Quick charging of a quantum battery with superposed trajectories

Abstract

We propose charging protocols for quantum batteries based on quantum superpositions of trajectories. Specifically, we consider that a qubit (the battery) interacts with multiple cavities or a single cavity at various positions, where the cavities act as chargers. Further, we introduce a quantum control prepared in a quantum superposition state, allowing the battery to be simultaneously charged by multiple cavities (the multiple-charger protocol) or a single cavity with different entry positions (the single-charger protocol). To assess the battery's performance, we evaluate the maximum extractable work, referred to as ergotropy. The primary discovery lies in the quick charging effect, wherein we prove that the increase in ergotropy stems from the quantum coherence initially present in the quantum control. Moreover, the induced “Dicke-type interference effect” in the single-charger protocol can further lead to a “perfect charging phenomenon”, enabling a complete conversion of the stored energy into extractable work across the entire charging process, with just two entry positions in superposition. Furthermore, we propose circuit models for these charging protocols and conduct proof-of-principle demonstrations on IBMQ and IonQ quantum processors. The results validate our theoretical predictions, demonstrating a clear enhancement in ergotropy.

Published by the American Physical Society 2024

Full text

PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
Editors’ Suggestion
Quick charging of a quantum battery with superposed trajectories
Po-Rong Lai ,1,*Jhen-Dong Lin,1,*Yi-Te Huang ,1Hsien-Chao Jan ,1and Yueh-Nan Chen 1,2,†
1Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort),
National Cheng Kung University, Tainan 701, Taiwan
2Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
(Received 28 July 2023; accepted 29 March 2024; published 7 May 2024)
We propose charging protocols for quantum batteries based on quantum superpositions of trajectories. Specif-
ically, we consider that a qubit (the battery) interacts with multiple cavities or a single cavity at various positions,
where the cavities act as chargers. Further, we introduce a quantum control prepared in a quantum superposition
state, allowing the battery to be simultaneously charged by multiple cavities (the multiple-charger protocol) or a
single cavity with different entry positions (the single-charger protocol). To assess the battery’s performance, we
evaluate the maximum extractable work, referred to as ergotropy. The primary discovery lies in the quick charg-
ing effect, wherein we prove that the increase in ergotropy stems from the quantum coherence initially present in
the quantum control. Moreover, the induced “Dicke-type interference effect” in the single-charger protocol can
further lead to a “perfect charging phenomenon”, enabling a complete conversion of the stored energy into ex-
tractable work across the entire charging process, with just two entry positions in superposition. Furthermore, we
propose circuit models for these charging protocols and conduct proof-of-principle demonstrations on IBMQ and
IonQ quantum processors. The results validate our theoretical predictions, demonstrating a clear enhancement
in ergotropy.
DOI: 10.1103/PhysRevResearch.6.023136
I. INTRODUCTION
Quantum batteries (QBs) have emerged as a popular
research topic, providing valuable insights into how ther-
modynamics functions at the quantum scale [1–4]. Recent
development has demonstrated that various quantum re-
sources such as entanglement [5,6] or coherence [7,8] can
enhance the performance of quantum batteries in terms
of charging [9–19], storage [20–26], and work extraction
[27–37], etc. One of the intriguing phenomena used to achieve
these enhancements is the collective effects triggered by a
group of QBs [10,11,21,38–42]. In his seminal paper [43],
Dicke characterized one of the collective effects, superra-
diance, by the quantum interference of emissions from an
ensemble of atoms. Recent investigations have also demon-
strated the utility of the time-reversed phenomenon, known as
superabsorption [39,40,44,45], on enhancing the capabilities
of QBs.
In this work, our focus lies on an interferometric ap-
proach known as “superpositions of trajectories” [46–57].
This approach utilizes a Mach-Zehnder interferometer (MZI)
and treats an atom’s space-time trajectories as a quantum
*These authors contributed equally to this work.
†yuehnan@mail.ncku.edu.tw
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and the published article’s title, journal citation, and DOI.
system, enabling the exploration of quantum interference
of these trajectories. A notable outcome of this approach
is the effective noise mitigation in various quantum infor-
mation tasks [50,51,53,56,57]. In our recent work [54], we
have further advanced the understanding by interpreting this
noise mitigation as a Zeno-type state freezing phenomenon
[58] within the framework of open quantum systems [59].
Additionally, we have demonstrated that this approach can
also manifest Dicke-type interference effects even when only
one single atom is involved. Building upon these insights,
this work aims to delve into the potential of leveraging
superpositions of trajectories to enhance the performance
of QBs.
We consider a qubit acting as the quantum battery, gain-
ing energy through interactions with cavities functioning as
chargers. To assess the QB’s performance, we focus on the
maximum extractable work, known as ergotropy [27]. The
ergotropy is bounded by the stored energy, i.e., the change
in the qubit’s internal energy, which can be regarded as a
consequence of energy conservation. To utilize superposed
trajectories, we propose two charging protocols using an MZI
setup akin to scenarios in Ref. [54]. The first one is called the
multiple-charger protocol, which consists of multiple identi-
cal chargers (cavities), and the QB can interact with these
chargers in a manner of quantum superposition via a mul-
tiport beam splitter. In principle, there are multiple output
beams of QB when it exits the interferometer. This enables
us to adjust the work extraction strategy for each output and
obtain an average ergotropy, also known as the daemonic
ergotropy [28,60]. The primary result of this protocol is an
“activation” of the ergotropy. Specifically, we demonstrate
2643-1564/2024/6(2)/023136(15) 023136-1 Published by the American Physical Society

