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Physical Review A

Extracting ergotropy from nonequilibrium steady states of an XXZ spin-chain quantum battery

B. Mojaveri, R. Jafarzadeh Bahrbeig, M. A. Fasihi

Published Apr 15, 2024• 22 citations
Publisher

Abstract

Coupling of a quantum battery (QB) to two thermal reservoirs brings it to a nonequilibrium steady-state (NESS). In this situation, although, the QB can be charged through the NESS heat current, no work may be extracted from it through unitary cyclic processes. We exemplify this statement by studying the two- and three-qubit $XXZ$ spin-chain QBs coupled weakly to two different thermal reservoirs in a symmetric configuration wherein each end of chain interacts with an individual reservoir. We show that the work can be extracted at positive temperature bias when two ends of a $XXZ$ chain are collectively coupled to the hot reservoir.

DOI
10.1103/physreva.109.042619

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Coupling of a quantum battery (QB) to two thermal reservoirs brings it to a nonequilibrium steady-state (NESS). In this situation, although, the QB can be charged through the NESS heat current, no work may be extracted from it through unitary cyclic processes. We exemplify this statement by studying the two- and three-qubit XXZXXZXXZ spin-chain QBs coupled weakly to two different thermal reservoirs in a symmetric configuration wherein each end of chain interacts with an individual reservoir. We show that the work can be extracted at positive temperature bias when two ends of a XXZXXZXXZ chain are collectively coupled to the hot reservoir.
Understanding the generation and activation of ergotropy, the maximum work extractable by cyclic unitary operations, is central to quantum battery research. Positive-temperature thermal states are passive, whereas nonequilibrium environments can sustain nonpassive steady states with nonzero ergotropy. However, the full secular approximation commonly used in Lindblad descriptions can suppress steady-state coherence and obscure its contribution to ergotropy. Using a Bloch-Redfield master equation beyond the full secular approximation, we investigate the ergotropy of the nonequilibrium steady state (NESS) of two exchange-coupled qubits connected to independent thermal reservoirs. The NESS ergotropy of two qubits with the same frequency is entirely coherent and grows quadratically with temperature bias near equilibrium. In contrast, detuned qubits can exhibit incoherent ergotropy. We derive an exact analytical criterion for its activation and show that activation requires the hotter reservoir to be coupled to the qubit with the higher bare frequency. Comparing NESSs obtained with and without full secularization reveals quantitative and qualitative differences in ergotropy that reflect changes in both its coherent and incoherent contributions. The results clarify the mechanisms of NESS ergotropy activation and demonstrate the importance of retaining steady-state coherence beyond the full secular approximation when characterizing stationary work resources.
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 .
When a quantum system is subject to a thermal gradient it may sustain a steady nonequilibrium heat current by entering into a so-called nonequilibrium steady state (NESS). Here we show that NESS constitute a thermodynamic resource that can be exploited to charge a quantum battery. This adds to the list of recently reported sources available at the nanoscale, such as coherence, entanglement, and quantum measurements. We elucidate this concept by showing analytic and numerical studies of a two-qubit quantum battery that is alternatively charged by an incoherent heat flow and discharged by application of a properly chosen unitary gate. The presence of a NESS for the charging step guarantees steady operation with positive power output. Decreasing the duration of the charging step results in a time-periodic steady state accompanied by increased efficiency and output power. The device is amenable to implementation with different nanotechnology platforms.
Abstract We introduce a cyclic quantum battery QB model, based on an interacting bipartite system, weakly coupled to a thermal bath. The working cycle of the battery consists of four strokes: system thermalization, disconnection of subsystems, ergotropy extraction, and reconnection. The thermal bath acts as a charger in the thermalization stroke, while ergotropy extraction is possible because the ensuing thermal state is no longer passive after the disconnection stroke. Focusing on the case of two interacting qubits, we show that phase coherence, in the presence of non-trivial correlations between the qubits, can be exploited to reach working regimes with efficiency higher than 50% while providing finite ergotropy. Our protocol is illustrated through a simple and feasible circuit model of a cyclic superconducting QB. Furthermore, we simulate the considered cycle on superconducting IBM quantum machines. The good agreement between the theoretical and simulated results strongly suggests that our scheme for cyclic QBs can be successfully realized in superconducting quantum hardware.
We investigate work extraction in open quantum batteries composed of interacting spin chains weakly coupled to engineered environments. Focusing on two- and four-qubit XX models initially prepared in thermal Gibbs states, we analyze how dissipation and dephasing, acting either locally or collectively, can generate and shape ergotropy during both transient and steady-state dynamics. By introducing a continuous interpolation between parallel and collective noise channels, we systematically characterize the impact of environmental structure on work extractability. We show that purely dissipative dynamics can activate finite ergotropy from completely passive thermal states, giving rise to temperature-dependent transient regimes where hotter initial states temporarily outperform colder ones in an ergotropic Mpemba-like fashion. In contrast, collective dissipation leads to steady states whose passivity crucially depends on the initial temperature and system size, a behavior we trace back to the emergence of non-trivial dark subspaces. Finally, we demonstrate that dephasing channels suppress both transient advantages and steady-state work extraction, highlighting the qualitative difference between dissipative and dephasing environments.
This study investigates the thermodynamic behavior of a two-qubit quantum system, where each qubit is coupled to an independent thermal reservoir, either bosonic or fermionic. Using a master equation approach, we analyze both steady-state and time-dependent ergotropy to understand how different reservoir statistics affect work extraction. In bosonic environments, ergotropy consistently declines with increasing temperature due to thermal noise. In contrast, fermionic reservoirs exhibit more complex behavior, with ergotropy enhanced by particle transport under non-equilibrium conditions. Our results reveal a threshold-like sensitivity to the chemical potential configuration, leading to qualitatively distinct regimes of energy storage performance. Time-resolved analyses show that the system's approach to steady state varies depending on the type of reservoir and the coupling strength between qubits. These insights highlight how carefully engineered reservoir properties and non-equilibrium driving can be leveraged to optimize quantum battery performance.