# Micromasers as Quantum Batteries

Vahid Shaghaghi<EMAIL_ADDRESS>Center for Nonlinear and Complex
Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi
dell’Insubria, via Valleggio 11, 22100 Como, Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy
Center for Theoretical Physics of Complex Systems, Institute for Basic Science
(IBS), Daejeon - 34126, Korea  Varinder Singh<EMAIL_ADDRESS>Center for
Theoretical Physics of Complex Systems, Institute for Basic Science (IBS),
Daejeon - 34126, Korea  Giuliano Benenti<EMAIL_ADDRESS>Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta
Tecnologia, Università degli Studi dell’Insubria, via Valleggio 11, 22100
Como, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via
Celoria 16, 20133 Milano, Italy  NEST, Istituto Nanoscienze-CNR, Piazza S.
Silvestro 12, 56127 Pisa, Italy  Dario Rosa<EMAIL_ADDRESS>Center for
Theoretical Physics of Complex Systems, Institute for Basic Science (IBS),
Daejeon - 34126, Korea  Basic Science Program, Korea University of Science and
Technology (UST), Daejeon - 34113, Korea

###### Abstract

We show that a micromaser is an excellent model of quantum battery. A highly
excited, pure, and effectively steady state of the cavity mode, charged by
coherent qubits, can be achieved, also in the ultrastrong coupling regime of
field-matter interaction. Stability of these appealing features against loss
of coherence of the qubits and the effect of counter-rotating terms in the
interaction Hamiltonian are also discussed.

_Introduction.—_ Given current trend to miniaturization of technology, with
devices operating at dimensions of the order of the nanoscale, quantum effects
must be taken into account. The possibility of using genuine quantum effects
to improve performances of such devices compared to their classical
counterparts — generically dubbed as quantum advantage — has in turn led to
enormous efforts and research, to find all tasks where a quantum advantage can
be obtained [1, 2, 3, 4, 5].

Any quantum machine needs energy to operate. Therefore, in parallel with
developments on building nanodevices, topic of quantum batteries (see [6, 7],
and references therein) has emerged and flourished in the last few years. A
quantum battery is a quantum mechanical system suitable to store energy in
some highly excited states, to be released on demand. Starting from the
seminal work of Alicki and Fannes, [8], several figures of merit, to
characterize the performance of a quantum battery, have been introduced and
studied. These include energy storage [9, 10, 11, 12, 13], work extraction [8,
14], fluctuations properties [15, 16, 17, 18] and charging power [19, 4, 20,
21, 22, 23, 24, 5, 25, 26, 27, 28, 29, 30, 31, 32], to name some of them.

In parallel with such theoretical developments, proposals to concretely build
quantum batteries have emerged, including models realized by spin chains [20]
and qubit systems charged by electromagnetic fields [21]. However, with some
very recent notable exceptions [33], achievements on actual experimental
realizations have been so far quite limited. As for any quantum information
protocol, the search for high-speed operations is vital. Such possibility is
naturally offered by circuit quantum electrodynamics (QED), one of the most
promising platforms for quantum hardware, in which one can address the so-
called ultrastrong coupling (USC) regime of light-matter interaction, where
the qubit-cavity interaction energy becomes comparable, or can even exceed the
bare frequencies of the uncoupled systems [34, 35].

In this paper we revisit, with an eye towards features persisting in the USC
limit, a very well-established model, which has been extensively implemented
and studied at the experimental level: the micromaser [36, 37, 38, 39, 40, 41,
42], where a stream of qubits (two-level atoms in cavity QED [43])
sequentially interact with a cavity mode with a high-quality factor. That is,
the radiation decay time is much larger than the characteristic time of the
qubit–field interaction, and the overall evolution is to a good approximation
coherent. We will show that micromasers can be regarded as excellent models of
quantum batteries. In particular, they display excellent performances in terms
of charging temporal stability and ergotropy [44].

