# Quantum battery based on dipole-dipole interaction and external driving
field

Wuji Zhang  Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China  Shuyue Wang  Center for Quantum
Sciences and School of Physics, Northeast Normal University, Changchun 130024,
China  Chunfeng Wu  Science, Mathematics and Technology, Singapore University
of Technology and Design, 8 Somapah Road, Singapore 487372, Singapore
Gangcheng Wang<EMAIL_ADDRESS>Center for Quantum Sciences and School
of Physics, Northeast Normal University, Changchun 130024, China

###### Abstract

The Dicke model is a fundamental model in quantum optics, which describes the
interaction between quantum cavity field and a large ensemble of two-level
atoms. In this work, we propose an efficient charging quantum battery achieved
by considering an extension Dicke model with dipole-dipole interaction and an
external driving field. We focus on the influence of the atomic interaction
and the driving field on the performance of the quantum battery during the
charging process and find that the maximum stored energy exhibits a critical
phenomenon. The maximum stored energy and maximum charging power are
investigated by varying the number of atoms. When the coupling between atoms
and cavity is not very strong, compared to the Dicke quantum battery, such
quantum battery can achieve more stable and faster charging. In addition, the
maximum charging power approximately satisfies a superlinear scaling relation
$P_{\rm max}\varpropto\beta N^{\alpha}$, where the quantum advantage
$\alpha=1.6$ can be reached via optimizing the parameters.

##  I Introduction

With the advancement of quantum technology, there is a growing interest in
schemes that utilize quantum effects to enable superior performance of future
technological devices [1, 2, 3, 4, 5, 6]. Recently, it has shown great success
in several practical fields, such as quantum computing, quantum cryptography,
and thermodynamic nanoscale device, which are expected to completely solve
data analysis in the communication process, optimize sensitive parameters to
improve network security, and provide more accurate temperature measurement
[7, 8, 9, 10, 11, 12]. Overall, the development of quantum technology promises
to offer more miniaturized and more precise devices. The devices with
potential quantum information processing have also been developed, but
strategies for storing and releasing energy in these devices remain a major
problem to be addressed [13, 14, 15].

The question of whether quantum effects can improve energy storage to meet the
current needs of quantum devices has been explored. The so-called quantum
battery (QB), a small quantum system for energy storage and extraction, has
been published in a seminal paper by Alicki and Fannes [16]. This provides an
efficient way to solve the energy constraints of quantum devices [17, 18, 19].
A central goal for such research field is to optimize the performance of QBs,
such as energy storage and charging rate [20, 21, 22, 23]. Generally speaking,
two distinct schemes for charging have been proposed, namely collective
charging and parallel charging [24, 25]. For collective charging scheme, all
cells are charged from the same charger; for parallel charging, each cell is
charged through its own charger independently. Some results have demonstrated
that collective charging shows better performance over parallel charging and
charging acceleration can be achieved [26, 27, 28, 29, 30]. The advantage of
collective charging over parallel charging is called the quantum charging
advantage [31, 32].

In pursuit of the quantum advantage and possible experimental realization, the
QB has been proposed in various models, such as two-level systems [33, 34],
three-level systems [35, 36, 37], two photons model [38], the superconducting
circuit model [39, 40, 41], Lipkin-Meshkov-Glick model [42], Sachdev-Ye-Kitaev
model [27, 43, 44], Heisenberg spin-chain model [45, 46], quantum cavity model
[47, 48], collision model [49, 50, 51], many-body localized model [52, 53, 54,
55], dissipation model [56, 57] and so on [58, 59]. One of the most well-known
examples is the Dicke model [60, 42, 61], which describes the interaction of
the ensemble of two-level atoms with the single-photon mode of a cavity.
Despite the relative simplicity of the Dicke model, it still exhibits several
interesting phenomena as the coupling strength increases to ultrastrong
coupling (USC) or even deep-strong coupling (DSC) [62, 63, 64]. Recently, QBs
based on the Dicke model have been introduced, assuming a conventional
coupling between atoms and photon radiation of a cavity. It has been reported
that a $\sqrt{N}$ acceleration can be achieved for Dicke QB under the
collective charging scheme [65, 66].

In the development stage of QBs, the spin-chain model placed in a optical
cavity has also received much attention [67, 68, 68]. The cavity can affect
the radiation of the atoms, inducing atom-atom interactions and collective
properties. It is worth noting that various protocols can be experimentally
employed to create an array of traps in the cavity to maintain the atoms at a
fixed distance [69, 70]. The atoms placed at a smaller distance are coupled to
each other by dipole-dipole interaction [71, 72, 73]. Previous studies have
rarely considered long-distance interactions of atoms, but long-distance
interactions between atoms can produce many interesting phenomena that cannot
be ignored in actual atomic chain models [74, 75]. These interactions are
dependent on the distance between atoms. Through the interaction, we can
regulate the energy levels and control whether atomic transitions occur [72,
76, 77, 78, 79]. A natural question arises: can the charging process be
accelerated compared to Dicke QB when considering long-distance interactions
between atoms?

