# Quick charging of a quantum battery with superposed trajecotries

Po-Rong Lai  Department of Physics and Center for Quantum Frontiers of
Research & Technology (QFort), National Cheng Kung University, Tainan 701,
Taiwan  Jhen-Dong Lin  Department of Physics and Center for Quantum Frontiers
of Research & Technology (QFort), National Cheng Kung University, Tainan 701,
Taiwan  Yi-Te Huang  Department of Physics and Center for Quantum Frontiers of
Research & Technology (QFort), National Cheng Kung University, Tainan 701,
Taiwan  Yueh-Nan Chen<EMAIL_ADDRESS>Department of Physics and
Center for Quantum Frontiers of Research & Technology (QFort), National Cheng
Kung University, Tainan 701, Taiwan

###### Abstract

We propose novel 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 or a single cavity with different
entry positions. To assess the battery’s performance, we evaluate the maximum
extractable work, referred to as ergotropy. Our main result is that the
proposed protocols can utilize quantum interference effects to speed up the
charging process. For the protocol involving multiple cavities, we observe a
substantial increase in ergotropy as the number of superposed trajectories
increases. In the case of the single-cavity protocol, we show that two
superposed trajectories (entry positions) are sufficient to achieve the upper
limit of the ergotropy throughout the entire charging process. 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.

##  I Introduction

Quantum batteries (QBs) have emerged as a popular research topic, providing
valuable insights into how thermodynamics functions at the quantum scale Uzdin
_et al._ (2015); Niedenzu  _et al._ (2018). Recent development has
demonstrated that various quantum resources such as entanglement Horodecki
_et al._ (2009); Bruß (2002) or coherence Streltsov  _et al._ (2017); Nairz
_et al._ (2003) can enhance the performance of quantum batteries in terms of
charging Binder  _et al._ (2015); Campaioli  _et al._ (2017); Ferraro  _et
al._ (2018); Le  _et al._ (2018); Rossini  _et al._ (2019); Crescente  _et
al._ (2020a, b); Ghosh  _et al._ (2021), storage Santos  _et al._ (2019);
Pirmoradian and Mølmer (2019); Liu  _et al._ (2019); Quach and Munro (2020);
Gherardini  _et al._ (2020); Rosa  _et al._ (2020) and work extraction Alicki
and Fannes (2013); Francica  _et al._ (2017); Manzano  _et al._ (2018);
Andolina  _et al._ (2019); Barra (2019); Francica  _et al._ (2020); Kamin  _et
al._ (2020); Monsel  _et al._ (2020); Ghosh  _et al._ (2020), etc. One of the
intriguing phenomena used to achieve these enhancements is the collective
effects triggered by a group of QBs Mei and Fang (1992); Higgins  _et al._
(2014); Campaioli  _et al._ (2017); Ferraro  _et al._ (2018); Pirmoradian and
Mølmer (2019); Quach  _et al._ (2022); Joshi and Mahesh (2022). In his seminal
paper Dicke (1954), Dicke characterized one of the collective effects,
superradiance, by the quantum interference of emissions from an ensemble of
atoms. Recent investigations have also demonstrated the utility of the time-
reversed phenomenon, known as superabsorption Higgins  _et al._ (2014); Yang
_et al._ (2021); Ueki  _et al._ (2022); Quach  _et al._ (2022), on enhancing
the capabilities of QBs.

In this work, our focus lies on an interferometric approach known as
“superpositions of trajectories” Chiribella and Kristjánsson (2019); Foo  _et
al._ (2020); Kristjánsson  _et al._ (2020); Duprey and Matzkin (2022); Ghafari
_et al._ (2019); Rubino  _et al._ (2021); Foo  _et al._ (2021); Chan  _et al._
(2022); Lin  _et al._ (2022); Lin and Chen (2023); Ku  _et al._ (2023); Lee
_et al._ (2023). This approach treats an atom’s space-time trajectories as a
quantum system, enabling the exploration of quantum interference of these
trajectories. A notable outcome of this approach is the effective noise
mitigation in various quantum information tasks Ghafari  _et al._ (2019);
Rubino  _et al._ (2021); Chan  _et al._ (2022); Ku  _et al._ (2023); Lee  _et
al._ (2023). In our recent work Lin  _et al._ (2022), we have further advanced
the understanding by interpreting this noise mitigation as a Zeno-like state
freezing phenomenon Misra and Sudarshan (1977) within the framework of open
quantum systems Breuer and Petruccione (2002). Additionally, we have
demonstrated that this approach can also manifest Dicke-like collective
effects even when only one single atom is involved. Building upon these
insights, the present 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, gaining energy through
interactions with cavities functioning as chargers. To assess the QB’s
performance, we focus on the maximum extractable work, known as ergotropy
Alicki and Fannes (2013). 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 interferometric setup akin to
scenarios in Ref. Lin  _et al._ (2022). The first one is called the multiple-
charger protocol, which consists of multiple identical chargers (cavities),
and the QB can interact with these chargers in a manner of quantum
superposition via a multi-port beam splitter. In principle, there are multiple
output beams of QB when it exits the interferometer. This enable us to adjust
the work extraction strategy for each output and obtain an average ergotropy,
also known as the daemonic ergotropy Francica  _et al._ (2017). The primary
result of this protocol is an “activation” of the ergotropy. Specifically, we
demonstrate 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 being
larger than its ground state population, is required to obtain a nontrivial
ergotropy. Therefore, one must wait until the battery reaches the inversion
point in order to obtain extractable work.

