Aging effects in free quantum Brownian motion

Abstract

The two-time correlation function of the displacement of a free quantum Brownian particle with respect to its position at a given time is calculated analytically in the framework of the Caldeira and Leggett ohmic dissipation model (linear coupling of the particle with a thermal bath with a continuum of modes, proportional to the frequency at low frequency). As a result, at any temperature, the displacement correlation function exhibits aging, i.e. it depends explicitly on both times involved and not only on their difference, even in the limit of large age (or waiting time), in contrast with a dynamic variable in equilibrium such as the particle velocity. The equilibrium quantum fluctuation-dissipation theorem (QFDT) has to be modified in order to relate the displacement response and correlation functions, since this latter quantity takes into account even those fluctuations of the displacement which take place during the waiting time. We describe the deviation from the equilibrium QFDT in terms of an effective temperature which depends both on the time difference and of the waiting time. The definition of this quantity heavily relies on the time-dependent diffusion coefficient. The behavior of the effective temperature as a function of the time difference for given values of the bath temperature and of the waiting time is analyzed.