On the other hand, in a recent work [47], a generalized Dicke QB has been
proposed based on the global entanglement interactions and the driving field,
which can achieve a much faster charging process than Dicke QB. To take
another step forward, we consider the direct dipole-dipole interaction that
depends on the atomic distance. In this study, we present the atoms with
dipole-dipole interaction that are placed in the driven optical cavity as a QB
system. We discuss the essential factors of the QB such as stored energy,
charging power and energy quantum fluctuation under various conditions. In
addition, we also introduce the quantum advantage of maximum charging power to
analyze the performance of QBs under the collective charging scheme.

The paper is organized as follows. In Sec. II, we propose the QB model,
charging protocols, and numerical approach. In Sec. III, we analyze the
influence of the external driving field and dipole-dipole interaction between
atoms on the performance at different coupling parameters. In Sec. IV, the
impact of the number of qubits in the model on the maximum stored energy and
maximum charging power under various coupling conditions and the quantum
advantage of maximum charging power are analyzed. Finally, we give a summary
that our QB can be more efficient than Dicke QB in Sec. V.

##  II Model of the battery and approach of charge

The proposed QB consists of $N$ identical two-level atoms coupled to a single-
mode optical cavity with an external driving field. Here we consider that two-
level atoms (qubits) are placed at a small distance, leading to the presence
of dipole-dipole interaction [80]. The Hamiltonian of this QB system reads (we
set $\hbar=1$ throughout this work)

$\hat{H}_{s}(t)=\hat{H}_{b}+\Theta(t)\hat{H}_{c},$ (1)

where $\Theta(t)=\theta(t)-\theta(t-T)$ with $\theta(t)$ being the Heaviside
step function. Such function represents a sudden switch on/off of the charging
process: $\Theta(t)=1$ for $0\leq t\leq T$ with $T$ being the charging
interval; $\Theta(t)=0$ elsewhere. The terms $\hat{H}_{b}$ and $\hat{H}_{c}$,
describing the battery and charger, can be written as

$\begin{split}\hat{H}_{b}=\omega_{0}&\hat{J}_{z},\\\
\hat{H}_{c}=\omega_{c}\hat{a}^{\dagger}\hat{a}+2g(\hat{a}^{\dagger}+\hat{a})&\hat{J}_{x}+\hat{H}_{\rm
int}+\hat{H}_{d},\\\ \end{split}$ (2)

where

$\begin{split}\hat{H}_{\rm int}&=\sum_{i\neq
j}^{N}\eta_{ij}(\hat{\sigma}_{i}^{-}\hat{\sigma}_{j}^{+}+\hat{\sigma}_{j}^{-}\hat{\sigma}_{i}^{+}),\\\
\hat{H}_{d}&=\Omega\cos(\omega_{d}t)(\hat{a}^{\dagger}+\hat{a}).\end{split}$
(3)

The notation $\omega_{0}$ denotes the energy splitting between the atomic
ground state $|\textit{g}\rangle$ and atomic excited state
$|\textit{e}\rangle$,
$\hat{J}_{\alpha}=\frac{1}{2}\sum_{i=1}^{N}\hat{\sigma}_{i}^{\alpha}$
($\alpha=x,y,z$) are the collective spin operators with
$\hat{\sigma}_{i}^{\alpha}$ being the Pauli operators acting on the $i$-th
site. The operators $\hat{J}_{\alpha}$ constitute a representation of the
$\mathfrak{su}(2)$ algebra, obeying the usual commutation relations:
$[\hat{J}_{\alpha},\hat{J}_{\beta}]=i\varepsilon_{\alpha\beta\gamma}\hat{J}_{\gamma}$
($\alpha,\beta,\gamma\in\\{x,y,z\\}$) with $\varepsilon_{\alpha\beta\gamma}$
being the Levi-Civita symbol. The notation $\hat{a}$ ($\hat{a}^{\dagger}$) is
the annihilation (creation) operator of the cavity field with frequency
$\omega_{c}$. Otherwise, $g$ denotes the uniform atom-cavity coupling
strength. The cavity is pumped by a coherent laser with amplitude $\Omega$ and
frequency $\omega_{d}$. The symbol $\eta_{ij}$ denotes the dipole-dipole
interaction between the $i$-th and the $j$-th atoms, which is given by [80,
81, 82]

