Bloch function description of a Bose-Einstein condensate in a finite optical lattice

School of Mathematics, Physics, Computing and Electronics, Macquarie University, North Ryde, New South Wales 2109,Australia.(February 1, 2008)We consider stationary and propagating solutions for a Bose-Einstein condensate in a periodicoptical potential with an additional confining optical or magnetic potential. Using an effectivemass approximation we express the condensate wavefunction in terms of slowly-varying envelopesmodulating the Bloch modes of the optical lattice. In the limit of a weak nonlinearity, we derive anonlinear Schr¨odinger equation for propagation of the envelope function which does not contain therapid oscillation of the lattice. We then consider the ground state solutions in detail in the regimeof weak, moderate and strong nonlinear interactions. We describe the form of solution which isappropriate in each regime, and place careful limits on the validity of each type of solution.I. INTRODUCTION