# Super-Optimal Charging of Quantum Batteries via Reservoir Engineering

Borhan Ahmadi  0000-0002-2787-9321<EMAIL_ADDRESS>International
Centre for Theory of Quantum Technologies, University of Gdansk, Jana
Bażyńskiego 1A, 80-309 Gdansk, Poland  Paweł Mazurek  0000-0003-4251-3253
<EMAIL_ADDRESS>Institute of Informatics, Faculty of Mathematics,
Physics and Informatics, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk,
Poland  Shabir Barzanjeh  Department of Physics and Astronomy, University of
Calgary, Calgary, AB T2N 1N4 Canada  Paweł Horodecki  International Centre for
Theory of Quantum Technologies, University of Gdansk, Jana Bażyńskiego 1A,
80-309 Gdansk, Poland

###### Abstract

Energy dissipation, typically considered an undesirable process, has recently
been shown to be harnessed as a resource to optimize the performance of a
quantum battery. Following this perspective, we introduce a novel technique of
charging in which coherent charger-battery interaction is replaced by a
dissipative interaction via an engineered shared reservoir. We demonstrate
that exploiting collective effects of the engineered shared reservoir allows
for extra optimization giving rise to optimal redistribution of energy, which
leads to a significant enhancement in the efficiency of the charging process.
The article unveils the intricacies of built-in detuning within the context of
a shared environment, offering a deeper understanding of the charging
mechanisms involved. These findings apply naturally to quantum circuit battery
architectures, suggesting the feasibility of efficient energy storage in these
systems. Moreover, the super-optimal charging offers a practical justification
for charger-battery configurations.

Introduction.—The role of energy transfer is of primary importance in
industry. In recent years, significant research has been devoted to examining
how to efficiently facilitate energy transfer between two quantum systems by
means of a direct interaction between the systems in various contexts such as
quantum heat engines, quantum thermal logic gates, quantum thermal
transistors, and quantum batteries [1, 2, 3, 4, 5, 6, 7]. Quantum batteries
[8] represent a cutting-edge advancement in the field of energy transfer and
storage, leveraging the principles of quantum mechanics to potentially
revolutionize how we catch and store energy in quantum devices [8, 6, 7, 9, 6,
10, 11, 12, 13, 14, 15, 4]. Unlike traditional batteries, quantum batteries
utilize quantum phenomena such as entanglement [6, 7, 9, 6] to facilitate
energy storage processes that could surpass the limitations of conventional
systems. Other approaches such as using quantum optimal control [10, 11],
quantum catalysis processes [15], superabsorption [16], phase transition [17]
and indefinite causal order [18] can also be used to improve the efficiency of
the charging process, where the charging time decreases as the battery’s
capacity increases, a stark contrast to classical batteries where larger
capacities typically mean longer charging times.

While these contributions have significantly advanced our understanding of
quantum batteries, the search for enhanced efficiency drives us to explore new
avenues for improving energy storage, most notably boosting its energetic
efficiency and power. A novel approach to optimize the efficiency of quantum
battery charging leverages reservoir engineering to induce nonreciprocity [19,
20, 21, 22, 23, 24, 25, 26, 27] by which, as recently shown, energy transfer
to the battery may be significantly boosted [28]. Reservoir engineering, a
crucial technique in quantum technology, involves the deliberate design and
manipulation of a system’s interaction with its environment to achieve
specific quantum states or dynamics. This method has found applications across
various fields, including quantum information processing [19, 29], quantum
simulation [30], and precision measurement [31]. By tailoring the
environmental interactions, researchers can counteract decoherence and
dissipation, traditionally seen as detrimental, and instead stabilize desired
quantum states or induce coherent dynamics [32].

In this work, we identify a novel scenario in which, unlike in Ref. [28],
nonreciprocity condition is not satisfied, and yet the sheer presence of an
engineered shared reservoir allows for significant enhancement of efficiency
of the quantum battery. We first demonstrate that the optimal charging for any
charging process is done by optimisation over characteristics of the external
energy source for a given charging device which is determined by a built-in
detuning of the charger-battery structure. We then show that the engineered
shared reservoir allows for further optimization of the charging process over
the internal structure of the charging device (parameters of the coupling with
the engineered reservoir), resulting in super-optimal charging. We provide the
microscopic description of this intriguing phenomenon through the master
equation formalism. We show that energy dissipation, typically considered a
damping and an undesirable process, can be harnessed as a resource to further
optimize the performance of a quantum battery. Specifically, replacing the
coherent (direct) interaction between the charger and the battery with a
dissipative (indirect) interaction enables a significant increase in transfer
of energy to the battery. Generally, creating coherent interaction is more
challenging than designing broadband lossy systems. At last, we show that
super-optimal charging allows for the energy of the battery in the charger-
battery setting to exceed that of the battery directly coupled to the external
laser (no charger mediating the process), providing further justification for
charger-battery configurations [33]. This research opens up possibilities for
exploring chiral and topological properties in systems utilizing dissipative
coupling.

Quantum Battery Model.—Figure 1 illustrates the schematic of the system in a
charger-battery configuration [33], including a harmonic oscillator serving as
the charger with a resonance frequency of $\omega_{a}$ and a local damping
rate $\kappa_{a}$. This charger interacts with another mode acting as the
battery, characterized by a frequency of $\omega_{b}$ and a damping rate
$\kappa_{b}$. The coherent interaction between the charger and the battery is
described by $J=|J|e^{i\phi}$. An external classical drive with a frequency
$\omega_{L}$ and amplitude $F$ provides energy to the charger during the
charging process.