By charging temporal stability we mean the ability of battery to keep, under
time evolution and after an initial transient time, a stable value of its mean
energy [15, 5, 17, 18]. In many models of quantum batteries, after an initial
regime, in which energy grows since battery is evolving towards excited
states, time evolution of mean energy undergoes temporal fluctuations. These
fluctuations are of course unwanted, since they require a very high precision
in controlling the charging time, in order to reach the target value of mean
energy. We will numerically show that a micromaser can be quickly charged to
reach an almost steady state, whose energy is controlled by the physical
parameters defining the model. By almost steady state we mean a state which,
although metastable, is characterized by very long lifetime. An almost steady
state solves the problem of temporal stability, since it is constant for very
long times.

By ergotropy it is meant the amount of energy that can be extracted from a
battery via unitary operations 111The notion of ergotropy assumes a perfect
knowledge of the battery state as well as the possibility of performing any
possible unitary operations. In many-body contexts, both these assumptions are
questionable. In consequence, the notion of ergotropy has been critically re-
analyzed and an improved definition, more suitable for many-body systems, has
been proposed in [52].. Often, part of energy is locked in correlations and
cannot be used. For example, when the battery state is a mixture it cannot be
transformed into the ground state by means of unitary transformations.
Therefore not all its energy can be extracted. On the opposite side, a pure
state has ergotropy which equals its mean energy. We will show that for a
micromaser the almost steady states mentioned above are approximately pure and
therefore, in principle, almost all their energy can be reversibly extracted.
This is a very surprising feature, since dynamics in micromasers involves a
trace over qubit degrees of freedom after each interaction.

The possibility of building pure states (steady or not) in micromasers has
been discussed in previous studies [37, 38, 39, 40, 41]. However, they relied
on several assumptions: very weak coupling between qubits and field and/or
highly fine-tuned values of these couplings. All these assumptions are
unwanted in a quantum battery: weak coupling implies slow charging and
consequently low values of charging power, while fine-tuned values require
very high precision in building the battery, thus making it hard to realize.
Our numerical results show that all these assumptions can be relaxed to a
large extent: the battery reaches an  almost pure steady state even when
entering in USC and without fine-tuning.

The combination of above features promotes micromasers to status of very
robust and reliables quantum batteries, thus making them as very promising
models for experimental realizations.

_Model and figures of merit.—_ We consider a quantum battery made up of a
quantized electromagnetic field in a cavity, modeled as a harmonic oscillator.
It is initially prepared in its ground state $\ket{0}$. The charging protocol
is realized via a stream of two-level systems (qubits) which sequentially
interact with the battery, thus realizing a micromaser. The initial state of
each qubit is

$\displaystyle\rho_{q}$ $\displaystyle=q\ket{g}\bra{g}+(1-q)\ket{e}\bra{e}+$
$\displaystyle+c\sqrt{q(1-q)}\left(\ket{e}\bra{g}+\ket{g}\bra{e}\right)\,,$
(1)

where $\ket{g}$ and $\ket{e}$ are ground and excited state of the qubit,
respectively. Parameters $q$ and $c$ control the degrees of population
inversion and coherence, respectively. The evolution of the system is
described, in interaction picture, by the Hamiltonian [46, 42]

$\hat{H}_{I}=g\left(\hat{a}\hat{\sigma}_{+}+\hat{a}^{\dagger}\hat{\sigma}_{-}+e^{i2\omega
t}\hat{a}^{\dagger}\hat{\sigma}_{+}+e^{-i2\omega
t}\hat{a}\hat{\sigma}_{-}\right)\,.$ (2)

where $\omega$ is the frequency of both qubit and field (we are considering
the resonant case in which they are equal, [42]); $\hat{a}^{\dagger}$,
$\hat{a}$ are creation/annihilation operators for the field and
$\hat{\sigma}_{+}$, $\hat{\sigma}_{-}$ are raising and lowering operators for
the qubit (respectively). Finally, $g$ is the coupling constant for
interaction between qubit and field and we take units such that $\hbar=1$. It
is customary to name the first two terms in Eq. (2) as rotating terms and the
last two terms as counter-rotating terms [42, 46].

