_{l=1}\tau^{z}_{l}\right),$
$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\frac{g}{2}\left[(1+\gamma)\sum^{M}_{k=1}\sigma^{x}_{k}\sum^{N}_{l=1}\tau^{x}_{l}+(1-\gamma)\sum^{M}_{k=1}\sigma^{y}_{k}\sum^{N}_{l=1}\tau^{y}_{l}\right],$

where $H_{\rm B}$, $H_{\rm C}$ and $H_{1}$ are the battery Hamiltonian, the
charger Hamiltonian, and the battery-charger interaction Hamiltonian,
respectively. The battery-spin and the charger-spin operators are denoted by
the Pauli matrices $\sigma^{\alpha}_{k}$ and $\tau^{\alpha}_{l}$ with
$\alpha\in\\{x,y,z\\}$, respectively. By convention, the battery cells are set
to be resonant with the charger spins with frequency $\omega_{0}$. In the
interaction Hamiltonian, $g$ is the coupling strength between batteries and
chargers and $\gamma$, $0\leq\gamma\leq 1$, is the anisotropic coefficient for
coupling. When $\gamma=0$, the battery-charger interaction reduces to an
isotropic XY interaction. When $\gamma=1$, it becomes an XX interaction with
equal-weighted counter-rotating terms. The same model with the general
Heisenberg XY interaction was applied to study the disentanglement dynamics of
the central spins Yuan  _et al._ (2007); Jing and Lü (2007); Jing  _et al._
(2007).

Using the collective angular momentum operators for the charger spins
$J_{z,\pm}\equiv\sum^{N}_{l=1}\tau^{z,\pm}_{l}$ with
$\tau^{\pm}_{k}\equiv(\tau^{x}_{k}\pm i\tau^{y}_{k})/2$, the charger
Hamiltonian can thus be rewritten as

$H_{\rm C}=\frac{\omega_{0}}{2}J_{z}.$ (2)

And the interaction Hamiltonian becomes

$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
g\left[\sum^{M}_{k=1}\left(\gamma\sigma^{-}_{k}+\sigma^{+}_{k}\right)\sum^{N}_{l=1}\tau^{-}_{l}+{\rm
H.c.}\right]$ (3) $\displaystyle=$ $\displaystyle
g\left[J_{-}\sum^{M}_{k=1}\left(\gamma\sigma^{-}_{k}+\sigma^{+}_{k}\right)+{\rm
H.c.}\right],$

which is valid due to the indistinguishability of all of the charger spins. To
study the charging performance, the quantum batteries and the spin chargers
are prepared as the spin-down (empty or passive) and spin-up (full or active)
states, respectively. Then the overall wavefunction reads,

$|\Psi(0)\rangle=\prod^{M}_{k=1}|0_{k}\rangle\otimes\prod^{N}_{l=1}|1_{l}\rangle=\prod^{M}_{k=1}|0_{k}\rangle\otimes|m=\frac{N}{2}\rangle.$
(4)

Note the eigenstates of the spin charger are denoted by $|m\rangle$’s, where
$m$’s are the eigenvalues running from $-N/2$ to $N/2$ due to Eq. (2).
$|m=N/2\rangle$ implies that the spin chargers are all in the excited state.

In the thermodynamical limit $N\rightarrow\infty$, the preceding Hamiltonian
can be further simplified by the Holstein-Primakoff (HP) transformation
Holstein and Primakoff (1940):

$J_{+}=b^{\dagger}\left(\sqrt{N-b^{\dagger}b}\right),\quad
J_{-}=\left(\sqrt{N-b^{\dagger}b}\right)b.$ (5)

It is important to remark that here the bosonic operators $b^{\dagger}$ and
$b$ do not directly correspond to the angular momentum operators $J_{+}$ and
$J_{-}$, respectively. According to Eq. (4), the initial state of the charger
spins is a completely polarized state $|m=N/2\rangle$, i.e., the fully-up
state, which is symmetric to another completely polarized state
$|m=-N/2\rangle$, i.e., the fully-down state. To describe a high-spin system
with a bosonic mode, it is often appropriate to use the HP transformation when
the state of the high-spin system does not significantly deviate from a
benchmark state, that could be either one of the completely polarized state.

Consequently, the interaction Hamiltonian in Eq. (3) is revised to be a spin-
boson Hamiltonian:

$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
g\left[\left(\sqrt{N-b^{\dagger}b}\right)b\sum^{M}_{k=1}\left(\gamma\sigma^{-}_{k}+\sigma^{+}_{k}\right)+{\rm
H.c.}\right]$ (6) $\displaystyle=$
$\displaystyle\tilde{g}\left[b\sum^{M}_{k=1}\left(\gamma\sigma^{-}_{k}+\sigma^{+}_{k}\right)+{\rm
H.c.}\right],$

where the coupling strength $\tilde{g}\approx\sqrt{N}g$ in the thermodynamical
limit. When $\gamma=0$, the transformed interaction Hamiltonian in Eq. (6)
describes the Tavis-Cummings (TC) model Tavis and Cummings (1968, 1969). It is
then used to provide a benchmark for our model in the limit of
$N\rightarrow\infty$, while our spin-charger model focuses on the situation
with a finite $N$ and most of our results are obtained by Eq. (3). When
$\gamma=1$, Eq. (6) describes the Dicke model Dicke (1954) as a platform for
superradiance. It can also be applied to a high-power charging protocol for
quantum battery Ferraro  _et al._ (2018).