The Hamiltonian describing this driven bipartite system $AB$ can be written,
in the rotating frame with respect to $\omega_{L}$, as:

$H=H_{AB}+H_{L}.$ (1)

The Hamiltonian $H_{AB}$ is given by $H_{AB}=H_{a}+H_{b}+H_{int}$, where
$H_{a}=\Delta a^{\dagger}a$ ($\hbar=1$) and $H_{b}=\Delta b^{\dagger}b$
represent the free Hamiltonians of the charger and the battery, respectively.
Here, $a$ ($b$) and $a^{\dagger}$ ($b^{\dagger}$) are the annihilation and
creation operators of the charger (battery), and $\Delta=\omega_{L}-\omega$
denotes the local detuning of the input laser beam from the optical charger
resonance frequency. For simplicity, we assume that the charger and the
battery are in resonance, i.e., $\omega_{a}=\omega_{b}=\omega$. The
interaction Hamiltonian, $H_{int}$, is described as
$Ja^{\dagger}b+J^{*}ab^{\dagger}$, and the term $H_{L}=F(e^{i\Delta
t}a+e^{-i\Delta t}a^{\dagger})$ accounts for the laser field applied to the
charger for feeding energy into the system (under rotating wave approximation
[34]). As we construct our charging system $AB$, a built-in eigen-frequencies
of the Hamiltonian $H_{AB}$ are established $\omega_{\pm}=\omega\pm|J|$ (see
Appendix A). This in turn creates a built-in detuning of

$\displaystyle\Delta_{in}=\pm|J|,$ (2)

where the subscript $in$ indicates its built-in origin. This suggests that in
a non-dissipative charging process, to optimize energy transfer to the battery
the detuning of the external field $\Delta$ be equal to this built-in detuning
$\Delta_{in}$. This phenomenon will be rigorously proved in the following
sections.

In addition, we also consider a dissipative interaction between the charger
and the battery, mediated by their mutual coupling to a shared reservoir
characterized by a coupling rate of $\Gamma$. This interaction is realized by
adiabatically eliminating the reservoir’s degrees of freedom. From a practical
perspective, the shared reservoirs could be an additional damped cavity or a
waveguide [21, 35]. The dynamics of the system can be described by Lindblad
master equation [21]

$\dot{\rho}=-i[H,\rho]+\Gamma\mathcal{D}_{z}[\rho]+\kappa_{a}\mathcal{D}_{a}[\rho]+\kappa_{b}\mathcal{D}_{b}[\rho],$
(3)

where $\mathcal{D}_{O}[\cdot]=O\cdot
O^{\dagger}-\frac{1}{2}\\{O^{\dagger}O,\cdot\\}$ represents the dissipative
super-operator resulting from the coupling to the reservoirs. Here
$\kappa_{a(b)}$ is the local decay rate of the charger (battery) and
$z=p_{a}a+p_{b}b$, with $p_{a}$ ($p_{b}$) denoting the coupling strength
between the charger (battery) with the shared reservoir with decay rate
$\Gamma$. Note that the second term in Eq. (3) represents the dissipation of
both modes into the shared reservoir, resulting in dissipative coupling
between the two modes. The last two terms account for local damping of each of
the modes due to their coupling ports or the surrounding environment [21, 36].
As we are going to explore, in presence of dissipation the optimal detuning is
not anymore the built-in detuning, but includes also corrections depending on
the type of the dissipation.

We define the internal energy of each subsystem as
$E_{i}=\omega\text{tr}{\big{[}\rho_{i}i^{\dagger}i}\big{]}$ with $i=a,b$ [37].
Given the assumption of zero temperature for all reservoirs, the states of
subsystems remain pure if the initial state of the whole system is a pure
product state [38]. This in turn implies that all the internal energy of the
battery can be extracted as ergotropy [39]. Therefore, in the following, we
will only discuss the internal energies of the battery and the charger.

Figure 1: Schematic representation of the charging process. A quantum charger
$A$ coherently interacts with a quantum battery $B$ with a coupling rate $J$.
Both the charger and the battery decay into a shared reservoir at the decay
rate $\Gamma$ establishing an effective dissipative interaction between them.
A laser field with amplitude $F$ is utilized to supply energy to the charger.
Here, $\kappa_{a}$ and $\kappa_{b}$ describe the local damping rates of each
mode.

Charging Dynamics.—We now analyze the local dynamics of the system using the
master equation (3). The evolution of the first and second moments is given by

$\displaystyle\frac{d\langle a\rangle}{dt}$
$\displaystyle=-\frac{\Gamma_{a}+\kappa_{a}+2i\Delta}{2}\langle
a\rangle-i\left(J+i\mu\frac{\Gamma}{2}\right)\langle b\rangle-iF,$ (4)
$\displaystyle\frac{d\langle b\rangle}{dt}$
$\displaystyle=-\frac{\Gamma_{b}+\kappa_{b}+2i\Delta}{2}\langle
b\rangle-i\left(J^{*}+i\mu^{*}\frac{\Gamma}{2}\right)\langle a\rangle,$ (5)
$\displaystyle\frac{d\langle a^{\dagger}a\rangle}{dt}$
$\displaystyle=-(\Gamma_{a}+\kappa_{a})\langle
a^{\dagger}a\rangle-2\Re\left\\{i\left(J+i\mu\frac{\Gamma}{2}\right)\langle
a^{\dagger}b\rangle\right\\}-2F\Im\left\\{\langle a\rangle\right\\},$ (6)
$\displaystyle\frac{d\langle b^{\dagger}b\rangle}{dt}$
$\displaystyle=-\left(\Gamma_{b}+\kappa_{b}\right)\langle
b^{\dagger}b\rangle+2\Re\left\\{i\left(J-i\mu\frac{\Gamma}{2}\right)\langle
a^{\dagger}b\rangle\right\\},$ (7) $\displaystyle\frac{d\langle
a^{\dagger}b\rangle}{dt}$
$\displaystyle=-\frac{\Gamma_{a}+\kappa_{a}+\Gamma_{b}+\kappa_{b}}{2}\langle
a^{\dagger}b\rangle-i\left(J^{*}+i\mu^{*}\frac{\Gamma}{2}\right)\langle
a^{\dagger}a\rangle,$
$\displaystyle\quad+i\left(J^{*}-i\mu^{*}\frac{\Gamma}{2}\right)\langle
b^{\dagger}b\rangle+iF\langle b\rangle,$ (8)

where $\mu=-p_{b}p^{*}_{a}$ and $\Gamma_{i}=\Gamma|p_{i}|^{2}$ with $i=a,b$.
The symbols $\Re$ and $\Im$ indicate the real and imaginary parts,
respectively. As long as both $p_{a}$ and $p_{b}$ are non-zero, we re-scale
them such that $|\mu|=1$, and the remaining factor is absorbed into $\Gamma$.