From Eq. (2) we derive the time evolution operator,
$\hat{U}_{I}(g,\,\omega,\,\tau)$, to be

$\displaystyle\hat{U}_{I}(g,\,\omega,\,\tau)\equiv\mathcal{T}\exp{-i\int_{0}^{\tau}\hat{H}_{I}(t)\mathrm{d}t},$
(3)

where $\mathcal{T}$ and $\tau$ are the time ordering operator and the
interaction time between a single incoming qubit and the battery,
respectively. We fix, without loosing generality, $\tau=1$. Accordingly,
$\hat{U}_{I}(g,\,\omega)\equiv\hat{U}_{I}(g,\,\omega,\,\tau=1)$ is the time
evolution operator.

Denoting with $\rho_{B}(k)$ the battery state after having interacted with $k$
qubits, we denote by $\rho(k)$ the product state of $\rho_{B}(k)$ and the
incoming $(k+1)$-th qubit, i.e. $\rho(k)=\rho_{B}(k)\otimes\rho_{q}$. The
system is then evolved by means of $\hat{U}_{I}(g,\,\omega)$, and
$\rho_{B}(k+1)$ is obtained as follows:

$\rho_{B}(k+1)=\Tr_{q}\left(\hat{U}_{I}(g,\,\omega)\rho(k)\hat{U}_{I}^{\dagger}(g,\,\omega)\right)\,,$
(4)

where $\Tr_{q}$ is the trace over qubit degrees of freedom. Although the
battery is initially in a pure state $\ket{0}$, by tracing out qubit degrees
of freedom it becomes mixed, in general. More concretely, one expects that the
purity of $\rho_{B}$, $\mathcal{P}(k)$, defined as
$\mathcal{P}(k)\equiv\mathrm{Tr}\left(\rho_{B}^{2}(k)\right)$, should decrease
from $1$ (which is the value for pure states) before the first collision to
almost vanish after many collisions.

The energy stored after $k$ collisions amounts to
$E(k)\equiv\Tr\left(\hat{H}_{B}\rho_{B}(k)\right)$, where
$\hat{H}_{B}\equiv\omega\hat{a}^{\dagger}\hat{a}$ is the battery Hamiltonian
and trace is taken over battery degrees of freedom. We now turn to
characterize the charging performance of the battery by measuring $E(k)$ and
$\mathcal{P}(k)$.

_Numerical results.—_ The time evolution operator, Eq. (3), gets enormously
simplified when counter-rotating terms in Eq. (2) can be omitted. As it is
known, [46, 42], counter-rotating terms can be omitted when weak-coupling
condition $\frac{g}{\omega}\ll 1$ is met. However, to build a battery which is
fast in charging, weak-coupling is not the most relevant limit. Hence, it will
be desirable to consider the case in which $g$ and $\omega$ are comparable,
such that $0.1<\frac{g}{\omega}<1$. This case can be treated, still without
the need of considering counter-rotating terms, by performing a simultaneous
modulation of the frequencies of field and qubit, as first observed in [47].
We now assume that such a modulation has been performed and consider the
celebrated Jaynes-Cummings (JC) operator [48],

$\hat{U}_{I}(g)=e^{-ig\left(\hat{a}\hat{\sigma}_{+}+\hat{a}^{\dagger}\hat{\sigma}_{-}\right)}.$
(5)

Figure 1: Performance of the micromaser quantum battery. FIG. 1: energy stored
(upper panel) and purity (lower panel) computed for $g=0.5,\,0.9$ and $q=0.25$
in JC model (denoted “JC nft” in legend) as well as in presence of finite
values of $\omega$ and non-vanishing contributions from counter-rotating
terms. As a comparison, we report also the charging performances for JC
batteries having the closest fine-tuned values for $q=1$,
$g=\frac{\pi}{\sqrt{39}}\approx 0.503$ and $g=\frac{\pi}{\sqrt{12}}\approx
0.906$ (denoted “JC ft” in legend). For purity, it is essentially impossible
distinguishing fine-tuned from non fine-tuned dynamics. FIG. 1: absolute
values of density matrix elements, $\absolutevalue{\rho_{nm}}^{2}$, after
$10^{6}$ collisions, for fine-tuned case $g=\frac{\pi}{\sqrt{12}}\approx
0.906$ (upper panel) and for non fine-tuned case $g=0.9$ (lower panel).

