Bosons with incommensurate potential and spin-orbit coupling.

We chart out the phases of ultracold "spin-half" bosons in a one-dimensional optical lattice in the presence of Aubry-André (AA) potential and with spin-orbit (SO) and Raman couplings. We investigate the superfluid (SF) and localized phases and demonstrate the existence of density wave phase for nearest-neighbor interaction (NNI) between the bosons. We show that the presence of SO coupling and AA potential leads to a spin-split momentum distribution of the bosons in the localized phase near the boundary with the SF phase, which can act as a signature of the SF-localized phase transition. We also obtain the level statistics of the bosons in the superfluid phase with finite NNI and demonstrate its change from Gaussian unitary ensemble (GUE) to Gaussian orthogonal ensemble (GOE) as a function of the Raman coupling. We discuss experiments which can test our theory. [ABSTRACT FROM AUTHOR]