Revisiting the damped quantum harmonic oscillator

TL;DR

This paper reexamines the quantum damped harmonic oscillator by employing a continuum reservoir model, exact diagonalisation, and thermofield techniques, providing new insights into its relaxation behavior and equilibrium states.

Contribution

It introduces a continuum reservoir model, applies an exact diagonalisation technique, and uses thermofield methods to analyze the quantum damped harmonic oscillator at finite temperature.

Findings

Oscillator relaxes to the mean-force Gibbs state.

Identification of two distinct natural frequencies.

Reconciliation of damping effects with exact Hamiltonian diagonalisation.

Abstract

We reanalyse the quantum damped harmonic oscillator, introducing three less than common features. These are (i) the use of a continuum model of the reservoir rather than an ensemble of discrete oscillators, (ii) an exact diagonalisation of the Hamiltonian by adapting a technique pioneered by Fano, and (iii) the use of the thermofield technique for describing a finite temperature reservoir. We recover in this way a number of well-known and some, perhaps, less familiar results. An example of the latter is an ab initio proof that the oscillator relaxes to the mean-force Gibbs state. We find that special care is necessary when comparing the damped oscillator with its undamped counterpart as the former has two distinct natural frequencies, one associated with short time evolution and the other with longer times.