Micromasers driven by JC operator have been heavily studied in literature,
starting from [37, 38, 39], both numerically and analytically. The most
important feature of the JC operator, in our context, is the existence of
trapping chambers. It was analytically shown in [37, 38, 40] that when $c=1$
and, more crucially, when $g$ is fine-tuned to satisfy

$g=\frac{Q}{\sqrt{m}}\pi,\qquad Q,m\in\mathbb{N},$ (6)

the system reaches a steady state given by a macroscopic superposition of
number states, $\ket{\psi}=\sum_{n}f_{n}\ket{n}$, involving at most $m$ number
states, with $f_{n}$ satisfying a recursion relation. From this property, $m$
defines the corresponding trapping chamber, since it bounds the largest energy
eigenstate entering in the expansion of the steady state. On the other hand,
to characterize a quantum battery it is important to consider both the cases
in which $g$ is not fine-tuned and to estimate the number of collisions
necessary to reach such a state.

Given any value $g$, clearly it can be written as

$g=\frac{Q}{\sqrt{m+\epsilon}}\pi,\qquad
Q,m\in\mathbb{N},\quad-0.5<\epsilon\leq 0.5,$ (7)

i.e. it is approximated by a certain fine-tuned value, with $\epsilon$
quantifying the error in considering such an approximation. Moreover, by
increasing $Q$ and correspondingly $m$ $\epsilon$ can be made arbitrarily
small, thus suggesting that, by taking $Q$ and $m$ very large, an almost fine-
tuned dynamics could be eventually reached. Such a picture, however, does not
give any information on the number of collisions necessary to reach such an
almost fine-tuned regime and, more important, on the behavior of the state
during evolution. In particular, it does not provide any information of the
role played on dynamics by other approximate trapping chambers, having $Q$ and
$m$ small. These questions are crucial when dealing with a quantum battery.

To this end, we have numerically computed the micromaser time evolution,
controlled by Eq. (4), for two non fine-tuned values of $g$, as well as for
their closest fine-tuned counterparts having Q = 1, i.e. for $g=\pi/\sqrt{m}$.
Results are reported in Fig. 1. For both fine-tuned and non fine-tuned values,
system reaches, after $\approx 30$ collisions, an effective steady value of
$E(k)$, which remains constant up to $10^{6}$ collisions. This state is
essentially pure, as demonstrated by $\mathcal{P}(k)$. While these results are
non surprising when $g$ is fine-tuned, and they are in agreement with
analytical results [37, 38, 40], it is remarkable that they hold in non fine-
tuned cases as well. In particular, when $g=0.5$ and $Q=1$, $\epsilon\approx
0.48$ is almost maximal. Nevertheless, battery reaches an effective steady
state which, for practical purposes, is stable and pure. These are wanted
properties for a model of quantum battery, which our results show to be
present up to, at least, $10^{6}$ collisions. To better investigate the
properties of these states, with and without fine-tuning of $g$, we have
studied the density matrix, $\rho_{B}$, after $10^{6}$ collisions. Results are
reported in Fig. 1. As expected, when $g$ is fine-tuned all non-vanishing
elements of the density matrix are strictly confined in the first trapping
chamber. What is non-trivial and interesting is the behavior when $g$ is not
fine-tuned: we see that the vast majority of non-vanishing elements are still
confined in the first trap. However, some non-vanishing elements are actually
escaping and a bubble is forming around matrix element $(30,\,30)$.

