Testing the validity of the local and global GKLS master equations on an exactly solvable model

TL;DR

This paper compares local and global GKLS master equations in a solvable quantum wire model, showing that each is valid in different regimes and highlighting their complementary nature for modeling open quantum systems.

Contribution

It provides a detailed benchmark of local and global master equations against exact solutions, clarifying their regimes of validity and limitations in open quantum system modeling.

Findings

LME predicts the correct steady state and heat currents in certain regimes.

GME fails when the secular approximation breaks down.

LME and GME are complementary tools suited for different parameter regimes.

Abstract

When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed \textit{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 \textit{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 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 \textit{complementary} tools, best suited to different parameter regimes.