FIG. 15.

Spin-dependent topological lattices. (a) Level diagram for three-level total angular-momentum f = 1 case with m_F states labeled, as is applicable for the common alkali atoms ⁷Li, ²³Na, ³⁹K, ⁴¹K, and ⁸⁷Rb. For reference, the diagram shows the decomposition of these optical fields into ${\sigma }_{\pm }$ and π, but as discussed in the text, this is not an overly useful way of considering this problem. (b) Laser geometry for a typical optical flux lattice (left) producing a real-space Berry curvature with nonzero average (right). (c) Experimental geometry for spin-dependent topological lattice (Sun et al., 2017). (d) Directly observed topological band structure (left) and computed (right). From Sun et al., 2017.