This result is confirming that in the presence of just partial trapping,
dynamics turns out to be extremely slow. For many collisions the first trap,
i.e. the case with $Q=1$, is enough to fully control the dynamics.
Furthermore, to get a reliable battery, it is important that battery charging
does not depend too strongly on fine-tuned values of $g$. Fig. 1(b) (lower
panel) shows that in case of non-fine-tuned $g$ there are alternate bubbles in
the system density matrix; however Fig. 1(a) confirms that for practical
purposes first trapping state in the density matrix is sufficient as it is a
long-lived trapping state, up to $10^{6}$ collisions. We also checked (not
shown) that results are quite stable for deviations away from $c=1$: for
values of $c$ not too far from $1$, the battery still reaches a stable
dynamics characterized by a high degree of purity, $\mathcal{P}\gtrapprox c$.

_Stability with counter-rotating terms.—_ Although by performing high-
amplitude, low-frequency modulations [47], the validity of the JC
approximation can be pushed well beyond the standard weak-coupling limit
($g/\omega\ll 1$), it is important to investigate the stability of the
physical features just described in the presence of the counter-rotating terms
in Eq. (2). Hence, without performing any modulations, we have studied the
time evolution of the battery by means of the full time evolution operator
$\hat{U}_{I}(g,\,\omega)$ in Eq. (3). Importance of counter-rotating terms
becomes more prominent when entering in USC regime, $\frac{g}{\omega}\sim
0.1$. Accordingly, we have checked that counter-rotating terms do not
significantly affects dynamics up to $\frac{g}{\omega}\sim 0.05$. More
interestingly, we have studied $E(k)$ and $\mathcal{P}(k)$ for
$\frac{g}{\omega}=0.1$ and larger values of $g$, i.e. in USC regime. Results
are reported in Fig. 1. We see that for $\frac{g}{\omega}\geq 0.5$ all
features described for JC evolution are lost: the system does not reach any
stability with energy increasing indefinitely and purity steadily decreasing.
The same is not true and turns out to be more interesting for
$\frac{g}{\omega}=0.1$. In this case, dynamics is different for $g=0.5$ and
$0.9$, thus showing that the actual behavior is controlled not only by
$g/\omega$ but also by $g$ itself. In particular, for $g=0.5$ we see that
trapping dynamics is still effective after $10^{6}$ collisions, although both
$E(k)$ and $\mathcal{P}(k)$ are negatively affected by counter-rotating terms.
On the other hand, when $g=0.9$ we see that stability is preserved only up to
$\approx 10^{4}$ collisions. All in all, our results show that trapping
properties of the JC operator are rather robust, even in USC limit and without
any modulation.

_Conclusions and outlook.—_ We have numerically shown that a micromaser,
charged by means of coherent qubits, can be thought as an excellent model of
quantum battery. The system reaches an effectively steady state which is
dynamically stable. More crucially, such state is essentially pure, which
means that all of its energy can be extracted, in principle, via unitary
operations. Finally, we have shown that our results are quite stable, even
when entering in USC regime, and they are generic, i.e., they do not require
fine-tuning. Combining all our results, we conclude that a micromaser, charged
in presence of an external modulation and at USC, behaves as a very reliable
model of quantum battery, even when modulation is not perfect and counter-
rotating terms cannot be fully ignored. Micromasers have been extensively
studied in literature, even at experimental level [49]. Hence, we think our
results show an explicit and promising model of quantum battery.

Most of recent theoretical developments have been tailored towards systems
involving many batteries and instances where collective quantum effects
improve performance, i.e. examples of quantum advantage. Micromasers are
single body systems but they can be combined in architectures to build many-
body setups, [50, 51]. Hence, our results show that they are promising
building blocks of many-body quantum batteries, an aspect that we plan to
explore in near future.

On a more theoretical ground, our results show that, for practical purposes,
pure steady states can be found in micromaser at USC and without any fine-
tuning. It would be extremely interesting to better investigate and
characterize these states and to find analytic arguments supporting their
existence and predicting their lifetime.

_Acknowledgments.—_ We thank Matteo Carrega and Dominik Šafránek for
discussions and collaboration on related projects. Vahid Shaghaghi, Varinder
Singh and Dario Rosa acknowledge support by the Institute for Basic Science in
Korea (IBS-R024-D1).

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