Purity Temperature Dependent for Coupled Harmonic Oscillators

TL;DR

This paper investigates how the purity of a system of two coupled harmonic oscillators varies with temperature, using path integral methods to analyze thermal equilibrium states and their dependence on physical parameters.

Contribution

It introduces a path integral approach to explicitly calculate the purity of coupled oscillators at thermal equilibrium, exploring temperature effects and parameter dependencies.

Findings

Purity decreases with increasing temperature.

Explicit formulas for purity in terms of system parameters.

Numerical results illustrating parameter influences.

Abstract

We consider the thermal aspect of a system composed of two coupled harmonic oscillators and study the corresponding purity. We initially consider a situation where the system is brought to a canonical thermal equilibrium with a heat-bath at temperature $T$. We adopt the path integral approach and introduce the evolution operator to calculate the density matrix and subsequently the reduced matrix density. It is used to explicitly determine the purity in terms of different physical quantities and therefore study some limiting cases related to temperature as well as other parameters. Different numerical results are reported and discussed in terms of the involved parameters of our system.