The Raman dressed spin-1 spin-orbit coupled quantum gas

The recently realized spin-orbit coupled quantum gases (Y.-J Lin {\it et al}., Nature 471, 83-86 (2011); P. Wang {\it et al}., PRL 109, 095301 (2012); L. W. Cheuk {\it et al}., PRL 109, 095302 (2012)) mark a breakthrough in the cold atom community. In these experiments, two hyperfine states are selected from a hyperfine manifold to mimic a pseudospin-1/2 spin-orbit coupled system by the method of Raman dressing, which is applicable to both bosonic and fermionic gases. In this work, we show that the method used in these experiments can be generalized to create any large pseudospin spin-orbit coupled gas if more hyperfine states are coupled equally by the Raman lasers. As an example, we study in detail a quantum gas with three hyperfine states coupled by the Raman lasers, and show when the state-dependent energy shifts of the three states are comparable, triple-degenerate minima will appear at the bottom of the band dispersions, thus realizing a spin-1 spin-orbit coupled quantum gas. A novel feature of this three minima regime is that there can be two different kinds of stripe phases with different wavelengths, which has an interesting connection to the ferromagnetic and polar phases of spin-1 spinor BECs without spin-orbit coupling.