Optimal Charging.—From the set of differential equations of motion (4)-(8), we
calculate the energy of the charger and the battery in the steady state

$E_{A}=\frac{4F^{2}\mathcal{K}_{bb}}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|^{2}},$
(9)
$E_{B}=\frac{4F^{2}\mathcal{J}(\Gamma,\alpha)}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|^{2}},$
(10)

where $\mathcal{K}_{bb}=4\Delta^{2}+(\Gamma_{b}+\kappa_{b})^{2}$,
$\mathcal{K}_{ab}=\left(2\Delta+i(\Gamma_{a}+\kappa_{a})\right)\left(2\Delta+i(\Gamma_{b}+\kappa_{b})\right)$,
$\mathcal{J}(x,\theta)=4|J|^{2}+x^{2}-4|J|x\cos{\theta}$ and
$\alpha+\phi=\pi/2$. From Eqs. (4)-(8) we can see that $\mathcal{J}(x,\theta)$
effectively incorporates the interactions facilitating energy transfer between
the charger and the battery, while $\mathcal{K}_{ij}$ describes processes
restricting energy transfers such as energy dissipation or detuning. We first
start the analysis by examining the conventional quantum battery scenario (the
absence of the shared reservoir). After setting $\Gamma=0$ in Eqs. (9) and
(10), the steady-state energies of the charger and the battery are,
respectively, determined as

$E_{A}=\frac{4F^{2}\mathcal{K}_{bb}^{\Gamma=0}}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{\Gamma=0}^{2}},$
(11)
$E_{B}=\frac{16F^{2}\mathcal{J}(0,\alpha)}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{\Gamma=0}^{2}}.$
(12)

We choose the initial states of both charger and battery to be ground states.
For the ideal charging process where there exists no local dissipations, i.e.
$\kappa_{a}=\kappa_{b}=0$, by optimizing $E_{B}$ over $\Delta$ we find that
the optimal detuning equals to the build-in detuning (2):
$\Delta_{opt}=\Delta_{in}$. In fact, in the case $\Delta=\Delta_{in}$ we have
$\mathcal{J}(i\Gamma,\phi)=\mathcal{K}_{ab}$, which gives rise to infinite
energy transfer to the battery. This phenomena of infinite energy transfer
indicates the crucial significance of the built-in detuning $\Delta_{in}$.
However, in realistic charging processes, local dissipations are inevitable.
Therefore, for nonzero local dissipation rates $\kappa_{a(b)}$, optimizing
$E_{B}$ over $\Delta$ the optimal charging process results in

$\Delta_{opt}=\pm\sqrt{\Delta^{2}_{in}-\frac{\kappa^{2}}{8}},$ (13)

where $\kappa^{2}=\kappa_{a}^{2}+\kappa_{b}^{2}$. Note that in this case,
infinite energy transfer to the battery is not feasible. This is because the
denominator in Eq. (12) takes the form
$|4J^{2}-(2\Delta+i\kappa_{a})(2\Delta+i\kappa_{b})|^{2}$, which is always
greater than zero due to the imaginary terms resulting from local dissipation
rates $\kappa_{a(b)}$. In Fig. 2 we plot the time evolution of $E_{B}(t)$ for
the optimal charging procedure (dashed blue line). Additionally, in Fig. 2 we
illustrate $E_{B}(t)$ for a non-optimal detuning case (green solid line) by
taking $\Delta=0$, to underscore the substantial influence of optimal detuning
on charging process. It is worth mentioning that Eq. (13) implies the
important fact that the charging laser filed comprehensively perceives not
solely the coherent interaction $J$, but also energy dumping to the local
baths. The interactions with the baths show their effects via the dissipation
rates $\kappa_{a(b)}$. Therefore, in presence of local dissipations, $\kappa$
is in fact the correction to the built-in detuning $\Delta_{in}$ (a fact
overlooked in our earlier work [15]). In other words, the optimal detuning is
influenced by the entirety of the interactions. We also notice that in the
conventional scenario $\Delta_{opt}$ depends only on the strength of the
coherent interaction, and not on its phase $\phi$.

We now move to the crucial question of the paper: can the presence of an
engineered shared reservoir enhance the optimal conventional charging process?
Let us first consider adding the engineered shared reservoir to the charging
process by setting $\Gamma\neq 0$, and optimizing $E_{B}$ in the stationary
limit. In this scenario we derive the optimal detuning to be again close to
the built-in detuning $\Delta_{in}$, with more convoluted corrections due to
interactions with both local and shared reservoirs (see Appendix A). In this
case, $\Delta_{opt}$ depends also on the interaction phase $\phi$. As
illustrated in Fig. 2, adding the shared reservoir does not provide a
straightforward advantage over the optimal conventional charging process.

Yet, there is another way to exploit the engineered shared reservoir, which we
outline below.

Figure 2: Optimal energy of the battery in the conventional procedure (dashed
blue line) and the shared reservoir-assisted procedure (solid red line) versus
$Jt$. Here we have $|J|=0.2$, $\kappa_{a}=0.05$, $\kappa_{b}=0.01$ and for the
shared reservoir-assisted procedure $\Gamma=0.4$, $p_{a}=p_{b}=1$,
$\phi=0,\pi$. In both procedures we selected the corresponding optimal
detuning $\Delta_{opt}$. The green solid line is for conventional procedure
when detuning is not optimal, for instance here $\Delta=0$, to emphasize the
huge effect of optimal detuning on the charging process.

