Superfluidity and Bogoliubov excitations of a striped spin-orbit coupled BEC induced by a weak optical lattice

We investigate the ground-state superfluidity and Bogoliubov excitations of a striped spin-orbit coupled Bose-Einstein condensate (BEC) with the momentum-space hopping induced by a weak optical lattice. The ground-state phase diagram and corresponding superfluid fraction are calculated for different system parameters. The Bogoliubov spectrum is studied and an exotic avoided band crossing between roton and Goldstone modes is revealed, which is very different from traditional spin-orbit coupled Bose-Einstein condensate with double-gapless Goldstone modes. We study the quantum depletion, which can be used to characterize the phase transitions.