# Energetics of a pulsed quantum battery

Charles Andrew Downing<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom  Muhammad
Shoufie Ukhtary  Department of Physics and Astronomy, University of Exeter,
Exeter EX4 4QL, United Kingdom  Research Center for Quantum Physics, National
Research and Innovation Agency (BRIN), South Tangerang 15314, Indonesia

###### Abstract

Abstract
The challenge of storing energy efficiently and sustainably is highly
prominent within modern scientific investigations. Due to the ongoing trend of
miniaturization, the design of expressly quantum storage devices is itself a
crucial task within current quantum technological research. Here we provide a
transparent analytic model of a two-component quantum battery, composed of a
charger and an energy holder, which is driven by a short laser pulse. We
provide simple expressions for the energy stored in the battery, the maximum
amount of work which can be extracted, both the instantaneous and the average
powers, and the relevant charging times. This allows us to discuss explicitly
the optimal design of the battery in terms of the driving strength of the
pulse, the coupling between the charger and the holder, and the inevitable
energy loss into the environment. We anticipate that our theory can act as a
helpful guide for the nascent experimental work building and characterizing
the first generation of truly quantum batteries.

Introduction. Battery research seeks to play a role in addressing some of the
most pressing issues affecting modern society, including the sustainable
generation, storage and transport of energy. These kinds of important problems
also arise in the quantum world, where the very nature of fields like quantum
information and quantum thermodynamics suggest the existence of a novel type
of energy storage device: a quantum battery Alicki2013 ; Campaioli2023 .

Early experimental work (mostly with spins and superconducting qubits) has
already started to pioneer the fundamental workings of quantum batteries,
including: the degradation of the stored energy over time, various charging
and discharging protocols, and the overall power performance Peterson2019 ;
Cimini2020 ; Quach2022 ; Joshi2022 ; Hu2022 ; Stevens2022 ; Maillette2022 .
Meanwhile, theoretical studies have sought to provide innovative ideas to
increase the efficiency and precision of quantum batteries, as well as to
exploit various inherently quantum advantages Ferraro2018 ; Andolina2018 ;
Farina2019 ; Crescente2020 ; NewCrescente2020 ; Santos2020 ; Carrega2020 ;
Santos2021 ; Gyhm2022 ; Shaghaghi2002 ; Catalano2023 .

Here we seek to provide a simple yet explanatorily powerful model of a
bipartite quantum battery driven by a short laser pulse. We model our battery
charger and battery holder as quantum harmonic oscillators, such that our
theory contributes to the emerging body of work on quantum continuous variable
batteries Friis2018 ; Huangfu2021 ; Downing2023 ; Centrone2021 ; Konar2022 .
Within an open quantum systems approach, we provide compact expressions
describing the energetic and power performance of the battery, as well as
brief formulae for the maximum amount of work which can be extracted
Alicki2013 ; Allahverdyan2004  and for various desirable charging times.
Interestingly, the behaviour of the pulsed quantum battery is intrinsically
governed by the presence of an exceptional point Berry2003 , which are
becoming increasingly influential in both classical and quantum optics
Miri2019 ; Downing2021 ; Bender2023 . We note that theoretical analyses of
other driving protocols can be found in Refs. Crescente2020 ; NewCrescente2020
amongst others.

Figure 1:  Sketch of the pulsed quantum battery. The bipartite quantum battery
is composed of a battery charger (associated with the bosonic operator $a$)
and a battery holder (operator $b$). Both parts are modelled as quantum
harmonic oscillators with the level spacing $\omega_{b}$, and they are coupled
at the strength $g$. The charger is driven by a pulse of strength $\Omega$,
and it suffers from dissipation at the rate $\gamma$.

Model. The total Hamiltonian operator $\hat{H}$ describing the bipartite
quantum battery can be split into four components [cf. the cartoon of Fig. 1]

$\hat{H}=\hat{H}_{a}+\hat{H}_{b}+\hat{H}_{a-b}+\hat{H}_{d}.$ (1)

The battery charger and battery holder are taken to be quantum harmonic
oscillators, governed by the twin terms (we set $\hbar=1$ here and throughout)

$\hat{H}_{a}=\omega_{b}a^{\dagger}a,\quad\quad\quad\hat{H}_{b}=\omega_{b}b^{\dagger}b,$
(2)

where $\omega_{b}$ is the energy level spacing of each oscillator. The
operator $a^{\dagger}$ creates an excitation in the battery charger and $a$
destroys an excitation, while $b^{\dagger}$ and $b$ do likewise in the battery
holder. The bosonic commutation relations $[a,a^{\dagger}]=1$ and
$[b,b^{\dagger}]=1$ are observed. The interaction between the charger and the
holder is modelled with the coupling term

