We investigate the problem of finding the local analog of the ergotropy, which is the maximum work that can be extracted from a system if we can only apply local unitary transformation acting on a given subsystem. In particular, we provide a closed formula for the local ergotropy in the special case in which the local system has only two levels, and we give analytic lower bounds and semidefinite programming upper bounds for the general case. As nontrivial examples of application, we compute the local ergotropy for an atom in an electromagnetic cavity with Jaynes-Cummings coupling and the local ergotropy for a spin site in an XXZ Heisenberg chain, showing that the amount of work that can be extracted with a unitary operation on the coupled system can be greater than the work obtainable by quenching off the coupling with the environment before the unitary transformation.
We investigate the problem of finding the local analog of the ergotropy, which is the maximum work that can be extracted from a system if we can only apply local unitary transformation acting on a given subsystem. In particular, we provide a closed formula for the local ergotropy in the special case in which the local system has only two levels, and we give analytic lower bounds and semidefinite programming upper bounds for the general case. As nontrivial examples of application, we compute the local ergotropy for an atom in an electromagnetic cavity with Jaynes-Cummings coupling and the local ergotropy for a spin site in an XXZ Heisenberg chain, showing that the amount of work that can be extracted with a unitary operation on the coupled system can be greater than the work obtainable by quenching off the coupling with the environment before the unitary transformation.
Maximum quantum work extraction is generally defined in terms of the ergotropy functional, no matter how experimentally complicated is the implementation of the optimal unitary allowing for it, especially in the case of multipartite systems. In this framework, we consider a quantum battery made up of many interacting subsystems and study the maximum extractable work via concurrent local unitary operations on each subsystem. We call the resulting functional parallel ergotropy. Focusing on the bipartite case, we first observe that parallel ergotropy outperforms work extraction via egoistic strategies, in which the first agent A extracts locally on its part the maximum available work and the second agent B, subsequently, extracts what is left on the other part. For the agents, this showcases the need of cooperating for an overall benefit. Second, from the informational point of view, we observe that the parallel capacity of a state can detect entanglement and compare it with the statistical entanglement witness that exploits fluctuations of stochastic work extraction. Additionally, we face the technical problem of computing parallel ergotropy. We derive analytical upper bounds for specific classes of states and Hamiltonians and provide recipes to obtain numerical upper bounds via semidefinite programming in the generic case. Finally, extending the concept of parallel ergotropy, we demonstrate that system's free-time evolution and application of local unitaries allow one to saturate the gap with the ergotropy of the whole system.
A fundamental problem in quantum thermodynamics is to properly quantify the work extractable from out-of-equilibrium systems. While for closed systems, maximum quantum work extraction is defined in terms of the ergotropy functional, this question is unclear in open systems interacting with an environment. The concept of local ergotropy has been proposed, but it presents several problems, such as it is not guaranteed to be nonincreasing in time. Here, we introduce the concept of extended local ergotropy by exploiting the free evolution of the system-environment compound. At variance with the local ergotropy, the extended local ergotropy is greater, is nonincreasing in time, and activates the potential of work extraction in many cases. We then concentrate on specific schemes in which we alternate repeated local unitaries and free system-environment evolution. We provide examples based on the Jaynes-Cummings model, presenting practical protocols and analytic results that serve as proof of principle for the aforementioned advantages.
Abstract In this mini-review, we discuss work extraction under restricted control, where only the system is accessible while other degrees of freedom remain uncontrolled. In this setting, local ergotropy is the maximum work extractable by cyclic unitaries on the system, and it can differ strongly from “switch-off” extraction after quenching the system-environment coupling. We review two settings where correlations and many-body effects shape local work. First, in a two-qubit quantum Rabi model, dissipation induces a Berezinskii–Kosterlitz–Thouless quantum phase transition. A gate-based charge-store-extract protocol uses a decoherence-free state to stabilize storage and makes strong system-bath coupling a resource, yielding local ergotropy above switch-off extraction and making its dynamics transition-dependent, also in relative fluctuations. Second, in a disordered XXZ chain where the first two spins form a battery coupled to an interacting spin bath, local ergotropy shows distinct long-time behavior across Anderson, many-body localized, and ergodic regimes, linking extractable work to localization transitions. Overall, the examples reviewed here show that local ergotropic observables quantify quantum-battery performance under local control and probe collective phenomena.
Abstract We explore the energy content of superpositions of single-excitation current states. Specifically, we focus on the maximum energy that can be extracted from them through local unitary transformations. The figure of merit we employ is the local ergotropy. We consider an XY spin-chain model and perform a complete analysis in the whole range of the system parameters. This way, we prove that superpositions of two current states in spatially closed spin networks are characterized by specific peaks in extractable energy, generally overcoming the ergotropy of each of the two separate current states characterized by a single winding number. The many-body state dynamics entails to ergotropy evolving in a controlled fashion. The implementation we suggest is based on a Rydberg-atom platform. Optimal transformations able to extract locally the maximum possible amount of energy are sorted out.
Work extraction is a fundamental aspect in thermodynamics. In the context of quantum physics, ergotropy quantifies the maximum amount of work that can be obtained from quantum system through cyclic unitary process. In this work, the steady-state ergotropy of two coupled qubit, each interacting locally with its individual boson or fermion reservoir, will be examined. In this work, both equilibrium and non-equilibrium scenarios for bosonic and fermionic environments interacting with the qubits will be considered. In scenarios where two coupled qubits embedded within equilibrium boson reservoirs, it has been observed that the temperature of the reservoirs and the inter-qubits interaction strangth act as detrimental factors in work extraction. In the case of fermionic equilibrium reservoirs, it will be observed that ergotropy grows monotonically with the reservoirs chemical potential. In the non-equilibrium boson reservoirs, the temperature difference between the two reservoirs is a destructive factor for ergotropy. In non-equilibrium fermion reservoirs, the situation is somewhat more complicated. For r base chemical potential values that are smaller than the qubit transition frequency, the behavior of ergotropy is non-monotonic. However, for base chemical potential values that are larger than the transition frequency, ergotropy grows monotonically with the reservoirs chemical potential difference. Also, we study the situation in which the coupled qubits are asymmetric. It is observed that the maximum work will be extracted in the situation where the coupled qubits within both boson and fermion reservoirs be symmetric .
A subsystem of an entangled ground state (GS) is in a mixed state. Thus, if we isolate this subsystem from its surroundings, we may be able to extract work applying unitary transformations, up to a maximal amount which is called ergotropy. Once this work has been extracted, the subsystem will still contain some bound energy above its local GS, which can provide valuable information about the entanglement structure. We show that the bound energy for half a free fermionic chain decays as the square of the entanglement entropy divided by the chain length, thus approaching zero for large system sizes, and we conjecture that this relation holds for all one-dimensional critical states.