$\eta_{ij}=-\frac{3}{4}\frac{\Gamma_{0}c^{3}}{\omega_{0}^{3}|(i-j)R|^{3}}\left(3\cos^{2}\alpha-1\right),$
(4)

where $c$ is the speed of light in vacuum, $\Gamma_{0}$ denotes the
spontaneous radiation rate of the atom, $R$ is the distance between two
adjacent atoms and $\alpha$ denotes the angle of the atomic distance to the
electric dipole moment (please see Fig. 1). In the following, we consider that
the array of interacting atoms is assumed to be in one-dimensional array with
$N$ sites located at equal distances from each other. Simultaneously, we use
the symbol $\eta$ to denote the interaction strength between nearest-neighbor
atoms.

Figure 1: Schematic diagram of $N$ qubits coupled to each other by dipole-
dipole interaction. The unique cavity with frequency $\omega_{c}$ is driven by
a coherent laser field with strength $\Omega$ and frequency $\omega_{d}$. $R$
denotes the distance between adjacent atoms. $\alpha$ denotes the angle
between the dipole moment ($\bm{\mu}$) and atomic position vector
($\bm{R_{i}}$). Each atom can be considered as a two-level system with the
ground state ($|\textit{g}\rangle$) and excited state ($|\textit{e}\rangle$).

From the above Hamiltonian, there are two charging mechanisms: One is to
charge directly with atomic interactions; and the other is to transfer energy
from the cavity field to atoms. It is worth mentioning that we attribute the
indirect charging of the driving field to the second charging method. Then we
further assume that atomic system is in its ground state
$|\textit{g},\textit{g},\textit{g},...,\textit{g}\rangle$ and the cavity is in
a Fock state with a specific photon number $n$, i.e., $|\textit{n}\rangle$.
Therefore, the overall initial state of battery and charger is

$|\varphi(0)\rangle_{s}=|\underbrace{\textit{g},\textit{g},\textit{g},...,\textit{g}}_{N}\rangle\otimes|\textit{n}\rangle.$
(5)

In our charging protocol, the QB begins to charge when $\Theta(t)$ equals one,
and then the evolution of the wave function over time reads

$|\varphi(t)\rangle_{s}=\mathcal{T}{\rm
exp}\left[{-i\int_{0}^{t}\hat{H}_{s}(t^{\prime})dt^{\prime}}\right]|\varphi(0)\rangle,$
(6)

where $\mathcal{T}$ denotes time ordering operator. Thus, the density matrix
of the QB in the initial state and final state at an arbitrary time can be
respectively written as
$\hat{\rho}_{s}(0)=|\varphi(0)\rangle_{s}\langle\varphi(0)|$ and
$\hat{\rho}_{s}(t)=|\varphi(t)\rangle_{s}\langle\varphi(t)|$. In the charging
process, the charger transfers its own energy to the battery, leading to the
internal atoms of the battery to transition from the ground state
$|\textit{g}\rangle$ to the excited state $|\textit{e}\rangle$. Therefore, the
stored energy in the QB at time $t$ is given by

$E_{b}(t)={\rm Tr}\left[\hat{\rho}_{b}(t)\hat{H}_{b}\right]-{\rm
Tr}\left[\hat{\rho}_{b}(0)\hat{H}_{b}\right].$ (7)

Here $\hat{\rho}_{b}(t)={\rm Tr}_{c}[\hat{\rho}_{s}(t)]$ is the reduced
density matrix of the QB. Remarkably, the stored energy $E_{b}(t)$ in which we
are concerned does not contain the Hamiltonian $H_{\rm int}$ since the
existence of atomic interactions induces either positive or negative
contributions to $E_{b}(t)$, which finally results in an uneven comparison.
Similarly, in order to quantify the rate of storing energy during the charging
process, we define the average charging power as follows

$P_{b}(t)=\frac{E_{b}(t)}{t}.$ (8)

When evaluating the performance of QBs in terms of stored energy and average
charging power, the number of atoms in the cavity will inevitably generate
energy fluctuations. To ensure the most stable charging process, we need to
further analyze the superiority of the charging process by energy quantum
fluctuation [38], which can be written as

$\begin{split}\Delta E_{b}(t)=&\omega_{0}\Big{[}\sqrt{{{\rm
Tr}(\hat{\rho}_{b}(t)\hat{H}_{b}^{2})}-[{\rm
Tr}(\hat{\rho}_{b}(t)\hat{H}_{b})]^{2}}\\\ &-\sqrt{{{\rm
Tr}(\hat{\rho}_{b}(0)\hat{H}_{b}^{2})}-[{\rm
Tr}(\hat{\rho}_{b}(0)\hat{H}_{b})]^{2}}\Big{]}.\\\ \end{split}$ (9)

