Riemannian geometry of optical responses

Geometry of quantum states has proved to be a useful concept for understanding responses of condensed matters to static electromagnetic fields. On the other hand, it has been challenging to relate quantum geometry with resonant optical responses. The main obstacle is that optical transitions are properties of a pair of states, while geometrical properties have been defined for a single state. Therefore, geometric interpretations of optical responses, when possible, have been limited to two-level systems, where the two states involved in transitions are not independent. In this talk, I will present a general theory of Riemannian geometry in resonant optical processes. An optical response related to the Riemannian curvature tensor is identified.