Measurement of the quantum geometric tensor and of the anomalous Hall drift

Topological physics relies on the structure of the eigenstates of the Hamiltonians. The geometry of the eigenstates is encoded in the quantum geometric tensor(1)-comprising the Berry curvature(2) (crucial for topological matter)(3) and the quantum metric(4), which defines the distance between the eigenstates. Knowledge of the quantum metric is essential for understanding many phenomena, such as superfluidity in flat bands(5), orbital magnetic susceptibility(6,7), the exciton Lamb shift(8) and the non-adiabatic anomalous Hall effect(6,9). However, the quantum geometry of energy bands has not been measured. Here we report the direct measurement of both the Berry curvature and the quantum metric in a two-dimensional continuous medium-a high-finesse planar microcavity(10)-together with the related anomalous Hall drift. The microcavity hosts strongly coupled exciton-photon modes (exciton polaritons) that are subject to photonic spin-orbit coupling(11) from which Dirac cones emerge(12), and to exciton Zeeman splitting, breaking time-reversal symmetry. The monopolar and half-skyrmion pseudospin textures are measured using polarization-resolved photoluminescence. The associated quantum geometry of the bands is extracted, enabling prediction of the anomalous Hall drift, which we measure independently using high-resolution spatially resolved epifluorescence. Our results unveil the intrinsic chirality of photonic modes, the cornerstone of topological photonics(13-15). These results also experimentally validate the semiclassical description of wavepacket motion in geometrically non-trivial bands(9,16). The use of exciton polaritons (interacting photons) opens up possibilities for future studies of quantum fluid physics in topological systems.