LAI, LIN, HUANG, JAN, AND CHEN PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
that when the battery is charged by a single cavity (without
utilizing superposed trajectories), the ergotropy remains zero
for a certain period, despite storing energy immediately after
interaction with the cavity. Thus, there exists a finite delay
before the battery can store “useful energy”, i.e., extractable
work. According to the definition of ergotropy, population
inversion, i.e., the excited state population of the qubit be-
ing larger than its ground state population, is required to
obtain a nontrivial ergotropy. Therefore, one must wait until
the battery reaches the inversion point to obtain extractable
work.
Remarkably, we demonstrate that by considering this
multiple-charger protocol, nonzero ergotropy can be obtained
right after the charging process begins. This implies that this
protocol enables the achievement of “quick charging” for the
QB, where the ergotropy can be activated before reaching the
inversion point. We observe that the ergotropy increases as
the number of superposed trajectories grows. In the limit of
an infinite number of superposed trajectories and considering
the strong-coupling regime (where the rotating-wave approxi-
mation is valid), we observe a “perfect charging result” with a
complete conversion of stored energy into extractable work
throughout the entire charging process. To gain a deeper
insight, we prove that the initial coherence of the quantum
control is the necessary resource for ergotropy enhancement.
Furthermore, we reveal that the degree of the ergotropy en-
hancement is proportional to the initial coherence.
The second protocol is coined the single-charger protocol
with only one cavity (charger) involved. In this protocol, the
QB can enter from different positions into the cavity, ex-
periencing different coupling strengths with the cavity. By
using the superposed trajectories, the QB can enter these
various positions simultaneously. As indicated in Ref. [54],
this particular setup can induce the Dicke-type interference
effect. Notably, we show that two entry positions in quantum
superposition are sufficient to trigger the “perfect charging
result” regardless of the coupling strength. Furthermore, we
investigate the potential work costs for introducing the quan-
tum control and the impact of scaling up the number of qubit
batteries. The results suggest that the single-charger proto-
col, harnessing the underlying Dicke-type interference effect,
emerges as a more advantageous design, offering benefits in
terms of both resource consumption and overall performance.
Additionally, we present quantum circuits designed for the
aforementioned charging protocols, requiring fewer than 20
two-qubit gates. We implement and execute these circuits
on both IonQ quantum processors (based on trapped ions)
and IBMQ quantum processors (based on superconducting
circuits). The experimental results obtained from these imple-
mentations further validate the increase in ergotropy, which is
consistent with our theoretical predictions.
The rest of the paper is organized as follows. In Sec. II,
we characterize the multiple-charger protocol. In Sec. III,we
further investigate the single-charger protocol. In Sec. IV,we
investigate the potential work cost for introducing the quan-
tum control. In Sec. V, we explore the impact of scaling up
the number of batteries for both protocols. In Sec. VI, we con-
sider the circuit implementations and present the experimental
results of the devices from IBMQ and IonQ. Finally, we draw
our conclusions in Sec. VII.
FIG. 1. The quantum battery Qis first sent into a multiport beam
splitter (MPBS1), which allows the quantum battery to travel along
Ndifferent trajectories (denoted by |jDwith j=1...N) in a man-
ner of quantum superposition. We consider two charging processes:
(a) the trajectories each lead to a charger (cavities) {Cj}, causing the
quantum battery to interact with all chargers simultaneously, (b) the
trajectories lead to a single charger Cbut at different positions {rj},
causing the quantum battery to interact with the charger with various
coupling strengths. Once the charging process is completed, a second
multiport beam splitter (MPBS2) is used to perform measurement on
the trajectories’ degree of freedom D. This measurement captures the
quantum interference effect between different trajectories and results
in Npossible reduced states ρj. We then extract work from each
ρj, where the maximum amount of extractable work is called the
ergotropy. The work extraction operations are described by unitary
operators Uj, which transform each of the batteries to a passive
state ϕj.
II. MULTIPLE-CHARGER PROTOCOL: QUICK
CHARGING THROUGH QUANTUM COHERENCE
In this section, we formulate the multiple-charger scenario,
which can be described by a MZI as shown in Fig. 1(a).The
MZI is constructed by two multiport beam splitters, MPBS1
and MPBS2. The first beam splitter (MPBS1) bifurcates the
battery’s trajectories, while the second one (MPBS2) enables
the quantum interference between these trajectories. In the
following, we demonstrate the quick charging effect through
the interferometric setup. Further, we prove that the ergotropy
enhancement originates from the quantum coherence of the
superposed trajectories.
Let us now delve into the detailed description of the charg-
ing protocol, which consists of three different components:
(i) A qubit Q, which acts as the quantum battery. (ii) N
identical single-mode cavities {Cj}j=1...N, which act as the
chargers. The QB moves at a speed vand gets charged when
it passes through one of the chargers. Suppose that the cav-
ity length is l. Then, the interaction time reads as τ=l/v.
To simplify our discussions, we assume that the cavity is
homogeneous such that the interaction strength between the
QB and the charger remains constant during the charging
process [61]. (iii) We characterize the trajectory degrees of
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QUICK CHARGING OF A QUANTUM BATTERY … PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
freedom as an N-dimensional qudit D, wherein we associate
Ndifferent trajectories inside the interferometer with Nbasis
states {|jD}j=1...N. When the QB takes the path labeled by
j, it interacts with the charger Cj. In other words, Dacts
as a quantum control that determines which charger the QB
interacts with. The total Hamiltonian involving these three
components can then be written as
Htot =
N

j=1|jj|D⊗HQCj.(1)
The Hamiltonian HQCjof the quantum battery Qand the
charger Cjcan be expressed as
HQCj=HQ+HCj+H
QC j,
HQ=¯h
2ωaˆσz=¯h
2ωcˆσz,
HCj=¯hωcˆa†
jˆaj,
H
QC j=¯hωcλˆσx(ˆaj+ˆa†
j).(2)
Here, ˆaj(ˆa†
j) annihilates (creates) a photon in Cjwith fre-
quency ωc,¯hωarepresents the energy splitting between the
ground state |gand the excited state |eof Q. The Pauli
operators are therefore given by ˆσz=|ee|−|gg|and
ˆσx=|eg|+|ge|. Moreover, the dimensionless constant λ
represents the coupling strength between Qand all the charg-
ers. Throughout this work, we focus on the resonant regime
ωa=ωc.
We first send the battery Qinto the multiport beam splitter
(MPBS1 in Fig. 1). In general, the beam splitter can prepare
the trajectories in a quantum superposition state so that Qcan
be charged by these Nchargers simultaneously. For simplicity,
we assume the superposition state of the trajectories is
|ψD=1
√N
N

j=1|jD.(3)
We consider that the battery and chargers are initialized in the
ground state and the single-photon Fock state, respectively.
Therefore, the total initial state reads as
|ψ(0)DQC =1
√N
N

j=1|jD⊗|gQ
N

j=1|1Cj.(4)
After Qinteracts with the chargers, according to Eq. (1), the
total states becomes
|ψ(τ)DQC =1
√N
N

j=1|jD⊗|φj(τ)QC ,(5)
where |φj(τ)QC is defined as
|φj(τ)QC =exp −iτ
¯hHQC j⎛
⎝|gQ
N

j=1|1Cj⎞
⎠.(6)
Finally, we make these trajectories interfere with one an-
other by using another beam splitter (MPBS2 in Fig. 1). In
principle, MPBS2 has Ndifferent outputs, which can be de-
scribed using a set of orthonormal projectors {Pk}kacting on
D, namely,
Pk=|ξkξk|D,
N

k=1
Pk=1,
ξk|ξk=δk,k∀k,k.
(7)
Therefore, the (unnormalized) reduced state for the system Q
with the output kreads as
σk(τ)=TrCD[Pk|ψ(τ)ψ(τ)|DQC Pk].(8)
Note that the probability of obtaining the outcome kis
pk(τ)=Tr[σk(τ)]. Thus, the normalized state conditioned on
the outcome kcan be written as ρk(τ)=σk(τ)/pk.
Throughout this work, we choose
|ξk=1ξk=1|D≡1
N
N

m,n=1|mn|D.(9)
According to the assumption that all chargers are identical as
well as the orthonormality of the projectors, in Appendix A,
we show that the explicit form of the rest of the projectors is
irrelevant, enabling us to further simplify the analysis.
Here, we evaluate the performance of a QB by consider-
ing the ergotropy, which quantifies the maximum extractable
work. Given a charged state of the QB ρ(τ), the ergotropy is
defined as
W[ρ(τ)] ≡Tr[ρ(τ)HQ]−min
UTr[Uρ(τ)U†HQ]
=Tr[ρ(τ)HQ]−Tr[ϕ(τ)HQ],(10)
where Urepresents the unitary operation for work extraction.
Also, ϕis known as the passive state [27] (associated with
ρ), which cannot provide useful work for all possible work
extraction operations U. According to Ref. [27], the passive
state of the battery can be written as
ϕ=s0|ee|+s1|gg|.(11)
Here, s0and s1denote the eigenvalues of ρwith s0<s1.Note
that the upper limit of the ergotropy is set by the stored energy
quantified by the difference in the internal energy of the QB
before and after charging, namely,
E(τ)≡
N

