Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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

Abstract

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Emergence of pseudospin monopoles from photoluminescence at 0 T.
Fig. 2: Broken time-reversal symmetry: emergence of half-skyrmion pseudospin textures, from photoluminescence at 9 T.
Fig. 3: Berry curvature and quantum metric distributions.
Fig. 4: Polariton anomalous Hall effect.

Data availability

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.

References

  1. 1.

    Berry, M. The quantum phase, five years after. In Geometric Phases in Physics (eds Wilczek, F. & Shapere, A.) 7–28 (World Scientific, 1989).

  2. 2.

    Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Hasan, M. Z. & Kane, C. L. Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  CAS  Article  Google Scholar 

  4. 4.

    Provost, J. & Vallee, G. Riemannian structure on manifolds of quantum states. Commun. Math. Phys. 76, 289–301 (1980).

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).

    ADS  CAS  Article  Google Scholar 

  6. 6.

    Gao, Y., Yang, S. A. & Niu, Q. Field induced positional shift of Bloch electrons and its dynamical implications. Phys. Rev. Lett. 112, 166601 (2014).

    ADS  Article  Google Scholar 

  7. 7.

    Piéchon, F., Raoux, A., Fuchs, J.-N. & Montambaux, G. Geometric orbital susceptibility: quantum metric without Berry curvature. Phys. Rev. B 94, 134423 (2016).

    ADS  Article  Google Scholar 

  8. 8.

    Srivastava, A. & Imamoglu, A. Signatures of Bloch-band geometry on excitons: nonhydrogenic spectra in transition-metal dichalcogenides. Phys. Rev. Lett. 115, 166802 (2015).

    ADS  Article  Google Scholar 

  9. 9.

    Bleu, O., Malpuech, G., Gao, Y. & Solnyshkov, D. D. Effective theory of nonadiabatic quantum evolution based on the quantum geometric tensor. Phys. Rev. Lett. 121, 020401 (2018).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  10. 10.

    Kavokin, A., Baumberg, J. J., Malpuech, G. & Laussy, F. P. (eds) Microcavities (Oxford Univ. Press, 2011).

  11. 11.

    Kavokin, A., Malpuech, G. & Glazov, M. Optical spin Hall effect. Phys. Rev. Lett. 95, 136601 (2005).

    ADS  Article  Google Scholar 

  12. 12.

    Terças, H., Flayac, H., Solnyshkov, D. D. & Malpuech, G. Non-Abelian gauge fields in photonic cavities and photonic superfluids. Phys. Rev. Lett. 112, 066402 (2014).

    ADS  Article  Google Scholar 

  13. 13.

    Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    ADS  CAS  Article  Google Scholar 

  14. 14.

    Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    ADS  CAS  Article  Google Scholar 

  15. 15.

    Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  16. 16.

    Sundaram, G. & Niu, Q. Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915–14925 (1999).

    ADS  CAS  Article  Google Scholar 

  17. 17.

    Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

    ADS  Article  Google Scholar 

  18. 18.

    Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  19. 19.

    Delplace, P., Marston, J. & Venaille, A. Topological origin of equatorial waves. Science 358, 1075–1077 (2017).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  20. 20.

    Zanardi, P., Giorda, P. & Cozzini, M. Information-theoretic differential geometry of quantum phase transitions. Phys. Rev. Lett. 99, 100603 (2007).

    ADS  Article  Google Scholar 

  21. 21.

    Liang, L., Peotta, S., Harju, A. & Törmä, P. Wave-packet dynamics of Bogoliubov quasiparticles: quantum metric effects. Phys. Rev. B 96, 064511 (2017).

    ADS  Article  Google Scholar 

  22. 22.

    Hauke, P., Lewenstein, M. & Eckardt, A. Tomography of band insulators from quench dynamics. Phys. Rev. Lett. 113, 045303 (2014).

    ADS  Article  Google Scholar 

  23. 23.

    Lim, L.-K., Fuchs, J.-N. & Montambaux, G. Geometry of Bloch states probed by Stückelberg interferometry. Phys. Rev. A 92, 063627 (2015).

    ADS  Article  Google Scholar 

  24. 24.

    Fläschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).

    ADS  Article  Google Scholar 

  25. 25.

    Wimmer, M., Price, H. M., Carusotto, I. & Peschel, U. Experimental measurement of the Berry curvature from anomalous transport. Nat. Phys. 13, 545–550 (2017).

    CAS  Article  Google Scholar 

  26. 26.

    Yu, M. et al. Experimental measurement of the quantum geometric tensor using coupled qubits in diamond. Natl Sci. Rev. nwz193 (2019).

  27. 27.

    Tan, X. et al. Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit. Phys. Rev. Lett. 122, 210401 (2019).

    ADS  CAS  Article  Google Scholar 

  28. 28.

    Bleu, O., Solnyshkov, D. D. & Malpuech, G. Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems. Phys. Rev. B 97, 195422 (2018).

    ADS  CAS  Article  Google Scholar 

  29. 29.

    Richter, S. et al. Exceptional points in anisotropic planar microcavities. Phys. Rev. A 95, 023836 (2017).

    ADS  Article  Google Scholar 

  30. 30.

    Nalitov, A. V., Solnyshkov, D. D. & Malpuech, G. Polariton topological insulator. Phys. Rev. Lett. 114, 116401 (2015).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  31. 31.

    Klembt, S. et al. Exciton–polariton topological insulator. Nature 562, 552–556 (2018).

    ADS  CAS  Article  Google Scholar 

  32. 32.

    St-Jean, P. et al. Lasing in topological edge states of a one-dimensional lattice. Nat. Photon. 11, 651–656 (2017).

    ADS  CAS  Article  Google Scholar 

  33. 33.

    Bleu, O., Malpuech, G. & Solnyshkov, D. D. Robust quantum valley Hall effect for vortices in an interacting bosonic quantum fluid. Nature Commun. 9, 3991 (2018).

    ADS  CAS  Article  Google Scholar 

  34. 34.

    Steger, M., Gautham, C., Snoke, D. W., Pfeiffer, L. & West, K. Slow reflection and two-photon generation of microcavity exciton–polaritons. Optica 2, 1–5 (2015).

    ADS  CAS  Article  Google Scholar 

  35. 35.

    Ballarini, D. et al. Macroscopic two-dimensional polariton condensates. Phys. Rev. Lett. 118, 215301 (2017).

    ADS  Article  Google Scholar 

  36. 36.

    Armitage, A. et al. Exciton polaritons in semiconductor quantum microcavities in a high magnetic field. Phys. Rev. B 55, 16395–16403 (1997).

    ADS  CAS  Article  Google Scholar 

  37. 37.

    Rahimi-Iman, A. et al. Zeeman splitting and diamagnetic shift of spatially confined quantum-well exciton polaritons in an external magnetic field. Phys. Rev. B 84, 165325 (2011).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

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".

Author information

Affiliations

Authors

Contributions

A.G., L.D. and D.B. designed the setup. A.G. realized the experiments with the help of V.A., M.D.G. and G.L. D.S. supervised the experimental part. K.W.W. and L.N.P. fabricated the sample. O.B., D.D.S. and G.M. performed the treatment of the experimental data. O.B. performed analytical calculations. G.M. and O.B. wrote the manuscript with input from all authors.

Corresponding authors

Correspondence to D. Sanvitto or G. Malpuech.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Ulf Peschel and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Experimental setup.

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.

Supplementary information

Supplementary Information

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing