1. Introduction
A quantum battery (QB) is a quantum system that can store and release energy as needed. Over the past decade, several types of QBs have been proposed, demonstrating advantages with respect to charging speed [
1,
2,
3,
4,
5,
6,
7,
8,
9] and work extraction [
5,
7,
10,
11,
12,
13,
14] over the classical analogues. In practice, a QB may interact with an environment, which could lead to losses that negatively impact its performance. Thus, any theoretical study should treat the QB as an open quantum system [
15,
16,
17]. Previously, open quantum network (OQN) models, in which the network sites (representing quantum systems) may be coupled to each other and to dissipative/decohering environments, have been used to study the dynamics of open QBs [
5,
6,
8,
18].
Over the years, a number of ways of protecting a quantum system from its environment have been proposed. One well-known approach is based on the use of a decoherence-free subspace (DFS) [
19,
20]—a subspace of the Hilbert space in which the dynamics is purely unitary. While in a DFS, the system’s dynamics is dissipationless and decoherence-free despite the coupling to an environment. As has been shown in Refs. [
21,
22,
23], if there exists a unitary operator that commutes with all elements in the system’s master equation exists, then the system will possess invariant subspaces. Among such subspaces, the one-dimensional subspaces are DFSs because the dynamics maps them onto themselves. Owing to this property, DFSs have many potential applications in quantum information and quantum computing [
24,
25,
26,
27,
28,
29].
Recently, researchers have used dark states living in DFSs for stabilizing and enhancing the performance of QBs and other OQNs [
18,
30,
31,
32]. In particular, we proposed an OQN model of a QB with site exchange symmetries that support the existence of dark states localized on the bulk sites of the network [
18]. We showed that it is possible to store an exciton in one of these dark states, without any exciton population transfer to the two surface sites which are connected to thermal baths at equal temperatures. In this way, it was possible to protect the QB from environment-induced excitation energy losses. Moreover, we showed that by attaching an additional bath to break the exchange symmetry of the QB, it is possible to discharge the exciton towards the designated exit site to be ultimately harnessed by a sink.
Although it was shown that our QB model is capable of operating as an excitonic QB (i.e., a QB that stores and discharges excitons), the question of whether or not the model is capable of operating as an energy battery (i.e., a QB whose energy does not decrease significantly during the storage phase and discharges energy to a sink with minimal loss to the baths) remained to be explored. Therefore, in this study, in addition to monitoring the exciton population dynamics, we calculate the various contributions to the total energy of the system during both the storage and discharge phases for a wide range of bath temperature gaps, bath reorganization energies, and site energies. The aim of the study is to identify the conditions which optimize the performance of the QB model for its use as an excitonic QB, energy QB, or both.
The paper is organized as follows. First, in
Section 2, we introduce the open QB model. In
Section 3, we explain how the dynamics of the model are simulated and how the populations/energies are calculated. In
Section 4, we present the time-dependent populations/energies of the model for different bath temperatures, bath reorganization energies, and site energies. The performance of the QB under the various conditions is also discussed. In
Section 5, we summarize our findings.
2. Open QB Model
Following Ref. [
18], we consider the same six-site para-benzene-shaped OQN (the numbering of the sites is depicted in
Figure 1b). Setting
, the Hamiltonian of the closed network is given by
where
corresponds to a singly excited state localized on site
n,
is the energy of site
n,
h is the electronic coupling strength between sites
n and
m, and
denotes that a cyclic summation over nearest-neighbour sites is performed. To construct the OQN, the two para-sites of the network are coupled to thermal baths, each composed of a set of independent harmonic oscillators. The sum of the bath and network–bath coupling Hamiltonians is given by
where
M is the number of harmonic oscillators in each bath,
and
are the mass-weighted momentum and position operators, respectively, of the
jth oscillator with frequency
, and
is the network–bath coupling strength between the
nth site and the
jth oscillator. The sites coupled to the baths are referred to as surface sites (SSs) while the remaining sites are referred to as bulk sites (BSs). In this study, we take the site energies of all BS to be equal to a value
, while considering different values of site energies of the SS.
The network defined by
and the aforementioned site energies possesses the following unitary symmetry operator [
33]
which satisfies
As a result,
shares the same eigenstates as
. Due to the existence of this symmetry operator, the system possesses two DFSs with the following DSs (
) and eigenvalues (
) [
22,
34]
If the system is initialized in the dark state
, it will undergo dissipationless dynamics, i.e., the dark state will be invariant under the effect of the evolution operator of the composite system and the site populations will be
where
is the projection operator corresponding to site
n and
denotes an ensemble average. Because an exciton can be stored indefinitely in this state, this phase is termed the
storage phase.
When a symmetry-breaking perturbation (SBP) is connected to the OQN, the symmetry operator and DSs no longer persist and the QB may begin to discharge a stored exciton. In this study, this is achieved by attaching a bath of
M harmonic oscillators to sites 2 and 3 simultaneously. The sum of the SBP and network-SBP coupling Hamiltonians is
where
,
,
, and
are the momentum operator, position operator, frequency, and network–SBP coupling strength of the
kth oscillator, respectively, and
. This phase is termed the
discharge phase.
