Low-temperature and long-time anomalies of a damped quantum particle

Abstract

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI(ω) ∞ω ^α at low frequencies, where α > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on α. For α > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for α < 1, an initially localized state remains localized for all times. For α ≥ 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for α ≤ 2 and at zero temperature for 1 ≤ α ≤ 2.