$\hat{H}_{a-b}=g\left(a^{\dagger}b+b^{\dagger}a\right),$ (3)

where the coupling frequency is of strength $g\geq 0$. The battery charger is
driven at the time $t=0$ by a short laser pulse

$\hat{H}_{d}=\Omega~{}\delta(t)\left(a^{\dagger}+a\right),$ (4)

which is of dimensionless strength $\Omega\geq 0$ and where $\delta(t)$ is
Dirac’s delta function (we support this choice of drive in the supplementary
data). We assume that the battery holder is sufficiently isolated from the
environment such that its energy losses are negligible, as should be the case
for any meaningful energy storage device. However, this isolation implies that
the battery holder is rather hard to transfer energy into in the first place.
Therefore we couple the battery holder to the battery charger, which is easy
to drive and hence it also feels its external environment such that it
noticeably dissipates [cf. the sketch of Fig. 1]. These losses from the
battery charger are incorporated into our model using a quantum master
equation, which is considered to be in Gorini–Kossakowski–Sudarshan–Lindblad
form as Breuer2002

$\partial_{t}\rho=\mathrm{i}\left[\rho,\hat{H}\right]+\frac{\gamma}{2}\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right),$ (5)

where the quantum battery system’s density matrix is $\rho$, the battery
charger dissipation rate $\gamma\geq 0$, and the total Hamiltonian operator
$\hat{H}$ is defined in Eq. (1). There are a triumvirate of measures of
primary interest for the proposed pulsed quantum battery: the energy $E$
stored in the battery holder, the corresponding instantaneous power
$\mathcal{P}$, and the average power $P$. These three quantities are given in
terms of the average population of the battery holder $\langle
b^{\dagger}b\rangle$, and are defined by

$E=\omega_{b}\langle b^{\dagger}b\rangle,$ (6)
$\mathcal{P}=\frac{\mathrm{d}}{\mathrm{d}t}E,$ (7) $P=\frac{E}{t}.$ (8)

In particular, we seek to find the optimal values of the model parameters $g$,
$\Omega$ and $\gamma$ [as represented pictorially in Fig. 1] such that desired
energetics (as judged by $E$, $\mathcal{P}$ and $P$) are achieved in the
smallest possible elapsed time after the battery charger is driven, such that
the charger can then be disconnected.

Figure 2:  Energetics of the pulsed quantum battery. Panel (a): the dynamical
energy $E$ stored in the battery (in units of $\omega_{b}\Omega^{2}$) as a
function of the time $t$ (in units of the inverse decay rate of the charger
$\gamma^{-1}$) [cf. Eq. (21)]. Coloured lines: results for increasingly large
strengths of the charger-battery coupling $g$, from below the exceptional
point (pink line) to exactly at the exceptional point (dashed red line) to
above the exceptional point (blue and cyan lines). Panel (b): the time
$t_{E}$, corresponding to the maximum of the stored energy $E$, as a function
of the coupling-to-dissipation ratio $g/\gamma$ is represented with the pink-
cyan line [cf. Eq. (22)]. Dashed lines: asymptotic results at small and large
couplings [cf. Eq. (23) and Eq. (24)]. Panel (c): the maximum energy
$E(t_{E})$ as a function of $g/\gamma$ is represented with the pink-cyan line
[cf. Eq. (25)]. Dashed lines: asymptotic results at small and large couplings
[cf. Eq. (26) and Eq. (27)]. Vertical grey lines in panels (b) and (c): guides
for the eye at the exceptional point of $g=\gamma/4$ [cf. Eq. (14)].

Dynamics. The first moments of the pulsed quantum battery follow from the
trace property $\operatorname{Tr}(\mathcal{O}\rho)=\langle\mathcal{O}\rangle$,
which is valid for any operator $\mathcal{O}$, and the quantum master equation
of Eq. (5). This averaging process leads to the equation of motion (see the
supplementary data for details)

$\mathrm{i}\partial_{t}\psi=\mathcal{H}\psi+\mathcal{S},$ (9)

where the first moments contained within $\psi$ and the dynamical matrix
$\mathcal{H}$ are together given by

