Initial value formulation of a quantum damped harmonic oscillator

Abstract

The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. In particular, we find that the dynamics must include a non-vanishing noise term to yield physical results. We show that the oscillator decoheres in time such that the late-time density operator is thermal, and find the parameter regime for which the fluctuation-dissipation relation is satisfied. We next develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results are in fact non-perturbative in the initial state.