  The ground state of a spin-orbit-coupled Bose gas in a one-dimensional optical lattice is known to exhibit a mixed regime, where the condensate wave function is given by a superposition of multiple Bloch-wave components, and an unmixed one, in which the atoms occupy a single Bloch state. The unmixed regime features two unpolarized Bloch-wave phases, having quasimomentum at the center or at the edge of the first Brillouin zone, and a polarized Bloch-wave phase at intermediate quasimomenta. By calculating the critical values of the Raman coupling and of the lattice strength at the transitions among the various phases, we show the existence of a tricritical point where the mixed, the polarized and the edge-quasimomentum phases meet, and whose appearance is a consequence of the spin-dependent interaction. Furthermore, we evaluate the excitation spectrum in the unmixed regime and we characterize the behavior of the phonon and the roton modes, pointing out the instabilities occurring when a phase transition is approached. 
Quantum phases and collective excitations of a spin-orbit-coupled   Bose-Einstein condensate in a one-dimensional optical lattice