$\psi=\begin{pmatrix}\langle a\rangle\\\ \langle
b\rangle\end{pmatrix},\quad\quad\quad\mathcal{H}=\begin{pmatrix}\omega_{b}-\mathrm{i}\frac{\gamma}{2}&g\\\
g&\omega_{b}\end{pmatrix},$ (10)

while the pulsed drive of the charger appears in the final term

$\mathcal{S}=\Omega\begin{pmatrix}\delta(t)\\\ 0\end{pmatrix}.$ (11)

The two complex eigenvalues $\epsilon_{\pm}$ of the non-Hermitian matrix
$\mathcal{H}$ defined in Eq. (10) are

$\epsilon_{\pm}=\omega_{b}-\mathrm{i}\frac{\gamma}{4}\pm G,$ (12)

where the renormalized coupling rate $G$ is defined by

$G=\sqrt{g^{2}-\left(\tfrac{\gamma}{4}\right)^{2}}.$ (13)

Notably, the pulsed quantum battery exhibits a quantum spectral degeneracy
Vidiella2023 ; Fox2023  at a certain exceptional point, which occurs when
$G=0$ (or equivalently when $g=g_{\mathrm{EP}}$) where

$g_{\mathrm{EP}}=\frac{\gamma}{4}.$ (14)

The existence of this exceptional point divides the response of the pulsed
quantum battery into two distinct behavioural regimes, a classification which
is used in what follows. At all times $t>0$, that is after the driving pulse
has been applied, the solution of the then Schrödinger-like equation of motion
of Eq. (9) reads

$\psi=A\begin{pmatrix}-\mathrm{i}G-\frac{\gamma}{4}\\\
-\mathrm{i}g\end{pmatrix}\mathrm{e}^{-\mathrm{i}\epsilon_{+}t}+B\begin{pmatrix}\mathrm{i}G-\frac{\gamma}{4}\\\
-\mathrm{i}g\end{pmatrix}\mathrm{e}^{-\mathrm{i}\epsilon_{-}t},$ (15)

where the eigenvalues $\epsilon_{\pm}$ are defined in Eq. (12). The two
constants $A$ and $B$ can be found by integrating Eq. (9) with respect to time
over the small temporal interval from $t=-\varepsilon$ to $t=+\varepsilon$,
which contains the pulse exactly at $t=0$ [cf. Eq. (11)]. Taking the limit of
$\varepsilon\to 0$ reveals that the values of the constants which satisfy the
boundary condition are

$A=-B,\quad\quad\quad\quad A=\frac{\Omega}{2G}.$ (16)

Therefore, the solution of Eq. (15) provides the dynamical behaviour of the
mean value of the operator $a$ as follows

$\langle a\rangle=\begin{cases}-\mathrm{i}\Omega\left[\cosh\left(\Gamma
t\right)-\frac{\gamma}{4\Gamma}\sinh\left(\Gamma
t\right)\right]\mathrm{e}^{-\frac{\gamma
t}{4}}\mathrm{e}^{-\mathrm{i}\omega_{b}t},&g<g_{\mathrm{EP}},\\\
-\mathrm{i}\Omega\left(1-\frac{\gamma t}{4}\right)\mathrm{e}^{-\frac{\gamma
t}{4}}\mathrm{e}^{-\mathrm{i}\omega_{b}t},&g=g_{\mathrm{EP}},\\\
-\mathrm{i}\Omega\left[\cos\left(Gt\right)-\frac{\gamma}{4G}\sin\left(Gt\right)\right]\mathrm{e}^{-\frac{\gamma
t}{4}}\mathrm{e}^{-\mathrm{i}\omega_{b}t},&g>g_{\mathrm{EP}},\end{cases}$ (17)

where we have made use of the exceptional point of Eq. (14) in order to
categorize the response of the system. In particular, below the exceptional
point of $g_{\mathrm{EP}}$ where the behaviour is non-oscillatory, we have
introduced the renormalized dissipation rate

$\Gamma=\sqrt{\left(\tfrac{\gamma}{4}\right)^{2}-g^{2}},$ (18)

as the counterpart to $G$, as was defined in Eq. (13). Above the exceptional
point the battery charger displays Rabi-like oscillations, while exactly at
$g=g_{\mathrm{EP}}$ some remarkable power function dynamics is instead
presented following Eq. (17). Similarly, the first moment of the battery
holder can be read off from Eq. (15) as

