  Impurity motion in one dimensional ultra cold quantum liquids confined in an
optical trap has attracted much interest recently. As a step towards its full
understanding, we construct a generating functional from which we derive the
position non equilibrium correlation function of a quantum Brownian particle
with general Gaussian non-factorizing initial conditions. We investigate the
slow dynamics of a particle confined in a harmonic potential after a position
measurement; the rapid relaxation of a particle trapped in a harmonic potential
after a quantum quench realized as a sudden change in the potential parameters;
and the evolution of an impurity in contact with a one dimensional bosonic
quantum gas. We argue that such an impurity-Luttinger liquid system, that has
been recently realized experimentally, admits a simple modeling as quantum
Brownian motion in a super Ohmic bath.