Remarkably, we demonstrate that by considering this multiple-charger protocol,
non-zero 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. Furthermore, 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 rotating-wave approximation, the
ergotropy even saturates to its upper bound (i.e., the stored energy)
throughout the charging process. This saturation indicates a complete
conversion of stored energy into extractable work. In addition, we reveal that
the enhancement of the average ergotropy stems from the increased average
purity of the QB’s output states, which can be regarded as a manifestation of
the Zeno-like phenomenon described in Ref. Lin  _et al._ (2022).

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, experiencing different coupling strengths with the
cavity. By using the superposed trajectories, the QB can enter these various
positions simultaneously. As indicated in Ref. Lin  _et al._ (2022), this
particular setup can induce the collective interference effect. It is
important to note that both the magnitude and relative phases of the coupling
strengths between the QB and the charger can influence the collective
interference effect and, consequently, the ergotropy. We show that two
superposed trajectories (positions) are sufficient to reach the upper limit of
the ergotropy, i.e., the stored energy, throughout the entire charging
process. This is achievable if the two coupling strengths possess the same
magnitude and are completely out of phase. Our analytical analysis
demonstrates that this phenomenon originates from the collective constructive
and destructive quantum interferences, which ensure that the output states
remain pure throughout the charging process.

Additionally, we present quantum circuits designed for the aforementioned
charging protocols, requiring fewer than twenty 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 implementations 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 consider the circuit implementations and
present the experimental results of the devices from IBMQ and IonQ . Finally,
we draw our conclusions in Sec. V.

##  II Multiple-charger protocol

Figure 1: The Quantum battery $Q$ is first sent into a multi-port beam
splitter (MPBS1), which allows the quantum battery to travel along $N$
different trajectories (denoted by $\ket{j}_{D}$, where $j=1\cdots N$ in the
following) in a manner of quantum superposition. We consider two charging
processes: (a) the trajectories each lead to a charger (cavities)
$\\{C_{j}\\}$, causing the quantum battery to interact with all chargers
simultaneously, (b) the trajectories lead to a single charger $C$ but at
different positions $\\{r_{j}\\}$, causing the quantum battery to interact
with the charger with various coupling strengths. Once the charging process is
completed, a second multi-port 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 $N$ possible reduced states $\rho_{j}$. We then extract work from
each $\rho_{j}$, where the maximum amount of extractable work is called the
ergotropy. The work extraction operations are described by unitary operators
$U_{j}$, which transforms each of the batteries to a passive state
$\varphi_{j}$.

We now formulate the multiple-charger scenario, which can be described by an
interferometric setup as shown in Fig. 1(a). The charging protocol consists of
three different components: (i) a qubit $Q$, which acts as the quantum
battery. (ii) $N$ identical single-mode cavities
$\left\\{C_{j}\right\\}_{j=1\cdots N}$, which act as the chargers. The QB
moves at a speed $v$ and gets charged when it passes through one of the
chargers. Suppose that the cavity length is $l$. Then, the interaction time
reads $\tau=l/v$. 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 Meschede  _et al._ (1985). (iii)
We characterize the trajectory degrees of freedom as an $N$ dimensional qudit
$D$, wherein we associate $N$ different trajectories inside the interferometer
with $N$ basis states $\left\\{\ket{j}_{D}\right\\}_{j=1\cdots N}$. When the
QB takes the path labeled by $j$, it interacts with the charger $C_{j}$. In
other words, $D$ acts as a quantum control that determines which charger the
QB interacts with. The total Hamiltonian involving these three components can
then be written as

$H_{\text{tot}}=\sum_{j=1}^{N}\ket{j}\bra{j}_{D}\otimes H_{QC_{j}}.$ (1)

The Hamiltonian $H_{QC_{j}}$ of the quantum battery $Q$ and the charger
$C_{j}$ can be expressed as

$\displaystyle H_{QC_{j}}$
$\displaystyle=H_{Q}+H_{C_{j}}+H_{QC_{j}}^{\prime},$ (2) $\displaystyle H_{Q}$
$\displaystyle=\frac{\hbar}{2}\omega_{a}\hat{\sigma}_{z}=\frac{\hbar}{2}\omega_{c}\hat{\sigma}_{z},$
$\displaystyle H_{C_{j}}$
$\displaystyle=\hbar\omega_{c}\hat{a}_{j}^{{\dagger}}\hat{a}_{j},$
$\displaystyle H_{QC_{j}}^{\prime}$
$\displaystyle=\hbar\omega_{c}\lambda\hat{\sigma}_{x}\left(\hat{a}_{j}+\hat{a}_{j}^{{\dagger}}\right).$

Here, $\hat{a}_{j}\left(\hat{a}_{j}^{{\dagger}}\right)$ annihilates (creates)
a photon in $C_{j}$ with frequency $\omega_{c}$, $\hbar\omega_{a}$ represents
the energy splitting between the ground state $\ket{g}$ and the excited state
$\ket{e}$ of $Q$. The Pauli operators are therefore given by
$\hat{\sigma}_{z}=\ket{e}\bra{e}-\ket{g}\bra{g}$ and
$\hat{\sigma}_{x}=\ket{e}\bra{g}+\ket{g}\bra{e}$. Moreover, the dimensionless
constant $\lambda$ represents the coupling strength between $Q$ and all the
chargers. Throughout this work, we focus on the resonant regime
$\omega_{a}=\omega_{c}$.

