Relaxation Phenomena in a System of Two Harmonic Oscillators

TL;DR

This paper investigates how quantum correlations develop in a system of two harmonic oscillators under repeated interactions, revealing exact evolution to equilibrium states and differences based on oscillator frequencies.

Contribution

It provides an exact, perturbation-free analysis of the relaxation process and the nature of equilibrium states in coupled harmonic oscillators.

Findings

Equal frequencies lead to a series of Maxwell-Boltzmann distributions reaching equilibrium.

Unequal frequencies result in a non-thermal equilibrium.

The evolution is derived through an iterative, exact method without perturbation.

Abstract

We study the process by which quantum correlations are created when an interaction Hamiltonian is repeatedly applied to a system of two harmonic oscillators for some characteristic time interval. We show that, for the case where the oscillator frequencies are equal, the initial Maxwell-Boltzmann distributions of the uncoupled parts evolve to a new equilibrium Maxwell-Boltzmann distribution through a series of transient Maxwell-Boltzmann distributions. Further, we discuss why the equilibrium reached when the two oscillator frequencies are unequal, is not a thermal one. All the calculations are exact and the results are obtained through an iterative process, without using perturbation theory.