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We first consider a generic two-band model which can be mapped to apseudospin on a Bloch sphere. We establish the link between the pseudospinorientation and the components of the quantum geometric tensor (QGT): themetric tensor and the Berry curvature. We show how the k-dependent pseudospinorientation can be measured in photonic systems with radiative modes. Weconsider the specific example: a 2D planar cavity with two polarizationeigenmodes, where the pseudospin measurement can be performed viapolarization-resolved photoluminescence. We also consider the s-band of astaggered honeycomb lattice for polarization-degenerate modes (scalar photons).The sublattice pseudospin can be measured by performing spatially resolvedinterferometric measurements. In the second part, we consider a morecomplicated four-band model, which can be mapped to two entangled pseudospins.We show how the QGT components can be obtained by measuring six angles. Thespecific four-band system we considered is the s-band of a honeycomb latticefor polarized (spinor) photons. We show that all six angles can indeed bemeasured in this system. We simulate realistic experimental situations in allcases. We find the photon eigenstates by solving Schrodinger equation includingpumping and finite lifetime, and then simulate the measurement of the relevantangles to finally extract realistic mappings of the k-dependent QGT components.

Measuring the Quantum Geometric Tensor in 2D Photonic and Polaritonic Systems