  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.