k=1
pk(τ)Tr[HQρk(τ)]−Tr(HQ|gg|)
=Tr[HQρ(τ)] −Tr(HQ|gg|),(12)
where ρ(τ)=kσk(τ). We can obtain WEbecause
Tr(HQ|gg|)Tr(HQϕ) in general. The inequality saturates
if and only if ϕ=|gg|, which implies that ρis a pure state.
Further, by employing the Bloch representation of the qubit
state, one can find that the extractable work increases as the
purity of the qubit also increases (see Appendix Bfor the
detailed derivations).
As aforementioned, in our charging protocol, there are N
different outputs (labeled as {k}). In principle, one can find the
optimal work extraction strategies for each output and obtain
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LAI, LIN, HUANG, JAN, AND CHEN PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
the average ergotropy, i.e.,
W=
k
pkW[ρk(τ)]
=
k
pk{Tr[ρk(τ)HQ]−Tr[ϕk(τ)HQ]},(13)
where ϕkdenotes the passive state associated with ρk.Fol-
lowing similar reasoning as mentioned earlier, the average
ergotropy is also upper bonded by the stored energy, and the
optimal extractable work can be obtained, i.e., W(τ)=E(τ),
if and only if {pk,ρ
k}forms a pure state decomposition of
ρ. In addition, the average extractable work increases as the
average purity for the ensemble {pk,ρ
k}increases (see Ap-
pendix Bfor the detailed derivations).
Let us begin with the strong coupling regime [62] (i.e.,
λ0.1), where the rotating-wave approximation can be em-
ployed. In this case, the interaction Hamiltonian of the QB
and the chargers in Eq. (2) can be reduced to the Jaynes-
Cummings model, namely,
˜
H
QC j=¯hωcλ(ˆσ+ˆaj+ˆσ−ˆa†
j),(14)
where ˆσ+=|eg|and ˆσ−=|ge|represent the creation and
annihilation operators of Q, respectively. We can then evaluate
Eq. (6) in this case:
|φj(τ)QC =−isin (ωcλτ )|eQ⊗ˆaj
N

j=1|1Cj
+cos (ωcλτ )|gQ
N

j=1|1Cj.(15)
Let us start from the simplest case with only one charger (i.e.,
N=1), where the reduced state of Qis expressed as
ρ(τ)=sin2(ωcλτ )|ee|+cos2(ωcλτ )|gg|.(16)
In this case, the stored energy is
E(τ)=¯hωcsin2(ωcλτ ),(17)
which oscillates with a period T=2π/(ωcλ). We now focus
on the time interval τ∈[0,T/4] (such that ωcλτ ∈[0,π/2]),
where the stored energy monotonically increases from 0 to its
maximum value ¯hωc. Note that ρis diagonalized under basis
{|e,|g}. Thus, according to Eq. (10), the criterion for obtain-
ing nonzero ergotropy is the moment that population inversion
occurs, where the excited-state population becomes larger
than the ground-state population (e|ρ(τ)|e>g|ρ(τ)|g).
The time dependence of the ergotropy can then be derived as
W(τ)=0if0τ<T
8,
¯hωc[2 sin2(ωcλτ )−1] if T
8τT
4.(18)
One can observe that in the duration τ∈[0,T/8], although
the stored energy Emonotonically increases, there is no ex-
tractable work, W=0, for the battery because ρremains a
passive state during this period.
We now consider the scenario involving Nchargers. When
the selective measurements satisfy Eqs. (7) and (9), the un-
normalized postmeasurement states can be written as (see
0π/4π/2
τ[ω−1
cλ−1]
0
1
E,W [¯hωc]
E(τ)
W[N=4]
W[N=2]
W[N=1]
FIG. 2. The stored energy Eand average ergotropy W(both in
units of ¯hωc) on time τ(in units of 1/ωc)forλ=0.05. The black
solid curve plots the stored energy while the dashed curves plot
the average ergotropy. From bottom to top, the blue, red, and green
dashed curves show the results for N=1, 2, and 4, respectively.
Appendix Afor detailed derivations)
σk=1(τ)=1
Nsin2(ωcλτ )|ee|+cos2(ωcλτ )|gg|,
σk=1(τ)=1
Nsin2(ωcλτ )|ee|.(19)
Here, for the case k=1, the postmeasurement state is passive
during the time period τ∈[0,TN] with the inversion time
TN=tan−1(√N)T/2π. Remarkably, for the cases of k= 1,
the postmeasurement states are exactly the excited state, im-
plying that the maximal extractable work ¯hωccan be obtained.
Therefore, the average ergotropy can be expressed as
W(τ)=W[σk=1(τ)] +(N−1)W[σk=1(τ)]
=¯hωcN−1
Nsin2(ωcλτ )if 0 τTN,
¯hωc[2 sin2(ωcλτ )−1] if TNτT
4.(20)
In Fig. 2, we present the time-dependent stored energy and
the average ergotropy for different values of N. In contrast to
the case of N=1, we observe nonzero average ergotropy for
the entire interval of interest because the states with k= 1are
nonpassive right after the QB-chargers interaction is turned
on (τ>0). Therefore, the protocol can be used for “quick
charging”, enabling immediate storage of useful work after the
charging process begins. Furthermore, the result indicates that
increasing Ndelays the inversion time TNand enhances the
average ergotropy before TN. To gain a deeper understanding,
we present the following proposition.
Proposition. The initial coherence of the control qudit
serves as a necessary resource for the ergotropy enhance-
ment. In addition, the degree of the enhancement is linearly
proportional to the coherence for our scenario. We pro-
vide the detailed proof and derivations in Appendix C.
In essence, we demonstrate that the lack of initial coher-
ence leads to the control-battery state remaining uncorrelated
throughout the entire charging process, thereby resulting in
no enhancement in the ergotropy. Further, when all the off-
diagonal terms of the control qudit are identical, the degree
of the enhancement is directly proportional to the amount of
the initial coherence. Consequently, it can be inferred that the
023136-4

