Quantum Numbers for Excitations of Bose-Einstein Condensates in 1D Optical Lattices

The excitation spectrum and the band structure of a Bose-Einstein condensate in a periodic potential are investigated. Analyses within full 3D systems, finite 1D systems, and ideal periodic 1D systems are compared. We find two branches of excitations in the spectra of the finite 1D model. The band structures for the first and (part of) the second band are compared between a finite 1D and the fully periodic 1D systems, utilizing a new definition of a effective wavenumber and a phase-slip number. The upper and lower edges of the first gap coincide well between the two cases. The remaining difference is explained by the existence of the two branches due to the finite-size effect.