# A quantum battery with quadratic driving

Charles Andrew Downing  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
<EMAIL_ADDRESS>
###### Abstract

Abstract
Quantum batteries are energy storage devices built using quantum mechanical
objects, which are developed with the aim of outperforming their classical
counterparts. Proposing optimal designs of quantum batteries which are able to
exploit quantum advantages requires balancing the competing demands for fast
charging, durable storage and effective work extraction. Here we study
theoretically a bipartite quantum battery model, composed of a driven charger
connected to an energy holder, within two paradigmatic cases of a driven-
dissipative open quantum system: linear driving and quadratic driving. The
linear battery is governed by a single exceptional point which splits the
response of the battery into two regimes, one of which induces a good amount
of useful work. Quadratic driving leads to a squeezed quantum battery, which
generates plentiful useful work near to critical points associated with
dissipative phase transitions. Our theoretical results may be realized with
parametric cavities or nonlinear circuits, potentially leading to the
manifestation of a quantum battery exhibiting squeezing.

Introduction
The Laws of Thermodynamics allow for a complete description of classical
thermal machines, from classical heat engines to refrigerators. However, the
ongoing trend for device miniaturization inevitably led to quantum effects
becoming important. This realization required the development of theories of
thermal energy conversion in the quantum regime Kosloff2014 ;
Bhattacharjee2021 . Shortly afterwards, an influential study of energy storage
and transduction at the nanoscale pioneered the concept of a quantum system
storing and releasing energy on demand: a quantum battery Alicki2013 ;
Campaioli2023 .

An archetypal quantum battery model consists of two parts, the battery holder
and the battery charger. The holder is essentially isolated from the external
environment in order to prevent energy loss, and hence it is modelled as a
dissipationless subsystem. In order to receive energy, the battery holder then
needs to be coupled to the the battery charger. The charger subsystem is
supposed to feel the environment – allowing it to be driven – but this comes
at the cost of the charger suffering from dissipation. After charging for some
finite period of time the battery holder is disconnected from the battery
charger so that energy storage, and eventually energy extraction on demand,
may occur Farina2019 . This fundamental bipartite quantum battery model, along
with its charging, storage and discharging performances, has been considered
theoretically in various guises over the last few years of intensive quantum
battery research Ferraro2018 ; Andolina2018 ; Le2018 ; Barra2019 ; Zhang2019 ;
Santos2019 ; Pirmoradian2019 ; Keck2019 ; Crescente2020 ; Santos2020 ;
Carrega2020 ; Santos2021 ; Xu2021 ; Dou2022 ; Barra2022 ; Carrasco2022 ;
Shi2022 ; Santos2023 .

Experimentally, superabsorption in a Dicke-style quantum battery has already
been reported Quach2022 , as has a detailed investigation of collective
charging in a variety of spin-based systems Joshi2022 . The realization of a
quantum battery based upon a superconducting qutrit, including a
characterization of its charging and self-discharging processes, has also been
achieved in some recent and rather ingenious experiments Hu2022 . Meanwhile,
the first steps in gauging the performance of quantum emitters and light
fields for the purpose of energy transfer have been performed empirically
Maillette2022 . Complementary studies of the energetics of various quantum
objects, including qubits Cimini2020 ; Stevens2022  and nuclear spins
Peterson2019 , demonstrate the promise of this nascent quantum technological
field.

In what follows, we consider theoretically the celebrated bipartite quantum
battery model, where the battery charger and battery holder are modelled as
two coupled quantum harmonic oscillators, in two different circumstances.
Firstly, we revisit the linear (one-photon) driving setup as discussed by
Farina and co-workers Farina2019 . This study both lays the theoretical
framework for the Gaussian quantum batteries considered and acts as a
comparator for the second case of interest: a quadratic (two-photon) driving
arrangement Leghtas2015 ; Wang2016 ; Pechal2019 ; Wustmann2019 ; Gaikwad2023 .
The latter case of parametric driving significantly alters the underlying
physics of the quantum battery due to the inducement of spectral collapse,
quantum squeezing, dynamic instabilities and dissipative phase transitions.
Most importantly, we reveal that the quadratic quantum battery allows for
abundant useful energy to be stored in it when the driving amplitude is near
to certain critical values.

Results
Bipartite quantum battery model
The total Hamiltonian operator $\hat{H}$ of the composite quantum battery
system can be decomposed into four parts,

$\hat{H}=\hat{H}_{c}+\hat{H}_{h}+\hat{H}_{c-h}+\hat{H}_{d},$ (1)

which accounts for the battery charger energy, the battery holder energy, the
charger-holder coupling and the coherent driving of the charger respectively.
Taking $\hbar=1$ throughout, these Hamiltonian contributions are – each in
turn – defined by

$\displaystyle\hat{H}_{c}$ $\displaystyle=\omega_{b}\,c^{\dagger}c,$ (2)
$\displaystyle\hat{H}_{h}$ $\displaystyle=\omega_{b}\,h^{\dagger}h,$ (3)
$\displaystyle\hat{H}_{c-h}$
$\displaystyle=g\,\left(c^{\dagger}h+h^{\dagger}c\right),$ (4)
$\displaystyle\hat{H}_{d}$
$\displaystyle=\Omega\mathrm{e}^{\mathrm{i}\theta}\mathrm{e}^{-\mathrm{i}\omega_{d}t}\,c^{\dagger}+\mathrm{h.c.\,}.$
(5)

The energy level spacings of the harmonic oscillators modelling the battery
charger and the battery holder are equal at $\omega_{b}$, the coupling
strength between them is $g$, and the laser driving the charger has an
amplitude $\Omega\geq 0$, phase $\theta$ and frequency $\omega_{d}$.
Excitations in the battery charger are created and destroyed by the operators
$c^{\dagger}$ and $c$ respectively, while the ladder operators $h^{\dagger}$
and $h$ track the transitions up and down the energy ladder of the battery
holder. Both flavours of operator are subject to bosonic commutation
relations, $[c,c^{\dagger}]=1$ and $[h,h^{\dagger}]=1$. Notably, the counter-
rotating terms $\propto ch$ and $\propto c^{\dagger}h^{\dagger}$ not appearing
in the coupling Hamiltonian of Eq. (4) have been dropped since they are small
for typical couplings $g$ satisfying $g\ll\omega_{b}$. Therefore, we do not
enter the so-called ultrastrong coupling regime, as commonly defined without
reference to losses Diaz2019 ; Boite2020 ; Toghill2022 ; Shaghaghi2002  (see
also Supplementary Notes 2 and 3, where we find that this rotating-wave
approximation is a good one for any coupling $g$ satisfying
$g\lesssim\omega_{b}/1000$). A graphical representation of this bipartite
battery arrangement, as captured mathematically by Eq. (1), is sketched in
Fig. 1 (a). This cartoon includes the three key parameters of the model: the
driving amplitude $\Omega$, the charger-holder coupling $g$, and the loss rate
of the charger $\gamma$.

While the battery holder is approximated as a dissipationless subsystem, we
consider the battery charger subsystem to suffer loss (as measured with the
decay rate $\gamma$, where $\gamma\geq 0$). Within an open quantum systems
approach Breuer2002 ; Downing2022 , we employ the
Gorini–Kossakowski–Sudarshan–Lindblad quantum master equation for the density
matrix $\rho$, in the standard form
$\partial_{t}\rho=\mathrm{i}[\rho,\hat{H}]+(\gamma/2)\mathcal{L}_{c}\left[\rho\right]$.
Here the Lindbladian superoperator $\mathcal{L}_{c}\left[\rho\right]=2c\rho
c^{\dagger}-c^{\dagger}c\rho-\rho c^{\dagger}c$ acts on $\rho$, while the
unitary dynamics of the closed system are provided by the Hamiltonian operator
$\hat{H}$ of Eq. (1). In arriving at this master equation, we have employed
the Born approximation (due to the assumption of weak interactions between the
system and the environment), the Markov approximation (due to the supposition
that the memory of the environment is much shorter than that of the system),
and the secular approximation (due to certain transition frequencies leading
to quickly-rotating terms which are neglectable) Breuer2002 .

