Spin-Orbit Coupled Weakly Interacting Bose-Einstein Condensates in Harmonic Traps

We investigate theoretically the phase diagram of a spin-orbit coupled Bose gas in two-dimensional harmonic traps. We show that at strong spin-orbit coupling the single-particle spectrum decomposes into different manifolds separated by $\hslash {\omega }_{\perp }$, where ${\omega }_{\perp }$ is the trapping frequency. For a weakly interacting gas, quantum states with Skyrmion lattice patterns emerge spontaneously and preserve either parity symmetry or combined parity-time-reversal symmetry. These phases can be readily observed in a spin-orbit coupled gas of ${}^{87}\mathrm{Rb}$ atoms in a highly oblate trap.

Figure 1

(a) Phase diagram of a trapped 2D BEC with a strong SO coupling $\tilde{\lambda }=20$, where the single-particle spectrum forms discrete manifolds. For the weak interaction considered here, only the LM is occupied. The phases I and II preserve, respectively, the parity and parity-time-reversal symmetries. There are several subphases indicated by A, ${\mathrm{A}}^{\prime }$, and B, which differ in the density profile and/or angular momentum. The mean-field density patterns in different phases of spin-up bosons are shown in Fig. 3. (b) and (c) Phase diagram at weak SO coupling. Here the phases are determined without the restriction to the LM approximation. The insets illustrate the density profiles of the two spin components in phases IA and IIA.

Figure 2

(a) Single-particle energy spectrum. The lines show the empirical Eq. (2). (b) The $W$ function for the lowest four single-particle states in the LM.

Figure 3

Density patterns of spin-up bosons in the different ground states at three intraspecies interactions $g(N-1)/(\hslash {\omega }_{\perp }{a}_{\perp }^2)$: (a), (d) 0.02, (b), (e) 0.1, and (c), (f) 0.2.

Figure 4

Spin texture $\mathbf{S}=(1/2){\Phi }^{+}\sigma \Phi$ corresponding to the state represented in Fig. 3f. The arrows represent the transverse spin vector ($S_x$, $S_y$) with color and length representing the orientation and the magnitude of the transverse spin. The contour plot shows the axial spin $S_z=(1/2)({\varphi }_{\uparrow }^2-{\varphi }_{\downarrow }^2)$.