Super-Optimal Charging.—Here we demonstrate that if instead of adding a shared
reservoir to the charging process, as discussed above, we replace the coherent
interaction by the shared reservoir, then the resulting dissipative
interaction between the charger and the battery will enable us to
significantly outperform the optimal conventional charging process. To better
understand the mechanism, we first provide the insight into the microscopic
dynamics underlying the concept of super-optimal charging by examining the
master equation (3) and then introduce our approach aimed at surpassing the
optimal conventional charging strategy discussed in the previous section. The
master equation (3) may be recast as follows

$\displaystyle\dot{\rho}$
$\displaystyle=-i[H,\rho]+(\Gamma_{a}+\kappa_{a})\mathcal{L}_{a}[\rho]+(\Gamma_{b}+\kappa_{b})\mathcal{L}_{b}[\rho]$
$\displaystyle+\Gamma\mathcal{L}_{a}^{b}[\rho]+\Gamma\mathcal{L}_{b}^{a}[\rho],$
(14)

where

$\mathcal{L}_{x}^{y}[\cdot]=:x\cdot
y^{\dagger}-\frac{1}{2}\\{x^{\dagger}y,\cdot\\}.$ (15)

Eq. (Super-Optimal Charging of Quantum Batteries via Reservoir Engineering)
implies that the shared reservoir dissipator $\mathcal{D}_{z}[\rho]$ from (3)
in practice affects the charging process in two distinct manners: firstly, via
$\Gamma_{x}\mathcal{D}_{x}[\rho]$ it causes energy dissipation of the charger
and the battery into the shared reservoir at rates $\Gamma_{a}$ and
$\Gamma_{b}$, respectively. Secondly, via $\Gamma\mathcal{D}^{y}_{x}[\rho]$ it
facilitates energy exchange between the charger and the battery at rate
$\Gamma$. Therefore, $\Gamma\mathcal{D}^{y}_{x}[\rho]$ provides a dissipative
interaction between the charger and the battery paving the way for more
transfer of energy. Notably, even with the direct coherent coupling turned off
($J=0$), energy exchange still remains feasible through the shared reservoir
with rate $\Gamma$. This observation leads to an intriguing phenomena in the
realm of charging quantum batteries, opening up new possibilities for energy
transfer manipulation, as discussed in the following. Note that in quantum
optics the same generator of the form (Super-Optimal Charging of Quantum
Batteries via Reservoir Engineering) is also used to explain the phenomenon of
collective resonance fluorescence [40].

Motivated by this, we explore a distinct scenario where the direct coherent
interaction is replaced by a dissipative interaction provided by a shared
reservoir (as a result, no built-in detuning anymore). This means that in the
equations of motion (4)-(8) we put $J=0$ and $\Gamma\neq 0$. Therefore in this
scenario the laser field exclusively interacts with the charger, leading to
resonance at $\omega_{L}=\omega$. The pivotal question arises: with the direct
coherent interaction replaced by a shared reservoir with the dissipation rate
$\Gamma$, what value of $\Gamma$ enables us to either to recover or to surpass
the optimal result obtained in the conventional procedure? Intuitively,
considering the equations of motion (4)-(8), since $J$ is always accompanied
with $i\Gamma/2$, the answer lies in selecting $\Gamma=2|J|$, as confirmed in
the following.

Replacing the coherent interaction $J$ by the shared reservoir with
dissipation rate $\Gamma$ yields the steady state energies of the charger and
the battery, respectively, as

$E^{S}_{A}=\frac{4F^{2}\mathcal{K}_{bb}}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{J=0}^{2}},$
(16)
$E^{S}_{B}=\frac{4F^{2}\mathcal{J}_{J=0}(\Gamma,\alpha)}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{J=0}^{2}},$
(17)

where superscript $S$ denotes the presence of the shared reservoir while no
coherent interaction on. From Eqs. (16) and (17) it can be shown that the
optimal detuning here is zero, $\Delta_{opt}=0$ (see also Eq. (16) of Appendix
A). This aligns with physical intuition, as deactivating the coherent
interaction confines the laser field influence solely to the charger,
disregarding the remainder of the system. Thus, by substituting $\Delta=0$
into Eqs. (16) and (17), we arrive at

$E^{S}_{A,opt}=\frac{4F^{2}\mathcal{K}_{bb}^{\Delta=0}}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{\Delta,J=0}^{2}},$
(18)
$E^{S}_{B,opt}=\frac{4F^{2}\mathcal{J}_{J=0}(\Gamma,\alpha)}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{\Delta,J=0}^{2}},$
(19)
$\xi^{S}_{opt}=\frac{4F^{2}(\mathcal{J}\left(\Gamma,\theta)+\mathcal{K}_{bb}\right)_{\Delta,J=0}}{|\mathcal{J}(i\Gamma,\phi)-\mathcal{K}_{ab}|_{\Delta,J=0}^{2}},$
(20)

where $\xi\equiv E_{A}+E_{B}$ denotes the total energy within the entire
system. In Fig. 3, we present a plot depicting the battery energy for
$\Gamma=0.4$, $J=0$ and $\Delta_{opt}=0$, comparing it with the optimal
conventional scenario in which $|J|=0.2$. Again it is seen that no benefits
are provided by the shared reservoir. We should stress that in Fig. 3 and the
following figures energy is plotted against $Jt$ where the scaling factor $J$
is taken from the conventional scenario. This choice in fact ensures a fair
comparison across both scenarios as long as we set $\Gamma$ in the shared
reservoir-assisted scenario to be twice the value of $|J|$ in the conventional
scenario. This is because in the equations of motion (4)-(8), $J$ always comes
with $i\Gamma/2$.

(a)

(b)

Figure 3: $(\textbf{a})$ Energies of the charger $E_{A}$ and $(\textbf{b})$
the battery $E_{B}$ versus $Jt$ for the optimal conventional scenario and the
shared reservoir-assisted scenario. In the conventional scenario we have
$F=0.1$, and $|J|=0.2$. In the shared reservoir-assisted scenario $F=0.1$,
$J=0$, $\Gamma=0.4$, $p_{a}=1$, $p_{b}=1$. In both procedures we have
$\kappa_{a}=0.05$, $\kappa_{b}=0.01$ and the corresponding optimal detuning
$\Delta_{opt}$.