In order to judge the energetic performance of the quantum battery there are
two crucial energies of interest, $E_{h}$ and $E_{h}^{\beta}$, which are both
associated with the battery holder Farina2019

$\displaystyle E_{h}$
$\displaystyle=\operatorname{Tr}\\{\hat{H}_{h}\rho_{h}\\},$ (6) $\displaystyle
E_{h}^{\beta}$
$\displaystyle=\operatorname{Tr}\\{\hat{H}_{h}\rho_{h}^{\beta}\\},$ (7)

here the reduced density matrix $\rho_{h}$ of the battery holder subsystem $h$
is defined by $\rho_{h}=\operatorname{Tr}_{c}\\{\rho\\}$, where the partial
trace over the battery charger subsystem $c$ has been taken. The quantity
$E_{h}$ in Eq. (6) measures the mean energy stored in the battery holder due
to the state $\rho_{h}$. However, not all of this energy $E_{h}$ may be useful
(for example, the system may be in thermal equilibrium). The state
$\rho_{h}^{\beta}$ appearing in Eq. (7) is the so-called passive state of the
battery holder, which has the property that no work can be extracted from it
cyclically under unitary evolution Pusz1977 ; Lenard1978 ; Friis2018 . The
passive state $\rho_{h}^{\beta}$ can be obtained by re-ordering the
eigenvalues of the Hamiltonian and density matrix appropriately
Allahverdyan2004 . The energies of Eq. (6) and Eq. (7) may then be simply
combined into an influential measure – the so-called ergotropy $\mathcal{E}$ –
like so

$\mathcal{E}=E_{h}-E_{h}^{\beta},$ (8)

which measures the upper bound of the useful energy stored within the battery
holder Allahverdyan2004 . The ergotropy $\mathcal{E}$ will be nonzero if the
state $\rho_{h}$ is non-passive, which can arise due to population inversion
or thanks to certain coherences for example. Understanding the dynamical and
steady state behaviors of the ergotropy $\mathcal{E}$ for the bipartite
quantum battery, with linear and quadratic drivings respectively, is the
principle goal of this theoretical study.

For the purposes of the following calculations, it is convenient to work with
the rotated density matrix $\rho\to\tilde{\rho}$, found via the transformation
$\tilde{\rho}=\mathrm{e}^{\mathrm{i}\omega_{d}t\left(c^{\dagger}c+h^{\dagger}h\right)}\rho\mathrm{e}^{-\mathrm{i}\omega_{d}t\left(c^{\dagger}c+h^{\dagger}h\right)}$.
This leads to the quantum master equation for $\tilde{\rho}$, complete with
the effective Hamiltonian operator $\hat{\mathcal{H}}$, as follows [cf. Eqs.
(2)–(5)]

$\displaystyle\partial_{t}\tilde{\rho}=$
$\displaystyle~{}\mathrm{i}\left[\tilde{\rho},\hat{\mathcal{H}}\right]+\frac{\gamma}{2}\mathcal{L}_{c}\left[\tilde{\rho}\right],$
(9) $\displaystyle\hat{\mathcal{H}}=$
$\displaystyle~{}\Delta\left(c^{\dagger}c+h^{\dagger}h\right)+g\left(c^{\dagger}h+h^{\dagger}c\right)$
$\displaystyle+\Omega\left(\mathrm{e}^{\mathrm{i}\theta}c^{\dagger}+\mathrm{e}^{-\mathrm{i}\theta}c\right),$
(10)

where we have introduced the driving-battery detuning frequency
$\Delta=\omega_{b}-\omega_{d}$. The rotation of the reduced density matrix
$\rho_{h}\to\tilde{\rho}_{h}$ is similarly governed by
$\tilde{\rho}_{h}=\mathrm{e}^{\mathrm{i}\omega_{d}t\,h^{\dagger}h}\rho_{h}\mathrm{e}^{-\mathrm{i}\omega_{d}t\,h^{\dagger}h}$,
such that the energies of interest [cf. Eq. (6) and Eq. (7)] can be
equivalently written as
$E_{h}=\operatorname{Tr}\\{\hat{H}_{h}\tilde{\rho}_{h}\\}$ and
$E_{h}^{\beta}=\operatorname{Tr}\\{\hat{H}_{h}\tilde{\rho}_{h}^{\beta}\\}$.
Furthermore, the properties of traces and partial traces allows for the
aforementioned energetic quantities, given in terms of the reduced density
matrices $\tilde{\rho}_{h}$ and $\tilde{\rho}_{h}^{\beta}$, to be reconfigured
in terms of the full (and rotated) density matrix $\tilde{\rho}$ like so:
$\operatorname{Tr}\\{\hat{H}_{h}\tilde{\rho}_{h}\\}=\operatorname{Tr}\\{\hat{H}_{h}\tilde{\rho}\\}$
and
$\operatorname{Tr}\\{\hat{H}_{h}\tilde{\rho}_{h}^{\beta}\\}=\operatorname{Tr}\\{\hat{H}_{h}\tilde{\rho}^{\beta}\\}$.
Given this identification, one can use the trace property
$\operatorname{Tr}\\{\mathcal{O}\tilde{\rho}\\}=\langle\mathcal{O}\rangle$,
which is valid for any operator $\mathcal{O}$, to simplify both energies of
interest $E_{h}$ and $E_{h}^{\beta}$. We consider the initial state of the
system (at $t=0$) to be the product of the local vacuum states of the battery
charger and battery holder respectively,
$\tilde{\rho}\left(t=0\right)=|0\rangle\langle 0|_{c}\otimes|0\rangle\langle
0|_{h}$, after which both battery components start to interact as the charging
process commences.

Figure 1:  The linear quantum battery and its steady state. Panel (a): a
sketch of the bipartite quantum battery (both parts are modelled as quantum
harmonic oscillators, with the level spacing $\omega_{b}$), composed of a
battery charger coupled to a battery holder with the coupling strength $g$
(green bar). The charger is driven coherently (cyan arrow) with an amplitude
$\Omega$, phase $\theta$ and frequency $\omega_{d}$, while it suffers loss
(yellow arrow) at the decay rate $\gamma$. Panel (b): the ergotropy
$\mathcal{E}$ in the steady state (in units of $\omega_{b}$) as a function of
$\Omega$ and $\Delta$ (both in units of $\gamma$) from Eq. (16). Here the
coupling $g=\gamma$. Panel (c): as for panel (b), but with
$g=\gamma/(2\sqrt{2})\simeq 0.35\gamma$. Panel (d): the detuning $\Delta$
associated with the maximum of the steady state ergotropy $\mathcal{E}$, as a
function of the coupling $g$ [cf. Eq. (17)]. Panel (e): a semi-logarithmic
plot of the maximum of the steady state ergotropy $\mathcal{E}$, in units of
$\omega_{b}$ and scaled by $(\gamma/\Omega)^{2}$, as a function of the
coupling $g$ [cf. Eq. (18)].

Pleasingly, one may exploit some neat properties of Gaussian systems
Ferraro2005 ; Olivares2012 ; Serafini2017  in order to fully determine the
passive state energy $E_{h}^{\beta}$ (see Supplementary Note 1 and elsewhere
Farina2019  for more details). We then finally arrive at compact expressions
for both Eq. (6) and Eq. (7), in terms of operator expectation values, as
follows

$\displaystyle E_{h}$ $\displaystyle=\omega_{b}\langle h^{\dagger}h\rangle,$
(11) $\displaystyle E_{h}^{\beta}$
$\displaystyle=\omega_{b}\left(\tfrac{\sqrt{\mathcal{D}}-1}{2}\right).$ (12)

Here we have introduced the dimensionless quantity $\mathcal{D}$, which
collects the first and second moments of the battery holder, that is objects
like $\langle h\rangle$ and $\langle h^{\dagger}h\rangle$, in the form

$\mathcal{D}=\Bigl{\\{}1+2\langle h^{\dagger}h\rangle-2\langle
h^{\dagger}\rangle\langle h\rangle\Bigl{\\}}^{2}-4\Big{|}\langle
hh\rangle-\langle h\rangle^{2}\Big{|}^{2}.$ (13)

Now the ergotropy $\mathcal{E}$ of the battery holder can be readily computed
via Eq. (8), along with Eq. (11)–(13). In what follows, we consider the
dynamical and steady state ergotropies for the composite quantum battery
system, firstly with the one-photon driving case of Eq. (5) and then for the
arguably more interesting quadratic case, featuring two-photon driving of the
battery charger.

