Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems

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