Figure 2: (a)-(c) Behavior of the stored energy $E_{b}(t)$ (in units of
$\hbar\omega_{0}$), and (d)–(f) behavior of the charging power $P_{b}(t)$ (in
units of $\hbar\omega_{0}^{2}$) as a function of the time $t$ (in units of
$\omega^{-1}_{0}$) in the case of the USC and DSC regime. Different timescales
have been used to properly show the maximum position of the related-
performance. The various curves in the figure represent the various
frequencies of driving field and all are for the QB with the number of qubits
$N$ =5. Figure 3: (a)-(c) Behavior of the stored energy $E_{b}(t)$ (in units
of $\hbar\omega_{0}$), (d)–(f) behavior of the charging power $P_{b}(t)$ (in
units of $\hbar\omega_{0}^{2}$), and (g)-(i) behavior of the energy quantum
fluctuation $\Delta E_{b}$ (in units of $\hbar\omega_{0}$) as a function of
the time $t$ (in units of $\omega^{-1}_{0}$) in the case of the USC and DSC
regime. Different timescales have been used to properly show the maximum
position of the related-performance. The various curves in the figure
represent different values of the selected $\Omega$ and all are for the QB
with the number of qubits $N$ = 5.

According to the previous studies [83], we focus on the resonant regime, as
off-resonant case ($\omega_{c}\neq\omega_{0}$) gives rise to a low probability
of transition, leading to less efficient energy transfer between the cavity
and battery [84]. Here and after, we assume the resonant condition, i.e.,
$\omega_{c}=\omega_{0}=1$. Furthermore, to determine the optimal QB as much as
possible, in Fig. 2 we illustrate the time evolution of the energy and
charging power for different values of the driving field frequency in various
coupling regimes. After comparing the results, it is evident that the
performance in the other two regimes is markedly inferior to that in the
resonant regime ($\omega_{d}=\omega_{0}$), in terms of both the maximum energy
storage and the maximum charging power. Therefore, given the situation
reported above, it is necessary to assume $\omega_{d}=\omega_{0}$ to achieve
greater energy and faster charging of the QB. Besides, we also take $\Omega$
and $\eta$ in the unit of $\omega_{0}$ for convenience. For dipole-dipole
interactions that decay with the cube of atomic distance, the long-range
interactions have negligible effects. Hence, the atomic interaction can be
ignored when separation distance is wider than four times of the distance
between two nearest-neighbor atoms in the numerical simulation.

Figure 4: (a)-(c) Behavior of the stored energy $E_{b}(t)$ (in units of
$\hbar\omega_{0}$), (d)–(f) behavior of the charging power $P_{b}(t)$(in units
of $\hbar\omega_{0}^{2}$), and (g)-(i) behavior of the energy quantum
fluctuation $\Delta E_{b}$ (in units of $\hbar\omega_{0}$) as a function of
the time $t$ (in units of $\omega_{0}^{-1}$) in the case of the USC and DSC
regime. Different timescales have been used to properly show the maximum
position of the related-performance. The various curves in the figure
represent the various strengths of atomic interaction and all are for the QB
with the number of qubits $N$ = 5.

Furthermore, it should be emphasized that our study of the QB is conducted
within a closed system. It may inevitably interact with the external
environment, which leads to the loss of photons in the cavity and the
relaxation of the atomic energy in the QB [85]. Therefore, in the following
text, we will ensure that the evolution time $t$ is significantly shorter than
the dissipation timescales $t_{\gamma}$ and $t_{r}$, so that we can safely
disregard the effect of dissipation. Based on the known experimental level of
operable qubits, this condition is typically fulfilled in the most
sophisticated circuit quantum electrodynamics devices [86, 87, 88, 89]. Not
only that, since the general condition $t_{r}>t_{\gamma}$, the stability of
the cavity becomes crucial in experiments [90, 91].

Notably, when the system enters the USC and DSC regimes, we need to
investigate the performance of the QB in both cases. To ensure consistency
with the approach used for the Dicke QB, we remark the existence of the
counter-rotating term in the Hamiltonian of the system, and the total
excitation number is not conserved according to the Hamiltonian of Eq. (2).
Thus, in principle, we can choose any large number of photons. However, in
practical finite-size numerical simulation, we need to introduce a maximum
photon number $N_{\rm ph}$ to ensure that the related-performance of the QB is
calculated accurately [66]. We have extensively examined the numerical
convergence of results (energy, charging power, and energy quantum
fluctuation) and demonstrated the error below $10^{-5}$ by selecting a
relatively large photon number $N_{\rm ph}=4N$ [38, 47].

