Phase-modulated superfluids of bosons in spin-orbit coupled optical lattice

We study the phase diagram of spin-orbit coupled ultra-cold bosons in a square lattice using the Gutzwiller method. In the superfluid regime, we show that the interplay between spin independent and spin-dependent tunnelings may give rise to a few different types of phase-modulated superfluids. The transitions between different superfluids are found to be the first-order. We investigate the rich~periodic structure of the phases of the superfluids, which may be directly probed using the spin structure factor. Different types of superfluids may also possess different excitation spectra.