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arXiv:2210.04028 · Physical Review A

Optimal control methods for quantum batteries

Francesco Mazzoncini, Vasco Cavina, Gian Marcello Andolina, Paolo Andrea Erdman, Vittorio Giovannetti

Published Mar 30, 2023• 86 citations
arXivHTMLPDF

Abstract

We investigate the optimal charging processes for several models of quantum batteries, finding how to maximize the energy stored in a given battery with a finite-time modulation of a set of external fields. We approach the problem using advanced tools of optimal control theory, highlighting the universality of some features of the optimal solutions, for instance the emergence of the well-known bang-bang behavior of time-dependent external fields. The technique presented here is general, and we apply it to specific cases in which the energy is either pumped into the battery by external forces (direct charging) or transferred into it from an external charger (mediated charging). In this paper we focus on particular systems that consist of coupled qubits and harmonic oscillators, for which the optimal charging problem can be explicitly solved using a combined analytical-numerical approach based on our optimal control techniques. However, our approach can be applied to more complex setups, thus fostering the study of many-body effects in the charging process.

DOI
10.1103/physreva.107.032218

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We investigate the optimal charging processes for several models of quantum batteries, finding how to maximize the energy stored in a given battery with a finite-time modulation of a set of external fields. We approach the problem using advanced tools of optimal control theory, highlighting the universality of some features of the optimal solutions, for instance the emergence of the well-known bang-bang behavior of time-dependent external fields. The technique presented here is general, and we apply it to specific cases in which the energy is either pumped into the battery by external forces (direct charging) or transferred into it from an external charger (mediated charging). In this paper we focus on particular systems that consist of coupled qubits and harmonic oscillators, for which the optimal charging problem can be explicitly solved using a combined analytical-numerical approach based on our optimal control techniques. However, our approach can be applied to more complex setups, thus fostering the study of many-body effects in the charging process.
Abstract Quantum control allows us to address the problem of engineering quantum dynamics for special purposes. While recently the field of quantum batteries has attracted much attention, optimization of their charging has not benefited from the quantum control methods. Here we fill this gap by using an optimization method. We apply for the first time the convergent iterative method for the control of the population of a bipartite quantum system in two cases, starting with a qubit-qubit case. The quantum charger-battery system is considered here, where the energy is pumped into the charger by an external classical electromagnetic field. Secondly, we systematically extend our investigation to a second case involving two harmonic oscillators in the Gaussian regime, presenting an original formulation of the method. In both cases, the charger is considered to be an open dissipative system, as its interaction with the drive may require a more pronounced exposure to general interaction with environment. A key consideration in our optimization strategy is the practical concern of turning the charging external field on and off. We find that optimizing the pulse shape yields a substantial enhancement in both the power and efficiency of the charging process compared to a sinusoidal drive. The harmonic oscillator configuration of quantum batteries is particularly intriguing, as the optimal driving pulse remains effective regardless of the environmental temperature. This study introduces a novel approach to quantum battery charging optimization, opening avenues for enhanced performance in real-world applications.
We consider the problem of maximizing the stored energy for a given charging duration in a quantum battery composed of a pair of spins-1/21/21/2 with Ising coupling starting from the spin-down state, using bounded transverse field control. We map this problem to an optimal control problem on a single qubit and by using optimal control theory we show that, although a single bang pulse can quickly achieve considerable charging levels for relatively large upper control bounds, higher levels of stored energy including complete charging are accomplished by a bang-singular-bang pulse sequence, where the intermediate singular pulse is an off pulse. If the control is restricted between zero and a maximum value, the initial and final bang pulses attain the maximum bound but have different durations, while if it is restricted between symmetric negative and positive boundaries, the bang pulses have the same duration but opposite boundary values. For both cases we provide transcendental equations from which the durations of the individual pulses in the optimal pulse-sequence can be calculated. For the case of full charging we surprisingly find that the three ``switching'' functions for the equivalent qubit problem become zero while the adjoint ket does not, consistent with optimal control theory.
We propose an approach for the optimization of the charging of harmonic oscillators (quantum batteries) coupled to a harmonic oscillator (charger), driven by a laser field. We demonstrate that energy transfer limitations can be significantly mitigated in the presence of catalyst systems, mediating between the charger and quantum batteries. We show that these catalyst systems, either qubits or harmonic oscillators, enhance the amount of energy transferred to quantum batteries, while they themselves store almost no energy. It eliminates the need for optimizing the frequency of the charging laser field whose optimal value in the bare setting depends on the coupling strengths between the charger and the batteries.
We present a systematic analysis and classification of several models of quantum batteries involving different combinations of two-level systems and quantum harmonic oscillators. In particular, we study energy-transfer processes from a given quantum system, termed a ``charger,'' to another one, i.e., the proper ``battery.'' In this setting, we analyze different figures of merit, including the charging time, the maximum energy transfer, and the average charging power. The role of coupling Hamiltonians which do not preserve the number of local excitations in the charger-battery system is clarified by properly accounting for them in the global energy balance of the model.
We study the charging process of open quantum batteries mediated by a common dissipative environment in two different scenarios. In the first case, we consider a quantum charger-battery model in the presence of a non-Markovian environment, where the battery can be properly charged in a strong coupling regime, without any external power and any direct interaction with the charger, i.e., a wireless-like charging happens. The environment plays a major role in the charging of the battery, while this does not happen in a weak coupling regime. In the second scenario, we show the effect of individual and collective spontaneous emission rates on the charging process of quantum batteries by considering a two-qubit system in the presence of Markovian dynamics. Our results demonstrate that open batteries can be satisfactorily charged in Markovian dynamics by employing an underdamped regime and/or strong external fields. We also present a robust battery by taking into account subradiant states and an intermediate regime. Moreover, we propose an experimental setup to explore the ergotropy in the first scenario.
Charger-mediated quantum batteries offer promising opportunities for efficient energy transfer and storage at the quantum level, with potential advantages beyond their classical counterparts. Here, we theoretically demonstrate that nonlinear charger-battery interactions can substantially accelerate the charging of quantum batteries. In the weak-driving regime, sufficiently strong nonlinear coupling can dominate over the effect of dephasing and significantly reduce the charging time. In the strong- and optimal-driving regimes, where the optimal driving strength maximizes the steady-state ergotropy in the linear-coupling regime, nonlinear coupling gives rise to a new optimum in the charging time beyond that of the linear-coupling regime, accompanied by nearly critically damped dynamics. Remarkably, even in the absence of external driving, nonlinear coupling alone can substantially reduce the charging time, providing an alternative route toward fast battery charging. Our results establish nonlinear charger-battery coupling as an effective resource for accelerating quantum-battery charging and broaden the scope of interaction-enabled quantum advantages in energy transfer and storage.