From nonequilibrium quantum Brownian motion to impurity dynamics in one-dimensional quantum liquids

Impurity motion in one-dimensional ultracold 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 nonequilibrium correlation function of a quantum Brownian particle with general Gaussian nonfactorizing 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, which has been recently realized experimentally, admits a simple modeling as quantum Brownian motion in a super-Ohmic bath.