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

From non equilibrium quantum Brownian motion to impurity dynamics in 1D quantum liquids