## (a) *T*_{c} of ^{87}Rb and non-interacting Bose gases in the harmonic trap

**Figure 9.** (a) *T*_{c} of ^{87}Rb and non-interacting Bose gases in the harmonic trap. (b) Shift of *T*_{c} of ^{87}Rb due to the interaction as a function of Ω/*E _{r}*. The non-interacting Bose gases are assumed to have the same atomic mass as

^{87}Rb. Here E_{r}=2\pi \times 2.2\ \mathrm{kHz}. The particle number is

*N*= 2.5

**×**10

^{5}, and the trapping frequency is \omega =2\pi \times 50\ \mathrm{Hz}.

**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.