(a)

(b)

Figure 4: $(\textbf{a})$ Energies of the charger $E_{A}$ and $(\textbf{b})$
the battery $E_{B}$ versus $Jt$ for the optimal conventional scenario and the
shared reservoir-assisted scenario. The quantity $\mathcal{G}_{A(B)}^{r}$
quantifies the redistribution gap caused by the engineered shared reservoir.
For the conventional scenario we have $F=0.1$, and $|J|=0.2$. For the shared
reservoir-assisted scenario $F=0.1$, $J=0$, $\Gamma=0.4$,
$p_{a}=(\kappa_{a}/\kappa_{b})^{1/4}$, $p_{b}=(\kappa_{a}/\kappa_{b})^{-1/4}$.
In both procedures we chose $\kappa_{a}=0.05$, $\kappa_{b}=0.01$ and the
corresponding optimal detuning $\Delta_{opt}$.

However, as we show below, leveraging the structure brought by the shared
reservoir, we are able to optimize the process by manipulating relative values
of $\Gamma_{a(b)}$. Examining equations of motion we see that changing
$\Gamma_{a}$ and $\Gamma_{b}$ by altering $p_{a}$ and $p_{b}$ can impact the
energy of the charger and the battery. Let us denote by $y$ the ratio
$y=\frac{|p_{a}|^{2}}{|p_{b}|^{2}}$, we then have $\Gamma_{a}/\Gamma_{b}=y$.
Now rewriting Eqs. (18)-(20) as functions of $x$, we find that both the total
energy of the setup and the energy of the battery are optimized for

$\displaystyle y=\sqrt{\frac{\kappa_{a}}{\kappa_{b}}},$ (21)

while energy of the charger reaches maximum in the limit $y\rightarrow 0$. We
see that the essence of optimization (21) is to properly reflect imbalance
between local dissipation strengths $\kappa_{a(b)}$ in the ratio of
$p_{a(b)}$. Consequently, we will be talking about a super-optimal charging
process in a scenario with shared reservoir optimized in this way. The
corresponding super-optimal stationary energies are given by:

$E^{S}_{A,sup-opt}=\frac{4F^{2}}{\kappa_{a}^{2}},$ (22) $\displaystyle
E^{S}_{B,sup-
opt}=\frac{4\Gamma^{2}F^{2}}{\kappa_{a}\kappa_{b}(2\Gamma+\sqrt{\kappa_{a}}\sqrt{\kappa_{b}})^{2}},$
(23) $\xi^{S}_{sup-
opt}=\frac{4F^{2}(\kappa_{a}\kappa_{b}^{2}+2\Gamma\kappa_{b}\sqrt{\kappa_{a}\kappa_{b}}+\Gamma^{2}(\kappa_{a}+\kappa_{b}))}{\kappa_{a}^{2}\kappa_{b}(2\Gamma+\sqrt{\kappa_{a}\kappa_{b}})^{2}}.$
(24)

As a direct result of the optimization, we observe a significant increase in
the total energy of the system $\xi^{S}$ (see Appendix B). This indicates that
the optimization helps preserve more energy in the charger-battery system
against the dissipation. Furthermore, the second intriguing result, as
detailed below, is that optimization not only enhances the total energy within
the entire system but allows for channelling, in the most efficient way, the
energy into the battery $B$ as well.

As shown in Fig. 4(a) applying further optimization, through manipulation of
$\Gamma_{a(b)}$ on the charging process, causes $E_{A,sup-opt}^{S}$ to be
substantially lower than $E_{A,opt}$, emphasizing again the significant
benefits of having an engineered shared reservoir in the charging process. At
the same time, as shown in Fig. 4(b), we observe that the super-optimized
energy of the battery $E_{B,sup-opt}^{S}$ (solid red line) significantly
surpasses the optimal conventional energy $E_{B,opt}$. This further justifies
the name super-optimal charging process, since it outperforms the optimal
conventional charging process. Comparing Fig. 3(a) with Fig. 4(a) and Fig.
3(b) with Fig. 4(b) the remarkable phenomena that we observe is the
redistribution of total energy within the entire system due to the
optimization, with a notable increase of energy in the desired
location—namely, the battery $B$.

As illustrated in Fig. 4, to quantify the redistribution gap caused by the
engineered shared reservoir, we also define the quantity
$\mathcal{G}_{A(B)}^{r}\equiv E^{S}_{A(B),sup-opt}-E_{A(B),opt}$. As we are
interested in the energy of the battery, we focus on

$\displaystyle\mathcal{G}_{B}^{r}$
$\displaystyle=4\Gamma^{2}F^{2}\left(\frac{4}{(\kappa_{a}+\kappa_{b})^{2}\left((\kappa_{a}-\kappa_{b})^{2}-4\Gamma^{2}\right)}+\frac{1}{\kappa_{a}\kappa_{b}\left(2\Gamma+\sqrt{\kappa_{a}\kappa_{b}}\right)^{2}}\right)$

and observe that in settings characterized by significant difference in local
dissipation rates $\kappa_{a}$ and $\kappa_{b}$, this function is positive.
This attests for charging scenarios in which we observe a guaranteed advantage
of shared reservoir over coherent couplings.

Figure 5: Three charging scenarios: (a) direct coupling of the battery to the
laser, (b) charger-mediated setting, with charger coherently coupled to the
battery, and (c) charger-mediated setting, with charger dissipatively coupled
to the battery. In (b) and (c) the presence of system $A$ (charger) isolates
the battery and assures smaller dissipation $\kappa_{b}<\kappa$.

In defense of charger-battery setups.— In fact, this advantage is significant
enough to address the motivational issues behind charger-battery charging
settings. Namely, Ref. [33] considered a separation of the charging setup into
a charger, whose role is to receive energy from the laser drive, and battery,
whose role is to store the energy, coupled to the charger. The battery, while
coupled coherently to the noisy charger, in experimental implementations can
be much more isolated and therefore experiencing much smaller energy
dissipation. However, in Ref. [41] the Authors pointed to the lack of
operational benefit of using this extended setting, compared to the simple
model of a battery interacting directly with the laser field.

