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arXiv:2303.16676 · Physical review. E

Trade-offs between precision and fluctuations in charging finite-dimensional quantum batteries

Pharnam Bakhshinezhad, Beniamin R. Jablonski, Felix C. Binder, Nicolai Friis

Published Jan 22, 2024• 18 citations
arXivHTMLPDF

Abstract

Within quantum thermodynamics, many tasks are modeled by processes that require work sources represented by out-of-equilibrium quantum systems, often dubbed quantum batteries, in which work can be deposited or from which work can be extracted. Here we consider quantum batteries modeled as finite-dimensional quantum systems initially in thermal equilibrium that are charged via cyclic Hamiltonian processes. We present optimal or near-optimal protocols for N identical two-level systems and individual d-level systems with equally spaced energy gaps in terms of the charging precision and work fluctuations during the charging process. We analyze the trade-off between these figures of merit as well as the performance of local and global operations.

DOI
10.1103/physreve.109.014131

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Within quantum thermodynamics, many tasks are modeled by processes that require work sources represented by out-of-equilibrium quantum systems, often dubbed quantum batteries, in which work can be deposited or from which work can be extracted. Here we consider quantum batteries modeled as finite-dimensional quantum systems initially in thermal equilibrium that are charged via cyclic Hamiltonian processes. We present optimal or near-optimal protocols for N identical two-level systems and individual d-level systems with equally spaced energy gaps in terms of the charging precision and work fluctuations during the charging process. We analyze the trade-off between these figures of merit as well as the performance of local and global operations.
One of the most fundamental tasks in quantum thermodynamics is extracting energy from one system and subsequently storing this energy in an appropriate battery. Both of these steps, work extraction and charging, can be viewed as cyclic Hamiltonian processes acting on individual quantum systems. Interestingly, so-called passive states exist, whose energy cannot be lowered by unitary operations, but it is safe to assume that the energy of any not fully charged battery may be increased unitarily. However, unitaries raising the average energy by the same amount may differ in qualities such as their precision, fluctuations, and charging power. Moreover, some unitaries may be extremely difficult to realize in practice. It is hence of crucial importance to understand the qualities that can be expected from practically implementable transformations. Here, we consider the limitations on charging batteries when restricting to the feasibly realizable family of Gaussian unitaries. We derive optimal protocols for general unitary operations as well as for the restriction to easier implementable Gaussian unitaries. We find that practical Gaussian battery charging, while performing significantly less well than is possible in principle, still offers asymptotically vanishing relative charge variances and fluctuations.
Abstract We consider a quantum battery modelled as a set of N independent two-level quantum systems driven by a time dependent classical source. Different figures of merit, such as stored energy, time of charging and energy quantum fluctuations during the charging process, are characterized in a wide range of parameters, by means of numerical approach and suitable analytical approximation scheme. Particular emphasis is put on the role of different initial conditions, describing the preparation state of the quantum battery, as well as on the sensitivity to the functional form of the external time-dependent drive. It is shown that an optimal charging protocol, characterized by fast charging time and the absence of charging fluctuations, can be achieved starting from the ground state of each two-level system, while other pure preparation states are less efficient. Moreover, we argue that a periodic train of peaked rectangular pulses can lead to fast charging. This study aims at providing a useful theoretical background in view of future experimental solid-state implementations.
We show that a cyclic unitary process can extract work from the thermodynamic equilibrium state of an engineered quantum dissipative process. Systems in the equilibrium states of these processes serve as batteries, storing energy. The dissipative process that brings the battery to the active equilibrium state is driven by an agent that couples the battery to thermal systems. The second law of thermodynamics imposes a work cost for the process; however, no work is needed to keep the battery in that charged state. We consider simple examples of these batteries and discuss situations in which the charged state has full population inversion, in which case the extractable work is maximal, and circumstances in which the efficiency of the process is maximal.
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 connection between the charging power of quantum batteries and the fluctuations of the extractable work. We prove that in order to have a nonzero rate of change of the extractable work, the state ${\ensuremath{\rho}}_{\mathcal{W}}$ of the battery cannot be an eigenstate of a ``free energy operator,'' defined by $\mathcal{F}\ensuremath{\equiv}{H}_{\mathcal{W}}+{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{log}({\ensuremath{\rho}}_{\mathcal{W}})$, where HW{H}_{\mathcal{W}}HW​ is the Hamiltonian of the battery and $\ensuremath{\beta}$ is the inverse temperature of a reference thermal bath with respect to which the extractable work is calculated. We do so by proving that fluctuations in the free energy operator upper bound the charging power of a quantum battery. Our findings also suggest that quantum coherence in the battery enhances the charging process, which we illustrate on a toy model of a heat engine.
Precision, robustness, and efficiency are central requirements for quantum technologies. We show that genuine quantum features combined with non-Gaussianity enable the simultaneous optimization of these properties in a quantum battery-charging process. Using a generalized Jaynes-Cummings interaction as a paradigmatic light-matter interaction model, we apply the Full Counting Statistics to characterize stochastic energy exchanges between a stack of qubits and a single-mode bosonic field. We demonstrate that a sequential charging protocol driven by a non-Gaussian quantum field yields high performance in charging precision, which remains maximal even under suboptimal operating conditions. Our results establish the use of non-Gaussian quantum-states in battery charging as a robust route to a quantum precision advantage over protocols based on Gaussian states, achieved through the suppression of detrimental quantum fluctuations.