Exact nonlinear Bloch-state solutions for Bose-Einstein condensates in a periodic array of quantum wells

A set of exact closed-form Bloch-state solutions to the stationary Gross-Pitaevskii equation are obtained for a Bose-Einstein condensate in a one-dimensional periodic array of quantum wells, i.e. a square-well periodic potential. We use these exact solutions to comprehensively study the Bloch band, the compressibility, effective mass and the speed of sound as functions of both the potential depth and interatomic interaction. According to our study, a periodic array of quantum wells is more analytically tractable than the sinusoidal potential and allows an easier experimental realization than the Kr\"onig-Penney potential, therefore providing a useful theoretical model for understanding Bose-Einstein condensates in a periodic potential.