With the initial state in Eq. (4), the energy stored in the battery during the
charging process is given by

$E(t)\equiv{\rm Tr}[\rho_{\rm B}(t)H_{\rm B}]-{\rm Tr}[\rho_{\rm B}(0)H_{\rm
B}],$ (7)

where $\rho_{\rm B}(t)\equiv{\rm Tr}_{\rm C}[|\Psi(t)\rangle\langle\Psi(t)|]$.
The average charging power is given by

$P(t)\equiv\frac{E(t)}{t}.$ (8)

The charging performance of a quantum battery can be measured respectively by
$E_{\rm max}$ and $P_{\rm max}$ as the maximum value for both $E(t)$ and
$P(t)$. In appendix A, we provide the dynamics of both $E(t)$ and $P(t)$ as
well as their optimized values for a special model in Eq. (1) with $M=2$ and
$\gamma=0$.

Figure 2: (Color online) The landscape of the maximum stored energy $E_{\rm
max}$ and the optimized average power $P_{\rm max}$ under the spin-charging
protocol with isotropic interaction $\gamma=0$. Results of $N/M=1$, $N/M=2$,
$N/M=10$, and $N/M=100$, are marked with orange diamonds, green squares, black
circles and blue crosses, respectively. The red dot-dashed line indicates the
benchmark result by the TC model, a conventional cavity-charging protocol. The
battery-charger coupling strength is set as $g/\omega_{0}=0.1$.

To distinguish the spin-charging protocol from the conventional cavity-
charging protocol, we first present a landscape about the maximum stored
energy $E_{\rm max}$ and the optimized average power $P_{\rm max}$ in Fig. 2
under the isotropic charging for various ratios of the charger-spin number $N$
and the battery-cell number $M$. The benchmark results of $E_{\rm max}$ and
$P_{\rm max}$ in the TC model are displayed by the red dot-dashed line with
$N/M=1$, where $N$ denotes the initial photon-number in the cavity mode. The
average charging power $P_{\rm max}$ is normalized in the unit of
$\sqrt{N}g\omega_{0}$ to be consistent with the rescaled coupling strength
$\tilde{g}$ in the thermodynamical limit Julià-Farré  _et al._ (2020). It is
found that the results in the spin-charging protocol are coincident with the
benchmark results, when the ratio becomes sufficiently large $N/M\geq 10$.
This asymptotic behavior can be understood by virtue of the HP transformation,
which maps the collective-spin ladder operators into the bosonic creation and
annihilation operators. Then with a sufficiently large $N$, a spin charger
becomes close to a cavity charger.

In Fig. 2, our spin-charger protocol departs significantly from the TC model
under a small ratio $N/M$. When $N/M=2$ (see the green squares), the maximum
stored energy $E_{\rm max}$ becomes gradually larger than the corresponding TC
result with an increasing $M$; while the optimized average powers $P_{\rm
max}$ are nearly invariant. When $N/M=1$ (see the orange diamonds), a clear
trade-off between $E_{\rm max}$ and $P_{\rm max}$ can be observed. In
comparison to the TC model, a much higher stored energy can be achieved at the
cost of a little lower optimized charing power when $M\geq 3$. And a larger
size of the spin-charger yields a larger $P_{\rm max}$ and a smaller $E_{\rm
max}$ under a fixed battery size.

Figure 3: (Color online) The landscape of $E_{\rm max}$ and $P_{\rm max}$
under the spin-charging protocol with various ratios of $N/M$: (a) $N/M=1$,
(b) $N/M=2$, (c) $N/M=3$, (d) $N/M=10$. Results under the anisotropic
coefficient $\gamma=0$, $0.2$, $0.6$, and $1$, are marked with orange
diamonds, green squares, black circles and blue crosses, respectively.
$g/\omega_{0}=0.1$.

Next we consider the impact of the anisotropic charging on the landscape of
$E_{\rm max}$ and $P_{\rm max}$. In Fig. 3, again we collect the data for
$M=1,2,\cdots,10$ with various ratios $N/M$. It is observed that all the data
follow almost the same linear pattern irrespective to the value of $N/M$.
Across all the four sub-figures, $E_{\rm max}$ is found to be proportional to
$P_{\rm max}$ with the slope coefficients gradually enhanced from $1.002$ for
$N/M=1$ to $1.307$ for $N/M=10$. A smaller anisotropic coefficient $\gamma$
yields a higher energy and a higher power. For example, in Fig. 3(a) with
$M=10$, when $\gamma=1$, $E_{\rm max}/\omega_{0}\approx 7.48$ and $P_{\rm
max}/(\sqrt{N}g\omega_{0})\approx 6.67$, and when $\gamma=0$, $E_{\rm
max}/\omega_{0}\approx 9.76$ and $P_{\rm max}/(\sqrt{N}g\omega_{0})\approx
9.24$. The results of a strong anisotropic interaction for $\gamma=1$ will
gradually deviate from the main line of $E_{\rm max}$ and $P_{\rm max}$ with
an increasing $N/M$. In Fig. 3(d), it is clear that both $E_{\rm max}$ and
$P_{\rm max}$ with $\gamma=1$ are considerably smaller than those with
$\gamma=0$.