QUICK CHARGING OF A QUANTUM BATTERY … PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
FIG. 3. (a) The stored energy Eand average ergotropy W(both in units of ¯hωc) as functions of time τ(in units of 1/ωc)forλ=0.5.
(b) The stored energy Eand the average ergotropy contributed by states of k=1,Wk=1. Two brown vertical lines at τ=0.63,1.00 indicate
the inversion points TNfor N=1 (blue) and N=2 (red). Here, Wk=1is 0 when N=7 (green) and N=∞(magenta). (c) The stored energy
Eand the average ergotropy contributed by states of k= 1,Wk=1. Except for N=1 (blue), these states contribute average ergotropy when
τ>0. (d) The dashed curves plot the change in average purity Pagainst time τ. The cutoff photon number is set to 9 in the above results.
quick charging effect stems from the initial coherence of the
quantum control. Furthermore, from the perspective of qubit-
only dynamics presented in Eq. (19), the enhancement can
be further linked to the Zeno-type state freezing effect [54],
for the postmeasurement state with k=1. Specifically, as the
number of superposed trajectories Nincreases, the excited-
state population of the postmeasurement state σk=1decreases,
indicating its convergence towards the ground state. Con-
sequently, the qubit dynamics associated with this specific
outcome decelerates. In the asymptotic limit (N→∞), the
population of the excited state for this outcome diminishes
to zero, resulting in the qubit frozen in the ground state.
Notably, as the postmeasurement state with k=1 approaches
the ground state, its purity also increases. Additionally, as
inferred from Eq. (19), the postmeasurement state with k= 1
stays in the (pure) excited state throughout the entire charging
process. Thus, the Zeno-type effect contributes to an over-
all increase in the average purity, consequently enhancing
the ergotropy as previously mentioned. Finally, with infinite
superposed trajectories, the postmeasurement states are pure
(namely, σk=1(τ)∝|gg|and σk=1(τ)∝|ee|). This signi-
fies a “perfect charging result”, where the stored energy can
be completely converted into extractable work over the entire
time interval, i.e, E(τ)=W(τ).
We now extend our scope of discussion into the ultrastrong
coupling regime [62], where the rotating-wave approximation
is no longer valid. In Fig. 3(a), we present the dynamics of
the average ergotropy and the stored energy. We can still ob-
serve the quick charging effect, a delay of the inversion point,
and an enhancement in average ergotropy as Nincreases. As
shown in Figs. 3(b) and 3(c), we further present the individual
contributions of the average ergotropy from the outputs k=1
and k= 1, which are, respectively, defined by
Wk=1=pk=1W[ρk=1(τ)] and Wk=1=
k=1
pkW[ρk(τ)].
(21)
We can observe that the inversion points come from the contri-
bution of k=1. Further, we can observe that its contribution
decreases as Nincreases. When N7, the contribution van-
ishes for the entire charging period. In Appendix A,we
provide analytical analysis, showing that the decrease in av-
erage ergotropy can also be attributed to the Zeno-type state
freezing effect. More specifically, we demonstrate that the
excited-state population for the postmeasurement state with
k=1 decreases when Nincreases. In the asymptotic limit
(N→∞), the state can be frozen in the ground state. We
further observe that the quick charging effect originates from
the contribution of k= 1 since the corresponding average
ergotropy becomes nonzero as soon as the charging process
begins when N>2.
However, in contrast to the previous results with the
rotating-wave approximation, the average ergotropy cannot
reach the upper bound even in the asymptotic limit, implying
that the postmeasurement states are not pure. In Fig. 3(d),we
present the average purity Passociated with the postmeasure-
ment states, which is defined by
P=
k
Tr[σk]Trσk
Tr[σk]2.(22)
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LAI, LIN, HUANG, JAN, AND CHEN PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
0.05 0.25 0.50
λ
0.5
1.0
Wmax
[N=1]
[N= 10]
[N= 100]
[N=∞]
0.05 0.25 0.50
λ
0.6
1.0
P
[N=1]
[N= 10]
[N= 100]
[N=∞]
(b)
(a)
FIG. 4. (a) Maximum average ergotropy Wmax defined in
Eq. (23) with respect to different coupling strength λ. The blue, red,
green, and magenta dashed lines represent the results for N=1, 10,
100, and ∞, respectively. (b) The average purity Pon dimensionless
coupling strength λ. The blue, red, green, and magenta dashed lines
represent the results for N=1, 10, 100, and ∞, respectively. The
cutoff photon number is set to 9 in the above results.
We can observe the overall average purity increasing as N
increases due to the Zeno-type effect, thereby leading to the
enhancement of average ergotropy. However, the average pu-
rity cannot reach unity even in the asymptotic limit. Thus, the
stored energy cannot be fully converted into extractable work.
In Figs. 4(a) and 4(b), we compare the maximum average
ergotropy with the average purity with respect to the coupling
strength λ, where the maximum average ergotropy is defined
by
Wmax =max
τ∈[0,T/4] W(τ).(23)
We can observe that as the coupling strength increases, the
average purity drops, hence leading to a decrease in the max-
imum average ergotropy.
III. SINGLE-CHARGER PROTOCOL
In this section, we discuss the single-charger protocol as
shown in Fig. 1(b), where the quantum battery can enter and
interact with a single cavity (charger) Cat different positions.
The charging protocol resembles the one in Sec. II, albeit with
two minor adjustments. First, we only use a single charger C
instead of multiple ones. Second, the N-dimensional qudit D
now acts as a quantum control that decides the QB’s position
of entry into C. More specifically, a path labeled by jguides
the QB to a designated position denoted as rjinside the cavity.
In this scenario, the QB experiences a varying coupling
strength with respect to the position inside that charger [63].
Therefore, the position-dependent QB-charger Hamiltonian
can be written as
Hj=HQ+HC+H
j,(24)
where HC=¯hωcˆa†ˆa. Here, H
jrepresents the interaction
Hamiltonian when QB is located at the position rj. Its explicit
form is given by
H
j=¯hωcλcos πrj
Lˆσx(ˆa+ˆa†),(25)
where Ldenotes the width of the charger and λrepresents
the maximal QB-charger coupling strength, which can be
achieved when rj=0orL. Thus, the total Hamiltonian for
this scenario can be expressed by
Htot =
N

j=1|jj|D⊗Hj.(26)
Similar to the previous consideration, we prepare the total
system in the following initial state:
|ψ(0)DQC =1
√N
N

j=1|jD⊗|gQ⊗|1C,(27)
and allow it to evolve according to the total Hamiltonian Htot,
namely,
|ψ(τ)DQC =1
√N
N

j=1|jD⊗|φj(τ)QC ,(28)
where |φj(τ)QC , in this case, is defined as
|φj(τ)QC =exp −iτ
¯hHj(|gQ⊗|1C).(29)
We also consider the projectors defined in Eq. (7) to character-
ize the measurements performed by using MPBS2, such that
the corresponding postmeasurement states read as
Pk|ψ(τ)DQC =|ξkD⊗
N

j=1
ck,j|φj(τ)QC ,(30)
where the coefficient ck,jis given by
ck,j=1
√Nξk|j.(31)
We now switch to the interaction picture, such that the
interaction Hamiltonian associated with the position rjcan be
expressed as
H
j(τ)=ei
¯h(HQ+HC)τH
je−i
¯h(HQ+HC)τ
=cos πrj
LH
I(τ),(32)
where H
I(τ) is the position-independent part and reads as
H
I(τ)=¯hωcλei
¯h(HQ+HC)τˆσx(ˆa+ˆa†)e−i
¯h(HQ+HC)τ.(33)
023136-6

QUICK CHARGING OF A QUANTUM BATTERY … PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
We can now characterize the time evolution with the propaga-
tor Uj,I(τ,0) in terms of the Dyson series, namely,
Uj,I(τ,0) =ˆ
Texp −i
¯hτ
0
H
j(t)dt
=∞

n=0
1
n!−i
¯hcos πrj
Ln
×τ
0
dt1···τ
0
dtnˆ
TH
I(t1)...H
I(tn),(34)
where ˆ
Tis the time-ordering operator with t1>t2>···>
tn. The postmeasurement state of the QB and the charger in
Eq. (30) can then be expressed by
N

j=1
ck,j|φj(τ)QC =
N

j=1
ck,jUj,I(τ,0)|gQ⊗|1C.(35)
Therefore, one can observe that the total evolution (includ-
ing the qudit, the QB, and the charger) is described by a
linear combination of the position-dependent propagators,
which characterizes the collective quantum interference effect
among different positions [54].
We now show that two superposed trajectories (positions)
can lead to the saturation of the ergotropy to its upper bound
with an appropriate adjustment of the Dicke-type interference
effect. More specifically, we consider that the two positions
satisfy r1+r2=L, such that
cos πr2
L=cos π−πr1
L=−cos πr1
L.(36)
Therefore, the coupling strengths share the same magnitude
but are completely out of phase. Furthermore, we consider
|ξk=1= 1
√2(|1D+|2D),
|ξk=2= 1
√2(|1D−|2D),(37)
and, according to Eq. (31), the coefficients in this case are
c1,1=c1,2=c2,1=−c2,2=1
2.(38)
Therefore, the time evolution of the postmeasurement state of
the QB and the charger for k=1 is given by
2

