Application of the time-dependent projection operator technique: The nonlinear quantum master equation

TLDR: In this article, the authors apply the time-dependent projection operator approach to a small system coupled to a reservoir and derive a closed nonlinear equation of motion for the reduced statistical operator of the small system.

Abstract: The time-dependent projection operator technique according to Willis and Picard (Phys. Rev. A 9 (1974) 1343) offers a unique quantum statistical description of two interacting subsystems. The technique is used here to go beyond the standard quantum master equation (QME) for a small system coupled to a reservoir. Applying the time-dependent projection operator approach, one is able to overcome a perturbational treatment of the system–reservoir coupling and can incorporate, (i) how the dynamics of the system may drive the reservoir out of the equilibrium, and (ii) how this nonequilibrium state reacts back on the system dynamics. The case of a reservoir of harmonic oscillators coupled linearly by its coordinates to the small system is studied in detail. The derivation of a closed nonlinear equation of motion for the reduced statistical operator of the small system is demonstrated. The method is used to describe the motion of a quantum particle in a molecular system, e.g. a tunneling electron or an exciton, which interacts strongly with its environment.