The anisotropic charging is associated mainly with the counter-rotating terms
of the interaction Hamiltonian in Eq. (3). During a unitary charging process,
the energy is coherently transferred from the charger spins to the battery
cells. Given the initial states in Eq. (4), it is then preferable to promote
the charging performance if the (empty) state of the battery can be exchanged
with the (full) state of the charger. Note the counter-rotating terms cannot
hold the energy conservation all the way. The charging dynamics is then mixed
with the extra energy-generation or energy-annihilation process in the overall
battery-charger system due to the non-commutativity between the interaction
Hamiltonian $H_{1}$ in Eq. (3) with a nonvanishing $\gamma$ and the free
Hamiltonian $H_{0}$ in Eq. (1). The isotropic charging is therefore more
favorable than the anisotropic charging. This result is in agreement with Ref.
Andolina  _et al._ (2018) about the negative effects of the counter-rotating
terms on the charging process.

Figure 4: (Color online) (a) $E_{\rm max}$ and (b) $P_{\rm max}$ as functions
of the battery-cell number $M$ with various anisotropic coefficient. The
results under $\gamma=0,0.2,0.6,1.0$ are plotted with the orange-dashed line,
the green-dotted line, the black dot-dashed line, and the blue-solid line,
respectively. The red circles indicate the benchmark result provided by the TC
model. Here $N/M=1$ and $g/\omega_{0}=0.1$.

The dependence on the battery size of the charging performance, including the
maximum stored energy $E_{\rm max}$ and the optimal average power $P_{\rm
max}$, is shown in Fig. 4. For simplicity, here the battery and the charger
are equivalent in their sizes, i.e., $N/M=1$. The cavity-charging protocol
based on the TC model is again compared to the spin-charger protocol. As
expected in Fig. 2 up to $M=10$, it is observed that the spin-charger protocol
under the isotropic battery-charger interaction $\gamma=0$ surpasses the
cavity-charging protocol in terms of $E_{\rm max}$, and more advantage can be
distinguished by an even larger $M$. In contrast, the spin-charger protocol
under $\gamma=0$ is slightly defeated by the benchmark protocol in terms of
$P_{\rm max}$. In both Figs. 4(a) and (b), the anisotropic interaction
degrades the charging performance, especially the charging power, in a
monotonic way with increasing $\gamma$. In Fig. 4(a), it is found that both
spin-charger and cavity-charger protocols attain almost the same $E_{\rm max}$
when $\gamma=0.2$.

##  III Quantum advantage of collective spin charging

Aside from the collective charging scheme studied in the previous section, an
alternative option to implement the energy transfer for a many-cell battery is
the parallel charging Campaioli  _et al._ (2017), by which each battery cell
is separably charged by an individual charger. In our spin-charging model, the
interaction Hamiltonian for the parallel scheme can be written as

$H_{\parallel}=\sum^{M}_{k=1}h_{k}\equiv\sum^{M}_{k=1}g\left[J^{(k)}_{-}\left(\gamma\sigma^{-}_{k}+\sigma^{+}_{k}\right)+{\rm
H.c.}\right],$ (9)

where $h_{k}$ indicates the $k$th quantum cell in parallel. Both the overall
maximum stored energy $E^{\parallel}_{\rm max}=Me_{\rm max}$ and the overall
optimal average power $P^{\parallel}_{\rm max}=Mp_{\rm max}$ grow linearly
with the number of battery cells $M$, where $e_{\rm max}$ and $p_{\rm max}$
represent the maximum stored energy and the optimal average power in a single
battery cell, respectively. We can use the following ratio Andolina  _et al._
(2019a)

$\eta\equiv\frac{P_{\rm max}}{P^{\parallel}_{\rm max}}$ (10)

to describe the quantum advantage of the collective charging scheme over the
parallel counterpart, if $\eta>1$.

Figure 5: (Color online) The ratio of the charging power under the collective-
charging scheme and that under the parallel-charging scheme $\eta$ as a
function of $M$. The results under $\gamma=0,0.2,0.6,1.0$ are plotted with the
orange-dashed line, the green-dotted line, the black dot-dashed line and the
blue-solid line, respectively. The red circles indicate the result of the TC
model. $N/M=1$ and $g/\omega_{0}=0.1$.

