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arXiv:2205.10212 · Physical review. E

Thermodynamically consistent master equation based on subsystem eigenstates

Siying Wang, Qinghong Yang, Fulin Zhang

Published Jan 9, 2023• 5 citations
arXivHTMLPDF

Abstract

Master equations under appropriate assumptions are efficient tools for the study of open quantum systems. For many-body systems, subsystems of which locally couple to thermal baths and weakly interact with each other, the local approach provides a more convenient description than the global approach. However, these local master equations are believed to generate inconsistencies with the laws of thermodynamics when intersubsystem interactions exist. Here we develop an alternative local master equation by virtue of similar approximations used in deriving the traditional Gorini-Kossakowski-Lindblad-Sudarshan master equation. In particular, we stick to using eigenstates of each subsystem to construct quantum jump operators, and the secular approximation is also employed to modify the intersubsystem interactions. Our results show that violations of thermodynamic laws will be avoided after correcting intersubsystem interactions. Finally, we study a two-qubit heat transfer model and this further shows the validity of our modified master equation.

DOI
10.1103/physreve.107.014108

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Master equations under appropriate assumptions are efficient tools for the study of open quantum systems. For many-body systems, subsystems of which locally couple to thermal baths and weakly interact with each other, the local approach provides a more convenient description than the global approach. However, these local master equations are believed to generate inconsistencies with the laws of thermodynamics when intersubsystem interactions exist. Here we develop an alternative local master equation by virtue of similar approximations used in deriving the traditional Gorini-Kossakowski-Lindblad-Sudarshan master equation. In particular, we stick to using eigenstates of each subsystem to construct quantum jump operators, and the secular approximation is also employed to modify the intersubsystem interactions. Our results show that violations of thermodynamic laws will be avoided after correcting intersubsystem interactions. Finally, we study a two-qubit heat transfer model and this further shows the validity of our modified master equation.
The study of open quantum systems often relies on approximate master equations derived under the assumptions of weak coupling to the environment. However when the system is made of several interacting subsystems such a derivation is in many cases very hard. An alternative method, employed especially in the modeling of transport in mesoscopic systems, consists in using local master equations (LMEs) containing Lindblad operators acting locally only on the corresponding subsystem. It has been shown that this approach however generates inconsistencies with the laws of thermodynamics. In this paper we demonstrate that using a microscopic model of LMEs based on repeated collisions all thermodynamic inconsistencies can be resolved by correctly taking into account the breaking of global detailed balance related to the work cost of maintaining the collisions. We provide examples based on a chain of quantum harmonic oscillators whose ends are connected to thermal reservoirs at different temperatures. We prove that this system behaves precisely as a quantum heat engine or refrigerator, with properties that are fully consistent with basic thermodynamics.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account non-secular effects, but, nevertheless, has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form (in particular, preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The non-secular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
Local master equations are a widespread tool to model open quantum systems, especially in the context of many-body systems.These equations, however, are believed to lead to thermodynamic anomalies and violation of the laws of thermodynamics.In contrast, here we rigorously prove that local master equations are consistent with thermodynamics and its laws without resorting to a microscopic model, as done in previous works.In particular, we consider a quantum system in contact with multiple baths and identify the relevant contributions to the total energy, heat currents, and entropy production rate.We show that the second law of thermodynamics holds when one considers the proper expression we derive for the heat currents.We confirm the results for the quantum heat currents by using a heuristic argument that connects the quantum probability currents with the energy currents, using an analogous approach as in classical stochastic thermodynamics.We finally use our results to investigate the thermodynamic properties of a set of quantum rotors operating as thermal devices and show that a suitable design of three rotors can work as an absorption refrigerator or a thermal rectifier.For the machines considered here, we also perform an optimization of the system parameters using an algorithm of reinforcement learning.
When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed locally on each component, i.e. ignoring the coherent couplings, which are later added ‘by hand’. Although simple, the resulting local master equation (LME) is known to be thermodynamically inconsistent. Otherwise, one may always obtain a consistent global master equation (GME) by working on the energy basis of the full interacting Hamiltonian. Here, we consider a two-node ‘quantum wire’ connected to two heat baths. The stationary solution of the LME and GME are obtained and benchmarked against the exact result. Importantly, in our model, the validity of the GME is constrained by the underlying secular approximation. Whenever this breaks down (for resonant weakly-coupled nodes), we observe that the LME, in spite of being thermodynamically flawed: (a) predicts the correct steady state, (b) yields with the exact asymptotic heat currents, and (c) reliably reflects the correlations between the nodes. In contrast, the GME fails at all three tasks. Nonetheless, as the inter-node coupling grows, the LME breaks down whilst the GME becomes correct. Hence, the global and local approach may be viewed as complementary tools, best suited to different parameter regimes.
We derive a time-dependent master equation for an externally driven system, where the subsystems weakly interact with each other and are locally connected to the thermal reservoirs. The nonadiabatic equation obtained here can be viewed as a generalization of the local master equation, which has already been extensively used in describing the dynamics of a boundary-driven system. In addition, we investigate the fundamental reason underlying the thermodynamic inconsistency generated by the local and nonadiabatic master equations. We also provide a criterion to assess the consistency of these two equations with the second law of thermodynamics. Finally, we numerically confirm our results by considering a toy model consisting of two qubits and two local heat baths.