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 noninteracting 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 nonequilibrium. 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.

Figure 1

A schematic of the setup we consider. We have a system of $N$ lattice sites with noninteracting particles on them. Each site of the system is coupled to its respective bath of a chosen temperature and chemical potential. Baths out of equilibrium can facilitate transport. The top diagram is general for the bosonic or fermionic case, and such situations have been realized experimentally in setups such as those given in the middle and bottom diagrams.

Figure 2

Bosonic model: steady-state properties. (a) Particle current and (b) the occupation number in the left site as a function of intersite hopping $g$ for the two-site boson problem. RQME shows near perfect agreement with exact results from QLE for all values of $g$, while LLQME and perturbation results [Eqs. (23a) and (23b)] are valid in their respective limits. The vertical line marks the position of $g=\varepsilon$, below which LLQME is valid. The parameter ${\mathrm{\Gamma }}_{1,2}=2{\gamma }_{1,2}^2/t_B$ is related to the system-bath coupling [see Eq. (10)]. Current is measured in units of ${\omega }_0$, and all energy variables are measured in units of $\hslash {\omega }_0$.

Figure 3

Bosonic model: thermalization. We show the time evolution of occupation of the lower-energy mode ${\widehat{A}}_1$, corresponding to energy ${\omega }_1={\omega }_0-g$, in equilibrium for the two-site boson problem, starting from an empty system, for $g=0.45$. The steady state (horizontal line) corresponds to the value of Bose distribution $n\left({\omega }_1\right)={\left[e^{\beta ({\omega }_1-\mu )}-1\right]}^{-1}$ with the equilibrium bath temperatures and chemical potentials. LLQME does not show thermalization because it is not valid for $g>\varepsilon$, while exact numerical results, RQME results, and the perturbation result [Eqs. (22a)] match and show thermalization. All parameters not explicitly specified are the same as in Fig. (2). Current is measured in units of ${\omega }_0$, and all energy variables are measured in units of $\hslash {\omega }_0$. Time is measured in units of ${\omega }_0^{-1}$.

Figure 4

Bosonic model: nonequilibrium time dynamics. We show the time evolution of (a) particle current and (b) occupation number of the left site for the two-site boson problem, starting from an empty system, for $g=0.45$. RQME results show good agreement with exact numerics. Since $g>\varepsilon$, LLQME does not match, while the perturbation result [Eqs. (22a) and (22b)] matches with RQME. The horizontal line shows the exact steady-state result obtained from QLE. All parameters not explicitly specified are the same as in Fig. (2). Current is measured in units of ${\omega }_0$, and all energy variables are measured in units of $\hslash {\omega }_0$. Time is measured in units of ${\omega }_0^{-1}$.

Figure 5

Fermionic model: steady-state properties. (a) Particle current vs voltage and (b) particle current vs intersite hopping in the two-site fermion problem. The vertical lines correspond to positions where potential difference $V={\mu }_1-{\mu }_2={\omega }_{\alpha }$, where ${\omega }_{\alpha }$ are the system energy levels. RQME shows nearly perfect agreement with exact results from QLE for all values of $g$, while LLQME and perturbation results [Eqs. (23a) and (23b)] are valid in their respective limits. The graphs demonstrate the effect of conductance quantization at low temperatures. The parameter ${\mathrm{\Gamma }}_{1,2}=2{\gamma }_{1,2}^2/t_B$. Current is measured in units of ${\omega }_0$, and all energy variables are measured in units of $\hslash {\omega }_0$.