##  III The influence of external driving field and dipole-dipole interaction
on the QB at different coupling strengths

In this section, we will separately analyze the influence of the driving field
amplitude, the atomic interaction, and the combined actions on the performance
of the QB in the USC ($0.1\omega_{0}\leq g\lesssim\omega_{0}$), and DSC
($g\geq\omega_{0}$) regimes. We expect higher energy storage and further
enhancement of the average charging power compared to the Dicke QB.

###  III.1 Influence of external driving field on the performance

We now consider that the distance between atoms is sufficiently large, and
dipole-dipole interaction can be ignored. Acting external driving field on the
system, the total Hamiltonian during the charging process reads

${\hat{H}_{s}}=\omega_{0}\hat{J}_{z}+\omega_{c}\hat{a}^{\dagger}\hat{a}+2g(\hat{a}^{\dagger}+\hat{a})\hat{J}_{x}+\Omega\cos(\omega_{d}t)(\hat{a}^{\dagger}+\hat{a}).$
(10)

In Fig. 3, we show the influence of variations in the driving field strength
on the stored energy, charging power, and energy quantum fluctuation in the
USC and DSC regimes. When the cavity-atom coupling is not very strong,
regardless of whether the external driving field strength is large or small
throughout the whole charging process, it can play a positive role in charging
power based on Eq. (10) in comparison to Dicke QB (green line). However, in
the USC regime, as the driving field amplitude increases, the stored energy
and average charging power also change significantly accordingly, but the
influence of this enhancement becomes smaller compared with that of the weak
USC case. In particular, when the amplitude raises at a fixed value, the
maximum charging power of the QB does not uniformly increase. More
interestingly, in the USC regime, the upgrade of the coupling strength allows
the QB to achieve the maximum stored energy more quickly than in the weak USC
regime.

Next, we discuss the coupling strength in the DSC regime, specifically up to
$g\geq\omega_{0}$. However, the changes in the effective amplitude will not
strongly impact the performance of the QB in the DSC regime. Despite this, the
corresponding performance of the QB has improved as a result of the increased
amplitude. Such situation can be explained as follows: when the coupling
between cavity and atoms is weak, the driving term dominates the dynamics;
nevertheless, as the coupling strength gradually increases, the indirect
charging dominated by the driving field has no advantages over the cavity-atom
charging. Thus, the QB does not change with the driving field amplitude in the
DSC regime. On the other hand, compared to the weak USC regime, the QB also
generates higher energy quantum fluctuation during charging. Although the
quantum fluctuation is inevitable and limited, this phenomenon should be
caused by the interaction between atoms and the cavity field, which is closely
related to the QB not being fully charged.

Figure 5: (a)-(c) Behavior of the stored energy $E_{b}(t)$ (in units of
$\hbar\omega_{0}$), (d)–(f) behavior of the charging power $P_{b}(t)$ (in
units of $\hbar\omega_{0}^{2}$), and (g)-(i) behavior of the energy quantum
fluctuation $\Delta E_{b}$ (in units of $\hbar\omega_{0}$) as a function of
the time $t$ (in units of $\omega^{-1}_{0}$) in the case of the USC and DSC
regime. Different timescales have been used to properly show the maximum
position of the related-performance. The various curves in the figure
represent different values of the selected $\Omega$, $\eta$ and all are for
the QB with the number of qubits $N$ = 5.

###  III.2 Influence of dipole-dipole interaction on the performance

We now assess the performance of the QB for the Dicke model with dipole-dipole
interaction between atoms. The total Hamiltonian based on Eq. (2) during the
charging process is given as

$\hat{H_{s}}=\omega_{0}\hat{J}_{z}+\omega_{c}\hat{a}^{\dagger}\hat{a}+2g(\hat{a}^{\dagger}+\hat{a})\hat{J}_{x}+\sum_{i\neq
j}^{N}\eta_{ij}(\hat{\sigma}_{i}^{-}\hat{\sigma}_{j}^{+}+\hat{\sigma}_{j}^{-}\hat{\sigma}_{i}^{+}).$
(11)

In Fig. 4, we show the influence of atomic interaction on stored energy,
charging power, and energy quantum fluctuation in the USC and DSC regimes. The
red and blue lines indicate the repulsive ($\eta>0$) and attractive ($\eta<0$)
interactions, respectively. For comparison, the performance of Dicke QB is
shown with solid green line. When the QB is in the weak USC regime, the
interactions always have disadvantages on the maximum stored energy, which can
be attributed to a portion of the energy is stored into the interaction
between atoms. So, we can identify that $\eta$ has the disadvantage of energy
storage over Dicke QB. However, for the charging power, different atomic
interactions have different effects, in which the attractive interaction
decreases and repulsive interaction increases the maximum charging power. It
can be explained that the repulsive interaction increases the effective atom-
field interaction; while the attractive interaction suppresses the effective
interaction [92]. On the other hand, quantum correlations during charging can
enhance the rate at which energy can be stored, usually denoted as a quantum
speed up [42]. Therefore, different $\eta$ lead to varying degrees of
increasing or decreasing the charging power.