Linear quantum battery
The situation with one-photon coherent driving, as sketched in Fig. 1 (a) and
as governed by the Hamiltonian operator $\hat{\mathcal{H}}$ of Eq. (10),
readily allows for analysis of the first and second moments (as is carried out
in Supplementary Note 2, see also calculations in the literature Farina2019
for the more specific case with $\Delta=0$). Since the system correlators
factorize perfectly, such that $\langle h^{\dagger}h\rangle=\langle
h^{\dagger}\rangle\langle h\rangle$ and $\langle hh\rangle=\langle
h\rangle\langle h\rangle$ for example, the key quantity $\mathcal{D}$ as
defined in Eq. (13) simply reduces to

$\mathcal{D}=1.$ (14)

This correlator factorization stems from the joint battery charger–battery
holder system evolving in a product state Ferraro2005 ; Olivares2012 ;
Serafini2017 . (Notably, the neat Eq. (14) is not met when there is nonzero
incoherent driving, which leads to unuseful energy storage in that particular
case, as discussed previously Farina2019 ). Given the form of Eq. (14), the
battery holder state must be pure due to the properties of Gaussian systems
Serafini2017 . Hence all of the stored energy in the battery holder is useful
because the passive state energy must be [cf. Eq. (12)]

$E_{h}^{\beta}=0.$ (15)

Therefore the ergotropy-to-battery-holder energy ratio for the linear quantum
battery is, most pleasantly, a perfect one: $\mathcal{E}/E_{h}=1$ [cf. Eq.
(8)].

Figure 2:  The linear quantum battery and its dynamics. Panel (a): the real
parts of the complex eigenvalues $\epsilon_{\pm}$, shifted by the drive-
battery detuning $\Delta$, as a function of the charger-holder coupling
strength $g$ (both in units of the charger damping rate $\gamma$) using Eq.
(19). Panel (b): as for panel (a), but for the imaginary parts. Dashed grey
lines: the exceptional point $g_{\mathrm{EP}}$ [cf. Eq. (21)]. Panel (c): the
ergotropy $\mathcal{E}$, scaled by $(\gamma/\Omega)^{2}$ and in units of
$\omega_{b}$, as a function of time $t$ (in units of the inverse of $\gamma$).
Thick colored lines: the results for various values of the coupling $g$ above
the exceptional point ($g\geq g_{\mathrm{EP}}$). Thin grey lines: intermediate
couplings as guides for the eye. Panel (d): as for panel (c), but for
couplings $g$ below the exceptional point ($g\leq g_{\mathrm{EP}}$), and
plotted as a semi-logarithmic plot. In panels (c) and (d), we consider the
case of zero detuning ($\Delta=0$), as governed by Eq. (23), Eq. (24) and Eq.
(25).

In the steady state ($t\to\infty$), the ergotropy $\mathcal{E}$, and
identically the energy $E_{h}$ of the battery holder, are simply given by [see
Eq. (11) and Supplementary Note 2]

$\lim_{t\to\infty}\mathcal{E}=\lim_{t\to\infty}E_{h}=\omega_{b}\frac{\left(g\Omega\right)^{2}}{\left(\Delta^{2}-g^{2}\right)^{2}+\left(\frac{\gamma}{2}\Delta\right)^{2}}.$
(16)

Quite intuitively, maximising the steady state ergotropy $\mathcal{E}$
requires the maximisation of the driving amplitude $\Omega$ and the
minimisation of the charger decay rate $\gamma$. There is a more complicated
competition between the charger-holder coupling strength $g$ and the driving-
battery detuning $\Delta$. Typical circumstances are plotted in Fig. 1 (b, c)
for example cases with the couplings $g=\gamma$ and
$g=\gamma/(2\sqrt{2})\simeq 0.35\gamma$ respectively. In these two panels, the
darker the colour the larger the steady state ergotropy $\mathcal{E}$,
showcasing how the ergotropic maximum depends nontrivially on the detuning
$\Delta$. Denoting the specific value of the detuning $\Delta$ corresponding
to the ergotropy maximum in the steady state as $\Delta^{\prime}$, one finds
the relation

$\Delta^{\prime}=\sqrt{g^{2}-\left(\tfrac{\gamma}{2\sqrt{2}}\right)^{2}}~{}\Theta\left(g-\tfrac{\gamma}{2\sqrt{2}}\right),$
(17)

where $\Theta(x)$ is the Heaviside step function. This formula suggests that
zero detuning is optimal for smaller couplings $g\leq\gamma/(2\sqrt{2})$ below
a critical value, while otherwise a nonzero detuning is preferable – as is
hinted at in Fig. 1 (b, c) and as shown explicitly in Fig. 1 (d). Inserting
Eq. (17) into Eq. (16) yields the most desirable steady state egrotopies
$\mathcal{E}$, depending upon the relative weightings between $g$ and
$\gamma$, as follows

$\lim_{t\to\infty}\mathrm{max}\\{\mathcal{E}\\}=\begin{cases}\omega_{b}\left(\frac{\Omega}{g}\right)^{2},&g\leq\frac{\gamma}{2\sqrt{2}},\\\
\omega_{b}\left(\frac{\Omega}{\gamma}\right)^{2}\frac{\left(4g\right)^{2}}{\left(2g\right)^{2}-\left(\frac{\gamma}{2}\right)^{2}},&g>\frac{\gamma}{2\sqrt{2}},\end{cases}$
(18)

as is plotted explicitly in Fig. 1 (e). Most notably, for smaller couplings
$g\leq\gamma/(2\sqrt{2})$ there is a simple quadratic relation for the largest
steady state egrotropy, as was discussed in detail previously Farina2019 . For
larger couplings $g\gg\gamma/(2\sqrt{2})$, the largest steady state egrotopy
becomes independent of the coupling $g$ as it tends towards
$\lim_{t\to\infty}\mathcal{E}=\omega_{b}(2\Omega/\gamma)^{2}$, which features
a natural competition between the drive amplitude $\Omega$ and the decay rate
$\gamma$.

The time-dependent behaviour of this linear quantum battery is mainly
determined by the complex eigenvalues $\epsilon_{\pm}$, which arise from the
dynamical matrix describing the first moments of the system (see Supplementary
Note 2), like so

$\epsilon_{\pm}=\Delta\pm
G-\mathrm{i}\frac{\gamma}{4},\quad\quad\quad\phi_{\pm}=\begin{pmatrix}1\\\
\frac{g}{\epsilon_{\pm}-\Delta}\end{pmatrix},$ (19)

where $\phi_{\pm}$ are the associated eigenvectors. Here we have introduced
the renormalized coupling strength $G$ and (for use later on) a closely
associated parameter known here as the renormalized decay rate $\Gamma$, which
are defined together via

$G=\sqrt{g^{2}-\left(\tfrac{\gamma}{4}\right)^{2}},\quad\quad\quad\Gamma=\sqrt{\left(\tfrac{\gamma}{4}\right)^{2}-g^{2}}.$
(20)

The real and imaginary parts of the complex eigenvalues $\epsilon_{\pm}$ are
plotted in Fig. 2 (a, b) using Eq. (19). Most interestingly, when the coupling
strength satisfies $g=g_{\mathrm{EP}}$ (dashed and grey vertical lines in both
panels), where

$g_{\mathrm{EP}}=\frac{\gamma}{4},$ (21)

the twin objects provided in Eq. (19) simultaneously coalesce, such that Eq.
(21) highlights the presence of an exceptional point in the dynamical system.
Exceptional points are known to be highly consequential in systems which may
be described as non-Hermitian in some sense, and typically they mark the
borderland between regimes with very different physical responses Berry2003 ;
Heiss2003 ; Ozdemir2019 ; Miri2019 ; Downing2021 .

