Reservoir effects on the temperature dependence of the relaxation to equilibrium of three simple quantum systems

The approach to thermal equilibrium of each of three simple quantum systems in interaction with a reservoir is analyzed by calculating the time evolution of an observable appropriate for each system. Two types of interaction with the reservoir are considered: a single-phonon modulation of the interaction matrix element and a multiphonon interaction arising from a polaronic transformation for a given single-phonon, but strong, modulation of energy or frequency. The methodology employed is a recent formalism based on a coarse-grained generalized master equation. Interesting results are obtained for the multiphonon case including a nonmonotonic dependence of the time-dependent observables in the multiphonon system as the temperature is varied. Such a result does not appear in the single-phonon case, i.e., for weak coupling. In addition to contributing towards the understanding of the detail in the approach to thermal equilibrium, the analysis has practical applications to the vibrational relaxation of molecules embedded in phonon baths and to the transport of charge in crystals subjected to electric fields strong enough to lead to the formation of Stark ladders.