Nevertheless, the results show that the influence of the atomic interaction on
the performance is not significant in the DSC regime. It is shown that the
advantage of the atomic interaction on the QB has become smaller and smaller,
indicating competition between atom-atom and atom-cavity interactions [92,
47]. The interaction between atoms plays a dominant charging mechanism
comparing to the atom-cavity interaction in the USC regime, making it a
crucial factor influencing the QB. However, in the DSC regime, the interaction
between atoms and the cavity field is strong enough to counteract the benefits
generated by the atomic interaction, resulting in minimal impact on
performance as $\eta$ changes. Correspondingly, we observe that the maximum
stored energy exhibits a critical behavior, that is, the QB system exists at
the critical point where the maximum energy storage changes significantly. We
already know that spin-spin interactions can produce ground-state quantum
phase transition (QPT) from Refs. [60, 93, 94, 92, 95]. Since we assume that
all the qubits in the initial state are oriented downward, the original
symmetry is broken which affects the ground state properties during charging.
The QPT also influences the energy charging and release of the QB (More
details are presented in Appendix A).

Figure 6: (a) Maximum stored energy $E_{\rm max}$ (in unit of
$\hbar\omega_{0}$), and (b) maximum charging power $P_{\rm max}$ (in unit of
$\hbar\omega^{2}_{0}$) of QB with the number of qubits $N$ for different
coupling regimes $g$. The other parameters are valued as $\Omega=1.0$,
$\eta=0.8$.

###  III.3 Influence of dipole-dipole interaction and external driving field
on the performance

Table 1: The scaling exponent $\alpha$ of the maximum charging power for different values of the cavity-spin coupling strength $g$, the effective amplitude of external driving field $\Omega$ and the interaction strength $\eta$. |  | $\eta=-1.0$ |  |  | $\eta=-0.5$ |  |  | $\eta=+0.5$ |  |  | $\eta=+1.0$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
| $g=0.1$ | $g=0.5$ | $g=2.0$ | $g=0.1$ | $g=0.5$ | $g=2.0$ | $g=0.1$ | $g=0.5$ | $g=2.0$ | $g=0.1$ | $g=0.5$ | $g=2.0$
$\Omega=0.1$ | 1.59 | 1.20 | 1.03 | 1.55 | 1.17 | 1.02 | 1.57 | 1.19 | 1.02 | 1.60 | 1.23 | 1.05
$\Omega=0.5$ | 1.47 | 1.43 | 1.33 | 1.53 | 1.42 | 1.32 | 1.55 | 1.44 | 1.34 | 1.49 | 1.45 | 1.34
$\Omega=1.0$ | 1.45 | 1.45 | 1.40 | 1.51 | 1.43 | 1.38 | 1.53 | 1.45 | 1.36 | 1.47 | 1.46 | 1.42
$\Omega=2.0$ | 1.40 | 1.47 | 1.47 | 1.43 | 1.45 | 1.45 | 1.45 | 1.48 | 1.44 | 1.37 | 1.48 | 1.46

In this section, we investigate the performance of the QB for the Dicke model
with an external driving field and the atomic interaction. Accordingly, the
Hamiltonian during the charging process reads

$\begin{split}\hat{H_{s}}=&\omega_{0}\hat{J}_{z}+\omega_{c}\hat{a}^{\dagger}\hat{a}+2g(\hat{a}^{\dagger}+\hat{a})\hat{J}_{x}+\Omega\cos(\omega_{d}t)(\hat{a}^{\dagger}+\hat{a})\\\
&+\sum_{i\neq
j}^{N}\eta_{ij}(\hat{\sigma}_{i}^{-}\hat{\sigma}_{j}^{+}+\hat{\sigma}_{j}^{-}\hat{\sigma}_{i}^{+}).\end{split}$
(12)