3. Simulation Details
Due to the large number of degrees of freedom in the composite system (i.e., QB, thermal baths, and SBP), a fully quantum dynamical simulation of the composite system would be computationally expensive. Thus, following Ref. [
18], we use a mixed quantum–classical dynamics method known as “Deterministic evolution of coordinates with initial decoupled equations” (DECIDE) [
35,
36], which treats the OQN quantum mechanically and the baths and SBP in a classical-like way. Previously, the DECIDE method has been successfully applied to a host of model systems over a large range of parameter regimes [
35,
37,
38,
39,
40], and is, therefore, expected to produce reliable results in this study. That being said, DECIDE may yield inaccurate results for systems with very slow thermal baths (i.e., when the bath cut-off frequency is much smaller than the subsystem energy gaps) or with very low bath temperatures, neither of which is the case in the present study.
To apply the DECIDE method, we must first apply the Wigner transform [
41] to the bath and SBP degrees of freedom. The resulting partially Wigner-transformed Hamiltonian of the composite system is
where
and
are the position and momentum variables of the baths and SBP, respectively. The parameter
is equal to 1 when the SBP is attached to the OQN and 0 otherwise. The coordinates of the OQN are taken to be
, while the coordinates of the baths and SBP are their positions and momenta. According to the DECIDE method, the coupled equations of motion for all of the coordinates are given by [
18]
where
, and when
,
for
and 0 otherwise. Here,
labels an arbitrary basis state, i.e., the matrix element of
is given by
. In this work,
.
We assume the initial state of the composite system to be factorized, i.e.,
, where
is the initial density operator of the network, and
and
are the initial Wigner-transformed densities of the heat baths and SBP, respectively (N.B.:
is omitted when the OQN is not attached to the SBP). The initial state of the network is taken to be the dark state
, where
is defined in Equation (5). The initial values of the OQN coordinates are always taken to be
. The bath oscillators are initialized in the thermal equilibrium state given by (setting
) [
42]
where
is the inverse temperature. The oscillators of the SBP are also initialized in a thermal equilibrium state with an analogous form. The initial positions and momenta of the bath and SBP oscillators are sampled from Equation (
10) and its analog for the SBP, respectively. The system–bath and system–SBP couplings are characterized by a Debye–Drude spectral density, i.e.,
. In this work, the spectral density is discretized to yield the following expressions for the coupling strengths
and frequencies
[
43,
44]:
where
is the bath reorganization energy and
is the bath cut-off frequency [
45].
Previously, the fourth-order Runge–Kutta method was used to integrate the DECIDE equations of motion in Equation (
9), yielding conserved total populations for the system under study [
18,
35]. However, in this work, we found that a combination of a smaller time step and a higher order integrator is needed for good energy conservation and more accurate calculations of the various contributions to the total energy. High-order methods such as the eighth-order Runge–Kutta method [
46] can yield accurate results with a relatively large time step, but it contains many integration stages which increase the simulation time drastically. Considering the trade-off between time step and number of integration steps, we employed the sixth-order Runge–Kutta method in this work. Using this integrator with a time step of 0.16 fs, the total energy drift is less than
over a 1 ps trajectory [see
Section S1 of the Supporting Information (SI)].
The time-dependent population of site
n is calculated via an ensemble average of the projection operator
, viz.,
where
are the initial coordinates of the baths and SBP. Similarly, the average total energy of the composite system is
In the above equations,
in the absence of the SBP and
when the SBP is attached to the QB. Using our numerical results, we have verified that
(i.e., population conservation) and
(i.e., energy conservation). For the purposes of our analysis, the total energy of the composite system may be decomposed into the following contributions: OQN energy (
), bath energy (
), and SBP energy (
). All simulation results are averaged over 10,000 trajectories, which ensures that the error bars are much smaller than the symbols in the figures.
5. Concluding Remarks
In this paper, we studied the population and energy transfer dynamics of an open quantum battery model, originally proposed in Ref. [
18], over a wide range of parameter regimes. In the battery’s storage phase, we demonstrated that, in addition to no population leakage, there is no energy leakage from the battery into the attached baths. During the discharge phase, the changes in the populations and OQN energy are influenced by the bath temperatures, bath reorganization energies, and site energies. When increasing the temperature of one bath (while kee** the temperature of the other bath constant), we observed an increase in the energy transferred from that bath to the OQN. We found that the right bath (i.e., the bath connected to the exit site) exerts a larger influence on the exit site population than that exerted by the left bath (i.e., the bath connected to site 1) on the site 1 population. Moreover, when increasing the reorganization energy of the right bath, we observed an increase in the exit site’s population and a decrease in OQN energy. On the other hand, varying the reorganization energy of the left bath does not have pronounced effect on the population and energy. Regarding the site energies, when the energy of the exit site is lower than those of the BS, the OQN energy decreases in most of the parameter regimes studied. Lowering a given site energy causes the corresponding site population to increase. When the site energies are equal, the site populations reach roughly equal values after 1 ps, despite the different initial populations.
The results of our parameter space exploration show that different parameter sets render the QB conducive to different applications. In practice, this would amount to designing the QB in such a way that its properties are consistent with those of the desired parameter set. For example, for an energy battery, one may desire that the QB gains energy from its environment during the discharge phase. As we have shown, if one sets the site energies of the SS to be larger than those of the BS, then the QB gains energy as the SS populations grow in time. On the other hand, for an excitonic battery, one may desire to maximize the population of the exit site, regardless of the change in the OQN energy. In such a case, one could lower the exit site energy or lower the right-bath temperature while maintaining the left-bath temperature at its original value. If one would like to simultaneously reduce the loss in OQN energy and increase the exit site population, one could either increase the exit site energy or increase the left bath temperature. Overall, our findings shed light on design principles that could be used to construct different types of QBs operating between two thermal reservoirs.