Topological physics relies on the structure of the eigenstates of the Hamiltonians. The geometry of the eigenstates is encoded in the quantum geometric tensor1—comprising the Berry curvature2 (crucial for topological matter)3 and the quantum metric4, which defines the distance between the eigenstates. Knowledge of the quantum metric is essential for understanding many phenomena, such as superfluidity in flat bands5, orbital magnetic susceptibility6,7, the exciton Lamb shift8 and the non-adiabatic anomalous Hall effect6,9. 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 microcavity10—together with the related anomalous Hall drift. The microcavity hosts strongly coupled exciton–photon modes (exciton polaritons) that are subject to photonic spin–orbit coupling11 from which Dirac cones emerge12, 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 photonics13,14,15. These results also experimentally validate the semiclassical description of wavepacket motion in geometrically non-trivial bands9,16. The use of exciton polaritons (interacting photons) opens up possibilities for future studies of quantum fluid physics in topological systems.
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The datasets generated and/or analysed during the current study are available in the Open Science Framework (OSF) repository at https://osf.io/s4rzu/?view_only=1cabd49416c04a9baed856dee3ae1ba9.
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We thank D. Colas for critical reading of the manuscript. This work was supported by the ERC project ElecOpteR (grant number 780757). We acknowledge the support of the project Quantum Fluids of Light (ANR-16-CE30-0021), of the ANR Labex Ganex (ANR-11-LABX-0014), and of the ANR program Investissements d’Avenir through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25). D.D.S. acknowledges the support of the IUF (Institut Universitaire de France). This work was partially supported by the FISR-CNR project "TECNOMED—Tecnopolo di nanotecnologia e fotonica per la medicina di precisione".
The authors declare no competing interests.
Peer review information Nature thanks Ulf Peschel and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Extended data figures and tables
Schematic of the polarization tomography experiment. The incoming pump laser (bottom right) is focused onto the sample held in the cryogenic superconductive magnet (bottom left). The emission is recollected, polarization filtered and the momentum space optically rebuilt at the entrance slits of a spectrometer (top) with energy resolution of 30 µeV (top left). The Zeeman splitting is highlighted in the inset.
The file contains five Supplementary Notes and six Supplementary Figures. The notes present the details of the data treatment, uncertainty estimates, and the analytical calculations. The figures show the examples of raw photoluminescence data and additional results on the Berry curvature and anomalous Hall effect.
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Gianfrate, A., Bleu, O., Dominici, L. et al. Measurement of the quantum geometric tensor and of the anomalous Hall drift. Nature 578, 381–385 (2020). https://doi.org/10.1038/s41586-020-1989-2
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