  In this paper we investigate the properties of Bose gases with Raman-induced
spin-orbit(SO) coupling. It is found that the SO coupling can greatly modify
the single particle density-of-state, and thus lead to non-monotonic behavior
of the condensate depletion, the Lee-Huang-Yang correction of ground-state
energy and the transition temperature of a non-interacting Bose-Einstein
condensate. The presence of the SO coupling also breaks the Galilean
invariance, and this gives two different critical velocities, corresponding to
the movement of the condensate and the impurity respectively. Finally, we show
that with SO coupling, the interactions modify the BEC transition temperature
even at Hartree-Fock level, in contrast to the ordinary Bose gas without SO
coupling. All results presented here can be directly verified in the current
cold atom experiments using Raman laser-induced gauge field.