This problem, and the role of super-optimal charging in solving it, is
illustrated in Fig. 5. Three charging scenarios are presented there: (a)
through a direct coupling of the battery to the laser, (b) in a charger-
mediated setting with the charger coherently coupled to the battery, and (c)
in a charger-mediated setting with the charger dissipatively coupled to the
battery. We ask the following question: can we obtain higher energies in the
stationary state of the battery, if we include a charger to intermediate
between the laser and the battery? In this procedure, we keep the dissipation
rate $\kappa$ of the party interacting with laser constant, and allow for
smaller values of the dissipation rate $\kappa_{b}$ of the battery in the
extended scenario, to account for its better isolation from the surroundings.

We compare stationary energies achieved with extended scenarios (b) $E_{B}$
(given by Eqs. (12) and (13)) and (c) $E^{S}_{B,sup-opt}$ (given by Eq. (23))
with the stationary energy of the scenario (a):

$E_{single-batt}=\frac{4F^{2}}{\kappa_{a}^{2}},$ (25)

and obtain

$\displaystyle\Delta E_{coh}$ $\displaystyle\equiv E_{B,opt}-E_{\text{single-
batt}}$
$\displaystyle=4F^{2}\left(-\frac{4J^{2}}{(\kappa+\kappa_{b})^{2}\left((\kappa-\kappa_{b})^{2}-4J^{2}\right)}-\frac{1}{\kappa^{2}}\right),$
(26) $\displaystyle\Delta E_{diss}$ $\displaystyle\equiv E^{S}_{B,\text{sup-
opt}}-E_{\text{single-
batt}}=\frac{4F^{2}\left(\frac{\Gamma^{2}\kappa}{\kappa_{b}\left(2\Gamma+\sqrt{\kappa}\sqrt{\kappa_{b}}\right)^{2}}-1\right)}{\kappa^{2}}.$
(27)

We plot these functions for typical parameter settings in Fig. 6. As shown,
with sufficient isolation of the battery (i.e., sufficiently small
$\kappa_{b}$), the super-optimal charging process is significantly more
efficient than the direct charging scenario. In fact, in the limit of perfect
isolation of the battery, the super-optimal charging allows for unbounded
values of energy stored $\lim_{\kappa_{b}\rightarrow 0}\Delta
E_{diss}=\infty$, while energy in the coherent setting is bounded:
$\lim_{\kappa_{b}\rightarrow 0}\Delta
E_{coh}=\frac{4F^{2}}{4J^{2}-\kappa^{2}}$. The charger-mediated setting with
coherent coupling is beneficial only in the regime of extremely small
$\kappa_{b}$. It is interesting to note that the stationary energy of single-
battery setting (Eq. (25)) is the same as the stationary energy of the charger
in the super-optimal charging (Eq. (22)). Therefore, extension of the noisy
single battery system via super-optimal setting effectively allows for
creating a charged battery system with energy not bounded by the dissipation
on the original charger. In turn, the stationary state of the charger is not
affected by the extension.

Figure 6: Advantage in stationary energy of the battery $B$ in extended
scenarios (corresponding to Fig. 5 (b) and (c)), compared to single-battery
scenario (Fig. 5(a)), expressed through $\Delta E_{coh}$ and $\Delta
E_{diss}$, respectively, for various dissipation rates $\kappa_{b}$. For
scenario (a) we choose $F=0.1$, for (b) we have $F=0.1$, $J=0.2$, $\Gamma=0$,
and for (c) we select $F=0.1$, $J=0$, $\Gamma=0.4$, and optimized values
$p_{a}=(\kappa_{a}/\kappa_{b})^{1/4}$, $p_{b}=(\kappa_{a}/\kappa_{b})^{-1/4}$.
In all scenarios we choose $\kappa=0.05$ and the corresponding optimal
detuning $\Delta_{opt}$. We have $\lim_{\kappa_{b}\rightarrow 0}\Delta
E_{diss}=\infty$ and $\Delta E_{coh}(0)=0.016$.

We conclude the characterization of super-optimal charging with a remark about
transient behaviour of this charging method. The effective coupling strength
of the dissipative interaction depends entirely on the value of $\Gamma$ (see
Eq. (Super-Optimal Charging of Quantum Batteries via Reservoir Engineering)).
Therefore, while super-optimal scenario for more isolated batteries relies on
reduced $\Gamma_{b}$ (as implied by reduced $\kappa_{b}$ via Eq. (21)), the
limit $\lim_{\kappa_{b}\rightarrow 0}$ does not impose divergent times for
reaching a given value of energy of the battery. Consequently, the advantage
of super-optimal charging over the single-battery setting can be observed
within the same time scales, and therefore offers significant practically-
achievable boost in capacity of quantum batteries.

Summary and Outlook.—In this work we investigate optimal strategies for
charging quantum batteries by a laser coherent field in a charger-battery
setting. We start from optimisation of external energy source (its frequency),
by examining the built-in detunings of the charging setup. We then move to
setups which introduce engineered shared reservoir into charging processes.
Our investigation reveals that replacing the coherent interaction with a
dissipative interaction via an engineered shared reservoir not only reproduces
the same optimal charging outcomes but also allows for further optimization
over the internal structure of the charging device (parameters of the coupling
with the engineered reservoir), giving rise to super-optimal charging. As a
result we observe that the total energy within the entire system attains a
higher value in the steady state. Above all, we show that the ultimate
advantage of a shared reservoir lies in its ability to facilitate a beneficial
redistribution of energy within the charger-battery system, with a notable
increase of energy in the desired location—namely the battery, presenting a
significant advancement in quantum battery charging. This advantage is
observable already in the transient regime. Therefore, we conclude that the
super-optimal charging achieved through careful optimization of the engineered
reservoir paves the way for transformative developments in high-performance
energy solutions.