j=1
c1,jUj,I(τ,0)|gQ⊗|1C
=1
2[U1,I(τ, 0) +U2,I(τ,0)] |gQ⊗|1C
=∞

neven
1
n!−i
¯hcos πr1
Ln
×τ
0
dt1···τ
0
dtnˆ
TH
I(t1)...H
I(tn)|gQ⊗|1C.
(39)
We can observe that in the Dyson series, all the odd terms
vanish, leading to the phenomenon of destructive interfer-
ence. This effect originates from the complete out-of-phase
nature of the coupling strengths for the two positions. As a
direct consequence, the QB remains in the ground state |g
throughout the entire process. Analogously, the postmeasure-
ment state for k=2 can be written as
2

j=1
c2,jUj,I(τ,0)|gQ⊗|1C
=1
2[U1,I(τ, 0) −U2,I(τ,0)] |gQ⊗|1C
=∞

nodd
1
n!−i
¯hcos πr1
Ln
×τ
0
dt1···τ
0
dtnˆ
TH
I(t1)...H
I(tn)|gQ⊗|1C.
(40)
In this case, all the even terms in the Dyson series vanish,
implying the QB is in the excited state |efor all τ>0
(with zero probability of obtaining the output k=2atτ=0).
Notably, because the postmeasurement QB states for these
two outputs are pure states (i.e., the average purity is one),
we can conclude that the stored energy can be fully converted
to the extractable work, i.e., E=W, throughout the whole
charging process. Note that the presented analysis does not
rely on the rotating-wave approximation, thus indicating that
the “perfect charging result” holds for all regimes of the QB-
charger coupling strength.
It is worth emphasizing that achieving the perfect charging
result only requires two superposed trajectories (positions).
This indicates that a single-charger protocol emerges as a
more experimentally feasible charging design with superior
performance. In the subsequent sections, we delve into poten-
tial energy costs associated with various charging protocols
and analyze the effects of scaling up the number of batteries.
Our findings can reinforce that the single-charger protocol
holds more advantages compared to the multiple-charger pro-
tocol.
IV. ENERGY COSTS OF THE QUANTUM CONTROL
As previously mentioned, the enhancement of ergotropy
in the proposed protocols hinges on quantum control and its
coherence. Notably, this may entail additional energy con-
sumption, especially when considering energetic quantum
control platforms such as the IBMQ and the IonQ experi-
ments discussed below. In this section, we investigate two key
aspects regarding potential energy consumption: The energy
costs associated with (1) establishing the initial coherence and
(2) the measurement.
A. Energy cost for creating coherence
Suppose that Dis initialized in a state ρDand HDis
the corresponding Hamiltonian. The coherence can be in-
duced by performing a unitary transformation V. According
to Ref. [64], the work cost for generating coherence can be
estimated by the change of D’s internal energy, expressed as
Wcoh =Tr[HD(VρDV†−ρD)].(41)
023136-7

LAI, LIN, HUANG, JAN, AND CHEN PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
0π/4π/2
τω
−1
c
λ
−1
0
0.5
ΔW[¯hω
c
]
ΔWfor 2 paths
ΔWfor 7 paths
ΔWfor infinite paths
0π/4π/2
τω
−1
c
λ
−1
0
0.5
ΔW[¯hω
c
]
ΔW[N=2]
ΔW[N=4]
ΔW[N=∞]
0π/4π/2
τω
−1
c
λ
−1
0
0.7
S,I
S(ρ)
I[N=2]
I[N=4]
I[N=∞]
0π/4π/2
τω
−1
c
λ
−1
0
0.7
S,I
S(ρ)
I[N=2]
I[N=7]
I[N=∞]
(a)
(b)
(c)
(d)
FIG. 5. Under the multiple-charger protocol, the numerical simulations for the von Neumann entropy of the premeasurement state
S(ρ) and quantum-classical mutual information Iwith (a) λ=0.05 and (c) λ=0.5, and the ergotropy gain Wwith (b) λ=0.05 and
(d) λ=0.5.
For the IBMQ and the IonQ experiments outlined in the fol-
lowing, we utilize a physical qubit as quantum control with
a Hamiltonian HD=¯hωDσz/2, where ωDis the correspond-
ing working frequency. The qubit is initialized in the ground
state |gD, and the target superposition (coherent) state is
(|gD+|eD)/√2, achievable through a Hadamard transform.
Consequently, the corresponding work cost is ¯hωD/2. One can
expect that the work cost increases proportionally with the
dimension of D, i.e., the number of superposed trajectories.
From this perspective, the single-charger protocol can be more
advantageous than the multiple-charger protocol, as achieving
perfect charging result in the former scenario requires only
two trajectories, while the latter demands an infinite number
of trajectories (and consequently infinite energy to create co-
herence) for achieving the perfect charging result.
B. Work cost for the measurement and the postselection
The proposed charging protocols also require the quan-
tum control to serve as Maxwell’s demon, where the energy
required to perform the measurement and reset the demon’s
memory shall be taken into account. To this end, we adopt the
framework proposed in Ref. [65], where the work costs for
performing the measurement and resetting the memory can be
characterized by the following relation:
WM
meas +WM
eras kBTI (42)
with kBbeing the Boltzmann constant and Trepresenting the
temperature of a thermal bath used to reset the memory. Thus,
the work costs for the measurement WM
meas together with the
memory erasure WM
eras are lower bounded by the quantum-
classical mutual information I, which is defined by
I=S(ρ)−
k
p(k)S(ρk),(43)
where Sdenotes the von Neumann entropy. Here, ρdenotes
the premeasurement quantum state of the battery, while p(k)
represents the probability of obtaining the outcome kwith a
corresponding postmeasurement state ρk. Note that IS(ρ),
and the upper limit is achieved when all the postmeasurement
states are pure, i.e., S(ρk)=0∀k.
According to Eq. (42), we estimate the cost for the
measurement through the quantum-classical mutual infor-
mation in the following analysis. In Fig. 5, we showcase
the quantum-classical mutual information and the ergotropy
gain corresponding to the multiple-charger protocol. The er-
gotropy gain Wis defined as the enhancement in the average
ergotropy for multiple superposed trajectories (N>1) com-
pared to the case with N=1:
W≡W[˜
N]−W[N=1]with ˜
N>1.(44)
In general, as we increase the number of superposed tra-
jectories, Ialso increases, signifying an increase in the work
cost associated with the measurement. Further, a comparison
between the top and bottom figures reveals that ergotropy
023136-8