We first send the battery $Q$ into the multiport beam splitter (MPBS1 in Fig.
1). In general, the beam splitter can prepare the trajectories in a quantum
superposition state so that $Q$ can be charged by these $N$ chargers
simultaneously. For simplicity, we assume the superposition state of the
trajectories is

$\ket{\psi}_{D}=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\ket{j}_{D}.$ (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

$\ket{\psi\left(0\right)}_{DQC}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\ket{j}_{D}\otimes\ket{g}_{Q}\bigotimes_{j=1}^{N}\ket{1}_{C_{j}}.$
(4)

After $Q$ interacts with the chargers, according to Eq. (1), the total states
becomes

$\ket{\psi\left(\tau\right)}_{DQC}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}\ket{j}_{D}\otimes\ket{\phi_{j}\left(\tau\right)}_{QC},$
(5)

where $\ket{\phi_{j}\left(\tau\right)}_{QC}$ is defined as

$\ket{\phi_{j}\left(\tau\right)}_{QC}=\exp\left(-i\frac{\tau}{\hbar}H_{QC_{j}}\right)\left(\ket{g}_{Q}\bigotimes_{j=1}^{N}\ket{1}_{C_{j}}\right).$
(6)

Finally, we make these trajectories interfere with one another by using
another beam splitter (MPBS2 in Fig. 1). In principle, MPBS2 has $N$ different
outputs, which can be described using a set of orthonormal projectors
$\\{P_{k}\\}_{k}$ acting on $D$, namely

$\displaystyle P_{k}=\ket{\xi_{k}}\bra{\xi_{k}}_{D},$ (7)
$\displaystyle\sum_{k=1}^{N}P_{k}=\openone,$
$\displaystyle\braket{\xi_{k}}{\xi_{k^{\prime}}}=\delta_{k,k^{\prime}}~{}\forall~{}k,k^{\prime}.$

Therefore, the (unnormalized) reduced state for the system $Q$ with the output
$k$ reads

$\displaystyle\sigma_{k}\left(\tau\right)=\text{Tr}_{CD}\left[P_{k}\ket{\psi\left(\tau\right)}\bra{\psi\left(\tau\right)}_{DQC}P_{k}\right].$
(8)

Note that the probability of obtaining the outcome $k$ is
$p_{k}(\tau)=\mathrm{Tr}\left[\sigma_{k}(\tau)\right]$. Thus, the normalized
state conditioned on the outcome $k$ can be written as
$\rho_{k}(\tau)=\sigma_{k}(\tau)/p_{k}$.

Throughout this work, we choose

$\displaystyle\ket{\xi_{k=1}}\bra{\xi_{k=1}}_{D}\equiv\frac{1}{N}\sum_{m,n=1}^{N}\ket{m}\bra{n}_{D}.$
(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 considering the ergotropy, which
quantifies the maximum extractable work. Given a charged state of the QB
$\rho(\tau)$, the ergotropy is defined as

$\displaystyle W(\rho(\tau))$
$\displaystyle\equiv\text{Tr}\left(\rho(\tau)H_{Q}\right)-\min_{U}~{}\text{Tr}\left(U\rho(\tau)U^{{\dagger}}H_{Q}\right)$
(10)
$\displaystyle=\text{Tr}\left(\rho(\tau)H_{Q}\right)-\text{Tr}\left(\varphi(\tau)H_{Q}\right),$

where $U$ represents the unitary operation for work extraction. Also,
$\varphi$ is known as the passive state Alicki and Fannes (2013) (associated
with $\rho$), which cannot provide useful work for all possible work
extraction operations $U$. According to Ref. Alicki and Fannes (2013), the
passive state of the battery can be written as

$\displaystyle\varphi=s_{0}\ket{e}\bra{e}+s_{1}\ket{g}\bra{g}.$ (11)

Here, $s_{0}$ and $s_{1}$ denote the eigenvalues of $\rho$ with $s_{0}<s_{1}$.
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

$\displaystyle E\left(\tau\right)$
$\displaystyle\equiv\sum_{k=1}^{N}p_{k}\left(\tau\right)\text{Tr}\left[H_{Q}\rho_{k}\left(\tau\right)\right]-\text{Tr}\left(H_{Q}\ket{g}\bra{g}\right)$
(12)
$\displaystyle=\text{Tr}\left[H_{Q}\rho(\tau)\right]-\text{Tr}\left(H_{Q}\ket{g}\bra{g}\right),$

where $\rho(\tau)=\sum_{k}\sigma_{k}(\tau)$. We can obtain $W\leq E$, because
$\mathrm{Tr}(H_{Q}\ket{g}\bra{g})\leq\mathrm{Tr}(H_{Q}\varphi)$ in general. In
addition, the inequality saturates if and only if $\varphi=\ket{g}\bra{g}$,
which implies that $\rho$ is a pure state.

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 the average ergotropy, i.e.,

$\displaystyle\overline{W}$ $\displaystyle=\sum_{k}p_{k}W(\rho_{k}(\tau))$
(13)
$\displaystyle=\sum_{k}p_{k}\left[\mathrm{Tr}\left(\rho_{k}(\tau)H_{Q}\right)-\mathrm{Tr}\left(\varphi_{k}(\tau)H_{Q}\right)\right],$

where $\varphi_{k}$ denotes the passive state associated with $\rho_{k}$.
Following 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., $\overline{W}(\tau)=E(\tau)$, if and only if
$\\{p_{k},\rho_{k}\\}$ forms a pure state decomposition of $\rho$.