The dynamical ergotropy $\mathcal{E}$ can be calculated using the framework
built in Supplementary Note 2, and the analysis is most easily done for the
case of zero drive-battery detuning ($\Delta=0$). We start by noting that in
the completely dissipationless battery charger limit (that is, when $\gamma\to
0$), one finds the oscillating ergotropy

$\lim_{\gamma\to
0}\mathcal{E}=\omega_{b}\left(\frac{2\Omega}{g}\right)^{2}\sin^{4}\left(\frac{gt}{2}\right),$
(22)

which clearly reaches its maximum value of $\omega_{b}(2\Omega/g)^{2}$ at the
time $t=\pi/g$ (and at all later times with the periodicity of $2\pi/g$).
Within this lossless regime, we have also obtained the result equivalent to
Eq. (22) but with the counter-rotating terms in the coupling Hamiltonian of
Eq. (4) also included, which may be useful for future considerations of
ultrastrongly coupled systems (see Supplementary Note 2). Otherwise, within
the full driven-dissipative theory, there are two regimes of interest (split
by an intermediate marginal case) due to the presence of the exceptional point
of Eq. (21). These cases may be described analytically with the expressions

$\displaystyle\mathcal{E}$
$\displaystyle=\omega_{b}\left(\frac{\Omega}{g}\right)^{2}\Bigl{\\{}1-\bigr{[}\cosh\left(\Gamma
t\right)+\tfrac{\gamma}{4\Gamma}\sinh\left(\Gamma
t\right)\bigr{]}\mathrm{e}^{-\frac{\gamma}{4}t}\Bigl{\\}}^{2},$ $\displaystyle
g<g_{\mathrm{EP}},$ (23) $\displaystyle\mathcal{E}$
$\displaystyle=\omega_{b}\left(\frac{4\Omega}{\gamma}\right)^{2}\Bigl{\\{}1-\bigr{[}1+\tfrac{\gamma
t}{4}\bigr{]}\mathrm{e}^{-\frac{\gamma}{4}t}\Bigl{\\}}^{2},$ $\displaystyle
g=g_{\mathrm{EP}},$ (24) $\displaystyle\mathcal{E}$
$\displaystyle=\omega_{b}\left(\frac{\Omega}{g}\right)^{2}\Bigl{\\{}1-\bigr{[}\cos\left(Gt\right)+\tfrac{\gamma}{4G}\sin\left(Gt\right)\bigr{]}\mathrm{e}^{-\frac{\gamma}{4}t}\Bigl{\\}}^{2},$
$\displaystyle g>g_{\mathrm{EP}},$ (25)

where the twin frequencies $G$ and $\Gamma$ are both defined in Eq. (20).
Remarkably, the physics of the exceptional point ensures damped-hyperbolic,
damped-algebraic and damped-trigonometric ergotropic behaviors are all
possible, depending upon the relative strength of the charger-holder coupling
$g$ as compared to the charger decay rate $\gamma$. Notably, we always
maintain the coupling regime $g\ll\omega_{b}$ within our driven-dissipative
theory in order to avoid the discarded counter-rotating terms in the original
Hamiltonian becoming non-negligible, as was discussed after Eq. (5).
Nevertheless, in the extreme limiting cases of very weak ($g\ll\gamma$) and
very strong ($g\gg\gamma$) couplings, as defined in comparison to the loss
parameter $\gamma$ only, the ergotropy $\mathcal{E}$ follows the even simpler
approximate expressions

$\displaystyle\lim_{g\ll\gamma}\mathcal{E}$
$\displaystyle\simeq\omega_{b}\left(\frac{\Omega}{g}\right)^{2}\Bigl{\\{}1-\mathrm{e}^{-\left(\frac{\gamma}{4}-\Gamma\right)t}\Bigl{\\}}^{2},$
(26) $\displaystyle\lim_{g\gg\gamma}\mathcal{E}$
$\displaystyle=\omega_{b}\left(\frac{\Omega}{g}\right)^{2}\Bigl{\\{}1-\cos(gt)\mathrm{e}^{-\frac{\gamma
t}{4}}\Bigl{\\}}^{2}.$ (27)

With vanishing coupling $g$, the rough expression of Eq. (26) demonstrates the
exponential rise in time of the erogtropy $\mathcal{E}$ towards its eventual
steady state value [cf. Eq. (16)]

$\lim_{t\to\infty}\mathcal{E}=\omega_{b}\left(\frac{\Omega}{g}\right)^{2},$
(28)

as delicately controlled by the time constant $(\gamma/4-\Gamma)^{-1}$ and its
double within Eq. (26). The very strong coupling $g$ result of Eq. (27)
describes periodic oscillations, optimally bounded by $0$ and
$\omega_{b}(2\Omega/g)^{2}$, which are gradually damped out towards the common
steady state ergotropy of Eq. (28), which is notably at a value of one quarter
of its dynamical maximum.

Figure 3:  The quadratic quantum battery and its dynamics. Panel (a): a sketch
of the bipartite quantum battery (both parts are modelled as harmonic
oscillators, with the level spacing $\omega_{b}$), composed of a battery
charger coupled to a battery holder at the coupling strength $g$ (green bar).
The charger is driven parametrically (two cyan arrows) with an amplitude
$\Omega$, phase $\theta$ and frequency $2\omega_{d}$, while it suffers loss
(yellow arrow) at the decay rate $\gamma$. Panel (b): a semi-logarithmic plot
of the ergotropy $\mathcal{E}$ (in units of $\omega_{b}$) as a function of
time $t$ (in units of the inverse of $\gamma$) [cf. Eq. (11) and Eq. (12) with
Eq. (44)]. Inset: as for panel (b), but with the ergotropy $\mathcal{E}$
scaled by its maximum value, and with the timescale considered extended. Panel
(c): a semi-logarithmic plot of the power $P=\mathcal{E}/t$ (in units of
$\omega_{b}\gamma$) as a function of time $t$. In panels (b) and (c), we
consider the case where $g=\gamma$ and $\Delta=\gamma/2$, so that from Eq.
(40) the first critical driving amplitude
$\Omega_{\mathrm{c}}^{(1)}=\sqrt{5}/2\gamma\simeq 1.118\gamma$, while the
second critical driving amplitude
$\Omega_{\mathrm{c}}^{(2)}=\sqrt{5/2}\gamma\simeq 1.581\gamma$ [cf. Eq. (41)].
The plot legend in panel (c) also applies to panel (b), and marks the various
driving amplitudes $\Omega$ considered.

The time-dependent ergotropy $\mathcal{E}$, for various couplings
$g>g_{\mathrm{EP}}$ above the exceptional point, is shown in Fig. 2 (c) using
the exact result of Eq. (25). This panel displays Rabi-style oscillations
(which are stronger with larger $g$) before an eventual decay into a steady
state with a rather moderate ergotropy $\mathcal{E}$. The red line in panel
(c) represents the border case, as governed by Eq. (24), where the coupling
$g=g_{\mathrm{EP}}$ exactly. For smaller couplings $g<g_{\mathrm{EP}}$ below
the exceptional point, the dynamics is similarly plotted in Fig. 2 (d) using
Eq. (23). Now Rabi-style oscillations are entirely absent, and the system
instead monotonically increases in time towards a plateau at a much larger
ergotropy $\mathcal{E}$ [cf. Eq. (28)], albeit at the cost of much larger
charging times. We have checked that the general physics of the linear quantum
battery, including the interesting exceptional point physics, is maintained
even after the inclusion of loss from the battery holder (see Supplementary
Note 2).