QUICK CHARGING OF A QUANTUM BATTERY … PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
(a)
(b)
(c)
(d)
E[M=1] W[M=1] E[M=3] W[M=3] E[M=5] W[M=5]
FIG. 6. The stored energy Eand average ergotropy Wfor different numbers (M) of QBs under two superposed trajectories (N=2) but
different charging protocols. (a) Multiple-charger protocol with the coupling strength λ=0.05. (b) Multiple-charger protocol with the coupling
strength λ=0.5. (c) Single-charger protocol with r1=0.1l,r2=0.9l, and the coupling strength λ=0.05. (d) Single-charger protocol with
r1=0.1l,r2=0.9l, and the coupling strength λ=0.5. Note that the rotating-wave approximation is considered for (a) and (c).
enhancement through our protocol requires nonzero quantum-
classical mutual information but not vice versa.Asshownin
Figs. 5(a) and 5(c), the quantum-classical mutual information
remains nonzero throughout the entire charging time. Never-
theless, for the strong coupling case [Fig. 5(b)], the ergotropy
gain diminishes after a finite charging period in cases with a
finite number of superposed trajectories. In such instances, the
information does not assist in work extraction, which suggests
that the energy cost for the measurement becomes wasted. A
similar situation can be observed for the ultrastrong coupling
case [Fig. 5(d)], where seven superposed trajectories are re-
quired so that nonzero ergotropy gain can persist throughout
the entire process. From this perspective, the single-charger
protocol emerges as a more energy-efficient choice due to
the “perfect charging effect”, where nonzero ergotropy gain
persists throughout the entirety of the charging process with
only two superposed trajectories.
V. IMPACTS ON SCALING UP THE NUMBER
OF BATTERIES
As reported in Ref. [11], collective charging for multiple
quantum batteries can further accelerate the charging pro-
cess. Here, we present numerical simulations for scaling up
the number of the qubit batteries, revealing the interplay be-
tween the collective charging effect and the quick charging
effect in the proposed protocols. Specifically, we modify the
qubit-charger Hamiltonians for the multiple-charger and the
single-charger protocols, which are, respectively, expressed
by
HQCj=¯hωcˆa†
jˆaj+ωaˆ
Jz+2ωcλˆ
Jx(ˆa†
j+ˆaj),
Hj=¯hωcˆa†ˆa+ωaˆ
Jz+2ωcλcos πrj
Lˆ
Jx(ˆa+ˆa†).
(45)
Here, ˆ
Jα=(¯h/2) N
iˆσα
irepresents the components of the
collective spin operator in terms of the Pauli operators for
the ith QB. Initially, the batteries reside in their ground state,
while each cavity charger harbors one photon. In Fig. 6,we
present the results with two superposed trajectories (N=
2) for both the multiple-charger [Figs. 6(a) and 6(b)] and
single-charger protocols [Figs. 6(c) and 6(d)]. As the num-
ber of batteries (M) increases, the charging of energy and
ergotropy accelerates. This observation aligns with previ-
ous studies, suggesting that the collective effect of multiple
quantum batteries can expedite charging processes [11]. In
addition, the phenomenon of quick charging persists for both
charging protocols, with ergotropy becoming available at the
outset of the charging process. Furthermore, the available er-
gotropy for the single-charger protocol is generally larger than
that for the multiple-charger protocol due to the additional
superposition-induced Dicke-type interference effect [66].
Notably, in the strong coupling case for the single-charger
protocol [Fig. 6(c)], the perfect charging effect persists in
the multiple-battery scenario. These findings further en-
hance the potential utility of the single-charging protocol and
the underlying superposition-induced Dicke-type interference
023136-9

LAI, LIN, HUANG, JAN, AND CHEN PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
(c)
DQC1
C2
X
Z
(a) Charging process Measurement
Preparation
D:|+ +|
C2:|1 1|
C1:|1 1|
Q:|00|exp[−iHQC1τ/ ]
exp[−iHQC2τ/ ](d)
(b)
exp[−iHQC1τ/ ]=XX [θ(τ)] YY [θ(τ)]YY [−θ(τ)] =
XX [θ(τ)]
Tr
Rz [θ(τ)]
H
HH
H
XX [−θ(τ)]
Tr
State (multiple-chargers)
FIG. 7. (a) Quantum circuit for multiple-chargers protocol. Here, D,Q,C1,C2represent the control qubit, battery qubit, first charger, and
second charger, respectively. “Tr” is trace out. (b) Decomposition of a controlled unitary in (a). (c) Qubit configuration used on ibmq_algiers.
(d) Decomposition of a XX gate into CNOT gates.
effect. Finally, we remark that for the ultrastrong coupling
case [Figs. 6(b) and 6(d)], the results indicate not only the
acceleration of the charging process but also an increase in the
amount of energy stored in the batteries. This originates from
the breakdown of excitation conservation when the coupling
strength between the batteries and the chargers is sufficiently
strong.
VI. IMPLEMENTATION ON QUANTUM DEVICES
In this section, we provide circuit models for the proposed
charging protocols and perform proof-of-concept experiments
on the quantum processors provided by IBMQ and IonQ,
which involves two superposed trajectories (N=2).
The quantum circuit for the multiple-charging setup is
described by Fig. 7(a). The circuit consists of four qubits,
representing the control qubit D, the quantum battery Q, and
the two charging cavities C1and C2, respectively. The circuit
can be divided into three parts: state preparation, charging
process, and measurements on the control qubit and quantum
battery. In the state preparation part, the qubits are prepared
in the initial state specified in Eq. (4) using single-qubit
gates. The charging process involves the utilization of two
controlled-unitary gates to simulate the simultaneous charging
of the qubit by the two cavities through Jaynes-Cummings
interactions. In Fig. 7(b), we present the decomposition of the
controlled unitaries into bit-flip (X) gates, controlled-zgates
(CZ), and Ising coupling gates [XX(θ) and YY (θ)], defined
as follows:
X=ˆσx,
CZ =|00|⊗1+|11|⊗ˆσz,
XX(θ)=cos(θ/2)1⊗1−isin(θ/2) ˆσx⊗ˆσx,
YY(θ)=cos(θ/2)1⊗1−isin(θ/2) ˆσy⊗ˆσy.(46)
Here, we map the charging time τinto the angle θusing
the following relation:
θ(τ)=ωcλτ /2.(47)
Finally, in the measurement part, we measure the control qubit
Din the xdirection, aligned with the projectors described
by Eq. (37). Furthermore, as indicated in Eq. (19), the QB’s
postmeasurement states are diagonalized under the energy
eigenstates. Consequently, we can only measure Qin the zdi-
rection to determine the stored energy as well as the ergotropy.
We utilize the ibmq_algiers and IonQ-Aria 1 devices. Note
that the qubit configuration for ibmq_algiers is illustrated in
Fig. 7(c), while the qubits in IonQ-Aria 1 are fully connected.
Also, since the Ising coupling gates are not native gates for
the IBMQ device, we need to further decompose them into
CNOT gates, which is shown in Fig. 7(d). Consequently, the
circuits for the IBMQ and IonQ devices consist of 20 and 12
two-qubit gates, respectively. Here, we use the Hadamard (H)
gate and the rotation-z[Rz(θ)] gate, which are defined by
H=1
211
1−1,
Rz(θ)=exp(−iˆσzθ/2).(48)
For the single-charger setup, the circuit model for the charging
process is presented in Fig. 8, which consists of 6 and 4 two-
qubit gates in the circuits for the IBMQ and the IonQ devices,
respectively.
Figures 9and 10 illustrate the results obtained from the
two protocols, with each data point representing the aver-
age of 1000 experiment repetitions. The experimental results
demonstrate a notable increase in average ergotropy, aligning
with the theoretical predictions. Furthermore, we can observe
that the deviation between the experimental and theoretical
results is correlated by the circuit size, primarily determined
by the number of two-qubit gates involved. Therefore, com-
paring the results from the IonQ and IBMQ devices, we find
that the experimental data from the IonQ device exhibit a
closer match to the theoretical curves compared to those from
the IBMQ device. In addition, the errors associated with the
D
Q
C
XX [2θ(τ)] YY [2θ(τ)]
Charging process(single-charger)
FIG. 8. Charging process of the single-charger setup, where the
coupling strengths of the two superposed trajectories have the same
magnitude but are out of phase.
023136-10