$\langle b\rangle=\begin{cases}-\Omega\>\frac{g}{\Gamma}\sinh\left(\Gamma
t\right)\mathrm{e}^{-\frac{\gamma
t}{4}}\mathrm{e}^{-\mathrm{i}\omega_{b}t},&g<g_{\mathrm{EP}},\\\
-\Omega\>\frac{\gamma t}{4}~{}\mathrm{e}^{-\frac{\gamma
t}{4}}\mathrm{e}^{-\mathrm{i}\omega_{b}t},&g=g_{\mathrm{EP}},\\\
-\Omega\>\frac{g}{G}\sin\left(Gt\right)\mathrm{e}^{-\frac{\gamma
t}{4}}\mathrm{e}^{-\mathrm{i}\omega_{b}t},&g>g_{\mathrm{EP}},\end{cases}$ (19)

which manifests a similar structure to the solutions of Eq. (17), including a
non-oscillatory to oscillatory transition at the exceptional point (where the
response is linear in time). The solution of Eq. (19) allows for the mean
population of the battery holder to be accessed thanks to the relation
$\langle b^{\dagger}\rangle\langle b\rangle=\langle b^{\dagger}b\rangle$,
which holds within this particular model because the joint charger–battery
holder system evolves in a product state Farina2019 ; Downing2023 . This
correlator factorization prescription allows for the energetics of the quantum
battery to be readily investigated.

Figure 3:  Instantaneous power of the pulsed quantum battery. Panel (a): the
instantaneous power $\mathcal{P}$ of the battery (in units of
$\omega_{b}\Omega^{2}\gamma$) as a function of the time $t$ (in units of the
inverse decay rate of the charger $\gamma^{-1}$) [cf. Eq. (29)]. Coloured
lines: results for increasingly large strengths of the charger-battery
coupling $g$, from below the exceptional point (pink line) to exactly at the
exceptional point (dashed red line) to above the exceptional point (blue and
cyan lines). Panel (b): the time $t_{\mathcal{P}}$, corresponding to the
maximum of the instantaneous power $\mathcal{P}$ of the battery, as a function
of the coupling-to-dissipation ratio $g/\gamma$ is represented with the pink-
cyan line [cf. Eq. (30)]. Dashed lines: asymptotic results at small and large
couplings [cf. Eq. (31) and Eq. (32)]. Panel (c): the maximum instantaneous
power $\mathcal{P}(t_{\mathcal{P}})$ as a function of $g/\gamma$ is
represented with the pink-cyan line. Dashed lines: asymptotic results at small
and large couplings [cf. Eq. (33) and Eq. (34)]. Vertical grey lines in panels
(b) and (c): guides for the eye at the exceptional point of $g=\gamma/4$ [cf.
Eq. (14)].

Stored energy. The energy $E$ stored in the battery holder follows from Eq.
(6) and the solutions of Eq. (19), due to the conjugation relation $\langle
b\rangle^{\ast}=\langle b^{\dagger}\rangle$. In the dissipationless limit,
this stored energy follows the sinusoidal squared formula

$\lim_{\gamma\to 0}E=\omega_{b}\>\Omega^{2}\sin^{2}\left(gt\right),$ (20)

which highlights the intuitive properties of greater amounts of stored energy
with stronger pulse strengths $\Omega$, and the reaching of an energetic
maximum at some finite time (in this case, a maximal energy of
$\omega_{b}\Omega^{2}$ is achieved after a time delay of $\pi/2g$ from the
action of the pulse). However, in any realistic quantum battery the battery
charger will suffer from nonzero dissipation $\gamma$, such that Eq. (20) is
superseded by

$E=\begin{cases}\omega_{b}\>\Omega^{2}\left(\frac{g}{\Gamma}\right)^{2}\sinh^{2}\left(\Gamma
t\right)\mathrm{e}^{-\frac{\gamma t}{2}},&g<g_{\mathrm{EP}},\\\
\omega_{b}\>\Omega^{2}\left(\frac{\gamma
t}{4}\right)^{2}\mathrm{e}^{-\frac{\gamma t}{2}},&g=g_{\mathrm{EP}},\\\
\omega_{b}\>\Omega^{2}\left(\frac{g}{G}\right)^{2}\sin^{2}\left(Gt\right)\mathrm{e}^{-\frac{\gamma
t}{2}},&g>g_{\mathrm{EP}},\end{cases}$ (21)

with $G$ and $\Gamma$ as defined by Eq. (13) and Eq. (18) respectively, and
where the exceptional point of Eq. (14) separates the responses. Notably, the
stored energy $E$ now decays with a time constant of $2/\gamma$, such that the
battery charger should be disconnected from the battery holder after some
finite amount of charging time to avoid complete degradation. Whilst
connected, the battery holder dynamical energy $E$ displays a non-oscillatory
(pink line) to oscillatory (blue and cyan lines) transition upon passing
through its exceptional point (dashed red line), as is demonstrated
graphically in Fig. 2 (a) [cf. Eq. (21)]. The maximum amount of stored energy
in the battery, over all charging time $t$, allows for the optimal energetic
charging time $t_{E}$ to be found, which satisfies
$\mathrm{max}_{t}\\{E(t)\\}=E(t_{E})$. The stationary points of Eq. (21) imply
the energetic times

