Out of equilibrium open quantum systems: a comparison of approximate Quantum Master Equation approaches with exact results

TL;DR

This paper compares the Redfield quantum master equation (RQME) with exact results for open quantum systems, showing RQME's accuracy in steady state and dynamics, and highlighting limitations of Lindblad equations in non-equilibrium scenarios.

Contribution

It demonstrates that the full RQME provides accurate results beyond Lindblad approximations for out-of-equilibrium quantum systems, supported by analytical and numerical comparisons.

Findings

RQME agrees with exact steady state results and numerics

Lindblad equations have limited validity in non-equilibrium

Analytical expressions for time dynamics and thermalization

Abstract

We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of non-interacting particles (bosons or fermions) on an arbitrary lattice of $N$ sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the $N=2$ case, we show that RQME gives results which agree with exact analytical results for steady state properties and with exact numerics for time-dependent properties, over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in non-equilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED and far-from-equilibrium quantum dot experiments.