Highly Controllable and Robust 2D Spin-Orbit Coupling for Quantum Gases

We report the realization of a robust and highly controllable two-dimensional (2D) spin-orbit (SO) coupling with a topological nontrivial band structure. By applying a retro-reflected 2D optical lattice, phase tunable Raman couplings are formed into the antisymmetric Raman lattice structure, and generate the 2D SO coupling with precise inversion and $C_4$ symmetries, leading to considerably enlarged topological regions. The lifetime of the 2D SO coupled Bose-Einstein condensate reaches several seconds, which enables exploring fine-tuning interaction effects. These essential advantages of the present new realization open the door to explore exotic quantum many-body effects and nonequilibrium dynamics with novel topology.

Figure 1

(a) Experimental setup. The $\mathbf{B}$ field along the $z$ axis generates the Zeeman splitting and the quantization axis of the atoms. The red and blue lines are lasers to construct the 2D lattice and Raman couplings. The $\lambda /2$ wave plates are used to generate two orthogonal polarization components of the lattices. The $\lambda /4$ wave plates are phase retarders, which are applied to form the antisymmetric Raman coupling lattices. The EOM is applied to tune the relative phase $\delta \phi$ between two Raman couplings ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. Insets: level structure and Raman coupling scheme. (b) The antisymmetric structure of the two Raman couplings ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ in real space. The grid represent the square optical lattice $V_{\mathrm{latt}}(x,y)$.

Figure 2

(a) TOF image of the SO coupled BEC with various phases $\delta \phi$. The images with $\delta \phi =-\pi ,0$ correspond to 1D SO coupling, and those with $\delta \phi =\pm \pi /2$ are for symmetric 2D SO coupling. (b) $W$ versus $\delta \phi$. The blue circles are experimental data, with the error bar for the standard statistical error. The red solid line is the theoretical curve without fitting. (c) The measured life time $\tau$ of 2D SO coupled BEC with $\delta \phi =\pi /2$. By fitting with an exponential decay, we obtain the lifetime as $\tau =2.52\pm 0.07\mathrm{s}$ (red line), $2.00\pm 0.09\mathrm{s}$ (green line), and $0.980\pm 0.05\mathrm{s}$ (blue line).

Figure 3

Phase transition of the 2D SO coupled BEC between the ST and the MG phase. (a) Magnetization histograms with varying ${\mathrm{\Omega }}_0$ for nearly pure condensate. (b) $\sqrt{⟨M_0^2⟩}$ as a function of ${\mathrm{\Omega }}_0$. The error bars give the standard statistical errors. The red solid line is an arbitrary fit. The dashed line shows the theoretical calculation of phase transition point.

Figure 4

Topological phase diagram of the lowest band. The regions I and IV are trivial with the Chern number ${\mathcal{C}}_1^{(0)}=0$. The areas II and III are topological and nontrivial with ${\mathcal{C}}_1^{(0)}=\pm 1$, respectively. The black solid lines are the topological phase boundaries in the present scheme, and the gray dash-dot lines are that of asymmetric Raman coupling in Ref. [39]. The circles with statistical error bars are the phase boundaries measured in the experiment. (a) $\delta -V_0$ plane with ${\mathrm{\Omega }}_0=0.5E_{\mathrm{r}}$. (b) $\delta -{\mathrm{\Omega }}_0$ plane with $V_0=4.0E_{\mathrm{r}}$. The measurements deviate from the calculation at small $V_0$ and large ${\mathrm{\Omega }}_0$, due to the higher band and heating effects. Insets: by considering the thermal effect and removing the high-band contribution from the experimental data, the phase boundaries are found to be fit to the theoretical calculation.