Kraus map for non-Markovian quantum dynamics driven by a thermal reservoir

Starting from unitary dynamics we study the evolution in time of a non-relativistic quantum system that exchanges energy with a thermal reservoir of harmonic oscillators. System and reservoir are assumed to be initially decorrelated. Reservoir correlation functions are factorized by means of a Kraus compliant version of Wick's theorem. As a result, the non-Markovian Kraus map for the system density operator can be completely expressed in terms of system potentials and reservoir pair correlation functions. An infinite hierarchy for the evolution operators of the Kraus map is derived. The system density operator is obtained as a time-ordered exponential containing a non-Markovian counterpart of the standard Markovian generator for dissipative dynamics. We specify a condition on this non-Markovian generator for return to thermal equilibrium. We set up a non-Markovian perturbation theory that preserves both trace and positivity of the system density operator.