The collective ratio $\eta$ as a function of $M$ under various anisotropic
coefficients $\gamma$ are displayed in Fig. 5 with $N/M=1$. It is noticeable
that $\eta$ for the spin-charger protocol is remarkably larger than that for
the cavity-charging protocol. In magnitude, the collective ratio $\eta$
roughly follows a power law of $M$:

$\eta\propto M^{\beta}.$ (11)

The exponent index $\beta$ is found to gradually approach an asymptotic value
by increasing $M$, which is near to unit when $\gamma=0$. In particular, when
$M=20$, $\beta\approx 0.81$; when $M=100$, $\beta\approx 0.87$; and when
$M=1000$, $\beta\approx 0.99$. In Fig. 5, it is also observed that the index
$\beta$ for $M=100$ decreases with an increasing $\gamma$. In particular, when
$\gamma=0,0.2,0.6,1.0$, $\beta\approx 0.87,0.84,0.79,0.48$, respectively. The
result under $\gamma=0$ is consistent with a previous spin-charger protocol
Andolina  _et al._ (2019a) with a large $M$. Note it was found that
$\beta=0.5$ for $M\gg 1$ in the Dicke quantum battery Ferraro  _et al._ (2018)
charged by a cavity mode, which is verified by the red circles plotted in Fig.
5 under the TC model. Then our spin-charger protocol demonstrates a superior
pow-law advantage than the cavity-charging protocols. The collective advantage
could be understood by the effect from the global operations Binder  _et al._
(2015). In particular, the interaction between the array of quantum battery
cells and the common chargers creates a quantum correlation giving rise to the
advantage of the charging power over the parallel battery.

Additionally, the performance of the parallel charging does not depend on
$\gamma$ as the collective charging when $N=M$. It is found to be completely
determined by the single-cell result. Consider one cell charged by one spin-
charger, which is ideally prepared as $|0_{\rm B}\rangle\otimes|1_{\rm
C}\rangle$. In the space supported by $\\{|1_{\rm B},1_{\rm C}\rangle$,
$|1_{\rm B},0_{\rm C}\rangle$, $|0_{\rm B},1_{\rm C}\rangle$, and $|0_{\rm
B},0_{\rm C}\rangle\\}$, the total Hamiltonian can be written as

$\displaystyle h_{1}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}}{2}(\sigma_{z}+\tau_{z})+g[\tau_{-}(\gamma\sigma_{-}+\sigma_{+})+{\rm
H.c.}]$ (16) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\omega_{0}&0&0&g\gamma\\\ 0&0&g&0\\\
0&g&0&0\\\ g\gamma&0&0&-\omega_{0}\\\ \end{array}\right).$

The induced charging dynamics is localized in the subspace spanned by $|1_{\rm
B},0_{\rm C}\rangle$ and $|0_{\rm B},1_{\rm C}\rangle$, irrelevant to the
anisotropic coefficient $\gamma$. And it is straightforward to find that the
stored energy $e(t)=\omega_{0}\sin^{2}(gt)$ and the average charging power
$p(t)=\omega_{0}[\sin^{2}(gt)]/t$ for a single battery cell. Subsequently, the
renormalized charging performance is given by $e_{\rm max}/\omega_{0}=1$ and
$p_{\rm max}/g\omega_{0}=0.72$.

##  IV Non-ideal charging

Regarding the systematic errors, it is physically relevant to consider the
charging performance with respect to the inner-coupling among spin chargers
and the non-ideal initial states for both battery and charger. The former is
associated with the situation where the charger spins are mutually coupled to
each other with Heisenberg XY interaction, that would be inevitable due to the
crosstalk among neighboring chargers. The latter might be induced by a finite
fidelity in the state initialization of the battery or the charger. In this
section, we only consider $\gamma=0$, i.e., the situation for the isotropic
Heisenberg XY interaction. Then the interaction Hamiltonian reads,

$H_{1}=\frac{g}{2}\left(\sum^{M}_{k=1}\sigma^{x}_{k}\sum^{N}_{l=1}\tau^{x}_{l}+\sum^{M}_{k=1}\sigma^{y}_{k}\sum^{N}_{l=1}\tau^{y}_{l}\right),$
(17)

which also serves as the main Hamiltonian in the rotating frame with respect
to $H_{0}$. We choose a small size of quantum battery with $M=2$ and $N=4$,
unless otherwise stated. One can compare directly the numerical results under
these non-ideal situations with the analytical solution given in appendix A.
Also, it is numerically found that with a fixed set of parameters, these non-
ideal impacts on our quantum battery with spin charger decline gradually with
the system size.

###  IV.1 The spin-charging protocol with inner-coupled chargers

This subsection is contributed to the consequence from the crosstalk between
charger spins. To simplify the discussion, the extra XY interactions among the
charger spins are all the same in the coupling strength. A similar
configuration for the spin-star network has been used to study the non-
Markovian quantum dynamics of the central spins Breuer (2004); Yuan  _et al._
(2007). The Hamiltonian for the spin chargers in the rotating frame with
respect to $H_{0}$ can then be written as

$H_{\rm C}=\frac{g_{1}}{2}\sum^{N}_{l\neq
j}\left(\tau^{x}_{l}\tau^{x}_{j}+\tau^{y}_{l}\tau^{y}_{j}\right),$ (18)

where $g_{1}$ is the charger-charger coupling strength. Using the collective
angular momentum operators, we have

$H_{\rm C}=g_{1}(J_{+}J_{-}+J_{-}J_{+}-\mathcal{I}),$ (19)

where $\mathcal{I}$ represents the identity operator, compensating the
interaction terms with $l=j$ in Eq. (18). Again the charging performance under
the full Hamiltonian $H=H_{1}+H_{\rm C}$ is measured by the stored energy
$E(t)$ the average charging power $P(t)$.

