On oscillator-bath system: Exact propagator, Reduced density matrix and Green's function

TL;DR

This paper derives the exact quantum propagator, reduced density matrix, and Green's function for an oscillator coupled to a bosonic bath, providing insights into its dynamics and equilibrium properties.

Contribution

It presents the exact form of the propagator and Green's function for a general oscillator-bath system, connecting to the Feynman-Vernon influence functional.

Findings

Exact propagator and reduced density matrix derived

Green's function connecting initial and time-evolved states obtained

Equilibrium weak coupling results for position, momentum, and energy

Abstract

The exact form of quantum propagator of a quantum oscillator interacting with a bosonic bath consisting of $N$ distinguished quantum oscillators with different frequencies is obtained in the Heisenberg picture. Reduced density matrix for oscillator is obtained. The kernel or Green's function connecting the initial density matrix of the oscillator to the density matrix in an arbitrary time is obtained and its connection to Feynman-Vernon influence functional is discussed. Weak coupling regime and squared mean values for position, momentum and energy of the oscillator are obtained in equilibrium.