Nonlinear Band Structure in Bose Einstein Condensates: The Nonlinear Schr\"odinger Equation with a Kronig-Penney Potential

TL;DR

This paper provides an analytical study of the band structure of Bose-Einstein condensates in a one-dimensional lattice modeled by the nonlinear Schrödinger equation with a Kronig-Penney potential, revealing stable regions and superfluid properties.

Contribution

It derives closed-form solutions for Bloch states in a nonlinear lattice and demonstrates the Kronig-Penney potential's effectiveness as an analytically tractable model for BECs in optical lattices.

Findings

Analytic expressions for Bloch states in nonlinear lattices.

Identification of stable and unstable regions in the band structure.

Confirmation that Kronig-Penney potential mimics sinusoidal lattices.

Abstract

All Bloch states of the mean field of a Bose-Einstein condensate in the presence of a one dimensional lattice of impurities are presented in closed analytic form. The band structure is investigated by analyzing the stationary states of the nonlinear Schr\"odinger, or Gross-Pitaevskii, equation for both repulsive and attractive condensates. The appearance of swallowtails in the bands is examined and interpreted in terms of the condensates superfluid properties. The nonlinear stability properties of the Bloch states are described and the stable regions of the bands and swallowtails are mapped out. We find that the Kronig-Penney potential has the same properties as a sinusoidal potential; Bose-Einstein condensates are trapped in sinusoidal optical lattices. The Kronig-Penney potential has the advantage of being analytically tractable, unlike the sinusoidal potential, and, therefore, serves as a good model for experimental phenomena.