Experimental Observation of a Topological Band Gap Opening in Ultracold Fermi Gases with Two-Dimensional Spin-Orbit Coupling

The recent experimental realization of synthetic spin-orbit coupling (SOC) opens a new avenue for exploring novel quantum states with ultracold atoms. However, in experiments for generating two-dimensional SOC (e.g., Rashba type), a perpendicular Zeeman field, which opens a band gap at the Dirac point and induces many topological phenomena, is still lacking. Here, we theoretically propose and experimentally realize a simple scheme for generating two-dimensional SOC and a perpendicular Zeeman field simultaneously in ultracold Fermi gases by tuning the polarization of three Raman lasers that couple three hyperfine ground states of atoms. The resulting band gap opening at the Dirac point is probed using spin injection radio-frequency spectroscopy. Our observation may pave the way for exploring topological transport and topological superfluids with exotic Majorana and Weyl fermion excitations in ultracold atoms.

Figure 1

Scheme of the atom-light interaction configuration for generating 2D synthetic SOC and an effective perpendicular Zeeman field simultaneously. (a) Schematic of energy levels of ${}^{40}\mathrm{K}$ and Raman transitions. Each of the three Raman lasers dresses one hyperfine ground state. (b) The experimental geometry and laser configuration. The Raman lasers 1-3 are initially prepared with the linear polarization along the $z$, $x$, and $y$ directions, respectively. Then the Raman lasers 1 and 2 pass through a $\lambda /2$ wave plate, which rotates their linear polarizations by the same angle $\theta$ synchronously. Cases I and II correspond to without and with the $\lambda /4$ wave plate after the $\lambda /2$ wave plate for the Raman laser 1.

Figure 2

The energy dispersions of dressed atoms measured by rf spin-injection spectroscopy. Columns (a1)–(a3) and (b1)–(b3) correspond to the energy-momentum dispersions of 2D SOC without (case I) and with (case II) effective perpendicular Zeeman field, respectively. The experimental parameters are ${\mathrm{\Omega }}_{12}=-4.97E_r$, ${\mathrm{\Omega }}_{13}=5.46E_r$, ${\mathrm{\Omega }}_{23}=6.46E_r$, ${\delta }_2=-0.5E_r$, ${\delta }_3=-1.8E_r$ and $\theta =45°$. (a1) and (b1) are theoretical results calculated using the Hamiltonian (1) with the experimental parameters. (a2) and (b2) are experimental results measured by rf spin-injection spectroscopy. The black dots represent the experimental data. The yellow circles in (a1) and (a2) indicate the Dirac points. In both rows, we only show the lowest two bands for better visualization of the Dirac points and the band gap opening. (a3) The cross-section drawings of (a1) and (a2) in the energy-$p_y$ coordinates for $p_x=0.45k_r$. Triangles are from experimental data. Solid and dashed lines are from theoretical calculations using the full [Eq. (1)] and effective [Eq. (2)] Hamiltonians, respectively. (b3) The cross-section drawings of (b1) and (b2).

Figure 3

Measurement of the Raman coupling strength by Rabi oscillations between two hyperfine ground states. Plot of the Raman coupling strengths ${\mathrm{\Omega }}_{12}^i$ (a) and ${\mathrm{\Omega }}_{13}^i$ (b) versus $\theta$. The green squares and purple diamonds correspond to cases I and II, respectively. NRC represents normalized Raman coupling. The solid and dashed lines are theoretical curves.

Figure 4

Tunable band gap at the Dirac point induced by the perpendicular Zeeman field. (a),(b) Plot of the energy differences between the lowest two bands in experiments. The stars and dots correspond to the Dirac point positions in experiments and theory, respectively. (c),(d) The band gap (c) and the position of the Dirac point (d) are plotted as the function of $\theta$. In both figures, the squares and diamonds represent experimental data. The solid and dashed lines represent theoretical calculations using the $3\times 3$ Hamiltonian (1) and the spin-half effective Hamiltonian (2), respectively. The experimental parameters are the same as Fig. 2.