$t_{E}=\begin{cases}\tfrac{\operatorname{arctanh}\left(\tfrac{4\Gamma}{\gamma}\right)}{\Gamma},&g<g_{\mathrm{EP}},\\\
\tfrac{4}{\gamma},&g=g_{\mathrm{EP}},\\\
\tfrac{\arctan\left(\tfrac{4G}{\gamma}\right)}{G},&g>g_{\mathrm{EP}},\end{cases}$
(22)

which is plotted as the pink-cyan line in Fig. 2 (b), as a function of the
coupling-to-dissipation ratio $g/\gamma$. Clearly, shorter optimal charging
times $t_{E}$ arise when the charger-holder coupling $g$ is stronger, as was
already implied in Fig. 2 (a). In the two limiting cases of very weak ($g\ll
g_{\mathrm{EP}}$) and very strong ($g\gg g_{\mathrm{EP}}$) coupling, Eq. (22)
reduces to its asymptotic forms

$\displaystyle\tfrac{4}{\gamma}\ln\left(\tfrac{\gamma}{2g}\right),$
$\displaystyle g\ll g_{\mathrm{EP}},$ (23)
$\displaystyle\tfrac{\pi}{2g}-\tfrac{\gamma}{4g^{2}},$ $\displaystyle g\gg
g_{\mathrm{EP}}.$ (24)

The latter expression of Eq. (24) eventually recovers the inverse-$g$
behaviour of the fully dissipationless result of Eq. (20), and it is marked
with a dashed blue line in Fig. 2 (b). The former expression of Eq. (23)
exudes a logarithmic character, as is distinguished by the dashed red line in
Fig. 2 (b), which suggests that the optimal charging time of very weakly
coupled batteries is weakly (logarithmically) divergent. The exact quantity of
energy $E(t_{E})$ stored in the quantum battery at the optimal time $t_{E}$
follows immediately from Eq. (21) and Eq. (22). Explicitly, we find the brief
formulae

$E(t_{E})=\begin{cases}\omega_{b}\>\Omega^{2}\>\mathrm{e}^{-\frac{\gamma}{2\Gamma}\operatorname{arctanh}\left(\tfrac{4\Gamma}{\gamma}\right)},&g<g_{\mathrm{EP}},\\\
\omega_{b}\>\Omega^{2}\>\mathrm{e}^{-2},&g=g_{\mathrm{EP}},\\\
\omega_{b}\>\Omega^{2}\>\mathrm{e}^{-\frac{\gamma}{2G}\arctan\left(\tfrac{4G}{\gamma}\right)},&g>g_{\mathrm{EP}},\end{cases}$
(25)

where the number $\mathrm{e}^{-2}\simeq 0.135$ arises. This energy $E(t_{E})$
is graphed by the pink-cyan line in Fig. 2 (c) as a function of the coupling-
to-dissipation ratio $g/\gamma$. Notably, this optimal energy is bounded by
zero and $\omega_{b}\Omega^{2}$ depending upon the size of the coupling
strength $g$. The asymptotics of this optimal energy are provided by the
simple expressions

$\displaystyle\omega_{b}\>\Omega^{2}\left(\tfrac{2g}{\gamma}\right)^{2},$
$\displaystyle g\ll g_{\mathrm{EP}},$ (26)
$\displaystyle\omega_{b}\>\Omega^{2}\left(1-\tfrac{\pi\gamma}{4g}\right),$
$\displaystyle g\gg g_{\mathrm{EP}}.$ (27)

as drawn with dashed red and dashed blue lines respectively in Fig. 2 (c).
These formulae showcase the quadratic-in–$g$ decrease in optimal energy with
very weak coupling, as well as the inverse-$g$ scaling with very strong
coupling towards the upper energetic bound of $\omega_{b}\Omega^{2}$.