Figure 6: (Color online) (a) The stored energy $E(t)$ and (b) the average
charging power $P(t)$ against the dimensionless time $gt$ under various ratio
of the charger-charger coupling strength and the battery-battery coupling
strength $g_{1}/g$. Results for $g_{1}/g=0.1$, $g_{1}/g=1$, and $g_{1}/g=10$
are plotted with the solid, the dot-dashed, and the dotted lines,
respectively.

In Fig. 6, we plot $E(t)$ and $P(t)$ with various $g_{1}/g$. The renormalized
maximum stored energy is significantly reduced by increasing the charger-
charger coupling. According to Eq. (32), it is found that $E_{\rm
max}/\omega_{0}=1.92$ with $M=2$, $N=4$ and $g_{1}/g=0$. When $g_{1}/g=0.1$,
it is observed that $E_{\rm max}/\omega_{0}=1.91$ [see the solid line in Fig.
6(a)]. When $g_{1}/g=1$ (see the dot-dashed line), $E_{\rm
max}/\omega_{0}=0.85$. The numerical simulation shows that when
$g_{1}/g\approx 7$, $E_{\rm max}$ declines to about one percent of the energy
for a full-charged state. And when $g_{1}/g=10$ (see the dotted line), the
battery will completely lose its capacity to store energy. Similar pattern
manifests in the dynamics of the average charging power $P(t)$. In contrast to
the performance of an ideal battery with $M=2$ and $N=4$, i.e., $P_{\rm
max}/(g\omega_{0})\approx 3.04$, that can be numerically obtained by Eq. (31),
a weaker charger-charger coupling $g_{1}/g=0.1$ renders almost the same result
$P_{\rm max}/(g\omega_{0})\approx 3.02$ [see the solid line in Fig. 6(b)]; but
a stronger coupling $g_{1}/g=1$ renders a remarkably lower result $P_{\rm
max}/(g\omega_{0})\approx 1.83$.

It is clear that the charger-charger interaction will damage the energy
transfer from the full-charged spins to the empty-charged spins. The extra
interaction of $H_{\rm C}$ in Eq. (18) does not violate the energy
conservation of the full battery-charger system. However, it will allow the
energy to exchange among charger spins. That will affect the energy
distribution of the whole system due to the comparative weights between
$H_{1}$ in Eq. (17) and $H_{\rm C}$ in Eq. (18). The deviation from the ideal
situation can be estimated by the ratio $g_{1}/g$. As $g_{1}$ grows to
overwhelm $g$, the battery cannot obtain any energy from the charger, but
stays at the initial state [see the dotted lines in Figs. 6(a) and (b)]. Thus
the strong inner-coupling among charger spins will effectively decouple the
battery and the charger.

One can also consider the crosstalk among the battery cells instead of that
among the charger spins by exchanging these two spin-systems. Then the extra
interaction of $H_{\rm C}$ in Eq. (18) is replaced by $H_{\rm
B}=(g_{2}/2)\sum_{k\neq
m}^{M}(\sigma_{k}^{x}\sigma_{m}^{x}+\sigma_{k}^{y}\sigma_{m}^{y})\simeq
g_{2}(J_{+}J_{-}+J_{-}J_{+})$, where $g_{2}$ denotes the inner-coupling
strength among battery cells and $J_{\pm}$ is now understood as
$\sum_{k=1}^{M}\sigma_{k}^{\pm}$. Then through a parallel analysis as in Fig.
6, one can anticipate that the charge performance will decline in the presence
of a weak $g_{2}$ due to the energy redistribution in the quantum battery. A
sufficiently strong $g_{2}$ will have the same effect as a dominant $g_{1}$ by
effectively separating the two spin-systems, since now $H_{1}$ becomes
ignorable. In addition, the inner-coupling energy by $H_{\rm B}$ will be
transferred to the quantum battery, given the initial empty-state for the
battery cells. This “self-charging” phenomenon is however beyond our
consideration.

###  IV.2 The spin-charging protocol with non-ideal initial states

The spin-charging protocol starts with the full-charged chargers and the
empty-charged batteries in the ideal situation. In this subsection, we discuss
the impacts from the charger prepared at the partially-charged state and the
battery with partially-charged cells, respectively.

