Energy and coherence loss rates in a one-dimensional vibrational system interacting with a bath

We consider the problem of the dynamics of a Gaussian wave packet in a one-dimensional harmonic ocsillator interacting with a bath. This problem arises in many chemical and biochemical applications related to the dynamics of chemical reactions. We take the bath-oscillator interaction into account in the framework of the Redfield theory. We obtain closed expressions for Redfield-tensor elements, which allows finding the explicit time dependence of the average vibrational energy. We show that the energy loss rate is temperature-independent, is the same for all wave packets, and depends only on the spectral function of the bath. We determine the degree of coherence of the vibrational motion as the trace of the density-matrix projection on a coherently moving wave packet. We find an explicit expression for the initial coherence loss rate, which depends on the wave packet width and is directly proportional to the intensity of the interaction with the bath. The minimum coherence loss rate is observed for a “coherent” Gaussian wave packet whose width corresponds to the oscillator frequency. We calculate the limiting value of the degree of coherence for large times and show that it is independent of the structural characteristics of the bath and depends only on the parameters of the wave packet and on the temperature. It is possible that residual coherence can be preserved at low temperatures.