Non-Markovian reduced propagator, multiple-time correlation functions, and master equations with general initial conditions in the weak-coupling limit

Abstract

In this paper we derive the evolution equation for the reduced propagator, an object that evolves vectors of the Hilbert space of a system $\mathcal{S}$ interacting with an environment $\mathcal{B}$ in a non-Markovian way. This evolution is conditioned to certain initial and final states of the environment. Once an average over these environmental states is made, reduced propagators permit the evaluation of multiple-time correlation functions of system observables. When this average is done stochastically the reduced propagator evolves according to a stochastic Schr\"odinger equation. In addition, it is possible to obtain the evolution equations of the multiple-time correlation functions which generalize the well-known quantum regression theorem to the non-Markovian case. Here, both methods, stochastic and evolution equations, are described by assuming a weak coupling between system and environment. Finally, we show that reduced propagators can be used to obtain a master equation with general initial conditions, and not necessarily an initial vacuum state for the environment. We illustrate the theory with several examples.