It is heuristic to first consider one of the charger spins in a thermal-
equilibrium state. Take a model in which two cells are charged by two charger
spins as an example. Assume the $2$nd charger spin is in a thermal state as

$\rho^{(2)}_{\rm C}(t=0)=\left(\begin{array}[]{cc}p_{1}&0\\\ 0&p_{0}\\\
\end{array}\right),$ (20)

where $p_{1}$ and $p_{0}$ determine the effective (positive or negative)
temperature and $p_{1}+p_{0}=1$. Thus the initial state of the whole system
can be written as $\rho_{\rm BC}(0)=|00\rangle\langle
00|\otimes(p_{1}|11\rangle\langle 11|+p_{0}|10\rangle\langle 10|)$. The time
evolution of the stored energy for the battery is given by

$\displaystyle E(t)$ $\displaystyle=$ $\displaystyle
2\omega_{0}\sin^{2}\left(\sqrt{2}gt\right)$ (21) $\displaystyle-
p_{0}\omega_{0}\left[2\sin^{2}\left(\sqrt{2}gt\right)-\sin^{2}(2gt)\right].$

It reduced to Eq. (31) when $p_{0}=0$. Clearly, the full charging cannot be
realized in the presence of a nonvanishing $p_{0}$.

Generally the initial-state preparation for both battery and charger is a
challenge in experiments Yang  _et al._ (2016). Regarding the non-ideal spin
chargers, one can write the wavefunction of the overall system as

$|\Psi^{\rm
C}_{l}\rangle=\prod_{k}|0_{k}\rangle\otimes|m=-\frac{N}{2}+l-1\rangle,$ (22)

which means $l-1$ charger spins are initially at their excited level, $1\leq
l\leq N$. Thus in general, the overall density matrix can be written as

$\rho_{\rm
BC}=p_{0}|\Psi(0)\rangle\langle\Psi(0)|+\sum^{N}_{l=1}p_{l}|\Psi^{\rm
C}_{l}\rangle\langle\Psi^{C}_{l}|,$ (23)

where $|\Psi(0)\rangle$ is the ideal initial state given in Eq. (4) and
$\sum^{N}_{l=0}p_{l}=1$, $p_{l}\geq 0$. Note Eq. (23) can be used to roughly
estimate the effect from the charger spins in a mixed state, such as the
thermal state in Eq. (20).

Regarding the non-ideal quantum battery, where the $k$th cell is prepared at
the excited state, the wavefunction of the overall system can be written as

$|\Psi^{\rm
B}_{k}\rangle=\sigma^{x}_{k}\prod_{q}|0_{q}\rangle\otimes|m=\frac{N}{2}\rangle,$
(24)

and then the initial density matrix is in general given by

$\rho_{\rm
BC}=\tilde{p}_{0}|\Psi(0)\rangle\langle\Psi(0)|+\sum^{M}_{k=1}\tilde{p}_{k}|\Psi^{\rm
B}_{k}\rangle\langle\Psi^{B}_{k}|,$ (25)

where $\sum^{M}_{k=0}\tilde{p}_{k}=1$ with $\tilde{p}_{k}\geq 0$.

Figure 7: (Color online) (a) The stored energy $E(t)$ and (b) the average
charging power $P(t)$ against the dimensionless time $gt$ under different
initial states. The probability distribution for the initial state of the non-
ideal chargers and the non-empty batteries are set as $p_{0}=0.6$,
$p_{1}=p_{2}=p_{3}=p_{4}=0.1$ and $\tilde{p}_{0}=0.6$,
$\tilde{p}_{1}=\tilde{p}_{2}=0.2$, respectively.

The charging dynamics for the stored energy $E(t)$ and the average charging
power $P(t)$ are plotted in Figs. 7(a) and (b), respectively with $p_{0}=0.6$,
$p_{1}=p_{2}=p_{3}=p_{4}=0.1$ for the non-ideal chargers and
$\tilde{p}_{0}=0.6$, $\tilde{p}_{1}=\tilde{p}_{2}=0.2$ for the non-empty
batteries. In contrast to the ideal situation (see the solid lines), the non-
ideal initial states of the charger impede the full-charging process, due to
the existence of $|\Psi^{\rm C}_{1}\rangle=|00,m=-2\rangle$ and $|\Psi^{\rm
C}_{2}\rangle=|00,m=-1\rangle$. Under the chosen probability distribution in
Fig. 7, the optimal stored energy and the charging power decline respectively
to about $84\%$ and $88\%$ of the corresponding ideal results. On the other
hand, the reduction of the charging performance under the non-empty battery is
caused by the quantum interference between the identical states $|\Psi^{\rm
B}_{1}\rangle=|01,m=2\rangle$ and $|\Psi^{\rm B}_{2}\rangle=|10,m=2\rangle$.
The charging process from the empty state $|00\rangle$ is then not synchronous
with that from the non-empty state $|01\rangle$ or $|10\rangle$. Comparing to
the non-ideal chargers (see the dotted lines), the non-empty batteries (see
the dot-dashed lines) give rise to an even worse behavior in terms of charging
performance. The maximum stored energy and the optimal average charging power
are about $70\%$ and $72\%$, respectively, of the results in the ideal
situation.

