Collective excitations of a periodic Bose condensate in the Wannier representation

We study the dispersion relation of the excitations of a dilute Bose-Einste in condensate confined in a periodic optical potential and its Bloch oscill ations in an accelerated frame. The problem is reduced to one-dimensionalit y through a renormalization of the s-wave scattering length and the solutio n of the Bogolubov-de Gennes equations is formulated in terms of the approp riate Wannier functions. Some exact properties of a periodic one-dimensiona l condensate are easily demonstrated: (i) the lowest band at positive energ y refers to phase modulations of the condensate and has a linear dispersion relation near the Brillouin zone centre; (ii) the higher bands arise from the superposition of localized excitations with definite phase relationship s; and (iii) the wavenumber-dependent current under a constant force in the semiclassical transport regime vanishes at the zone boundaries. Early resu lts by Slater [Phys. Rev. 87, 807 (1952)] on a soluble problem in electron energy bands are used to specify the conditions under which the Wannier fun ctions may be approximated by an on-site tight-binding orbitals of harmonic oscillator form. In this approximation the connections between the low-lyi ng excitations in a lattice and those in a harmonic well are easily visuali zed. A analytical results are obtained in the tight-binding scheme and are illustrated with simple numerical calculations for the dispersion relation and semiclassical transport in the lowest energy band, at values of the sys tem parameters which are relevant to experiment.