Acknowledgements.—BA acknowledges support from National Science Center, Poland
within the QuantERA II Programme (No 2021/03/Y/ST2/00178, acronym ExTRaQT)
that has received funding from the European Union’s Horizon 2020. BA also
acknowledges Robert Alicki for helpful discussions. SB acknowledges funding by
the Natural Sciences and Engineering Research Council of Canada (NSERC)
through its Discovery Grant, funding and advisory support provided by Alberta
Innovates through the Accelerating Innovations into CarE (AICE) – Concepts
Program, and support from Alberta Innovates and NSERC through Advance Grant.

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##  Appendix A Global View Analysis

To have a global understanding of the dynamics of the charging process, we
rewrite the system Hamiltonian

$H=\omega_{A}a^{\dagger}a+\omega_{B}b^{\dagger}b+(Ja^{\dagger}b+J^{*}b^{\dagger}a)+F(e^{i\omega_{L}t}a+e^{-i\omega_{L}t}a^{\dagger}),$
(28)

in the properly selected basis, which depends on the value of $J$. We
introduce super-mode operators

$\displaystyle C_{+}=\frac{a+e^{i\phi}b}{\sqrt{2}},$ (29) $\displaystyle
C_{-}=\frac{a-e^{i\phi}b}{\sqrt{2}},$ (30)

with eigen-frequencies $\omega_{\pm}=\omega\pm|J|$, which satisfy commutation
relations $[C_{+},C_{-}]=[a,b]=0$ and
$[C_{\pm},C_{\pm}^{\dagger}]=[a,a^{\dagger}]=[b,b^{\dagger}]=1$. We make
$C_{\pm}$ depend on the phase of the coupling $J=|J|e^{i\phi}$. From Eqs. (29)
and (30) we have

$\displaystyle a=\frac{C_{+}+C_{-}}{\sqrt{2}},$ (31) $\displaystyle
b=e^{-i\phi}\frac{C_{+}-C_{-}}{\sqrt{2}},$ (32)

and hence we obtain

$\displaystyle H$
$\displaystyle=\left(\Delta+|J|\right)C_{+}^{\dagger}C_{+}+\left(\Delta-|J|\right)C_{-}^{\dagger}C_{-}+\frac{F}{\sqrt{2}}\left(e^{i\omega_{L}t}C_{+}+e^{-i\omega_{L}t}C_{+}^{\dagger}\right)+\frac{F}{\sqrt{2}}\left(e^{i\omega_{L}t}C_{-}+e^{-i\omega_{L}t}C_{-}^{\dagger}\right).$
(33)

At the same time, the master equation given in Eq. (3) of the manuscript can
be rewritten as

$\dot{\rho}=-i[H,\rho]+\Gamma\mathcal{D}_{z}[\rho]+\kappa_{a}\mathcal{D}_{z_{a}}[\rho]+\kappa_{b}\mathcal{D}_{z_{b}}[\rho],$
(34)

where

$z_{a}=:p_{a}^{a}C_{+}+p_{b}^{a}C_{-},$ (35)

with $p_{a}^{a}=\frac{1}{\sqrt{2}}$, $p_{b}^{a}=\frac{1}{\sqrt{2}}$ based on
Eq. (31),

$z_{b}=:p_{a}^{b}C_{+}+p_{b}^{b}C_{-},$ (36)

with $p_{a}^{b}=\frac{e^{-i\phi}}{\sqrt{2}}$,
$p_{b}^{b}=-\frac{e^{-i\phi}}{\sqrt{2}}$ based on Eq. (32), and

$z=p_{a}\frac{C_{+}+C_{-}}{\sqrt{2}}+p_{b}e^{-i\phi}\frac{C_{+}-C_{-}}{\sqrt{2}}=p_{a}^{z}C_{+}+p_{b}^{z}C_{-}.$
(37)

In the basis defined by $p_{a}^{z}=\frac{p_{a}+e^{-i\phi}p_{b}}{\sqrt{2}}$ and
$p_{b}^{z}=\frac{p_{a}-e^{-i\phi}p_{b}}{\sqrt{2}}$, it is evident that local
dissipators $\mathcal{D}_{a}$ and $\mathcal{D}_{b}$, expressed in the basis of
$a$ and $b$, consistently translate into global dissipators
$\mathcal{D}_{z_{a}}$ and $\mathcal{D}_{z_{b}}$ in the new basis. Conversely,
by setting $p_{a}=\pm p_{b}e^{-i\phi}$, we ensure that the dissipator
$\mathcal{D}_{z}$, which is global in the basis of $a$ and $b$, becomes local
in the basis of $C{+}$ and $C_{-}$.

To formulate equations of motion in the basis of $C_{+}$ and $C_{-}$, we
introduce $\mu_{z}=-p_{a}^{z}(p_{b}^{z})$,
$\mu_{a}=-p_{a}^{a}(p_{b}^{a})=-\frac{1}{2}$, and
$\mu_{b}=-p_{a}^{b}(p_{b}^{b})^{*}=\frac{1}{2}$. Subsequently, leveraging the
linearity of the master equation and accounting for the adjusted amplitudes
$F$ associated with both degrees of freedom, we arrive at:

