"A battery powers a device" can be read as "work stored in the battery is being transported to the device." In quantum batteries, the total amount of stored work can be measured by ergotropy, which is the maximal work extractable by unitary operations. Transporting ergotropy is fundamentally different from transporting energy, and here we find that ergotropy can be gained even when the transmission channel is strictly energy conserving. We show that, generically, ergotropy transport is lossy whenever the two systems start uncorrelated. In contrast, for a large class of correlated initial states, transport can be gainful. Furthermore, a single correlated state can be used multiple times, allowing to transport without losses an order of magnitude more work than the battery capacity. Correlations are thus a useful resource for ergotropy transport, and we quantify how this resource is consumed during gainful transport.
"A battery powers a device" can be read as "work stored in the battery is being transported to the device." In quantum batteries, the total amount of stored work can be measured by ergotropy, which is the maximal work extractable by unitary operations. Transporting ergotropy is fundamentally different from transporting energy, and here we find that ergotropy can be gained even when the transmission channel is strictly energy conserving. We show that, generically, ergotropy transport is lossy whenever the two systems start uncorrelated. In contrast, for a large class of correlated initial states, transport can be gainful. Furthermore, a single correlated state can be used multiple times, allowing to transport without losses an order of magnitude more work than the battery capacity. Correlations are thus a useful resource for ergotropy transport, and we quantify how this resource is consumed during gainful transport.
Work extraction protocol is always a significant issue in the context of quantum batteries, in which the notion of ergotropy is used to quantify a particular amount of energy that can be extracted through unitary processes. Given the total amount of energy stored in a quantum system, quantifying wasted energy after the ergotropy extraction is a question to be considered when undesired coupling with thermal reservoirs is taken into account. In this paper, we show that some amount of energy can be lost when we extract ergotropy from a quantum system and quantified by the exergy of passive states. Through a particular example, one shows that ergotropy extraction can be done by preserving the quantum correlations of a quantum system. Our study opens the perspective for new advances in open system quantum batteries able to explore exergy stored as quantum correlations.
The study of quantum thermodynamics has led to the development of quantum batteries. These devices use quantum advantages to store and extract useful energy from physical systems. Ergotropy is the maximum work that can be extracted from a quantum system by cyclic unitary operations. When external thermal baths couple with the quantum battery, there is energy loss due to thermal effects on the system. In some cases, a part of the total energy available in the system cannot be stored as ergotropy. Therefore, it is important to consider the amount of residual energy that cannot be extracted as useful work from quantum batteries by unitary processes. To better understand the amount of energy lost during work extraction, it is necessary to examine the constraint of unitary processes. The system exergy represents the maximum amount of work that can be extracted from the system while bringing it into equilibrium with a thermal bath. It can be separated into two parts: ergotropy and residual energy. Thus, the present chapter describes the relationship between exergy and its potential benefits and effects on the performance of quantum batteries.
Energy can be stored in quantum batteries by electromagnetic fields as chargers. In this paper, the performance of a quantum battery with single and double chargers is studied. It is shown that by using two independent charging fields, prepared in coherent states, the charging power of the quantum battery can be significantly improved, though the average number of embedded photons are kept the same in both scenarios. The results reveal that for the case of initially correlated states of the chargers, the amount of extractable energy, measured by ergotropy, is more than initially uncorrelated ones, with appropriate degrees of field intensity. Though the correlated chargers lead to greater reduction in the purity of the quantum battery, more energy and in turn more ergotropy are stored in this case. In addition, we study the battery-charger mutual information and von Neumann entropy and by using their relation we find that both quantum and classical correlations are generated between the quantum battery and chargers. We also study quantum consonance of the battery as the nonlocal coherence among its cells and find some qualitative relations between the generation of such correlations and the capability of energy storage in the quantum battery.
Going beyond isolated system dynamics, we examine how local and spatially correlated reservoirs influence the work extraction in quantum batteries. By employing a one-dimensional spin-1/2 model coupled to baths via dephasing and amplitude-damping noise, we demonstrate that correlations in reservoirs can significantly enhance battery's performance compared to local noise. In the dephasing scenario, we prove that correlated reservoirs produce a finite amount of extractable work, or ergotropy, during the transient regime when a two-cell battery is initialized in a product state while local noise yields vanishing ergotropy at all times, despite nonvanishing stored energy in both cases. Numerical simulations confirm that this advantage persists across larger system sizes and for both entangled and product initial states. We also find that the dynamics of quantum coherence closely mirror those of ergotropy, highlighting coherence as a key resource underlying the enhanced performance of quantum batteries. Further, we observe that the fraction of stored energy extracted from quantum batteries displays a sharper contrast between the correlated and local reservoirs. Moreover, for dephasing noise, this fraction remains independent of system size, whereas in the amplitude damping case, it exhibits a clear system-size dependence within the transient regime, highlighting distinct operational behaviors under different noise models. In addition, we reveal that when the battery dynamics is governed by an effective Hamiltonian with long-range interactions, it yields higher ergotropy compared to short-range interactions, emphasizing the advantages of reservoir engineering for efficient device design.
Ergotropic-gap relations provide thermodynamic witnesses of quantum correlations, but a direct dynamical connection between extractable work and entanglement remains largely unexplored in multipartite non-Gaussian bosonic systems. Here, we establish such a connection for a bosonic quantum battery driven by a coherent-state superposition. Despite the infinite-dimensional Hilbert space, the dynamics retains a simple structure that allows the relevant work and correlation measures to be obtained exactly. We show that the loss of extractable work associated with the passive part of the battery energy is directly determined by charger-battery entanglement. This relation also separates the charging power into contributions from energy transfer and from changes in the battery spectrum. At the same time, multipartite correlations are progressively redistributed from charger-involving correlations to correlations internal to the battery. For a symmetric resonant protocol, complete energy transfer coincides with charger-battery disentanglement and fully extractable stored energy, while multipartite correlations within the battery are maximal. Individual cells, however, remain mixed, revealing a distinction between global and local work extraction. We further show that, in the regime considered, locally extractable work is entirely supported by energetic coherence. Under single-photon loss, the analytical structure survives, but environmental correlations separate charger disentanglement from maximal work extraction.
Ergotropy provides a fundamental measure of the extractable work from a quantum system and, consequently, of the maximal useful energy, or charge, stored within it. Understanding how this quantity can be manipulated and transformed efficiently is crucial for advancing quantum energy management technologies. Here, we introduce and formalize the concepts of isoergotropic states and ergotropy-preserving operations, which reorganize the internal structure of ergotropy while keeping its total value unchanged. These ideas are illustrated for both discrete (two-level systems) and continuous-variable systems (single-mode Gaussian states). In each case, we show how ergotropy-preserving operations redistribute the respective coherent-incoherent and displacement-squeezing components. We further examine the thermodynamic exchanges accompanying ergotropy-preserving operations, including variations in energy and entropy, and demonstrate that these transformations can be dynamically implemented through standard beam-splitter-type interactions with an auxiliary system. Finally, we discuss the practical implications of isoergotropic states and operations in optimizing charging protocols and mitigating charge loss in open quantum batteries.