Quantum diffusion in dissipative systems

The motion of a quantum-mechanical particle driven by a random fluctuating force is studied. Dissipation is introduced by coupling the particle to a continuous distribution of harmonic oscillators. A general expression for the mean-square displacement 〈$x^2$(t)〉 is derived. It is shown that for long times t, one gets a crossover from a nondiffusive behavior 〈$x^2$(t)〉∝$t^3$ for the dissipationless case to a diffusive one 〈$x^2$(t)〉∝t when dissipation is introduced. The case where the random driving force vanishes is also examined. Here the long-time behavior depends on the initial conditions of the oscillators and one gets 〈$x^2$(t)〉∝lnt if they are initially in their ground state (i.e., at zero temperature) and 〈$x^2$(t)〉∝Tt if they are initially at a finite temperature T.