To gain some analytical insight, we now consider the rotating-wave
approximation, which is usually valid for coupling strength $\lambda\leq 0.1$
Kockum  _et al._ (2019), so that the interaction Hamiltonian of the QB and the
chargers in Eq. (2) can be reduced to the Jaynes-Cummings model, namely

$\tilde{H}_{QC_{j}}^{\prime}=\hbar\omega_{c}\lambda\left(\hat{\sigma}_{+}\hat{a}_{j}+\hat{\sigma}_{-}\hat{a}_{j}^{{\dagger}}\right),$
(14)

where $\hat{\sigma}_{+}=\ket{e}\bra{g}$ and $\hat{\sigma}_{-}=\ket{g}\bra{e}$
represent the creation and annihilation operators of $Q$, respectively. We can
then evaluate Eq. (6) in this case:

$\displaystyle\ket{\phi_{j}\left(\tau\right)}_{QC}=$
$\displaystyle-i\sin\left(\omega_{c}\lambda\tau\right)\ket{e}_{Q}\otimes\hat{a}_{j}\bigotimes_{j^{\prime}=1}^{N}\ket{1}_{C_{j^{\prime}}}$
(15)
$\displaystyle+\cos\left(\omega_{c}\lambda\tau\right)\ket{g}_{Q}\bigotimes_{j=1}^{N}\ket{1}_{C_{j}}.$

Let us start from the simplest case with only one charger (i.e., $N=1$), where
the reduced state of $Q$ is expressed as

$\rho(\tau)=\sin^{2}\left(\omega_{c}\lambda\tau\right)\ket{e}\bra{e}+\cos^{2}\left(\omega_{c}\lambda\tau\right)\ket{g}\bra{g}.$
(16)

In this case, the stored energy is

$E(\tau)=\hbar\omega_{c}\sin^{2}\left(\omega_{c}\lambda\tau\right),$ (17)

which oscillates with a period $T=2\pi/(\omega_{c}\lambda)$. We now focus on
the time interval $\tau\in[0,T/4]$ (such that
$\omega_{c}\lambda\tau\in[0,\pi/2]$), where the stored energy monotonically
increases from $0$ to its maximum value $\hbar\omega_{c}$. Note that $\rho$ is
diagonalized under basis $\\{\ket{e},\ket{g}\\}$. Thus, according to Eq. (10),
the criterion for obtaining non-zero ergotropy is the moment that population
inversion occurs, where the excited state population becomes larger than the
ground state population ($\bra{e}\rho(\tau)\ket{e}>\bra{g}\rho(\tau)\ket{g}$).
The time-dependence of the ergotropy can then be derived as

$W(\tau)=\begin{cases}0&\textrm{if}~{}0\leq\tau<\frac{T}{8},\\\
\hbar\omega_{c}(2\sin^{2}\left(\omega_{c}\lambda\tau\right)-1)&\textrm{if}~{}\frac{T}{8}\leq\tau\leq\frac{T}{4}.\end{cases}$
(18)

One can observe that in the duration $\tau\in[0,T/8]$, although the stored
energy $E$ monotonically increases, there is no extractable work, $W=0$, for
the battery because $\rho$ remains a passive state during this period.

We now consider the scenario involving $N$ chargers. When the selective
measurements satisfy Eq. (7) and Eq. (9), the unnormalized post-measurement
states can be written as (see Appendix A for detailed derivations):

$\displaystyle\sigma_{k=1}\left(\tau\right)=$
$\displaystyle\frac{1}{N}\sin^{2}\left(\omega_{c}\lambda\tau\right)\ket{e}\bra{e}+\cos^{2}\left(\omega_{c}\lambda\tau\right)\ket{g}\bra{g},$
(19) $\displaystyle\sigma_{k\neq 1}\left(\tau\right)=$
$\displaystyle\frac{1}{N}\sin^{2}\left(\omega_{c}\lambda\tau\right)\ket{e}\bra{e}.$

Here, for the case $k=1$, the post-measurement state is passive during the
time period $\tau\in\left[0,T_{N}\right]$ with the inversion time
$T_{N}=\tan^{-1}(\sqrt{N})T/2\pi$. Remarkably, for the cases of $k\neq 1$, the
post-measurement states are exactly the excited state, implying that the
maximal extractable work $\hbar\omega_{c}$ can be obtained. Therefore, the
average ergotropy can be expressed as

$\displaystyle\overline{W}\left(\tau\right)$ (20)
$\displaystyle=W(\sigma_{k=1}(\tau))+(N-1)W(\sigma_{k\neq 1}(\tau))$
$\displaystyle=\begin{cases}\hbar\omega_{c}\frac{N-1}{N}\sin^{2}\left(\omega_{c}\lambda\tau\right)&\text{if}~{}0\leq\tau\leq
T_{N}\\\
\hbar\omega_{c}\left(2\sin^{2}\left(\omega_{c}\lambda\tau\right)-1\right)&\text{if}~{}T_{N}\leq\tau\leq\frac{T}{4}.\end{cases}$

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 non-zero average ergotropy for the entire interval of interest because
the states with $k\neq 1$ are non-passive right after the QB-chargers
interaction is turned on ($\tau>0$). 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 $N$ delays
the inversion time $T_{N}$ and enhances the average ergotropy before $T_{N}$.
According to Eq. (20), this enhancement originates from the increase of the
purity for the output $k=1$ as its excited state population decreases with
$N$. This result aligns with the Zeno-like state freezing effect described in
Ref. Lin  _et al._ (2022). In the asymptotic limit ($N\rightarrow\infty$), we
can further obtain a pure state decomposition, i.e.,
$\sigma_{k=1}(\tau)\propto\ket{g}\bra{g}$ $\sigma_{k\neq
1}(\tau)\propto\ket{e}\bra{e}$, implying that the stored energy can be fully
converted into extractable work for the whole time interval, i.e,
$E(\tau)=\overline{W}(\tau)$.

