Spin-Orbit Coupling and Quantum Spin Hall Effect for Neutral Atoms without Spin Flips

We propose a scheme which realizes spin-orbit coupling and the quantum spin Hall effect for neutral atoms in optical lattices without relying on near resonant laser light to couple different spin states. The spin-orbit coupling is created by modifying the motion of atoms in a spin-dependent way by laser recoil. The spin selectivity is provided by Zeeman shifts created with a magnetic field gradient. Alternatively, a quantum spin Hall Hamiltonian can be created by all-optical means using a period-tripling, spin-dependent superlattice.

Figure 1

Spin-dependent tunneling in an optical lattice tilted by a magnetic field gradient. When the two spin states have opposite magnetic moments, the role of absorbtion and emission of the two photons is exchanged. The result is that the two states have tunneling matrix elements with opposite phases, leading to opposite synthetic magnetic fields and realizing spin-orbit coupling and the quantum spin Hall effect.

Figure 2

Superlattice scheme for realizing the quantum Hall and quantum spin Hall effect. A superlattice with 3 times the spatial period as the fundamental lattice leads to three distinguishable sites $A$, $B$, $C$. For the quantum spin Hall effect, the superlattice operates at a magic wavelength where the ac Stark effect is opposite for the two spin states. For rubidium, this is achieved at a wavelength of 790 nm. When filled with fermions the quantum Hall and spin quantum Hall effects are realized.