Figure 4:  Average power of the pulsed quantum battery. Panel (a): the average
power $P$ of the battery (in units of $\omega_{b}\Omega^{2}\gamma$) as a
function of the time $t$ (in units of the inverse decay rate of the charger
$\gamma^{-1}$) [cf. Eq. (8) with Eq. (21)]. Coloured lines: results for
increasingly large strengths of the charger-battery coupling $g$, from below
the exceptional point (pink line) to exactly at the exceptional point (dashed
red line) to above the exceptional point (blue and cyan lines). Panel (b): the
time $t_{P}$, corresponding to the maximum of the average power $P$ of the
battery, as a function of the coupling-to-dissipation ratio $g/\gamma$ is
represented with the pink-cyan line [cf. Eq. (36) and Eq. (37)]. Dashed lines:
asymptotic results at small and large couplings [cf. Eq. (41) and Eq. (42)].
Panel (c): the maximum average power $P(t_{P})$ as a function of $g/\gamma$ is
represented with the pink-cyan line [cf. Eq. (8) with Eq. (36) and Eq. (37)].
Dashed lines: asymptotic results at small and large couplings [cf. Eq. (43)
and Eq. (44)]. Vertical grey lines in panels (b) and (c): guides for the eye
at the exceptional point of $g=\gamma/4$ [cf. Eq. (14)].

Instantaneous power. The instantaneous power $\mathcal{P}$ of the pulsed
quantum battery is obtained from Eq. (7) and the energetic results provided
previously. In the idealized limit of zero dissipation, this instantaneous
power is described by the simple sinusoidal formula [cf. Eq. (20)]

$\lim_{\gamma\to 0}\mathcal{P}=\omega_{b}\>\Omega^{2}g\sin\left(2gt\right).$
(28)

This expression demonstrates the existence of an upper power bound of
$\omega_{b}\Omega^{2}g$, which is first reached at a time $\pi/4g$ after the
occurrence of the driving pulse. The full driven-dissipative system has an
instantaneous power instead encapsulated by [cf. Eq. (21)]

$\mathcal{P}=\begin{cases}\omega_{b}\>\Omega^{2}\left(\frac{g}{\Gamma}\right)^{2}\left[\Gamma\sinh\left(2\Gamma
t\right)-\frac{\gamma}{2}\sinh^{2}\left(\Gamma
t\right)\right]\mathrm{e}^{-\frac{\gamma t}{2}},&g<g_{\mathrm{EP}},\\\
\omega_{b}\>\Omega^{2}\frac{\gamma}{2}\left(\frac{\gamma
t}{4}\right)\left[1-\frac{\gamma t}{4}\right]\mathrm{e}^{-\frac{\gamma
t}{2}},&g=g_{\mathrm{EP}},\\\
\omega_{b}\>\Omega^{2}\left(\frac{g}{G}\right)^{2}\left[G\sin\left(2Gt\right)-\frac{\gamma}{2}\sin^{2}\left(Gt\right)\right]\mathrm{e}^{-\frac{\gamma
t}{2}},&g>g_{\mathrm{EP}},\end{cases}$ (29)

which features the common time constant $2/\gamma$, which controls the
exponential decay. The instantaneous power $\mathcal{P}$ is plotted as
function of time in Fig. 3 (a), presenting the critical behaviour of the model
due to the distinct behaviour below (pink line), at (dashed red line) and
above (cyan and blue lines) the exceptional point of Eq. (14). The trend of
larger instantaneous power $\mathcal{P}$ with increasing large charger-holder
coupling $g$, which peak at ever shorter timescales, is most apparent from the
plot. The maximum of the instantaneous power $\mathcal{P}$ over all time $t$
suggests the optimal charging time $t_{\mathcal{P}}$, which is consistent with
$\mathrm{max}_{t}\\{\mathcal{P}(t)\\}=\mathcal{P}(t_{\mathcal{P}})$. Finding
the relevant stationary points of Eq. (29) leads to the optimal instantaneous
power times $t_{\mathcal{P}}$, where

$t_{\mathcal{P}}=\begin{cases}\tfrac{\operatorname{arctanh}\left(\tfrac{8\Gamma\gamma-\sqrt{\gamma^{4}-256g^{4}}}{3\gamma^{2}-16g^{2}}\right)}{\Gamma},&g<g_{\mathrm{EP}},\\\
\tfrac{2\left(2-\sqrt{2}\right)}{\gamma},&g=g_{\mathrm{EP}},\\\
\tfrac{\arctan\left(\tfrac{\sqrt{256g^{4}-\gamma^{4}}-8G\gamma}{16g^{2}-3\gamma^{2}}\right)}{G},&g>g_{\mathrm{EP}},\end{cases}$
(30)

where the number $2\left(2-\sqrt{2}\right)\simeq 1.17$ appears. The charging
times $t_{\mathcal{P}}$ of Eq. (30) are given as the pink-cyan line in Fig. 3
(b) as a function of the coupling-to-dissipation ratio $g/\gamma$. Notably,
the upper bound of $\ln(4)/\gamma\simeq 1.39/\gamma$ is met with vanishing
coupling $g$, while for very strong coupling $g$ the optimal time
$t_{\mathcal{P}}$ is vanishing, with an inverse-$g$ relationship [cf. Eq.
(28)]. These limiting cases are well-described by the asymptotics