Figure 2: The stored energy $E$ and average ergotropy $\overline{W}$ (both in
units of $\hbar\omega_{c}$) on time $\tau$ (in units of $1/\omega_{c}$) for
$\lambda=0.05$. 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 $N=1$, $N=2$ and $N=4$, respectively.
Figure 3: (a) The stored energy $E$ and average ergotropy $\overline{W}$ (both
in units of $\hbar\omega_{c}$) as functions of time $\tau$ (in units of
$1/\omega_{c}$) for $\lambda=0.5$. (b) The stored energy $E$ and the average
ergotropy contributed by states of $k=1$ $\overline{W}_{k=1}$. Two brown
vertical lines at $\tau=0.63,1.00$ indicate the inversion points $T_{N}$ for
$N=1$(blue) and $N=2$(red). Here, $\overline{W}_{k=1}$ is 0 when $N=7$(green)
and $N=\infty$(magenta). (c) The stored energy $E$ and the average ergotropy
contributed by states of $k\neq 1$ $\overline{W}_{k\neq 1}$ on time $\tau$.
Except for $N=1$ (blue), these states contribute average ergotropy when
$\tau>0$. (d) The dashed curves plot the change in average purity
$\mathcal{P}$ against time $\tau$. The cutoff photon number is set to 9 in the
above results.

We now extend our scope of discussion into the ultrastrong–coupling regime
Kockum  _et al._ (2019), 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 observe the quick charging effect, a delay of the
inversion point, and an enhancement in average ergotropy as $N$ increases. As
shown in Figs. 3 (b) and (c), we further present the individual contributions
of the average ergotropy from the outputs $k=1$ and $k\neq 1$, which are
respectively defined by

$\displaystyle\overline{W}_{k=1}=p_{k=1}W(\rho_{k=1}(\tau))$ (21) and
$\displaystyle\overline{W}_{k\neq 1}=\sum_{k\neq 1}p_{k}W(\rho_{k}(\tau)).$

We can observe that the inversion points come from the contribution of $k=1$.
Further, we can observe that its contribution decreases as $N$ increases. When
$N\geq 7$, the contribution vanishes for the entire charging period. In
Appendix A, we provide analytical analysis, showing that the decrease in
average ergotropy can also attribute to the Zeno-like state freezing effect.
More specifically, we demonstrate that the excited state population decreases
when $N$ increases. In the asymptotic limie ($N\rightarrow\infty$), the state
can be frozen in the ground state. We further observe that the quick charging
effect originates from the contribution of $k\neq 1$, since the corresponding
average ergotropy becomes non-zero 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 post-measurement states are not pure.

In Fig. 3(d), we present the average purity $\mathcal{P}$ associated with the
post-measurement states, which is defined by

$\displaystyle\mathcal{P}=\sum_{k}\text{Tr}[\sigma_{k}]\text{Tr}\left[\left(\frac{\sigma_{k}}{\text{Tr}[\sigma_{k}]}\right)^{2}\right].$
(22)

We can observe the overall average purity increasing as $N$ increases, thereby
leading to the enhancement of average ergotropy. However, the average purity
cannot reach unity even in the asymptotic limit. Thus, the stored energy
cannot be fully converted into extractable work. In Figs. 4 (a) and (b), we
compare the maximum average ergotropy with the average purity with respect to
the coupling strength $\lambda$, where the maximum average ergotropy is
defined by

$\overline{W}_{\textrm{max}}=\max_{\tau\in[0,T/4]}\overline{W}(\tau).$ (23)

We can observe that as the coupling strength increases, the average purity
drops, hence leading to a decrease in the maximum average ergotropy.

Figure 4: (a) Maximum average ergotropy $\overline{W}_{\textrm{max}}$ defined
in Eq. (23) with respect to different coupling strength $\lambda$. The blue,
red, green and magenta dashed lines represent the results for $N=1$, $N=10$,
$N=100$, and $N=\infty$, respectively. (b)The average purity $\mathcal{P}$ on
dimensionless coupling strength $\lambda$. The blue, red, green and magenta
dashed lines represent the results for $N=1$, $N=10$, $N=100$, and $N=\infty$,
respectively. The cutoff photon number is set to 9 in the above results.

##  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) $C$ at 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 $j$ guides the QB to a designated
position denoted as $r_{j}$ inside the cavity.

In this scenario, the QB experiences a varying coupling strength with respect
to the position inside that charger Shanks  _et al._ (2013). Therefore, the
position-dependent QB-charger Hamiltonian can be written as

$H_{j}=H_{Q}+H_{C}+H_{j}^{\prime},$ (24)

where $H_{C}=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}$. Here, $H_{j}^{\prime}$
represents the interaction Hamiltonian when QB is located at the position
$r_{j}$. Its explicit form is given by

$H_{j}^{\prime}=\hbar\omega_{c}\lambda\cos{\left(\pi\frac{r_{j}}{L}\right)}\hat{\sigma}_{x}\left(\hat{a}+\hat{a}^{{\dagger}}\right),$
(25)

where $L$ denotes the width of the charger and $\lambda$ represents the
maximal QB-charger coupling strength, which can be achieved when $r_{j}=0$ or
$r_{j}=L$. Thus, the total Hamiltonian for this scenario can be expressed by

$H_{\text{tot}}=\sum^{N}_{j=1}\ket{j}\bra{j}_{D}\otimes H_{j}.$ (26)

Similar to the previous consideration, we prepare the total system in the
following initial state:

$\ket{\psi(0)}_{DQC}=\frac{1}{\sqrt{N}}\sum^{N}_{j=1}\ket{j}_{D}\otimes\ket{g}_{Q}\otimes\ket{1}_{C},$
(27)

and allow it to evolve according to the total Hamiltonian $H_{\text{tot}}$,
namely

