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

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