Markovian master equations for quantum thermal machines: local versus global approach

Abstract

The study of quantum thermal machines, and more generally of open quantum systems, often relies on master equations. Two approaches are mainly followed. On the one hand, there is the widely used, but often criticized, local approach, where machine sub-systems locally couple to thermal baths. On the other hand, in the more established global approach, thermal baths couple to global degrees of freedom of the machine. There has been debate as to which of these two conceptually different approaches should be used in situations out of thermal equilibrium. Here we compare the local and global approaches against an exact solution for a particular class of thermal machines. We consider thermodynamically relevant observables, such as heat currents, as well as the quantum state of the machine. Our results show that the use of a local master equation is generally well justified. In particular, for weak inter-system coupling, the local approach agrees with the exact solution, whereas the global approach fails for non-equilibrium situations. For intermediate coupling, the local and the global approach both agree with the exact solution and for strong coupling, the global approach is preferable. These results are backed by detailed derivations of the regimes of validity for the respective approaches.or more concretely of an electrical current [34,[42][43][44], the refrigeration of a quantum degree of freedom [45][46][47][48][49][50], the creation of entanglement [51][52][53], the determination of low temperatures [54], or the design of thermal transistors [55] and autonomous quantum clocks [56].The standard description of these systems crucially relies on Markovian master equations to predict the relevant observables, such as heat currents and power. Two main approaches are followed in the literature. The first is a local approach, where the thermal baths couple locally to sub-systems of the machine. The second is a global approach, where thermal baths couple to the global eigenmodes of the machine. As the two approaches are conceptually different, there has been considerable debate about which one should be used in order to accurately describe thermal machines, and more generally out-of-equilibrium systems. Since the global approach describes equilibrium situations accurately (see below), while the local in some cases does not, there has been incentive to use the global approach out of equilibrium as well. Furthermore, the local approach is often believed to be more phenomenological in nature [13,14,19,28,57] and it was even argued that it is unphysical in certain regimes [27,58,59].The goal of the present work is to discuss these questions in depth. We will consider a system for which the full unitary dynamics of the machine and the thermal baths can be solved exactly. This allows us to evaluate the performance of local and global master equations for the machine against the exact dynamics. In addition, we give detailed derivations of the local and the global approaches and discuss the involved approximations. Specifically, ...

Quantum current in dissipative systems

“…This is typically expressed by the thermal state not being a steady-state solution of the local ME when the temperatures of the baths are equal. This brings about thermodynamically inconsistent behavior [45][46][47][48][49] such as non-zero particle or heat flow between two thermal baths at equal temperatures [46], or, when the temperatures are different, spontaneous heat flow against the temperature gradient [47]. Figure 1.…”

“…We conclude this subsection by emphasizing that caution must be maintained when using the global ME for computing the heat flow for small K. As discussed above, the secular approximation, indispensable for the applicability of the global ME, is compromised when K approaches 0. Given the results in [48,49] for the Caldeira-Leggett model, it is reasonable to expect that the local ME would provide a more reliable description in that regime. This issue can be decisively settled only upon solving the global, system-plus-baths dynamics in the limit of infinitely large baths, doing which, however, appears to be unfeasible [28].…”

Work production of quantum rotor engines

“…Note that the intrinsic work expressions would deviate more in the strong coupling regime, G ? κ. Backaction would then enter the game, as the local master equation (2) would no longer be valid and a global master equation should be used instead [41][42][43].…”

Non-additive dissipation in open quantum networks out of equilibrium

“…showing that local additive Lindblad models may perform well in non-equilibrium scenarios when the network nodes are nearly degenerate [52,53], but fail for large detunings [20].…”