$\ket{\psi(\tau)}_{DQC}=\frac{1}{\sqrt{N}}\sum^{N}_{j=1}\ket{j}_{D}\otimes\ket{\phi_{j}(\tau)}_{QC},$
(28)

where $\ket{\phi_{j}(\tau)}_{QC}$, in this case, is defined as

$\ket{\phi_{j}(\tau)}_{QC}=\exp\left(-i\frac{\tau}{\hbar}H_{j}\right)\left(\ket{g}_{Q}\otimes\ket{1}_{C}\right).$
(29)

We also consider the projectors defined in Eq. (7) to characterize the
measurements performed by using MPBS2, such that the corresponding post-
measurement states reads

$P_{k}\ket{\psi(\tau)}_{DQC}=\ket{\xi_{k}}_{D}\otimes\sum^{N}_{j=1}c_{k,j}\ket{\phi_{j}(\tau)}_{QC},$
(30)

where the coefficient $c_{k,j}$ is given by

$c_{k,j}=\frac{1}{\sqrt{N}}\langle\xi_{k}|j\rangle.$ (31)

We now switch to the interaction picture, such that the interaction
Hamiltonian associated with the position $r_{j}$ can be expressed as

$\displaystyle H_{j}^{\prime}(\tau)=$ $\displaystyle
e^{\frac{i}{\hbar}(H_{Q}+H_{C})\tau}~{}H_{j}^{\prime}~{}e^{-\frac{i}{\hbar}(H_{Q}+H_{C})\tau}$
(32) $\displaystyle=$
$\displaystyle\cos\left(\pi\frac{r_{j}}{L}\right)H_{I}^{\prime}(\tau),$

where $H_{I}^{\prime}(\tau)$ is the position-independent part and reads

$H_{I}^{\prime}(\tau)=\hbar\omega_{c}\lambda
e^{\frac{i}{\hbar}(H_{Q}+H_{C})\tau}\hat{\sigma}_{x}(\hat{a}+\hat{a}^{\dagger})e^{-\frac{i}{\hbar}(H_{Q}+H_{C})\tau}.$
(33)

We can now characterize the time evolution with the propagator
$U_{j,I}(\tau,0)$ in terms of the Dyson series, namely

$\displaystyle U_{j,I}(\tau,0)=$
$\displaystyle\hat{\mathcal{T}}\exp\left[\frac{-i}{\hbar}\int_{0}^{\tau}H_{j}^{\prime}(t^{\prime})dt^{\prime}\right]$
(34) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\left[-\frac{i}{\hbar}\cos\left(\pi\frac{r_{j}}{L}\right)\right]^{n}$
$\displaystyle\times\int_{0}^{\tau}dt_{1}\cdots\int_{0}^{\tau}dt_{n}\hat{\mathcal{T}}H_{I}^{\prime}(t_{1})\cdots
H_{I}^{\prime}(t_{n}),$

where $\hat{\mathcal{T}}$ is the time-ordering operator with
$t_{1}>t_{2}>\cdots>t_{n}$. The post-measurement state of the QB and the
charger in Eq. (30) can then be expressed by

$\sum_{j=1}^{N}c_{k,j}\ket{\phi_{j}(\tau)}_{QC}=\sum_{j=1}^{N}c_{k,j}U_{j,I}(\tau,0)\ket{g}_{Q}\otimes\ket{1}_{C}.$
(35)

Therefore, one can observe that the total evolution (including 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 Lin  _et al._ (2022).

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 collective interference effect. More specifically, we consider that the
two positions satisfy $r_{1}+r_{2}=L$, such that

$\cos\left(\pi\frac{r_{2}}{L}\right)=\cos\left(\pi-\pi\frac{r_{1}}{L}\right)=-\cos\left(\pi\frac{r_{1}}{L}\right).$
(36)

Therefore, the coupling strengths share the same magnitude but are completely
out of phase. Furthermore, we consider

$\displaystyle\ket{\xi_{k=1}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{1}_{D}+\ket{2}_{D}),$ (37)
$\displaystyle\ket{\xi_{k=2}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{1}_{D}-\ket{2}_{D}),$

and, according to Eq. (31), the coefficients in this case are

$c_{1,1}=c_{1,2}=c_{2,1}=-c_{2,2}=\frac{1}{2}.$ (38)

Therefore, the time evolution of the post-measurement state of the QB and the
charger for $k=1$ is given by

$\displaystyle\sum_{j=1}^{2}$ $\displaystyle
c_{1,j}U_{j,I}(\tau,0)\ket{g}_{Q}\otimes\ket{1}_{C}$ (39) $\displaystyle=$
$\displaystyle\frac{1}{2}[U_{1,I}(\tau,0)+U_{2,I}(\tau,0)]\ket{g}_{Q}\otimes\ket{1}_{C}$
$\displaystyle=$
$\displaystyle\sum_{n~{}\textrm{even}}^{\infty}\frac{1}{n!}\left[-\frac{i}{\hbar}\cos\left(\pi\frac{r_{1}}{L}\right)\right]^{n}$
$\displaystyle\times\int_{0}^{\tau}dt_{1}\cdots\int_{0}^{\tau}dt_{n}\hat{\mathcal{T}}H_{I}^{\prime}(t_{1})\cdots
H_{I}^{\prime}(t_{n})\ket{g}_{Q}\otimes\ket{1}_{C}.$

We can observe that in the Dyson series, all the odd terms vanish, leading to
the phenomenon of destructive interference. 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 $\ket{g}$
throughout the entire process. Analogously, the post-measurement state for
$k=2$ can be written as

