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We show how a quantum optical measurement scheme based on heterodynedetection can be used to explore geometrical and topological properties ofcondensed matter systems. Considering a 2D material placed in a cavity with acoupling to the environment, we compute correlation functions of the photonsexiting the cavity and relate them to the hybrid light-matter state within thecavity. Different polarizations of the intracavity field give access to allcomponents of the quantum geometric tensor on contours in the Brillouin zonedefined by the transition energy. Combining recent results based on themetric-curvature correspondence with the measured quantum metric allows us tocharacterize the topological phase of the material. Moreover, in systems where$S_z$ is a good quantum number, the procedure also allows us to extract thespin Chern number. As an interesting application, we consider a minimal modelfor twisted bilayer graphene at the magic angle, and discuss the feasibility ofextracting the Euler number.

Quantum optics measurement scheme for quantum geometry and topological invariants