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

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.