In Fig. 5, we show the behavior of the stored energy $E_{b}$, average charging
power $P_{b}$, and energy quantum fluctuation $\Delta E_{b}$ for different
values of $\Omega$ and $\eta$ based on Eq. (12) in the USC and DSC regimes
during the charging process. For comparison purposes, the case without driving
field and atomic interaction is also depicted by the solid green line. In the
weak USC regime, the maximum energy storage and the maximum charging power
become weakened compared to the single driving field $\Omega$ action
previously shown in Fig. 3, while the charging process becomes more stable and
then the time for fully charging is indeed reduced accordingly; however, the
QB exceeds the single atomic interaction $\eta$ previously shown in Fig. 4 in
terms of the maximum energy storage and the maximum charging power, but the
time to complete charging is extended. With the increase in the coupling
strength $g$, the combined actions of the two also affect the performance of
the QB to varying degrees. Similarly, in the DSC regime, changing the
effective amplitude of the driving field $\Omega$ and the atomic interaction
strength $\eta$ has minimal impact on the related-performance of the QB. It
can also be explained that the cavity-atom charging outperforms the charging
dominated by the driving field (atomic interaction) with the gradual increase
in coupling strength. After comparison, we find the improvement of the
charging performance is the most obvious in the weak USC regime. Of the four
curves listed, when $\Omega=1.0,\eta=0.8$, the performance of the QB has a
significant advantage, whereas the other curves are not comparable, indicating
the need for appropriate adjustment of $\Omega$ and $\eta$ to achieve the
optimal performance. Therefore, combined with practical experimental
implementations, it is essential to rationally design the QB in the weak USC
regime to achieve greater energy storage and faster charging rate.

##  IV Enhancement of collective charging

In this section, we further discuss how the performance of the QB depends on
the number of qubits in the presence of both driving field and atomic
interaction. In Sec. III, the selected parameters, namely $\Omega=1.0$ and
$\eta=0.8$, have a significant impact on all aspects of the performance in the
weak USC regime. So we take these values as the basic parameters to study how
the number of qubits $N$ affects the charging process. In Fig. 6, we
illustrate the dependence of the energy storage and charging power on the
number of qubits. Therefore, we can easily observe that the maximum energy
storage of the QB increases with the number of qubits $N$. At the same time,
we can fit the data to find that the maximum energy storage of the current QB
is proportional to the number of qubits ($E_{\rm max}\varpropto N$), which
closely resembles the conclusion previously described in the literature [48,
66].

The trend of the charging power follows that of the stored energy in Fig. 6.
However, relying solely on the tendency of the graph line is far from
insufficient, and we need to introduce another parameter to compare the
performance of the QB. Since the charging power of the QB is limited by the
charging Hamiltonian added to the battery in the external quenching charging
protocol, we can apply the quantum advantage $\alpha$ proposed in Refs. [66,
47]. The scaling index $\alpha$ essentially reflects the property of
collective charging of the QB. We already know that the quantum advantage
$\alpha$ is an upper bound on the quadratic function of the maximum
participation number of the collective charging [25, 21]. Furthermore, in the
Dicke QB, the maximum charging power $P_{\rm max}$ is proportional to
$N^{\frac{3}{2}}$, suggesting that the maximum charging power of our QB may
also exhibit a scaling relationship with the number of qubits $N$. We assume
that the following form can be used for the maximum charging power ($P_{\rm
max}\varpropto\beta N^{\alpha}$). By properly adjusting the parameters, we can
bring the $\alpha$ to 1.60 in the range considered in Tab. 1 (More details are
reported in Appendix B). This result is consistent with the conclusion
previously reported in the reference [47]. On the other hand, setting $g=0.1$,
$\Omega=0.1$, $\eta=1$ corresponds to a smaller driving field and larger
atomic interaction, which could effectively reduce energy oscillation to
achieve a relatively stable charging process in the weak USC regime. So we
fully predict that the QB can store more energy and achieve higher charging
efficiency.

##  V Conclusion

We consider a QB described by introducing the external driving field and
dipole-dipole interaction between atoms in the Dicke model. When only the
external driving field exists, the QB always shows better stored energy and
charging power as the effective amplitude increases; while considering only
dipole-dipole interaction between atoms, the performance of the QB is
influenced by the atomic interaction in a different way. However, when both
external driving field and atomic interaction exist, we choose some parameters
and find the increase advantage of $\eta$ and $\Omega$ is not the same, which
requires to properly design the parameters to obtain the optimized QB.
Furthermore, we have observed that the maximum stored energy exhibits a
critical behavior. The critical atomic interactions and coupling strengths
have been analyzed by QPT. We also further examine the impact of the number of
qubits $N$ on the performance of the QB. Upon comparison with the traditional
Dicke QB, we obtain a quantum advantage of the maximum charging power,
approximately satisfying $P_{max}=\beta N^{\alpha}$ and find the quantum
advantage of the maximum charging power of our QB can reach 1.60 by adjusting
the corresponding parameters appropriately. So we can fully believe in the
implementation of greater energy storage, faster charging rate, and a more
stable process for QB. It is hoped that our proposed scheme can provide
valuable references for the future development of the QB.

