Aging effects in free quantum Brownian motion

The two-time correlation function Cxx(t,t′) of the displacement x(t)−x(t0) of a free quantum Brownian particle with respect to its position at a given time t0 is calculated analytically in the framework of the Caldeira and Leggett ohmic dissipation model. As a result, at any temperature T,Cxx(t,t′) exhibits aging, i.e. it depends explicitly on both times t and t′ and not only on the time difference τ=t−t′, even in the limit of large age t′(t0⩽t′⩽t), 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 response function χxx(t,t′) to Cxx(t,t′), since this latter quantity takes into account even those fluctuations of the displacement which take place during the waiting time tw=t′−t0. We describe the deviation from QFDT in terms of an effective inverse temperature βeff(τ,tw). The behaviour of this quantity as a function of τ for given values of T and tw is analysed. In the classical limit it is shown that βeff(τ,tw)=βD(τ)/[D(τ)+D(tw)], where D(t) denotes the time-dependent diffusion coefficient.