Quadratic quantum battery
The perspectives for a highly ergotropic quantum battery can arguably be
improved by instead considering parametric driving Minganti20166 ; Puri2017 ;
Candia2023 ; Vidiella2023 . The quadratic nature of the resultant Hamiltonian
ensures that the ergotropy-to-battery holder energy ratio can be readily
calculated using the same Gaussian theoretical framework as for the linear
quantum battery [cf. Eq. (11), Eq. (12) and Eq. (13)], while the two-photon
driving should lead to larger population inversions.

Let us replace the original driving Hamiltonian $\hat{H}_{d}$ of Eq. (5) with
a two-photon drive given by

$\hat{H}_{d}=\frac{\Omega}{2}\left(\mathrm{e}^{\mathrm{i}\theta}\mathrm{e}^{-2\mathrm{i}\omega_{d}t}c^{\dagger}c^{\dagger}+\mathrm{e}^{-\mathrm{i}\theta}\mathrm{e}^{2\mathrm{i}\omega_{d}t}cc\right),$
(29)

which nicely fits into the full Hamiltonian operator $\hat{H}$ of Eq. (1).
This quadratic quantum battery model is sketched in Fig. 3 (a), where the
parametric drive is of amplitude $\Omega\geq 0$, phase $\theta$ and frequency
$2\omega_{d}$. The analysis leading to the quantum master equation of Eq. (9)
also holds for the quadratic driving case, after the replacement of the
effective Hamiltonian operator $\hat{\mathcal{H}}$ of Eq. (10) with

$\displaystyle\hat{\mathcal{H}}=~{}$
$\displaystyle\Delta\left(c^{\dagger}c+h^{\dagger}h\right)+g\left(c^{\dagger}h+h^{\dagger}c\right)$
$\displaystyle+\frac{\Omega}{2}\mathrm{e}^{\mathrm{i}\theta}c^{\dagger}c^{\dagger}+\frac{\Omega}{2}\mathrm{e}^{-\mathrm{i}\theta}cc.$
(30)

The two-photon driving appearing within Eq. (A quantum battery with quadratic
driving) implies the presence of quantum squeezing Loudon1987 , in stark
contrast to the one-photon driving case described previously in Eqs. (2)–(5).
The quadratic Hamiltonian operator $\hat{\mathcal{H}}$ provided in Eq. (A
quantum battery with quadratic driving) may be diagonalized exactly by bosonic
Bogoliubov transformation Tsallis1978 ; Colpa1978  into the two-mode form

$\hat{\mathcal{H}}=\sum_{\tau=\pm}\omega_{\tau}\beta_{\tau}^{\dagger}\beta_{\tau},$
(31)

where the mode index $\tau=\pm$ characterizes the two Bogoliubov
eigenfrequencies $\omega_{\pm}$, as defined by the expression

$\omega_{\pm}=\sqrt{\Delta^{2}+g^{2}-\tfrac{\Omega^{2}}{2}\pm\sqrt{\Omega^{2}\left(\tfrac{\Omega^{2}}{4}-g^{2}\right)+4g^{2}\Delta^{2}}}.$
(32)

The twin Bogoliubov operators $\beta_{\pm}$ appearing in Eq. (31) satisfy the
standard bosonic commutation rule $[\beta_{\pm},\beta_{\pm}^{\dagger}]=1$, and
they are composed of all four operators of the problem ($c$, $h$,
$c^{\dagger}$, and $h^{\dagger}$) as follows

$\beta_{\pm}=\frac{1}{\sqrt{2}}\begin{pmatrix}\cosh\left(\mu_{\pm}\right)\\\
\cosh\left(\nu_{\pm}\right)\\\
\mathrm{e}^{\mathrm{i}\theta}\sinh\left(\mu_{\pm}\right)\\\
\mathrm{e}^{\mathrm{i}\theta}\sinh\left(\nu_{\pm}\right)\end{pmatrix}^{\mathrm{T}}\cdot\begin{pmatrix}c\\\
h\\\ c^{\dagger}\\\ h^{\dagger}\end{pmatrix},$ (33)

where the Bogoliubov coefficients in Eq. (33) may be found from the hyperbolic
tangent relations

$\displaystyle\tanh\left(\mu_{\pm}\right)$
$\displaystyle=\frac{\Omega\left(\Delta+\omega_{\pm}\right)}{\left(\Delta+\omega_{\pm}\right)^{2}-g^{2}},$
(34) $\displaystyle\tanh\left(\nu_{\pm}\right)$
$\displaystyle=\frac{\Omega\left(\Delta-\omega_{\pm}\right)}{\left(\Delta+\omega_{\pm}\right)^{2}-g^{2}}.$
(35)

We also note that inverting the operators of Eq. (33) allows for the
nontrivial vacuum state populations of the battery charger and battery holder
to be calculated (see Supplementary Note 3), while the squeezing promised by
the counter-rotating driving terms is considered later on. Most interestingly,
the two Bogoliubov eigenfrequencies $\omega_{\pm}$ of Eq. (32) are not wholly
real for all values of the three Hamiltonian parameters $\Delta$, $\Omega$ and
$g$. This suggests a spectral collapse Kirton2018 ; Salado2021  within the
purely Hamiltonian theory of Eq. (A quantum battery with quadratic driving) in
certain parameter regimes. The phase diagram implied by the stability of the
solely Hamiltonian operator $\hat{\mathcal{H}}$ approach is plotted in Fig. 4
(a), where the boundaries (grey lines) are defined by the three equations [cf.
when Eq. (32) become complex]

$\displaystyle\Omega$
$\displaystyle=\Big{|}\Delta-\tfrac{g^{2}}{\Delta}\Big{|}\Theta\left(\tfrac{\Delta}{g}-\Xi\right),$
(36) $\displaystyle\Omega$
$\displaystyle=g\left(\sqrt{1+2\tfrac{\Delta}{g}}-\sqrt{1-2\tfrac{\Delta}{g}}\right)\Theta\left(\tfrac{1}{2}-\tfrac{\Delta}{g}\right),$
(37) $\displaystyle\Omega$
$\displaystyle=g\left(\sqrt{1+2\tfrac{\Delta}{g}}+\sqrt{1-2\tfrac{\Delta}{g}}\right)\Theta\left(\tfrac{1}{2}-\tfrac{\Delta}{g}\right)\Theta\left(\tfrac{\Delta}{g}-\Xi\right),$
(38)

where the dimensionless number $\Xi=\sqrt{\sqrt{5}-2}\simeq 0.486$. The
diagram of Fig. 4 (a) hints at a classification where the stable regions
(blue, real $\omega_{\pm}$) are associated with convergent-in-time dynamics,
while the unstable regions (white, complex $\omega_{\pm}$) should not be able
to support a steady state due to their divergent-in-time dynamics. The
introduction of dissipation into the system via the quantum master equation of
Eq. (9) upgrades the simple Hamiltonian dynamics of Eq. (31) and necessarily
updates the phase diagram of Fig. 4 (a) in a more physically meaningful manner
(for example, the imaginary parts of complex eigenfrequencies can then be
interpreted as being related to inverse lifetimes), as we now explore with
proper reference to the steady state of the full driven-dissipative system.

