Partial thermalization in the quantum chain of harmonic oscillators

TL;DR

This paper investigates how a quantum chain of harmonic oscillators approaches thermal equilibrium, showing that degeneracies cause fluctuations around the Gibbs distribution, which can be suppressed by lifting degeneracies, leading to thermalization.

Contribution

It demonstrates that degeneracies in the energy spectrum cause fluctuations in local states, and removing these degeneracies promotes thermalization in an integrable quantum system.

Findings

Energy propagates along the chain at phonon velocity

Degeneracies cause fluctuations around Gibbs distribution

Lifting degeneracies suppresses fluctuations and promotes thermalization

Abstract

This preprint contains the English translation of the paper "Relaxation dynamics of a quantum chain of harmonic oscillators", which was published by the author in 1980 in the Ukrainian Physical Journal. Comments describing its motivations, background ideas and results are presented as well. This paper addressed to the problem of approach to the thermal equilibrium in an isolated macroscopic quantum system, which was studied on the example of the quantum chain of weakly interacting harmonic oscillators. In the initial state, macroscopic energy was supplied to one oscillator (atom) in the chain. Subsequent evolution of the quantum state of the whole chain was determined due to the model integrability. The main subject of interest was the time evolution of the reduced density operators characterizing the quantum state of a particular atom. On the short time-scale, the energy perturbation expands along the chain with the velocity of the fastest phonon mode. On the long-time scale, the single-atom density operators display strong fluctuations around the canonical Gibbs distribution. These fluctuations are caused by the degeneracy of the energy level differences (presence of resonances) in the model of coupled harmonic oscillators. After lifting this degeneracy, fluctuations are suppressed providing, that the reduced density operator of each atom in the chain becomes very close to the Gibbs distribution at almost any time moment.