## The phase diagrams of ^{87}Rb (a) and ^{23}Na (b)

**Figure 1.** The phase diagrams of ^{87}Rb (a) and ^{23}Na (b). The two pseudo-spin states are *m* = 0, −1 states of *F* = 1 hyperfine states. On the right of the red dashed line, the single-particle spectrum has one local minimum, and there is only one plane-wave phase (PW3). On the left of the red dashed line, the single-particle spectrum has two local minima, and the horizontal blue line separates two plane-wave condensates at different momenta (PW1 and PW2). Within the region rounded by two blue lines it is the stripe phase, in which the condensate coherently occupies two different momenta. Here E_{r}=2 \pi \times 2.2\ \mathrm{kHz} and 2 \pi \times 8.35\ \mathrm{kHz} for ^{87}Rb and ^{23}Na, respectively.

**Abstract**

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 leads to the non-monotonic behaviour of the condensate depletion, the Lee–Huang–Yang correction of ground-state energy and the transition temperature of a non-interacting Bose–Einstein condensate (BEC). 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 the SO coupling, the interactions modify the BEC transition temperature even at the Hartree–Fock level, in contrast to the ordinary Bose gas without the SO coupling. All results presented here can be directly verified in the current cold atom experiments using the Raman laser-induced gauge field.