Figure 4:  The quadratic quantum battery and its phase diagram. Panel (a): the
Hamiltonian phase diagram of the quantum battery, as a function of the drive-
battery detuning $\Delta$ and the two-photon driving amplitude $\Omega$ (both
in units of the charger-holder coupling strength $g$). The stable (blue)
region is associated with wholly real Hamiltonian eigenfrequencies
$\omega_{\pm}$, and the unstable (white) region corresponds to at least one
complex eigenfrequency [cf. Eq. (32)]. Grey lines: boundaries coming from Eq.
(36), Eq. (37) and Eq. (38). Panel (b): the Liouvillian phase diagram of the
quantum battery in the steady state ($t\to\infty$), as a function of $\Delta$
and $\Omega$ (both in units of the charger damping rate $\gamma$). The white
region corresponds to the situation where no steady state can be formed, while
the coloured region describes when a steady state is established. The borders
are given by Eq. (40) and Eq. (41). Color bar: the battery holder energy
$E_{h}$ in the steady state (in units of the battery holder energy level
spacing $\omega_{b}$) from Eq. (39). In this panel, we consider the case of
$g=\gamma$, and we cap the energy at the maximum value of $E_{h}=4\omega_{b}$
in the color bar. Panel (c): a logarithmic plot of the salient steady state
energies as a function of $\Omega$ (in units of $\gamma$). The ergotropy
$\mathcal{E}$ (thick red line) [cf. Eq. (8)], the battery holder energy
$E_{h}$ (medium cyan line) [cf. Eq. (39)], and the battery holder energy in
the passive state $E_{h}^{\beta}$ (thin yellow line) [cf. Eq. (12) with Eq.
(45)] are all shown. Dashed grey line: the critical driving amplitude
$\Omega_{\mathrm{c}}^{(1)}$. In this panel, we consider $\Delta=\gamma/2$ and
$g=\gamma$, so that $\Omega_{\mathrm{c}}^{(1)}=\sqrt{5}\gamma/2\simeq
1.118\gamma$ and $\Omega_{\mathrm{c}}^{(2)}=\sqrt{5/2}\gamma\simeq
1.581\gamma$ from Eq. (40) and Eq. (41) respectively.

Within an open quantum systems approach, the steady state energy of the
battery holder $E_{h}$ [cf. Eq. (11)] may be found from the first and second
moments of the system (see Supplementary Note 3 for the full theory). This
leads to the analytic expression

$\lim_{t\to\infty}E_{h}=\omega_{b}\frac{\Omega^{2}}{\left(2\Delta\right)^{2}+\left(\frac{\gamma}{2}\right)^{2}-\Omega^{2}}\frac{\Delta^{4}+\frac{g^{4}}{2}-\Delta^{2}\left[\frac{g^{2}}{2}+\Omega^{2}-\left(\frac{\gamma}{2}\right)^{2}\right]}{\Delta^{4}+g^{4}-\Delta^{2}\left[2g^{2}+\Omega^{2}-\left(\frac{\gamma}{2}\right)^{2}\right]},$
(39)

which may be compared to Eq. (16), the analogous result with one-photon
driving. Notably, the denominators of both the first and second terms in the
product of fractions comprising Eq. (39) may each become zero at some point in
parameter space, suggesting two critical driving amplitudes of the system –
which we call $\Omega_{\mathrm{c}}^{(1)}$ and $\Omega_{\mathrm{c}}^{(2)}$.
These critical points are given by [cf. Eq. (36)–(38) from the Hamiltonian
theory]

$\displaystyle\Omega_{\mathrm{c}}^{(1)}$
$\displaystyle=\sqrt{\left(\tfrac{\gamma}{2}\right)^{2}+\left(2\Delta\right)^{2}},$
(40) $\displaystyle\Omega_{\mathrm{c}}^{(2)}$
$\displaystyle=\sqrt{\left(\tfrac{\gamma}{2}\right)^{2}+\Delta^{2}\left(1-\tfrac{g^{2}}{\Delta^{2}}\right)^{2}},$
(41)

and help to define the Liouvillian phase diagram of the quadratic quantum
battery in the steady state, via the duo of equations
$\Omega=\Omega_{\mathrm{c}}^{(1)}\Theta(1/\sqrt{3}-\Delta/g)$ and
$\Omega=\Omega_{\mathrm{c}}^{(2)}\Theta(\Delta/g-1/\sqrt{3})$. The steady
state battery holder energy $E_{h}$ is plotted in Fig. 4 (b), where the
situation without a steady state is represented by the white region. Clearly,
approaching the critical line formed using Eq. (40) and Eq. (41) leads to a
transition in the steady state response of the quantum battery – above this
dynamical instability the energetics are unbounded since the mean battery
populations are themselves unbounded. This is because the energetic drive into
the battery more than compensates the loss into the external environment,
leading to a dramatic rise of bosonic excitations up the infinite energy
ladders comprising the quadratic quantum battery. The energy formula of Eq.
(39) becomes much simpler in the two limiting cases of small detuning
($\Delta\to 0$) and small battery charger-holder coupling ($g\to 0$), where

$\displaystyle\lim_{t\to\infty}E_{h}$
$\displaystyle=\frac{\omega_{b}}{2}\frac{\Omega^{2}}{\left(\frac{\gamma}{2}\right)^{2}-\Omega^{2}},\quad$
$\displaystyle\Delta\to 0,$ (42) $\displaystyle\lim_{t\to\infty}E_{h}$
$\displaystyle=\omega_{b}\frac{\Omega^{2}}{\left(2\Delta\right)^{2}+\left(\frac{\gamma}{2}\right)^{2}-\Omega^{2}},\quad$
$\displaystyle g\to 0.$ (43)

The left-hand vertical axis of Fig. 4 (b) is explained by Eq. (42), complete
with its finishing point at $\Omega=\gamma/2$, above which no steady state is
formed. Meanwhile, Eq. (43) describes the weakly coupled battery result (not
shown in Fig. 4 (b), which takes $g=\gamma$ as an example case) which sees a
reduction in the number of critical points from two to one, located at
$\Omega_{\mathrm{c}}^{(1)}$ only [cf. Eq. (40)]. The phase diagram of Fig. 4
(b) illustrates the critical nature of the quadratic quantum battery, where
significant energies can be obtained near to dynamical instabilities governed
by critical points, which may be starkly contrasted to the linear quantum
battery which is instead dominated by exceptional point physics.

The quadratic nature of the Hamiltonian operator $\hat{\mathcal{H}}$ given by
Eq. (A quantum battery with quadratic driving) ensures that the result of Eq.
(12), quantifying the energy of the battery holder in the passive state
$E_{h}^{\beta}$, also holds. In particular, the boundary conditions of the
system imply that the first moments of the system are all zero (see
Supplementary Note 3 for details). Therefore that the key quantity
$\mathcal{D}$, as defined in Eq. (13), reduces to the solely second moments
form [cf. Eq. (15) for the linear quantum battery]

$\mathcal{D}=\left(1+2\langle h^{\dagger}h\rangle\right)^{2}-4\langle
h^{\dagger}h^{\dagger}\rangle\langle hh\rangle.$ (44)

Notably, since in general $\mathcal{D}\neq 1$ the quadratic quantum battery is
associated with a nonzero passive state energy ($E_{h}^{\beta}\neq 0$), which
acts to reduce the ergotropy $\mathcal{E}$ following the definition of Eq.
(8). In the crucial steady state regime ($t\to\infty$), the explicit form of
Eq. (44) is derivable exactly as (see Supplementary Note 3)

$\fixTABwidth{T}\scalebox{0.82}{\mbox{$\displaystyle\lim_{t\to\infty}\mathcal{D}=\frac{\left[\left(2\Delta\right)^{2}+\left(\frac{\gamma}{2}\right)^{2}\right]^{2}\left[\left(\Delta^{2}-g^{2}\right)^{2}+\Delta^{2}\left(\frac{\gamma}{2}\right)^{2}\right]-\Omega^{2}\left[\left(2\Delta\right)^{2}+\left(\frac{\gamma}{2}\right)^{2}\right]\left[2\Delta^{4}+g^{4}-\Delta^{2}\left(2g^{2}+\left(\frac{\gamma}{2}\right)^{2}\right)\right]-\Delta^{2}\Omega^{4}\left[7\Delta^{2}-4g^{2}+\left(\frac{\gamma}{2}\right)^{2}\right]-\Delta^{2}\Omega^{6}}{\left[\left(2\Delta\right)^{2}+\left(\frac{\gamma}{2}\right)^{2}-\Omega^{2}\right]^{2}\left[\Delta^{4}-\Delta^{2}\left[2g^{2}+\Omega^{2}-\left(\frac{\gamma}{2}\right)^{2}\right]+g^{4}\right]}$}}.$
(45)

This analytic result ensures that the steady state ergotropy $\mathcal{E}$ has
been analytically obtained with the aid of the analytic expression of the
holder energy $E_{h}$ [cf. Eq. (39)] alongside the exact passive state holder
energy $E_{h}^{\beta}$ [cf. Eq. (12) with Eq. (45)]. The full expression for
the steady state ergotropy $\mathcal{E}$ is particularly compact in the
limiting case of small detuning ($\Delta\to 0$), where we find it reduces to

$\lim_{t\to\infty}\mathcal{E}=\frac{\omega_{b}}{2}\left(\frac{\left(\frac{\gamma}{2}\right)^{2}}{\left(\frac{\gamma}{2}\right)^{2}-\Omega^{2}}-\sqrt{\frac{\left(\frac{\gamma}{2}\right)^{2}}{\left(\frac{\gamma}{2}\right)^{2}-\Omega^{2}}}\right),$
(46)

which exhibits a single critical point at $\Omega_{\mathrm{c}}^{(1)}=\gamma/2$
[cf. Eq. (40)]. In the limit of small charger-holder coupling $g$, one finds
that the full expression reduces to the intuitive result of zero steady state
ergotropy ($\lim_{t\to\infty}\mathcal{E}=0$ for the case of $g\to 0$).