QUICK CHARGING OF A QUANTUM BATTERY … PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
0π/4π/2
τ[ω−1
cλ−1]
0
1
E,W [¯hωc]
E[Sim.]
E[IBMQ Exp.]
E[IonQ Exp.]
W[Sim.]
W[IBMQ Exp.]
W[IonQ Exp.]
FIG. 9. The stored energy Eand average ergotropy W(both in
units of ¯hωc) on time τ(in units of 1/ωc)forλ=0.05,N=2.
The black solid and dashed curves represent the stored energy and
average ergotorpy predicted by numerical simulations. The green
circles and blue “x”s represent experimental results performed on
ibmq_algiers. The red triangles and magenta diamonds represent
experimental results performed on IonQ Aria 1. Each data point is
obtained after averaging 1000 experimental repetitions.
single-charger protocol are smaller in magnitude than those of
the multiple-charger protocol. Finally, we remark that when
observing the experimental results for the stored energy, the
errors for the cases τ=0 and π/2 are larger compared to the
case τ=π/4. This can be attributed to the depolarizing noise
(i.e., gate error), which is modeled by incoherently mixing the
ideal system’s state with the maximally mixed state [67]. At
the times τ=0 and π/2, the ideal battery states are, respec-
tively, |gand |e, where the corresponding stored energies
are E=0 and ¯hωa. When mixing the maximally mixed state
for these cases, the stored energy, respectively, become larger
and smaller than the ideal results. However, for the ideal
case, the battery is already in the maximally mixed state, i.e.,
(|gg|+|ee|)/2 at the time τ=π/4. Consequently, the
0π/4π/2
τ[ω−1
cλ−1]
0
1
E,W [¯hωc]
E,W [Sim.]
E[IBMQ Exp.]
E[IonQ Exp.]
W[IBMQ Exp.]
W[IonQ Exp.]
FIG. 10. The stored energy Eand average ergotropy W(both
in units of ¯hωc) on time τ(in units of 1/ωc) with r1=0.1l
and r2=0.9l. The black curve represents the stored energy and
average ergotropy predicted by numerical simulations. The green
circle and blue “x”s represent the experimental results performed
on ibmq_algiers. The red triangles and magenta diamonds represent
experimental results performed on IonQ Aria 1. Each data point is
obtained after averaging 1000 experimental repetitions.
depolarizing noise does not change the battery’s state, leaving
the stored energy unchanged. Therefore, by taking the gate
error into account, the results for τ=0 and π/2 demonstrate
a larger error compared to the result for τ=π/4.
VII. SUMMARY AND OUTLOOK
In this work, we utilize superposition of trajectories to
propose two charging protocols (i.e., the multiple-charger
protocol and the single-charger protocol) for quantum bat-
teries (QBs), leading to quick charging effect of extractable
work (ergotropy). We prove that the increase in ergotropy
arises from the initial coherence of the quantum control. In
addition, upon examining the potential resource consumption,
we determine that the single-charger protocol, demonstrating
the perfect charging effect through the utilization of collec-
tive interference, presents a superior scenario compared to
the multiple-charger protocol. We further illustrate that this
advantage persists even when scaling up the number of QBs.
Moreover, we investigate the circuit implementations utilizing
IonQ and IBMQ devices, providing experimental data that
further support the enhanced extractable work, thus validating
our theoretical predictions.
As a possible future direction, we could extend our charg-
ing protocols to a related framework called indefinite causal
order [68–73], which allows for the control of quantum op-
eration ordering through a quantum switch. Based on this
framework, we could consider the scenarios, where the or-
dering of the charging process becomes indefinite [74–77].
This exploration could shed light on the potential benefits and
implications of incorporating indefinite causal order into our
proposed protocols, further advancing the field of quantum
battery charging.
ACKNOWLEDGMENTS
We acknowledge the NTU-IBM Q Hub, IBM quantum ex-
perience, and Cloud Computing Center for Quantum Science
& Technology at NCKU for providing us a platform to imple-
ment the experiment. This work is supported by the National
Center for Theoretical Sciences and National Science and
Technology Council, Taiwan, Grants No. NSTC 112-2123-M-
006-001 and No. NSTC 112-2119-M-006-010.
APPENDIX A: EFFECTS OF THE CHOICE OF
PROJECTORS ON THE POSTMEASUREMENT
QUANTUM BATTERY STATES
Here, we prove that the choice of projectors for
|ξk=1ξk=1|Dis irrelevant to the postmeasurement states
σk=1(τ) as long as they are orthonormal to |ξk=1ξk=1|D.
Let us start from Eq. (8), which can be expanded into
σk(τ)=1
N
N

j,f=1ξk|jf|ξkTrC[|φj(τ)φf(τ)|QC ].
(A1)
023136-11

LAI, LIN, HUANG, JAN, AND CHEN PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
We switch to the interaction picture, where the interaction
Hamiltonian reads as
H
QC j(τ)=ei
¯h(HQ+HCj)τH
QC je−i
¯h(HQ+HCj)τ.(A2)
The time-dependent part of postmeasurement states can be
generally expressed by
|φj(τ)QC =∞

n=0
αn(τ)|nj,(A3)
with
|nj=(ˆσx)n|e⊗(ˆa†
j)nˆaj
√n!
N

m=1|1Cm.(A4)
Note that in the following analysis, we show the explicit
expression of the time-dependent coefficient αn(τ) does not
affect the result. Thus, we will keep them unspecified. Now,
we can show that
TrC[|φj(τ)φf(τ)|QC ]
=∞
n=0|αn(τ)|2σn
x|ee|σn
xfor j=f,
|α1(τ)|2|gg|for j= f.(A5)
Inserting this into Eq. (A1), we can obtain
σk(τ)=∞

n=0|αn(τ)|2σn
x|ee|σn
x
1
N
N

j=f|ξk|j|2
+|α1(τ)|2|gg|1
N
N

j=f
N

f=1ξk|jf|ξk.(A6)
For the case of k=1, according to Eq. (7), we can obtain
ξk=1|j= 1
√N∀j=1,...,N.(A7)
Therefore, the corresponding postmeasurement state can be
written as
σk=1(τ)=1
N
∞

n=0|αn(τ)|2σn
x|ee|σn
x
+N−1
N|α1(τ)|2|gg|.(A8)
Note that in the asymptotic limit (N→∞), we observe
σk=1(τ)∝|gg|, indicating the Zeno-type state freezing ef-
fect [54].
Recall that for the case of k= 1, we require the projec-
tors to be orthonormal to that of k=1, i.e., ξk=1|ξk=1=0,
implying that the ket |ξk=1can be expressed by
|ξk=1=
N

j=1
βk,j|jD,with
N

j=1
βk,j=0∀k= 1.
(A9)
Therefore, the postmeasurement states can be written as
σk=1(τ)=1
N
∞