$\displaystyle\sum_{j=1}^{2}$ $\displaystyle
c_{2,j}U_{j,I}(\tau,0)\ket{g}_{Q}\otimes\ket{1}_{C}$ (40) $\displaystyle=$
$\displaystyle\frac{1}{2}[U_{1,I}(\tau,0)-U_{2,I}(\tau,0)]\ket{g}_{Q}\otimes\ket{1}_{C}$
$\displaystyle=$
$\displaystyle\sum_{n~{}\textrm{odd}}^{\infty}\frac{1}{n!}\left[-\frac{i}{\hbar}\cos\left(\pi\frac{r_{1}}{L}\right)\right]^{n}$
$\displaystyle\times\int_{0}^{\tau}dt_{1}\cdots\int_{0}^{\tau}dt_{n}\hat{\mathcal{T}}H_{I}^{\prime}(t_{1})\cdots
H_{I}^{\prime}(t_{n})\ket{g}_{Q}\otimes\ket{1}_{C}.$

In this case, all the even terms in the Dyson series vanish, implying the QB
is in the excited state $\ket{e}$ for all $\tau>0$ (with zero probability of
obtaining the output $k=2$ at $\tau=0$). Because the post-measurement 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., $E=\overline{W}$, throughout the whole charging
process. Note that the presented analysis does not rely on the rotating wave
approximation, thus indicating the saturation of the ergotropy to its upper
limit generally holds for all regimes of the QB-charger coupling strength.

##  IV 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 ($N=2$).

Figure 5: (a) Quantum circuit for multiple-chargers protocol. Here,
$D,Q,C_{1},C_{2}$ represents the control qubit, battery qubit, first charger
and second charger, respectively. (b) Decomposition of a controlled unitary in
(a). (c) Qubit configuration used on ibmq$\\_$algiers. (d) Decomposition of a
$XX$ gate into CNOT gates.

The quantum circuit for the multiple-charging setup is described by Fig. 5
(a). The circuit consists of 4 qubits, representing the control qubit $D$, the
quantum battery $Q$, and the two charging cavities $C_{1}$ and $C_{2}$,
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. 5 (b), we present the decomposition of the
controlled–unitaries into bit-flip ($X$) gates, controlled-z gates ($CZ$), and
Ising coupling gates [$XX(\theta)$ and $YY(\theta)$], defined as follows:

$\displaystyle X=\hat{\sigma}_{x},$ (41) $\displaystyle
CZ=\ket{0}\bra{0}\otimes\openone+\ket{1}\bra{1}\otimes\hat{\sigma}_{z},$
$\displaystyle
XX\left(\theta\right)=\cos(\theta/2)\openone\otimes\openone-i\sin(\theta/2)\hat{\sigma}_{x}\otimes\hat{\sigma}_{x},$
$\displaystyle
YY\left(\theta\right)=\cos(\theta/2)\openone\otimes\openone-i\sin(\theta/2)\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}.$

Here, we map the charging time $\tau$ into the angle $\theta$ using the
following relation

$\theta(\tau)=\omega_{c}\lambda\tau/2.$ (42)

Finally, in the measurement part, we measure the control qubit $D$ in the
x-direction, aligned with the projectors described by Eq. (37). Furthermore,
as indicated in Eq. (19), the QB’s post-measurement states are diagonalized
under the energy eigenstates. Consequently, we can only measure $Q$ in the
z-direction 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. 5 (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. 5 (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
[$R_{z}(\theta)$] gate, which are defined by

$\displaystyle H=\frac{1}{2}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},$ (43)
$\displaystyle R_{z}(\theta)=\exp(-i\hat{\sigma}_{z}\theta/2).$

For the single-charger setup, the circuit model for the charging process is
presented in Fig. 6, which consists of 6 and 4 two-qubit gates in the circuits
for the IBMQ and the IonQ devices, respectively.

Figure 6: 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.

Figures 7 and 8 illustrate the results obtained from the two protocols, with
each data point representing the average 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, comparing 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 single-charger protocol are smaller in
magnitude than those of the multiple-chargers protocol.

Figure 7: The stored energy $E$ and average ergotropy $\overline{W}$ (both in
units of $\hbar\omega_{c}$) on time $\tau$ (in units of $1/\omega_{c}$) for
$\lambda=0.05,N=2$. 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. Figure 8: The stored energy $E$
and average ergotropy $\overline{W}$ (both in units of $\hbar\omega_{c}$) on
time $\tau$ (in units of $1/\omega_{c}$) with $r_{1}=0.1l$ and $r_{2}=0.9l$.
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.

##  V Summary and Outlook

In this work, we utilize superposition of trajectories to propose two charging
protocols for quantum batteries (QBs), leading to improved storage of
extractable work. The first protocol, called the multiple-charger protocol,
allows for simultaneous interaction between the QB and multiple chargers.
Leveraging the interference effect, we observe an increase in extractable
work, attributed to the Zeno-like state freezing effect. The second protocol,
referred to as the single-charger protocol, enables the QB to interact with a
single charger from multiple positions concurrently. We demonstrate that the
collective interference effect generated by this setup allows the QB to
achieve the maximum extractable work throughout the entire charging period.
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 charging protocols to a
related framework called indefinite causal order Rubino  _et al._ (2017);
Goswami  _et al._ (2018); Ebler  _et al._ (2018); Zhao  _et al._ (2020);
Loizeau and Grinbaum (2020); Chiribella  _et al._ (2021), which allows for the
control of quantum operation ordering through a quantum switch. Based on this
framework, we could consider the scenarios, where the ordering of the charging
process becomes indefinite Simonov  _et al._ (2022). This exploration could
shed new light on the potential benefits and implications of incorporating
indefinite causal order into our proposed protocols, further advancing the
field of quantum battery charging.

