Master equation for open quantum systems: Zwanzig-Nakajima projection technique and the intrinsic bath dynamics

TL;DR

This paper generalizes the Zwanzig-Nakajima projection technique to derive a non-Markovian master equation for open quantum systems, incorporating intrinsic bath dynamics and demonstrating the importance of dynamic correlations for accurate system evolution.

Contribution

It introduces a coupled chain of equations for system and bath density matrices, leading to a generalized master equation that includes nonlinear terms from bath dynamics, extending prior Markovian approaches.

Findings

The nonlinear term vanishes in the Markovian limit.

Lowest order interaction is insufficient for accurate coherence evolution.

Application to a dephasing model confirms the approach's validity.

Abstract

The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath $\varrho_{B}(t)$ and the system $\varrho_{S}(t)$ are written down. A formal solution of the equation for $\varrho_{B}(t)$, having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in $\varrho_{S}(t)$, is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of $\varrho_{B}(t)$ and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.