These results could also be theoretically anticipated since the non-ideal
states are passive states from which no work can be extracted Farina  _et al._
(2019). The mixture of the non-ideal states and the ideal states affects
detrimentally to the charging performance.

##  V Conclusion

In this work, we investigate the charging capacity and power of a quantum
battery in a spin-charger protocol, which is established in a spin-spin-
environment model through the general Heisenberg XY interaction. In comparison
to the conventional cavity-charger protocol, the charging performance of our
spin-charger protocol is examined with respect to its dependence on the size
of both battery and charger and the anisotropic coefficient of the battery-
charger coupling. Exploiting the collective coupling between battery cells and
charger spins, it is interesting to find that: (1) a full charging can be
achieved when the charger spins are coupled to the same number of the battery
cells, that is beyond the reach of the conventional cavity-charging protocols;
(2) the influences from the finite spin-charger size on the maximum stored
energy and the optimized average charging power of quantum battery are
opposite to each other, i.e., a larger N gives rise to a higher $P_{max}$ but
a lower $E_{max}$; (3) the anisotropic battery-charger coupling will cut down
the charging performance. We also investigate the quantum advantage of the
collective-charging scheme over the parallel-charging scheme. It can be
demonstrated by a superior power law than the cavity-charging protocol with
respect to the battery size. Checking the performance of the quantum battery
under spin-charging in the non-ideal situations, it is found that a
sufficiently strong inner-coupling between the charger spins can be used to
separate the battery and charger systems. Our spin-charging protocol with the
advantage of high-capacity and high-power collective charging can be practiced
in a solid-state platform and sheds new light on the intersection between the
spin-spin-environment model and quantum thermodynamics.

## Acknowledgements

We acknowledge grant support from the National Science Foundation of China
(Grants No. 11974311 and No. U1801661), and Zhejiang Provincial Natural
Science Foundation of China (Grant No. LD18A040001).

##  Appendix A Two-spin battery coupled to $N$-spin charger

In this appendix, we concentrate on the charging performance of the battery
system consisted of two spins coupled to $N\geq M=2$ charger spins in an
isotropic Heisenberg XY model, i.e., $\gamma=0$. The total Hamiltonian in Eq.
(1) becomes

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{1},$ (26)
$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle H_{\rm B}+H_{\rm
C}=\frac{\omega_{0}}{2}(\sigma^{z}_{1}+\sigma^{z}_{2})+\frac{\omega_{0}}{2}J_{z},$
$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
g\left[J_{-}(\sigma^{+}_{1}+\sigma^{+}_{2})+J_{+}(\sigma^{-}_{1}+\sigma^{-}_{2})\right],$

which is equivalent to performing a rotating-wave approximation to the
interaction Hamiltonian with general Heisenberg XY interaction. The model
could be solved in the rotating frame with respect to $H_{0}$. Given the
initial state in Eq. (4), the Hamiltonian could be expressed in the subspace
spanned by $|\lambda_{1}\rangle=|11,N/2-2\rangle$,
$|\lambda_{2}\rangle=|10,N/2-1\rangle$,
$|\lambda_{3}\rangle=|01,N/2-1\rangle$, and
$|\lambda_{4}\rangle=|00,N/2\rangle$ as

$H_{I}=g\left(\begin{array}[]{cccc}0&\sqrt{2N-2}&\sqrt{2N-2}&0\\\
\sqrt{2N-2}&0&0&\sqrt{N}\\\ \sqrt{2N-2}&0&0&\sqrt{N}\\\
0&\sqrt{N}&\sqrt{N}&0\\\ \end{array}\right).$ (27)

In contrast, the Hamiltonian for the cavity-charging protocol in the TC model,
where the battery spins are charged by a single-mode cavity, is written as

$\tilde{H}=\omega_{0}b^{\dagger}b+\frac{\omega_{0}}{2}\sum^{M}_{k=1}\sigma^{z}_{k}+\tilde{g}\left(b\sum^{M}_{k=1}\sigma^{+}_{k}+{\rm
H.c.}\right).$ (28)

Here $b$ ($b^{\dagger}$) denotes the annihilation (creation) operator for the
resonant mode and $\tilde{g}$ is the coupling strength between the quantum
spin batteries and the mode $b$. Given the initial state
$|00\rangle\otimes|2\rangle_{b}$, the rotating Hamiltonian with respect to
$\omega_{0}b^{\dagger}b+(\omega_{0}/2)\sum_{k}\sigma^{z}_{k}$ could be written
as

$\tilde{H}_{I}=\tilde{g}\left(\begin{array}[]{cccc}0&\sqrt{2}&\sqrt{2}&0\\\
\sqrt{2}&0&0&1\\\ \sqrt{2}&0&0&1\\\ 0&1&1&0\\\ \end{array}\right),$ (29)

where the bases are ordered by $|00,2\rangle$, $|01,1\rangle$, $|10,1\rangle$,
and $|11,0\rangle$. In the thermodynamical limit with $N\rightarrow\infty$
Julià-Farré  _et al._ (2020); Andolina  _et al._ (2019a), the two Hamiltonian
in Eqs. (27) and (29) are equivalent to each other by virtue of the Holstein-
Primakoff transformation under the setting $\tilde{g}\approx\sqrt{N}g$.

