Time correlation functions for mixed quantum-classical systems

Abstract

We consider the time correlation function of observables pertaining to a (quantum subsystem +bath), where the bath is coupled to a reservoir with many degrees of freedom. Integrating over the coordinates of this reservoir and assuming no initial correlations between the (quantum subsystem+bath) and the reservoir, we obtain an expression for the time correlation function that contains an influence functional. We then take the semiclassical and Fokker–Planck limits while modeling the reservoir with an Ohmic continuum of harmonic oscillators coupled bilinearily to the coordinates of the bath. The semiclassical limit is taken using a variant of Pechukas’ stationary phase analysis of the reduced propagator that yields a time correlation function written in terms of connected “classical” paths. These paths are got by solving the concatenation of several short-time interval Pechukas equations; as a result, the determination of these paths is more feasible than the determination of the “classical” path associated with a single long-time interval Pechukas equation. This concatenation includes the dissipative and stochastic forces associated with a classical Brownian particle. We then use decoherence arguments derived from an inspection of the influence functional to eliminate the phase interference structure of the bath. This elimination yields a mixed quantum-classical time correlation function that can be evaluated using nonadiabatic mixed quantum-classical dynamics schemes similar to those proposed recently by Webster and Tully.