Limits of weak damping of a quantum harmonic oscillator

Abstract

In this Brief Report we analyze the limit of very weak damping of a quantum-mechanical Brownian oscillator. It is shown that the propagator for the reduced density operator of the oscillator can be written as a double path integral of the same form as that obtained in the high-temperature limit. As a direct consequence, we can write a Fokker-Planck equation for the reduced density operator of the weakly damped oscillator (at any temperature) involving only the damping and a generalized diffusion constant in momentum space.