$\displaystyle\frac{d\langle C_{+}\rangle}{dt}=$
$\displaystyle-\frac{\Gamma|p_{a}^{z}|^{2}+\frac{\kappa_{a}+\kappa_{b}}{2}+2i(\Delta+|J|)}{2}\langle
C_{+}\rangle+\left(\frac{\Gamma}{2}\mu_{z}^{*}+\frac{\kappa_{b}-\kappa_{a}}{4}\right)\langle
C_{-}\rangle-i\frac{F}{\sqrt{2}},$ (38) $\displaystyle\frac{d\langle
C_{-}\rangle}{dt}=$
$\displaystyle-\frac{\Gamma|p_{b}^{z}|^{2}+\frac{\kappa_{a}+\kappa_{b}}{2}+2i(\Delta-|J|)}{2}\langle
C_{-}\rangle+\left(\frac{\Gamma}{2}\mu_{z}+\frac{\kappa_{b}-\kappa_{a}}{4}\right)\langle
C_{+}\rangle-i\frac{F}{\sqrt{2}},$ (39) $\displaystyle\frac{d\langle
C_{+}^{\dagger}C_{+}\rangle}{dt}=$
$\displaystyle-\Big{(}\Gamma|p_{a}^{z}|^{2}+\frac{\kappa_{a}+\kappa_{b}}{2}\Big{)}\langle
C_{+}^{\dagger}C_{+}\rangle+2\Re\Big{\\{}\Big{(}\frac{\Gamma}{2}\mu_{z}^{*}+\frac{\kappa_{b}-\kappa_{a}}{4}\Big{)}\langle
C_{+}^{\dagger}C_{-}\rangle\Big{\\}}-2\Im\Big{\\{}\frac{F}{\sqrt{2}}\langle
C_{+}\rangle\Big{\\}},$ (40) $\displaystyle\frac{d\langle
C_{-}^{\dagger}C_{-}\rangle}{dt}=$
$\displaystyle-\Big{(}\Gamma|p_{b}^{z}|^{2}+\frac{\kappa_{a}+\kappa_{b}}{2}\Big{)}\langle
C_{-}^{\dagger}C_{-}\rangle+2\Re\Big{\\{}\Big{(}\frac{\Gamma}{2}\mu_{z}^{*}+\frac{\kappa_{b}-\kappa_{a}}{4}\Big{)}\langle
C_{+}^{\dagger}C_{-}\rangle\Big{\\}}-2\Im\Big{\\{}\frac{F}{\sqrt{2}}\langle
C_{-}\rangle\Big{\\}},$ (41) $\displaystyle\frac{d\langle
C_{+}^{\dagger}C_{-}\rangle}{dt}=$
$\displaystyle-\frac{\Gamma(|p_{a}^{z}|^{2}+|p_{b}^{z}|^{2})+\kappa_{a}+\kappa_{b}-4i|J|}{2}\langle
C_{+}^{\dagger}C_{-}\rangle+\Big{(}\frac{\Gamma}{2}\mu_{z}+\frac{\kappa_{b}-\kappa_{a}}{4}\Big{)}\langle
C_{+}^{\dagger}C_{+}\rangle+\Big{(}\frac{\Gamma}{2}\mu_{z}+\frac{\kappa_{b}-\kappa_{a}}{4}\Big{)}\langle
C_{-}^{\dagger}C_{-}\rangle$ $\displaystyle-i\frac{F}{\sqrt{2}}\langle
C_{+}\rangle+i\frac{F}{\sqrt{2}}\langle C_{-}\rangle.$ (42)

By selecting $\phi=0$ the expressions for $p_{a}^{z}$ and $p_{b}^{z}$ become
$p_{a}^{z}=\frac{p_{a}+p_{b}}{\sqrt{2}}$ and
$p_{b}^{z}=\frac{p_{a}-p_{b}}{\sqrt{2}}$. Specifically, when $p_{a}=p_{b}$, we
obtain $p_{a}^{z}=p_{a}$ and $p_{b}^{z}=0$, resulting in $\mu_{z}=0$.
Referring to Eq. (41), this implies that the global mode $C_{-}$ experiences a
decoherence-free evolution concerning the shared reservoir, mimicking a
scenario with no shared reservoir. And choosing $\phi=\pi$ it is the global
mode $C_{+}$ that experiences a decoherence-free evolution concerning the
shared reservoir.

However, for $\phi\neq 0$, the correction to the built-in detuning is more
complicated due to the presence of the shared reservoir:

$\displaystyle\Delta_{opt}=-\frac{6^{1/3}\left(8\Delta^{2}_{in}-2\Gamma^{2}-\Gamma_{a}^{2}-\Gamma_{b}^{2}-2\Gamma_{a}\kappa_{a}-\kappa_{a}^{2}-2\Gamma_{b}\kappa_{b}-\kappa_{b}^{2}\right)-(K-M)^{2/3}}{2\times
6^{2/3}(K-M)^{1/3}},$ (43)

where

$K=:36|J|\Gamma\cos{\phi}\left(\Gamma_{a}+\Gamma_{b}+\kappa_{a}+\kappa_{b}\right),$
(44)
$M=:6^{1/2}\left(\left(8\Delta^{2}_{in}-2\Gamma^{2}-\Gamma_{a}^{2}-\Gamma_{b}^{2}-2\Gamma_{a}\kappa_{a}-\kappa_{a}^{2}-2\Gamma_{b}\kappa_{b}-\kappa_{b}^{2}\right)^{3}-216\Delta^{2}_{in}\Gamma^{2}\left(\Gamma_{a}-\Gamma_{b}-\kappa_{a}-\kappa_{b}\right)^{2}\cos{\phi}^{2}\right)^{1/2}.$
(45)

Putting $\Gamma=0$ we obtain the optimal detuning given in Eq. (13) of the
main text.

##  Appendix B Total Energy Comparison: Before and After Optimization

Replacing the coherent interaction $J$ by the shared reservoir with
dissipation rate $\Gamma$ and then performing the optimization A significant
change in both the overall and local energies of the system is observed. In
Fig. 7 the total energy of the system $\xi^{S}$, in the steady state, before
(blue line) and after (red line) the optimization are illustrated. As is seen
a significant increase is observed due to the optimization. Therefore,
optimization preserves more energy, in the charger-battery system, against
dissipation.

Figure 7: $\xi^{S}$ and $\xi^{S}_{opt}$ versus $\Gamma$. For $\xi^{S}$ we have
the optimized $p_{a}=p_{b}=1$ and for $\xi^{S}_{opt}$ we have
$p_{a}=(\kappa_{a}/\kappa_{b})^{1/4}$, $p_{b}=(\kappa_{a}/\kappa_{b})^{-1/4}$.
In both cases we have $\Delta_{opt}=0$, $\kappa_{a}=0.05$, and
$\kappa_{b}=0.01$.