Time-reversal invariant bilayer 2D synthetic lattices

Recently it has been theoretically proposed and experimentally demonstrated that a 1D spin-orbit coupled optical lattice can emulate a 2D synthetic lattice with a uniform magnetic field, where atomic spins represent the synthetic dimension. The time-reversal symmetry in this system is broken by the magnetic flux. Here we propose the time-reversal symmetry may be restored by considering a bilayer spin-orbit coupled optical lattice with opposite Raman coupling between layers. We show how such layer-spin coupling modifies chiral edge states, fractal Hofstadter butterfly, and layer-spin correlation.