Starting from the state $|\Psi(0)\rangle=|\lambda_{4}\rangle=|00,N/2\rangle$,
the time evolution of the whole system driven by the Hamiltonian in Eq. (27)
is

$|\Psi(t)\rangle=e^{-iH_{I}t}|\Psi(0)\rangle=\sum^{4}_{k=1}c_{k}(t)|\lambda_{k}\rangle,$
(30)

where

$\displaystyle c_{1}(t)$ $\displaystyle=$
$\displaystyle-\frac{2\sqrt{2(\xi^{2}+4)(\xi^{2}-2)}}{3\xi^{2}}\sin^{2}\left(\frac{\xi
gt}{2}\right),$ $\displaystyle c_{2}(t)$ $\displaystyle=$ $\displaystyle
c_{3}(t)=-i\sqrt{\frac{\xi^{2}+4}{6\xi^{2}}}\sin(\xi gt),$ $\displaystyle
c_{4}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{3\xi^{2}}\left[(\xi^{2}+4)\cos(\xi
gt)+2(\xi^{2}-2)\right].$

with $\xi\equiv\sqrt{2(3N-2)}$.

Figure 8: (Color online) The stored energy $E(t)$ of the two-battery-cell
system against the dimensionless time $gt$ with various numbers of charger
spins: (a) $N=2$ with $E_{\rm max}/\omega_{0}=2$, (b) $N=20$ with $E_{\rm
max}/\omega_{0}=1.81$, (c) $N=200$ with $E_{\rm max}/\omega_{0}=1.78$. (d)
presents the benchmark result obtained in the TC model with $N=2$ and $E_{\rm
max}/\omega_{0}=16/9$. Figure 9: (Color online) The average charging power
$P(t)$ of the two-battery-cell system against the dimensionless time $gt$ with
various numbers of charger spins: (a) $N=2$, (b) $N=20$, (c) $N=200$. (d)
presents the benchmark result obtained in the TC model with $N=2$.

According to Eq. (7), the time evolution of the energy stored in the double-
spin quantum battery is given by

$\displaystyle E(t)$ $\displaystyle=$
$\displaystyle\omega_{0}\left[|c_{1}(t)|^{2}-|c_{4}(t)|^{2}\right]+\omega_{0}$
(31) $\displaystyle=$
$\displaystyle\frac{(\xi^{2}+4)\omega_{0}}{9\xi^{4}}\left[(\xi^{2}-8)\cos(\xi
gt)-(7\xi^{2}-8)\right]$ $\displaystyle\times$ $\displaystyle\left[\cos(\xi
gt)-1\right].$

Then it is straightforward to find the maximum stored energy measuring the
capacity of the quantum battery,

$E_{\rm
max}=\frac{16(\xi^{4}+2\xi^{2}-8)}{9\xi^{4}}\omega_{0}=\frac{16N(N-1)}{(3N-2)^{2}}\omega_{0},$
(32)

at $t=\bar{t}=\pi/(\xi g)$ for the first time. When $N\gg 1$, $E_{\rm
max}\rightarrow(16/9)\omega_{0}$. This result is consistent with that obtained
by the cavity-charging protocol in the TC model Ferraro  _et al._ (2018). We
plot the stored energy $E(t)$ with various $N$ in both spin-charging protocol
and the cavity-charging protocol in Fig. 8. As found in Eqs. (31) and (32), it
is observed that $E(t)$ shows an ever faster sinusoid evolution with an
increasing $N$, and then the optimized time $\bar{t}$ when $E(t)$ attains the
maximum value $E_{\rm max}$ for the first time is consequently reduced. More
importantly, $E_{\rm max}/\omega_{0}$ gradually approaches $16/9$, the value
for the cavity-charging protocol as depicted in Fig. 8(d). The spin-charging
protocol with a finite size thus activates a higher-capacity quantum battery
in terms of both $E(t)$ and $E_{\rm max}$.

The average charging power $P(t)$ can be obtained through numerical simulation
according to Eq. (8). In all the sub-figures in Fig. 9, it is shown that in
general after the power $P(t)$ increases to a maximum value $P_{\rm max}$,
that could be conveniently obtained by an optimization procedure. And then it
will asymptotically decay with time. This pattern is irrespective to the
charging protocols, although a larger number of spin-charger $N$ brings more
fluctuations during the evolution. The renormalized optimal charging-power is
found to be $P_{\rm max}/(\sqrt{N}g\omega_{0})\approx 1.45$ when $N=2$ [see
Fig. 9(a)], and it is slightly enhanced to $1.53$ when $N=200$ [see Fig.
9(c)]. The latter is close to the result obtained in the cavity-charging
protocol as shown in Fig. 9(d).

Combining Figs. 8 and 9, a trade-off relation with respect to the charger size
$N$ thus emerges between the maximum stored energy and the optimal average
power.

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