Condensation Transition of Ultracold Bose Gases with Rashba Spin-Orbit Coupling

We study the Bose-Einstein condensate phase transition of three-dimensional ultracold bosons with isotropic Rashba spin-orbit coupling. Investigating the structure of Ginzburg-Landau free energy as a function of the condensate density, we show, within the Bogoliubov approximation, that the condensate phase transition is first order with a jump in the condensate density. We calculate the transition temperature and the jump in the condensate density at the transition for large spin-orbit coupling, where the transition temperature depends linearly on the density of particles. Finally, we discuss the feasibility of producing the phase transition experimentally.

Figure 1

Functions $\alpha (x)$ (solid line) and $\beta (x)$ (dashed line) [see Eq. (13)] for $0>x>-1$. The inset shows the behavior of the functions in the lower right corner, where $-x\gtrsim 1$.

Figure 2

The first order phase transition from the normal to condensed phase at $\Gamma =16/3$. The lines show the Ginzburg-Landau free energy, measured with respect to the free energy of the normal phase. The top line is for $T>T_c$, where $\Gamma <16/3$; the middle line is at $T_c$, where $\Gamma =16/3$; and the bottom line is for $T<T_c$, where $\Gamma >16/3$.