  We consider a model of non-Markovian Quantum Brownian motion that consists of
an harmonic oscillator bilinearly coupled to a thermal bath, both via its
position and momentum operators. We derive the master equation for such a model
and we solve the equations of motion for a generic Gaussian system state. We
then investigate the resulting evolution of the first and second moments for
both an Ohmic and a super-Ohmic spectral density. In particular, we show that,
irrespective of the specific form of the spectral density, the coupling with
the momentum enhances the dissipation experienced by the system, accelerating
its relaxation to the equilibrium, as well as modifying the asymptotic state of
the dynamics. Eventually, we characterize explicitly the non-Markovianity of
the evolution, using a general criterion which relies on the positivity of the
master equation coefficients.
