Extension of Bogoliubov theory to quasi-condensates

TL;DR

This paper extends Bogoliubov theory to low-dimensional Bose gases, providing a rigorous framework for calculating physical properties without divergences, applicable in one, two, and three dimensions.

Contribution

It introduces a density-phase based extension of Bogoliubov theory that handles low-dimensional degenerate Bose gases with weak interactions and low density fluctuations.

Findings

Exact expressions for the equation of state and ground state energy.

Calculation of first and second order correlation functions.

Method applicable in 1D, 2D, and 3D systems without divergences.

Abstract

We present an extension of the well-known Bogoliubov theory to treat low dimensional degenerate Bose gases in the limit of weak interactions and low density fluctuations. We use a density-phase representation and show that a precise definition of the phase operator requires a space discretisation in cells of size $l$. We perform a systematic expansion of the Hamiltonian in terms of two small parameters, the relative density fluctuations inside a cell and the phase change over a cell. The resulting macroscopic observables can be computed in one, two and three dimensions with no ultraviolet or infrared divergence. Furthermore this approach exactly matches Bogoliubov's approach when there is a true condensate. We give the resulting expressions for the equation of state of the gas, the ground state energy, the first order and second order correlations functions of the field. Explicit calculations are done for homogeneous systems.