## Acknowledgments

Chunfeng Wu is supported by the National Research Foundation, Singapore and
A*STAR under its Quantum Engineering Programme (NRF2021-QEP2-02-P03).

##  Appendix A Critical behavior of maximum energy storage

The QPT is the phenomenon of phase transition occurring at zero temperature
due to quantum fluctuations, contrasting thermal phase transitions which are
caused by thermal disturbances. It represents a sudden burst of ground state
properties in many-body quantum systems with external parameters. The QPT
exists widely in magnetic materials, ferroelectric materials, metal-insulator
transition systems, and is an important foundation for a deeper understanding
of condensed matter physics.

Figure 7: (a)-(c) Behavior of maximum stored energy $E_{\rm max}$ (in units of
$\hbar\omega_{0}$), maximum charging power $P_{\rm max}$ (in units of
$\hbar\omega^{2}_{0}$) and the phase diagram $\langle\hat{J}_{z}\rangle/(N/2)$
as the function of the atomic interaction $\eta$ and the coupling strength
$g$. All plots correspond to the case of $N=5$.

For our QB system, atom-atom interactions can significantly change the maximum
stored energy of the QB in the USC regime. However, in the DSC regime, such
influence is negligible. In particular, the maximum stored energy exhibits a
critical behavior in the USC regime (There is a critical regime in the system,
and the maximum stored energy of the QB changes significantly around the
critical regime). Since in our QB charging scheme, the energy storage of the
system mainly depends on the state of all qubits in the cavity. Therefore, we
discuss the QPT in our analysis. To clarify the behavior, we treat the QB as
only two independent parameters, namely the atomic interaction $\eta$ and the
coupling strength between atoms and the cavity $g$.

In Fig. 7, since the maximum energy storage $E_{\rm max}$ and the maximum
charging power $P_{\rm max}$ are measured during the evolution process, it is
not clear whether they can illustrate the QPT. To better illustrate the QPT
behavior, we introduce an ordered parameter, namely the $z$ component of the
average magnetization $\hat{J}_{z}$. Therefore, the curve from the
ferromagnetic phase ($\langle\hat{J}_{z}\rangle/(N/2)=-1$) to the Mott phase
($\langle\hat{J}_{z}\rangle/(N/2)=0$) is depicted in Fig. 7. This means that
when $\langle\hat{J}_{z}\rangle/(N/2)=-1$, the system is well-ordered where
all the qubits are in the spin-down state of the ferromagnetic phase. However,
when the parameter changes so that $\langle\hat{J}_{z}\rangle/(N/2)=0$, the
system begins to deviate from the ferromagnetic phase to reach the Mott phase.
At this point, the order of the system is broken and presents the
antiferromagnetic phase. We already know that $\hat{J}_{z}$ is the layout
difference operator between the excited state and the ground state, and that
its value is related to the number of atoms in each state. The Mott phase
represents an equal number of atoms in both states; while the ferromagnetic
phase means that all atoms are in the ground state. As the ground state we set
is all spin-down, the QB in the ferromagnetic phase is detrimental for storing
energy. Correspondingly, the maximum stored energy changes significantly at
the critical point of the QPT.

##  Appendix B Collective enhancement of the charging power

In order to further discuss how the charging process of the QB depends on the
number of qubits when the charger remains fixed, we now analyze the scaling of
maximum charging power $P_{\rm max}$ based on Eq. (8) as a function of the
number of the qubits $N$. In Fig. 6, it can be found that the maximum stored
energy and the maximum charging power have a clear correlation with $N$ as it
increases. We recall that in Ref. [66], it has been shown that the energy
scales extensively with $N$, while the power shows a superlinear behavior with
$N$. The behavior of the maximum charging power reflects the acceleration
effect of the collective scheme and shows the quantum advantage. Although it
has been pointed out in the literature [25] that the increased quantum
advantage associated with the increasing number of cells in the cavity is
impossible to be achieved at any physically reasonable Hamiltonian (The
Hamiltonian with the most $N$-body interactions). However, with increased
interactions between QB during collective charging, the quantum advantage can
be achieved. We already know the maximum charging power of the Dicke QB
($P\propto N^{\frac{3}{2}}$), so naturally assume that the maximum charging
power takes the following form

$P_{\rm max}\varpropto\beta N^{\alpha},$ (13)

By taking the logarithm, we use linear fitting to obtain the scaling exponent
$\alpha$

$\log(P_{\rm max})=\alpha\log(N)+\log(\beta).$ (14)

If $\alpha$ has greatly surpassed $1.5$ of the corresponding Dicke QB, further
demonstrating the superiority of our QB.

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