## Acknowledgement

We acknowledge the NTU-IBM Q Hub and the IBM quantum experience for providing
us a platform to implement the experiment. This work is supported by the
National Center for Theoretical Sciences and National Science and Technology
Council, Taiwan, Grant Nos. MOST 111-2123-M-006-001and NSTC
111-2627-M-006-008.

##  Appendix A Effects of the choice of projectors on the post-measurement
quantum battery states

Here, we prove that the choice of projectors for $\ket{\xi_{k\neq
1}}\bra{\xi_{k\neq 1}}_{D}$ is irrelevant to the post-measurement states
$\sigma_{k\neq 1}(\tau)$ as long as they are orthonormal to
$\ket{\xi_{k=1}}\bra{\xi_{k=1}}_{D}$.

Let us start from Eq. (8), which can be expanded into

$\displaystyle\sigma_{k}(\tau)=$ (44)
$\displaystyle\frac{1}{N}\sum^{N}_{j,f=1}\braket{\xi_{k}}{j}\braket{f}{\xi_{k}}\text{Tr}_{C}\left[\ket{\phi_{j}(\tau)}\bra{\phi_{f}(\tau)}_{QC}\right]$

We switch to the interaction picture, where the interaction Hamiltonian reads

$H^{\prime}_{QC_{j}}(\tau)=e^{\frac{i}{\hbar}(H_{Q}+H_{C_{j}})\tau}H^{\prime}_{QC_{j}}e^{-\frac{i}{\hbar}(H_{Q}+H_{C_{j}})\tau}.$
(45)

The time-dependent part of post-measurement states can be generally expressed
by

$\displaystyle\ket{\phi_{j}\left(\tau\right)}_{QC}=\sum_{n=0}^{\infty}\alpha_{n}(\tau)\ket{n_{j}},$
(46)

with

$\ket{n_{j}}=(\hat{\sigma}_{x})^{n}\ket{e}\otimes\frac{(\hat{a}_{j}^{{\dagger}})^{n}\hat{a}_{j}}{\sqrt{n!}}\bigotimes_{m=1}^{N}\ket{1}_{C_{m}}.$
(47)

Note that in the following analysis, we show the explicit expression of the
time-dependent coefficient $\alpha_{n}(\tau)$ does not affect the result.
Thus, we will keep them unspecified. Now, we can show that

$\displaystyle\text{Tr}_{C}\left[\ket{\phi_{j}\left(\tau\right)}\bra{\phi_{f}\left(\tau\right)}_{QC}\right]$
(48)
$\displaystyle=\begin{cases}\sum_{n=0}^{\infty}{|\alpha_{n}(\tau)|}^{2}\sigma_{x}^{n}\ket{e}\bra{e}\sigma_{x}^{n}&\text{for~{}~{}}j=f\\\
{|\alpha_{1}(\tau)|}^{2}\ket{g}\bra{g},&\text{for~{}~{}}j\neq f.\end{cases}$

Inserting this into Eq. (44), we can obtain

$\displaystyle\sigma_{k}(\tau)=\sum_{n=0}^{\infty}{|\alpha_{n}(\tau)|}^{2}\sigma_{x}^{n}\ket{e}\bra{e}\sigma_{x}^{n}\frac{1}{N}\sum_{j=f}^{N}{|\braket{\xi_{k}}{j}|}^{2}$
(49)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+{|\alpha_{1}(\tau)}|^{2}\ket{g}\bra{g}\frac{1}{N}\sum_{j\neq
f}^{N}\sum_{f=1}^{N}\braket{\xi_{k}}{j}\braket{f}{\xi_{k}}.$

For the case of $k=1$, according to Eq. (7), we can obtain

$\displaystyle\braket{\xi_{k=1}}{j}=\frac{1}{\sqrt{N}}~{}~{}~{}~{}\forall~{}j=1,\cdots,N.$
(50)

Therefore, the corresponding post-measurement state can be written as

$\displaystyle\sigma_{k=1}(\tau)=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{\infty}{|\alpha_{n}(\tau)|}^{2}\sigma_{x}^{n}\ket{e}\bra{e}\sigma_{x}^{n}$
(51) $\displaystyle+\frac{N-1}{N}{|\alpha_{1}(\tau)|}^{2}\ket{g}\bra{g}.$

Note that in the asymptotic limit ($N\rightarrow\infty$), we observe
$\sigma_{k=1}(\tau)\propto\ket{g}\bra{g}$, indicating the Zeno-like state
freezing effect Lin  _et al._ (2022).

Recall that for the case of $k\neq 1$, we require the projectors to be
orthonormal to that of $k=1$, i.e., $\braket{\xi_{k\neq 1}}{\xi_{k=1}}=0$,
implying that the ket $\ket{\xi_{k\neq 1}}$ can be expressed by

$\displaystyle\ket{\xi_{k\neq 1}}=\sum_{j=1}^{N}\beta_{k,j}\ket{j}_{D},$ (52)
$\displaystyle\text{with~{}}\sum_{j=1}^{N}\beta_{k,j}=0~{}~{}~{}\forall~{}k\neq
1.$

Therefore, the post-measurement states can be written as

$\displaystyle\sigma_{k\neq 1}(\tau)=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{\infty}{|\alpha_{n}(\tau)|}^{2}\sigma_{x}^{n}\ket{e}\bra{e}\sigma_{x}^{n}$
(53) $\displaystyle-\frac{1}{N}{|\alpha_{1}(\tau)|}^{2}\ket{g}\bra{g}.$

This concludes the proof that the post-measurement states are independent of
the explicit expression of the coefficients $\beta_{k,j}$ associated with the
projectors.

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