Re-examining the quadratic approximation in theory of a weakly interacting Bose gas with condensate: the role of nonlocal interaction potentials

TL;DR

This paper re-examines the quadratic approximation in the Bogoliubov model for a weakly interacting Bose gas, highlighting the importance of nonlocal interaction potentials and their effects on the existence of solutions and excitation spectra.

Contribution

It provides a detailed analysis of the quadratic approximation equations for various interaction potentials, emphasizing the role of nonlocal potentials in obtaining solutions and understanding excitation gaps.

Findings

No solutions for local potentials, but solutions exist for nonlocal potentials.

Quadratic terms can significantly affect the chemical potential near experimental conditions.

The excitation spectrum can develop a gap due to quadratic contributions.

Abstract

We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its $c$-number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.