$\displaystyle\tfrac{\ln\left(4\right)}{\gamma}-16\left[1-\ln\left(2\right)\right]\tfrac{g^{2}}{\gamma^{3}},$
$\displaystyle g\ll g_{\mathrm{EP}},$ (31)
$\displaystyle\tfrac{1}{4g}\left(\pi-\tfrac{\gamma}{g}\right),$ $\displaystyle
g\gg g_{\mathrm{EP}}.$ (32)

which are plotted by the dashed red and dashed blue lines respectively in Fig.
3 (b), which exposes these key functional relationships. The progression of
$\mathcal{P}(t_{\mathcal{P}})$, the instantaneous power at the optimal time
$t_{\mathcal{P}}$ for the pulsed quantum battery, with increasing $g/\gamma$
is likewise plotted in Fig. 3 (c) [using Eq. (29) with Eq. (30)]. The limiting
cases are described by the compact expressions

$\displaystyle\omega_{b}\>\Omega^{2}\tfrac{g^{2}}{\gamma},$ $\displaystyle
g\ll g_{\mathrm{EP}},$ (33)
$\displaystyle\omega_{b}\>\Omega^{2}g\left(1-\tfrac{\pi+2}{8}\tfrac{\gamma}{g}\right),$
$\displaystyle g\gg g_{\mathrm{EP}}.$ (34)

as denoted by the dashed red line and dashed blue line in Fig. 3 (c). In
particular, the very weak coupling result and lower bound vanishes
quadratically with $g$ [cf. Eq. (33)], while the very strong coupling bound of
$\omega_{b}\Omega^{2}g$ is arrived at inverse-linearly in $g$ [cf. Eq. (34)].

Average power. The average power $P$ of the pulsed quantum battery is defined
by Eq. (8), and it is made explicit by the energetic expressions given in Eq.
(21). Upon completely neglecting dissipation from the battery charger
($\gamma\to 0$), the average power $P$ is simply given by [cf. Eq. (20)]

$\lim_{\gamma\to 0}P=\omega_{b}\>\Omega^{2}\frac{\sin^{2}\left(gt\right)}{t},$
(35)

which is indeed zero at $t=0$ and later presents exact nodes at regular time
intervals of $\pi/g$. The dynamic behaviour of Eq. (35) exhibits a rise to a
global maxima at some time $t_{P}$, before gradually decaying inverse-linearly
with time while oscillating. Working in the dimensionless variable $z=gt_{P}$,
this maximum of average power $P(t_{P})$ can be found from the transcendental
equation $\tan(z)=2z$, where the range $0<z<\pi/2$ emerges because the first
maxima has the largest peak. Calculating a series expansion about the point
$z=\pi/2$ up to 1st order and solving the resulting quadratic equation yields
the approximate root $z=(2\pi+\sqrt{9\pi^{2}-60})/10\simeq 1.17$. Therefore,
this idealized dissipationless case presents a maximum in average power at the
time $t_{P}\simeq 1.17/g$, when the average power is $P(t_{P})\simeq
0.72\omega_{b}\Omega^{2}g$. We plot the average powers $P$ as a function of
time for the full driven-dissipative cases in Fig. 4 (a), using Eq. (8) with
Eq. (21). As usual, the situations with the coupling $g$ below (pink line), at
(dashed red line) and above (blue and cyan lines) the exceptional point at
$g_{\mathrm{EP}}$ are graphed [cf. Eq. (14)]. Intuitively, larger average
powers $P$ arise with stronger couplings $g$, which are peaked at increasingly
short periods of time. Above the exceptional point ($g>g_{\mathrm{EP}}$),
nodes regularly occur at integer multiples of $\pi/G$ since no energy is
stored in these special cases due to the Rabi-like oscillations [cf. Eq.
(21)]. The average power $P$ displays a global maximum at some particular time
$t_{P}$, which is given by the solution of one of the following turning point
equations (depending on the coupling strength $g$)