n=0|αn(τ)|2σn
x|ee|σn
x
−1
N|α1(τ)|2|gg|.(A10)
This concludes the proof that the postmeasurement states are
independent of the explicit expression of the coefficients βk,j
associated with the projectors.
APPENDIX B: RELATION BETWEEN THE AVERAGE
ERGOTROPY AND THE AVERAGE PURITY
Here, we show that an increase in average purity implies an
increase in the average ergotropy. In this work, we mainly fo-
cus on a qubit battery, whose quantum state can be expressed
as
ρ=1
2(1+
v·
σ),(B1)
where 
v=(vx,vy,vz)Tdenotes the Bloch vector and 
σ=
(ˆσx,ˆσy,ˆσz)Trepresents a vector of Pauli matrices. In this
representation, the passive state is given by
ϕ=1
2(1−|
v|ˆσz).(B2)
This corresponds to rotating the Bloch vector of ρinto a
vector that aligns with the negative-zaxis of the Bloch sphere.
The ergotropy can then be expressed as
W(ρ)=¯hωc
2(vz+|
v|).(B3)
Given that the purity of ρis (1 +|
v|2)/2, we can con-
clude that an increase in purity leads to an increase in
ergotropy. This argument also holds for the average ergotropy,
where the average ergotorpy and the average purity are
expressed as
W=
k
pkW(ρk)=¯hωc
2vz+
k
pk|
vk|,
P=1+
k
pk|
vk|22.(B4)
Thus, we can conclude that an increase in average purity also
suggests an increase in the average ergotropy.
APPENDIX C: PROOF OF THE PROPOSITION: QUANTUM
COHERENCE IS A NECESSARY RESOURCE FOR THE
ERGOTROPY GAIN
We now delve into the details of the proof outlined in the
main text. We start by considering an N-dimensional control
qudit Dinitialized in a general quantum state:
ρD=
N

j=1
xj|jj|D+
j=f
yjf |jf|D,(C1)
where xjand yjf represent the diagonal and off-diagonal
(coherence) terms of ρD, respectively. For the multiple-
charger protocol, the temporal evolution of the total state is
written as
ρDQC =
N

j=1
xj|jj|D⊗|φj(τ)φj(τ)|QC
+
j=f
yjf |jf|D⊗|φj(τ)φf(τ)|QC ,(C2)
023136-12

QUICK CHARGING OF A QUANTUM BATTERY … PHYSICAL REVIEW RESEARCH 6, 023136 (2024)
where |φj(τ)QC is described in Eq. (A3). By tracing out the
chargers (cavities), we obtain
ρDQ =
N

j=1
xj|jj|D⊗[α0|ee|Q+(1 −α)|gg|Q]
+
j=f
yjf |jf|D⊗β|gg|Q,(C3)
where α=∞
n,even |αn(τ)|2,1−α=∞
n,odd |αn(τ)|2, and
β=|α1(τ)|2.
The key insight is that if the initial control qudit is in-
coherent, i.e., yjf =0∀j= f, the control and the battery
ρDQ becomes completely uncorrelated, namely, ρDQ =ρD⊗
ρQwith ρD=N
j=1xj|jj|Dand ρQ=[α|ee|Q+(1 −
α)|gg|Q]. Consequently, irrespective of the number of tra-
jectories Nor the measurement basis, we arrive at the same
battery state ρQ(τ), which is also identical to the scenario
with a single charger N=1. This implies that in the absence
of coherence, the battery fails to exhibit the quick charging
effect. The proof is completed, and we assert that coherence
serves as an indispensable resource for the quick charging
phenomenon in the multiple-charger protocol. Furthermore,
employing the same methodology, we can easily conclude that
the initial coherence of the control qudit is equally essential in
the single-charger protocol.
We now focus on our scenarios, where the off-diagonal
terms of ρDare identical. We introduce a parameter 0 1
for tuning the initial coherence of the control qudit and make
the amount of coherence become proportional to , i.e., xj=
1/N∀jand yjf =/N∀j= f. Therefore, the correspond-
ing state is given by
ρD=1
N
j=1|jj|D+
N
j=f|jf|.(C4)
In this case, the unnormalized postmeasurement states are
expressed as
σk=1=1
N[α|ee|+(1 −α)|gg|]+
N(N−1)β|gg|,
σk=1=1
N[α|ee|+(1 −α)|gg|]−
Nβ|gg|.(C5)
To reveal the connection between the coherence and the
ergotropy enhancement, we introduce the ergotropy gain for
˜
Nsuperposed trajectories, which is defined as
W≡W˜
N−W[N=1]with ˜
N>1.(C6)
As indicated in the main text, in the region where the quick
charging effect occurs (i.e., W>0), the nonzero ergotropy
is contributed by σk=1, such that
W[˜
N]=N−1
N¯hω(2α+β −1).(C7)
The ergotropy gain can then be expressed as
W=N−1
N¯hωβ  +N−1
N¯hω(2α−1) for α1
2,
N−1
N¯hωβ  −1
N¯hω(2α−1) for α1
2.(C8)
Therefore, we can conclude that the ergotropy gain is linearly
proportional to the initial coherence in the control qudit.
[1] R. Uzdin, A. Levy, and R. Kosloff, Equivalence of quantum heat
machines, and quantum-thermodynamic signatures, Phys. Rev.
X5, 031044 (2015).
[2] W. Niedenzu, V. Mukherjee, A. Ghosh, A. G. Kofman, and G.
Kurizki, Quantum engine efficiency bound beyond the second
law of thermodynamics, Nat. Commun. 9, 165 (2018).
[3] G. Gemme, M. Grossi, D. Ferraro, S. Vallecorsa, and M.
Sassetti, IBM quantum platforms: A quantum battery perspec-
tive, Batteries 8, 43 (2022).
[4] H.-L. Shi, S. Ding, Q.-K. Wan, X.-H. Wang, and W.-L. Yang,
Entanglement, coherence, and extractable work in quantum bat-
teries, Phys. Rev. Lett. 129, 130602 (2022).
[5] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,
Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
[6] D. Bruß, Characterizing entanglement, J. Math. Phys. 43, 4237
(2002).
[7] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quan-
tum coherence as a resource, Rev. Mod. Phys. 89, 041003
(2017).
[8] O. Nairz, M. Arndt, and A. Zeilinger, Quantum interference
experiments with large molecules, Am. J. Phys. 71, 319 (2003).
[9] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quanta-
cell: powerful charging of quantum batteries, New J. Phys. 17,
075015 (2015).
[10] F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S.
Vinjanampathy, and K. Modi, Enhancing the charging power of
quantum batteries, Phys.Rev.Lett.118, 150601 (2017).
[11] D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M.
Polini, High-power collective charging of a solid-state quantum
battery, Phys.Rev.Lett.120, 117702 (2018).
[12] T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A. Pollock,
Spin-chain model of a many-body quantum battery, Phys. Rev.
A97, 022106 (2018).
[13] D. Rossini, G. M. Andolina, and M. Polini, Many-body
localized quantum batteries, Phys. Rev. B 100, 115142
(2019).
[14] D. Rossini, G. M. Andolina, D. Rosa, M. Carrega, and M.
Polini, Quantum advantage in the charging process of Sachdev-
Ye-Kitaev batteries, Phys.Rev.Lett.125, 236402 (2020).
[15] A. Crescente, M. Carrega, M. Sasetti, and D. Ferraro, Charging
and energy fluctuations of a driven quantum battery, New J.
Phys. 22, 063057 (2020).
[16] A. Crescente, M. Carrega, M. Sassetti, and D. Ferraro, Ultrafast
charging in a two-photon Dicke quantum battery, Phys. Rev. B
102, 245407 (2020).
[17] S. Ghosh, T. Chanda, S. Mal, and A. Sen(De), Fast charging of
a quantum battery assisted by noise, Phys.Rev.A104, 032207
(2021).
[18] J.-Y. Gyhm, D. Šafránek, and D. Rosa, Quantum charging ad-
vantage cannot be extensive without global operations, Phys.
Rev. Lett. 128, 140501 (2022).
[19] C. Rodríguez, D. Rosa, and J. Olle, Artificial intelligence dis-
covery of a charging protocol in a micromaser quantum battery,
Phys. Rev. A 108, 042618 (2023).
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