We consider the steady state energetics of the quadratic quantum battery in
Fig. 4 (c), as a function of the driving amplitude $\Omega$, for the example
case where the detuning $\Delta=\gamma/2$ and the charger-holder coupling
$g=\gamma$. The ergotropy $\mathcal{E}$ (thick red line) [cf. Eq. (8)], the
battery holder energy $E_{h}$ (medium cyan line) [cf. Eq. (39)], and the
battery holder energy in the passive state $E_{h}^{\beta}$ (thin yellow line)
[cf. Eq. (12) with Eq. (45)] are all shown explicitly in their steady state
forms. As the driving amplitude $\Omega$ approaches the critical point at
$\Omega_{\mathrm{c}}^{(1)}=\sqrt{5/2}\gamma\simeq 1.118\gamma$ (dashed grey
line) [cf. Eq. (40)] all three energetic quantities increase without bound,
leading to an abundance of useful energy being stored in the battery. This
boundlessness occurs since a steady state is no longer supportable within this
driven-dissipative theory when the drive into the battery overcompensates the
loss into the outside environment. In practice, such a seemingly runaway
solution may be prevented by either truncating the infinite energy ladders
associated with the harmonic oscillators forming the quantum battery model,
leading instead to a saturation at some large energy (as is discussed later
on), or by introducing anharmonicities. Importantly, there is no analogous
critical points in the linear quantum battery, and hence there is no such
optimal driving associated with a huge ergotropic response.

Figure 5:  The quadratic quantum battery and its steady state. Panel (a): a
logarithmic plot of the battery holder energy $E_{h}$ in the steady state (in
units of the holder energy level spacing $\omega_{b}$) as a function of the
two-photon driving amplitude $\Omega$ (in units of the charger decay rate
$\gamma$). Thick line: the analytic result of Eq. (39). Thin lines: results
obtained by truncating the harmonic oscillators modelling the battery to
$N$-level systems. Panel (b): the purity $\mathcal{P}$, in the steady state
and as a function of $\Omega$, for a battery comprised of $N$-level systems.
Panel (c): the Liouvillian gap $\Lambda$ as a function of $\Omega$. Panel (d):
the negativity $\mathcal{N}$, in the steady state and as a function of
$\Omega$. Panel (e): a logarithmic plot of the battery holder quadrature
variances $\sigma_{X,h}^{2}$ (thick cyan line) and $\sigma_{P,h}^{2}$ (thick
yellow line) in the steady state and as a function of $\Omega$. The product of
the standard deviations $\sigma_{X,h}\sigma_{P,h}$ (medium dashed green line)
is also shown, as is a guide for the eye at the Robertson-Schrödinger minimum
uncertainty of $1/2$ (horizontal, dashed grey line). In the figure, the
charger-holder coupling strength $g=\gamma$ and the drive-battery detuning
$\Delta=\gamma/2$, so that from Eq. (40) the first critical driving amplitude
$\Omega_{\mathrm{c}}^{(1)}=\sqrt{5}\gamma/2\simeq 1.118\gamma$ (vertical,
solid grey lines in all panels).

The time-dependent properties of the ergotropy $\mathcal{E}$ are considered in
Fig. 3 (b). There we again take the example quadratic quantum battery with the
charger-holder coupling $g=\gamma$ and the detuning $\Delta=\gamma/2$, so that
from Eq. (40) the first critical driving amplitude
$\Omega_{\mathrm{c}}^{(1)}\simeq 1.118\gamma$, while the second critical
driving amplitude from Eq. (41) is $\Omega_{\mathrm{c}}^{(2)}\simeq
1.581\gamma$. Increasingly strong driving amplitudes $\Omega$ are considered
up to $\Omega_{\mathrm{c}}^{(1)}$ [see the plot legend in Fig. 3 (c) for the
cases considered], leading to dramatic ergotropic improvements at larger
timescales $\gamma t\gg 1$, as foreshadowed by the steady state analysis
provided in Fig. 4 (c). The inset of Fig. 3 (b) rescales the ergotropy
$\mathcal{E}$ by its maximum value over all times, showing the typical
timescales required for the battery to become fully charged (in an ergotropic
sense). Notably, full charging is achieved asymptotically ($t\to\infty$)
except for when the driving amplitude is furthest away (thin yellow line) from
$\Omega_{\mathrm{c}}^{(1)}$, where a dynamical ergotropic value is higher than
its steady state value [a circumstance which commonly occurs for the linear
quantum battery, see Fig. 2 (c)]. Finally, the equivalent charging powers
$P=\mathcal{E}/t$ for the ergotropies considered in Fig. 3 (b) are shown in
Fig. 3 (c). This panel (c) shows that the power $P$ increases with time from
zero up to some maximum power, which is seen to occur around the optimal time
$t\simeq 2.6/\gamma$ for the cases considered. Hence the maximum power does
not correspond to when the battery is fully charged [in the sense of the inset
to Fig. 3 (b)]. A logarithmic timescale is used in Fig. 3 (c) to confirm the
eventual loss of power when the steady state ergotropy is finally reached at
large timescales. Notably, we have made sure that the general physics of the
quadratic quantum battery, as described in Fig. 3 and Fig. 4 in particular, is
essentially unchanged after including additional dissipation coming from the
battery holder itself (see Supplementary Note 3).

The steady state properties of the quadratic quantum battery are further
considered in Fig. 5. In the figure, the charger-holder coupling strength
remains at $g=\gamma$ and the drive-battery detuning remains at
$\Delta=\gamma/2$, so that the critical point $\Omega_{\mathrm{c}}^{(1)}\simeq
1.118\gamma$ (vertical, solid grey lines in all panels of Fig. 5). In
particular, we are interested in the results obtained by truncating the two
harmonic oscillators modelling the battery charger and the battery holder to
finite $N$-level systems, in order to better understand where the energetic
unboundedness suggested in Fig. 4 (c) arises from.

In Fig. 5 (a), we consider the battery holder energy $E_{h}$ in the steady
state (in units of the battery energy level spacing $\omega_{b}$) as a
function of the two-photon driving amplitude $\Omega$ (in units of the charger
decay rate $\gamma$). The analytic (and $N\to\infty$ asymptotic) result of Eq.
(39) is represented by the thick salmon-pink line, showcasing the
aforementioned unboundedness as $\Omega\to\Omega_{\mathrm{c}}^{(1)}$.
Otherwise, the truncated oscillator results are denoted by thin pink lines
(the extreme cases for $N=3$ and $N=20$ levels are distinguished with light
pink and red lines respectively). The data in Fig. 5 (a) confirms the
increasing impact of the first critical driving amplitude
$\Omega_{\mathrm{c}}^{(1)}$ with an increasing number of levels $N$, and
highlights the eventual plateauing of the battery holder energy $E_{h}$ due to
the necessary saturation of the finite level systems making up the truncated
battery. These results support the main findings presented in Fig. 4 (c) for
infinite-level quantum harmonic oscillators, and imply that the energetic
unboundedness is a result of always being able to occupy higher and higher
levels of an untruncated harmonic oscillator, as opposed to occurring due to
an unphysical runaway or some unreasonable approximation.