$\displaystyle 2x$
$\displaystyle=\tanh\left(x\right)\left[1+\tfrac{\gamma}{2\Gamma}x\right],$
$\displaystyle g<g_{\mathrm{EP}},$ (36) $\displaystyle 2y$
$\displaystyle=\tan\left(y\right)\left[1+\tfrac{\gamma}{2G}y\right],$
$\displaystyle g>g_{\mathrm{EP}},$ (37)

which are written in the dimensionless variables $x=\Gamma t_{P}$ and
$y=Gt_{P}$, and where $0<y<\pi/2$. The solutions of Eq. (36) and Eq. (37) can
be approximated by the analytical forms

$t_{P}\simeq\begin{cases}\tfrac{\zeta-8\left(\zeta-2\right)\left(\frac{g}{\gamma}\right)^{3/2}}{\gamma},&g<g_{\mathrm{EP}},\\\
\tfrac{2}{\gamma},&g=g_{\mathrm{EP}},\\\
\tfrac{Z-\frac{2Z-1}{4\sqrt{2}}\left(\frac{\gamma}{g}\right)^{3/4}}{g},&g>g_{\mathrm{EP}},\end{cases}$
(38)

where two transcendental numbers $\zeta$ and $Z$ have arisen, where

$\displaystyle\zeta$ $\displaystyle=2.512862...,$ (39) $\displaystyle Z$
$\displaystyle=1.165561....$ (40)

The exact optimal times $t_{P}$ are plotted as the pink-cyan line in Fig. 4
(b) as a function of the coupling-to-dissipation ratio $g/\gamma$, using the
solutions of Eq. (36) and Eq. (37). In particular, the asymptotics of this
important temporal quantity may be found from Eq. (38) as follows

$\displaystyle\tfrac{\zeta}{\gamma},$ $\displaystyle g\ll g_{\mathrm{EP}},$
(41) $\displaystyle\tfrac{Z}{g},$ $\displaystyle g\gg g_{\mathrm{EP}},$ (42)

which are plotted as the dashed red and dashed blue lines respectively in Fig.
4 (b). Notably, with very small coupling $g$ the optimal time $t_{P}$ tends to
a constant (in units of $\gamma$), while for very large couplings $t_{P}$
drops off inverse-linearly with $g$. The maximum average power $P(t_{P})$, as
a function of the decisive ratio $g/\gamma$, is graphed in Fig. 4 (c) with the
pink-cyan line. The scaling dependencies can be found explicitly through the
limiting behaviours of Eq. (41) and Eq. (42) as

$\displaystyle\tfrac{\sinh^{2}\left(\frac{\zeta}{4}\right)\mathrm{e}^{-\frac{\zeta}{2}}}{\zeta}\>\omega_{b}\Omega^{2}\tfrac{\left(4g\right)^{2}}{\gamma},$
$\displaystyle g\ll g_{\mathrm{EP}},$ (43)
$\displaystyle\tfrac{\sin^{2}\left(Z\right)}{Z}\>\omega_{b}\Omega^{2}g,$
$\displaystyle g\gg g_{\mathrm{EP}},$ (44)

where the two numerical prefactors are
$\sinh^{2}(\zeta/4)\exp(-\zeta/2)/\zeta\simeq 0.051$ and $\sin^{2}(Z)/Z\simeq
0.72$ respectively. Importantly, there is a quadratic scaling with $g$ at
small coupling and a linear relationship with $g$ at large couplings. Taken
together, Fig. 2, Fig. 3 and Fig. 4 provide a comprehensive description of the
energetics of the pulsed quantum battery, while brief discussions of the
associated ergotropy and energy fluctuations are provided in the supplementary
data.

Conclusion. We have proposed a driven-dissipative quantum theory of a pulsed
quantum battery. Our model features several desirable features, including
brief analytical expressions for various energetic and temporal measures of
the quality of the battery, which highlight the optimal setup of the system.
We hope that our theory provides some utility for the expected deluge of
experiments involving energy storage in quantum objects in the oncoming years
Quach2022 ; Joshi2022 ; Hu2022 ; Stevens2022 ; Maillette2022 , as well as
stimulating further theoretical research into quantum technologies exploiting
quantum continuous variables Killoran2019 ; Fukui2022 .

Acknowledgments
Funding: CAD is supported by the Royal Society via a University Research
Fellowship (URF/R1/201158) and by Royal Society Enhanced Research Expenses
which support MSU. CAD also gratefully acknowledges an Exeter-FAPESP SPRINT
grant with the Universidade Federal de São Carlos. Discussions: CAD thanks R.
Bachelard, A. Cidrim, A. C. Santos and C. J. Villas-Boas for fruitful
discussions and for their generous hospitality during his visits to UFSCar.

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