Band structure, elementary excitations, and stability of a Bose-Einstein condensate in a periodic potential

TL;DR

This paper analyzes the band structure and stability of a Bose-Einstein condensate in a one-dimensional periodic potential, revealing the presence of swallow tail loops and their physical interpretation, along with stability criteria under various conditions.

Contribution

It provides a detailed analysis of the band structure, including the occurrence of swallow tail loops, and investigates the stability of solutions using both Gross-Pitaevskii and hydrodynamic approaches.

Findings

Swallow tail loops occur at zone boundaries and centers.

Stability regions depend on interaction strength and potential.

Hydrodynamic approach yields long-wavelength stability criteria.

Abstract

We investigate the band structure of a Bose-Einstein condensate in a one-dimensional periodic potential by calculating stationary solutions of the Gross-Pitaevskii equation which have the form of Bloch waves. We demonstrate that loops ("swallow tails") in the band structure occur both at the Brillouin zone boundary and at the center of the zone, and they are therefore a generic feature. A physical interpretation of the swallow tails in terms of periodic solitons is given. The linear stability of the solutions is investigated as a function of the strength of the mean-field interaction, the magnitude of the periodic potential, and the wave vector of the condensate. The regions of energetic and dynamical stability are identified by considering the behavior of the Gross-Pitaevskii energy functional for small deviations of the condensate wave function from a stationary state. It is also shown how for long-wavelength disturbances the stability criteria may be obtained within a hydrodynamic approach.