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Electromagnetic and gravitational responses of photonic Landau levels

Naturevolume 565pages173179 (2019) | Download Citation

Abstract

Topology has recently become a focus in condensed matter physics, arising in the context of the quantum Hall effect and topological insulators. In both of these cases, the topology of the system is defined through bulk properties (‘topological invariants’) but detected through surface properties. Here we measure three topological invariants of a quantum Hall material—photonic Landau levels in curved space—through local electromagnetic and gravitational responses of the bulk material. Viewing the material as a many-port circulator, the Chern number (a topological invariant) manifests as spatial winding of the phase of the circulator. The accumulation of particles near points of high spatial curvature and the moment of inertia of the resultant particle density distribution quantify two additional topological invariants—the mean orbital spin and the chiral central charge. We find that these invariants converge to their global values when probed over increasing length scales (several magnetic lengths), consistent with the intuition that the bulk and edges of a system are distinguishable only for sufficiently large samples (larger than roughly one magnetic length). Our experiments are enabled by applying quantum optics tools to synthetic topological matter (here twisted optical resonators). Combined with advances in Rydberg-mediated photon collisions, our work will enable precision characterization of topological matter in photon fluids.

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References

  1. 1.

    Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

  2. 2.

    Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010).

  3. 3.

    Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

  4. 4.

    Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

  5. 5.

    Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993).

  6. 6.

    Avron, J. E., Seiler, R. & Zograf, P. G. Viscosity of quantum Hall fluids. Phys. Rev. Lett. 75, 697–700 (1995).

  7. 7.

    Klevtsov, S. & Wiegmann, P. Geometric adiabatic transport in quantum Hall states. Phys. Rev. Lett. 115, 086801 (2015).

  8. 8.

    Can, T., Chiu, Y. H., Laskin, M. & Wiegmann, P. Emergent conformal symmetry and geometric transport properties of quantum Hall states on singular surfaces. Phys. Rev. Lett. 117, 266803 (2016).

  9. 9.

    Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

  10. 10.

    Duca, L. et al. An Aharonov-Bohm interferometer for determining Bloch band topology. Science 347, 288–292 (2015).

  11. 11.

    Fläschner, N. et al. Experimental reconstruction of the berry curvature in a floquet bloch band. Science 352, 1091–1094 (2016).

  12. 12.

    Li, T. et al. Bloch state tomography using Wilson lines. Science 352, 1094–1097 (2016).

  13. 13.

    Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

  14. 14.

    Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983).

  15. 15.

    Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296–300 (2016).

  16. 16.

    Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016).

  17. 17.

    Tarnowski, M. et al. Characterizing topology by dynamics: Chern number from linking number. Preprint at https://arxiv.org/abs/1709.01046 (2018).

  18. 18.

    Banerjee, M. et al. Observed quantization of anyonic heat flow. Nature 545, 75–79 (2017).

  19. 19.

    Mittal, S., Ganeshan, S., Fan, J., Vaezi, A. & Hafezi, M. Measurement of topological invariants in a 2D photonic system. Nat. Photon. 10, 180–183 (2016).

  20. 20.

    Ningyuan, F., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).

  21. 21.

    Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

  22. 22.

    Owens, C. et al. Quarter-flux hofstadter lattice in a qubit-compatible microwave cavity array. Phys. Rev. A 97, 013818 (2018).

  23. 23.

    Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

  24. 24.

    Tai, E. et al. Microscopy of the interacting Harper–Hofstadter model in the two-body limit. Nature 546, 519–523 (2017).

  25. 25.

    Stuhl, B. K., Lu, H.-I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).

  26. 26.

    Ma, R., Owens, C., LaChapelle, A., Schuster, D. I. & Simon, J. Hamiltonian tomography of photonic lattices. Phys. Rev. A 95, 062120 (2017).

  27. 27.

    Lim, H.-T., Togan, E., Kroner, M., Miguel-Sanchez, J. & Imamoğlu, A. Electrically tunable artificial gauge potential for polaritons. Nat. Commun. 8, 14540 (2017).

  28. 28.

    Schine, N., Ryou, A., Gromov, A., Sommer, A. & Simon, J. Synthetic landau levels for photons. Nature 534, 671–675 (2016).

  29. 29.

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

  30. 30.

    Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

  31. 31.

    Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

  32. 32.

