Extension of Bogoliubov theory to quasicondensates

TLDR

The paper extends Bogoliubov theory to low‑dimensional degenerate Bose gases under weak interactions and low density fluctuations. The authors use a density‑phase representation with spatial discretization into cells of size ℓ and perform a systematic expansion of the Hamiltonian in the small parameters of intra‑cell density fluctuations and inter‑cell phase changes. The extended theory yields finite, divergence‑free expressions for macroscopic observables—including the equation of state, ground‑state energy, and first‑ and second‑order correlation functions—in one, two, and three dimensions, matching standard Bogoliubov results when a true condensate exists, and is illustrated for homogeneous systems.

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 discretization 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, and the first order and second order correlation functions of the field. Explicit calculations are done for homogeneous systems.