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The Journal of Chemical Physics

Simple model for realizing coherent ergotropy in open quantum systems

Guo-hao Xu, Jiarui Zeng, Yao Yao

Published Apr 1, 2026• 0 citations
Publisher

Abstract

The advantage of quantum heat engines turns out to be the ability to utilize the quantum resources, such as the quantum coherence, which is, however, suppressed by the irreducible thermal dissipation of baths. Herein, we construct a two-qubit system that is coupled in different directions to the thermal bath with a single pseudomode, which can be regarded as a simple model for the establishment of the coherent contribution of ergotropy in the non-Markovian environment. It is found that the presence of coupling anisotropy gives rise to the long-standing quantum coherence that is essential for the coherent ergotropy, while with merely isotropic coupling, the incoherent contribution of ergotropy dominates. The stronger the anisotropic couplings, the greater the coherent ergotropy, and a completely coherent ergotropy can be observed in both antiferromagnetic and ferromagnetic cases. In addition, this significant ability to extract work from quantum coherence can be extended to multiple-qubit systems. Our model, therefore, provides a proof-of-principle pathway for the utilization of quantum coherence in heat engines.

DOI
10.1063/5.0325259

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The advantage of quantum heat engines turns out to be the ability to utilize the quantum resources, such as the quantum coherence, which is, however, suppressed by the irreducible thermal dissipation of baths. Herein, we construct a two-qubit system that is coupled in different directions to the thermal bath with a single pseudomode, which can be regarded as a simple model for the establishment of the coherent contribution of ergotropy in the non-Markovian environment. It is found that the presence of coupling anisotropy gives rise to the long-standing quantum coherence that is essential for the coherent ergotropy, while with merely isotropic coupling, the incoherent contribution of ergotropy dominates. The stronger the anisotropic couplings, the greater the coherent ergotropy, and a completely coherent ergotropy can be observed in both antiferromagnetic and ferromagnetic cases. In addition, this significant ability to extract work from quantum coherence can be extended to multiple-qubit systems. Our model, therefore, provides a proof-of-principle pathway for the utilization of quantum coherence in heat engines.
We show that it is possible to have nonzero ergotropy in the steady states of an open quantum system consisting of qubits that are collectively coupled to a thermal bath at a finite temperature. The dynamics of our model leads the qubits into a steady state that has coherences in the energy eigenbasis when the system consists of more than a single qubit. We observe that even though the system does not have inverted populations, it is possible to extract work from the coherences and we analytically show that in the high-temperature limit, ergotropy per unit energy is equal to the l_{1} norm of coherence for the two-qubit case. Further, we analyze the scaling of coherence and ergotropy as a function of the number of qubits in the system for different initial states. Our results demonstrate that one can design a quantum battery that is charged by a dissipative thermal bath in the weak-coupling regime.
This paper explores quantum heat engines based on qubit and qutrit working media interacting with thermal environments through generalized amplitude damping (GAD) channels. We investigate how quantum channels can be employed to model heat absorption, dissipation, and work extraction in open quantum thermal machines, and derive the conditions required for positive work extraction. The effects of quantum correlations, emission probability, population redistribution, and system--environment interactions on the thermodynamic performance of the engine are systematically analyzed across different operational regimes. In addition, we examine the ergotropy of qubit and qutrit systems under dissipative dynamics to understand how environmental effects influence the maximum extractable work. Our results demonstrate that multilevel quantum systems exhibit enhanced work extraction capability and improved robustness against decoherence compared to two-level systems, providing further insight into the role of dissipative dynamics and quantum resources in realistic quantum thermodynamic devices.
We study the ergotropy and the thermodynamic potentials of a qubit battery charged by repeated interactions with clusters of thermal ancillas, qubits or oscillators, through a composite system-bath coupling that generates steady-state coherence in the energy basis. A single parameter encoding the bath quantum statistics yields the coherence and ergotropy in closed form for any cluster size. For large clusters, the ergotropy becomes independent of the bath statistics and, at optimal couplings, reaches a universal fraction of the equilibrium internal energy of the qubit at the bath temperature. Since the charged state has no population inversion, all extractable work comes from coherence. The ergotropy is flat at low temperatures. We trace this plateau to the third law, and show that the ergotropy is proportional to the equilibrium enthalpy of the qubit, whose slope vanishes with the heat capacity. For a battery composed of multiple qubits, the coherence grows exponentially with the number of qubits while the ergotropy grows linearly, both remaining a nearly constant fraction of their respective maxima. The charged state is a non-equilibrium steady state with finite entropy at zero temperature, sustained by the work of switching the collisional interaction on and off. The cost is nonzero whenever the parallel coupling is nonzero, and, unlike the ergotropy, it depends on the bath statistics.
We theoretically identify the noise-induced coherent contribution to the ergotropy of a four-level quantum heat engine coupled to a unimodal quantum cavity. We utilize a protocol where the passive state's quasiprobabilities can be analytically identified from the population-coherence coupled reduced density matrix. The reduced density matrix elements are evaluated using a microscopic quantum master equation formalism. Multiple ergotropies within the same coherence interval, each characterized by a positive and pronounced coherent contribution, are observed. These ergotropies are a result of population inversion as well as quasiprobability-population inversion, controllable through the coherence measure parameters. The optimal flux and power of the engine are found to be at moderate values of ergotropy with increasing values of noise-induced coherence. The optimal power at different coherences is found to possess a constant ergotropy.
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.
Constraints on work extraction are fundamental to our operational understanding of the thermodynamics of both classical and quantum systems. In the quantum setting, finite-time control operations typically generate coherence in the instantaneous energy eigenbasis of the dynamical system. Thermodynamic cycles can, in principle, be designed to extract work from this nonequilibrium resource. Here, we isolate and study the quantum coherent component to the work yield in such protocols. Specifically, we identify a coherent contribution to the ergotropy (the maximum amount of unitarily extractable work via cyclical variation of Hamiltonian parameters). We show this by dividing the optimal transformation into an incoherent operation and a coherence extraction cycle. We obtain bounds for both the coherent and incoherent parts of the extractable work and discuss their saturation in specific settings. Our results are illustrated with several examples, including finite-dimensional systems and bosonic Gaussian states that describe recent experiments on quantum heat engines with a quantized load.