Plane and Stripe Wave Phases of a Spin-Orbit Coupled Bose-Einstein Condensate in an Optical Lattice with a Zeeman Field

Abstract: A weakly interacting, spin-orbit coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane and stripe wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane wave phase, while it is a poor approximation in the stripe wave phase. While the plane wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.