The behavior of the purity $\mathcal{P}$ of the quadratic quantum battery, a
measure of the degree of mixedness of the quantum state $\rho$ via the formula
$\mathcal{P}=\operatorname{Tr}(\rho^{2})$, is likewise shown in Fig. 5 (b) for
the steady state. The purity is bounded by $\mathcal{P}=1$ for a pure state
and $\mathcal{P}=1/d$ (where $d$ is the dimension of the relevant Hilbert
space) for a maximally mixed state. With vanishing driving $\Omega$, the
system remains in its vacuum state and is hence completely pure. However for
non-vanishing driving $\Omega$ the key quantity $\mathcal{D}$ [cf. Eq. (13)]
is non-unity, and the degree of mixedness rapidly increases as the driving
amplitude $\Omega$ is increased towards the critical point
$\Omega_{\mathrm{c}}^{(1)}$, eventually leading to a maximally mixed quantum
state and significant loss of coherences. This impure behaviour is in stark
contrast to the completely pure linear quantum battery, which remains in a
product state such that it exhibits the property $\mathcal{D}=1$.

The underlying nature of the critical driving amplitude
$\Omega_{\mathrm{c}}^{(1)}$ can be revealed by considering the response of the
Liouvillian gap $\Lambda$. This important quantity is defined as the gap
between zero and the real part of the largest (nonzero) eigenvalue in the
Liouvillian superoperator spectrum Kessler2012 ; Cai2013 ; Minganti2018 , and
is displayed in Fig. 5 (c). The closing of the Liouvillian gap (which is
possible in the truly thermodynamic limit, where $N\to\infty$) at some
critical value of a system parameter is typically associated with a
dissipative phase transition Fitzpatrick2017 ; Rodriguez2017 ; Fink2018 . In
the case of the quadratic quantum battery, the crucial parameter again seems
to be the critical driving amplitude $\Omega_{\mathrm{c}}^{(1)}$ (more
evidence supporting this conjecture is provided in Supplementary Note 3, which
suggests an algebraic scaling of the Liouvillian gap size with the system size
$N$). No such closing of the Liouvillian gap is possible for the one-photon
driving case considered previously for the linear quantum battery, since in
that case there is always a well-defined steady state [cf. Eq. (16) with Eq.
(39)] and as such no dynamical instability is present.

The fact that the key quantity $\mathcal{D}\neq 1$ [cf. Eq. (13)] for the
quadratic quantum battery raises the possibility that quantum entanglement is
playing a role. To investigate this, we consider the negativity $\mathcal{N}$,
a common entanglement measure defined using the absolute sum of the negative
eigenvalues of the partial transpose of the density matrix $\rho$
Zyczkowski1998 ; Vidal2002 . Zero negativity implies an unentangled state,
while increasingly nonzero negativity suggests an increasingly entangled
state. We plot the negativity $\mathcal{N}$, again in the steady state and as
a function of the driving amplitude $\Omega$, in Fig. 5 (d). The plot shows
that the state is completely unentangled with vanishing driving $\Omega\to 0$,
since it is simply the trivial vacuum state. However, with increasingly strong
driving $\Omega$ the entanglement of the state rises to a certain maximum,
before falling once again towards zero (in the thermodynamic $N\to\infty$
limit) near to the critical point $\Omega_{\mathrm{c}}^{(1)}$. This trend
occurs since, for any truncated oscillator case, the state at some large
enough driving $\Omega$ is simply that with the highest level filled. This
type of entanglement behaviour is entirely missing in the linear quantum
battery, which always remains in an unentangled product state.

Finally, the phenomena of quantum squeezing within the quadratic quantum
battery may be analyzed through the two battery holder quadrature variances
$\sigma_{X,h}^{2}$ and $\sigma_{P,h}^{2}$ Loudon1987 . These dimensionless
quantities, defined via the twin relations of
$\sigma_{X,h}^{2}=\langle\hat{X}_{h}^{2}\rangle-\langle\hat{X}_{h}\rangle^{2}$
(thick cyan line) and
$\sigma_{P,h}^{2}=\langle\hat{P}_{h}^{2}\rangle-\langle\hat{P}_{h}\rangle^{2}$
(thick yellow line), are considered (in the steady state and as a function of
the driving amplitude $\Omega$) in Fig. 5 (e). These variance definitions rely
on the generalized battery holder quadratures
$\hat{X}_{h}=(\mathrm{e}^{\mathrm{i}\theta/2}h^{\dagger}+\mathrm{e}^{-\mathrm{i}\theta/2}h)/\sqrt{2}$
and
$\hat{P}_{h}=\mathrm{i}(\mathrm{e}^{\mathrm{i}\theta/2}h^{\dagger}-\mathrm{e}^{-\mathrm{i}\theta/2}h)/\sqrt{2}$,
which obey the familiar commutation relation
$[\hat{X}_{h},\hat{P}_{h}]=\mathrm{i}$. The product of the standard deviations
$\sigma_{X,h}\sigma_{P,h}$ (medium dashed green line) is also shown in Fig. 5
(e), as is a guide for the eye at the Robertson-Schrödinger minimum
uncertainty of $1/2$ (horizontal, dashed grey line). Most notably, the quasi-
momentum variance $\sigma_{P,h}^{2}$ displays steady state squeezing for a
range of driving amplitudes $\Omega$, up to a certain value $\Omega\simeq
0.813\gamma$ (which is notably below the critical point residing at
$\Omega_{\mathrm{c}}^{(1)}\simeq 1.118\gamma$). Such quadrature squeezing of
the quadratic quantum battery directly originates from the parametric field
driving and thus is entirely absent for the coherent field driving case of the
linear quantum battery. Squeezing may be interesting for modern applications
in quantum sensing and for quantum information processing, while here it is
interesting to note that the quadrature variances display asymptotes at
$\Omega_{\mathrm{c}}^{(1)}$, the onset of the dynamical instability discussed
earlier. Within wider quantum battery research, squeezing has been recently
studied in the context of a coherent squeezing charging mechanism and an
incoherent squeezed thermal bath Centrone2021 , as well as battery charging
with local squeezing Konar2022 , which point at its utility within quantum
technological research.

Conclusion
In conclusion, we have studied theoretically the prototypical bipartite form
of a Gaussian quantum battery. We began by revisiting the case of a linearly
driven battery charger, where we highlighted the crucial role of exceptional
point physics in the ergotropic response of the battery. We then considered
the arguably more interesting case of quadratic driving, where we found
critical points instead play a decisive role in the battery energetics,
including by influencing several unconventional features such as the spectral
collapse of the Hamiltonian, quantum squeezing, dynamic instabilities and
dissipative phase transitions. Our proposed quadratic quantum battery exhibits
various desirable features, including storing only relatively small amounts of
useless energy, allowing for the possibility for storing (theoretically
unbounded) amounts of ergotropy, and requiring reasonable charging times to
achieve significant energy storage. We hope that our theoretical proposal for
a quadratic quantum battery can soon be realized with contemporary quantum
platforms such as photonic cavities Rota2019 ; Marty2021  and quantum circuits
Macklin2015 ; Nigg2017 , so that a squeezed battery may become a viable
candidate for an energy storage device within the next generation of quantum
technology. On the theory side, it should also be interesting to consider the
scaling up of the quadratic quantum battery to include multiple battery cells
Campaioli2017 ; Gyhm2022  and cooperative effects Martin2021 ; Jaseem2023 .

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. Discussions: We thank V. A. Saroka for fruitful
discussions.

Author contributions
CAD conceived of the study and wrote the first version of the manuscript, with
revisions from MSU. Both CAD and MSU performed the calculations and gave final
approval for publication.

Competing interests
The authors declare no competing interests.

ORCID
C. A. Downing: 0000-0002-0058-9746.
M. S. Ukhtary: 0000-0001-5197-7354.

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