Spatial and temporal coherence of a Bose-condensed gas

TL;DR

This paper investigates the temporal coherence of a three-dimensional Bose-condensed gas at finite temperature, using a number-conserving Bogoliubov approach and kinetic equations to analyze phase spreading and coherence loss over time.

Contribution

It introduces a combined theoretical framework using Bogoliubov and kinetic equations to describe phase dynamics and coherence loss in Bose-condensed gases.

Findings

Phase variance has ballistic and diffusive components with temperature-dependent coefficients.

Diffusion coefficient scales inversely with system volume.

Long-time phase variance relates to initial energy fluctuations and quantum ergodic theory.

Abstract

The central problem of this chapter is temporal coherence of a three-dimensional spatially homogeneous Bose-condensed gas, initially prepared at finite temperature and then evolving as an isolated interacting system. A first theoretical tool is a number-conserving Bogoliubov approach that allows to describe the system as a weakly interacting gas of quasi-particles. This approach naturally introduces the phase operator of the condensate: a central actor since loss of temporal coherence is governed by the spreading of the condensate phase-change. A second tool is the set of kinetic equations describing the Beliaev-Landau processes for the quasi-particles. We find that in general the variance of the condensate phase-change at long times $t$ is the sum of a ballistic term $\propto t^2$ and a diffusive term $\propto t$ with temperature and interaction dependent coefficients. In the thermodynamic limit, the diffusion coefficient scales as the inverse of the system volume. The coefficient of $t^2$ scales as the inverse volume squared times the variance of the energy of the system in the initial state and can also be obtained by a quantum ergodic theory (the so-called eigenstate thermalisation hypothesis).