    Zaletel, M. P., Mong, R. S. K. & Pollmann, F. Topological characterization of fractional quantum Hall ground states from microscopic Hamiltonians. Phys. Rev. Lett. 110, 236801 (2013).

  33. 33.

    Kane, C. L. & Fisher, M. P. A. Quantized thermal transport in the fractional quantum Hall effect. Phys. Rev. B 55, 15832–15837 (1997).

  34. 34.

    Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

  35. 35.

    Abanov, A. G. & Gromov, A. Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field. Phys. Rev. B 90, 014435 (2014).

  36. 36.

    Gromov, A., Cho, G. Y., You, Y., Abanov, A. G. & Fradkin, E. Framing anomaly in the effective theory of the fractional quantum Hall effect. Phys. Rev. Lett. 114, 016805 (2015).

  37. 37.

    Mitchell, N., Nash, L., Hexner, D., Turner, A. & Irvine, W. Amorphous topological insulators constructed from random point sets. Nat. Phys. 14, 380–385 (2018).

  38. 38.

    Kohmoto, M. Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343–354 (1985).

  39. 39.

    Brouder, C., Panati, G., Calandra, M., Mourougane, C. & Marzari, N. Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007).

  40. 40.

    Wen, X. G. Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states. Phys. Rev. B 41, 12838–12844 (1990).

  41. 41.

    Wen, X. G. & Zee, A. Shift and spin vector: new topological quantum numbers for the Hall fluids. Phys. Rev. Lett. 69, 953–956 (1992).

  42. 42.

    Klevtsov, S. Random normal matrices, Bergman kernel and projective embeddings. J. High Energy Phys. 1, 133 (2014).

  43. 43.

    Banerjee, M. et al. Observation of half-integer thermal Hall conductance. Nature 559, 205–210 (2018).

  44. 44.

    Cooper, N. R., Wilkin, N. K. & Gunn, J. M. F. Quantum phases of vortices in rotating Bose-Einstein condensates. Phys. Rev. Lett. 87, 120405 (2001).

  45. 45.

    Regnault, N. & Jolicoeur, Th. Quantum Hall fractions in rotating Bose-Einstein condensates. Phys. Rev. Lett. 91, 030402 (2003).

  46. 46.

    Hafezi, M., Lukin, M. D. & Taylor, J. M. Non-equilibrium fractional quantum Hall state of light. New J. Phys. 15, 063001 (2013).

  47. 47.

    Jia, N. et al. A strongly interacting polaritonic quantum dot. Nat. Phys. 14, 550–554 (2018).

  48. 48.

    Peyronel, T. et al. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature 488, 57–60 (2012).

  49. 49.

    Umucalılar, R. O. & Carusotto, I. Fractional quantum Hall states of photons in an array of dissipative coupled cavities. Phys. Rev. Lett. 108, 206809 (2012).

  50. 50.

    Grusdt, F., Letscher, F., Hafezi, M. & Fleischhauer, M. Topological growing of Laughlin states in synthetic gauge fields. Phys. Rev. Lett. 113, 155301 (2014).

  51. 51.

    Dutta, S. & Mueller, E. J. Coherent generation of photonic fractional quantum Hall states in a cavity and the search for anyonic quasiparticles. Phys. Rev. A 97, 033825 (2018).

  52. 52.

    Kapit, E., Hafezi, M. & Simon, S. H. Induced self-stabilization in fractional quantum Hall states of light. Phys. Rev. X 4, 031039 (2014).

  53. 53.

    Lebreuilly, J. et al. Stabilizing strongly correlated photon fluids with non-Markovian reservoirs. Phys. Rev. A 96, 033828 (2017).

  54. 54.

    Wu, Y.-H., Tu, H.-H. & Sreejith, G. J. Fractional quantum Hall states of bosons on cones. Phys. Rev. A 96, 033622 (2017).

  55. 55.

    Avron, J. E. Odd viscosity. J. Stat. Phys. 92, 543–557 (1998).

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Acknowledgements

We thank C. Kane and M. Levin for conversations. This work was supported by DOE grant DE-SC0010267 for apparatus construction and data collection and MURI grant FA9550-16-1-0323 for analysis.

Author information

Author notes

    • Michelle Chalupnik

    Present address: Department of Physics, Harvard University, Cambridge, MA, USA

Affiliations

  1. James Franck Institute and the Department of Physics, University of Chicago, Chicago, IL, USA

    • Nathan Schine
    • , Michelle Chalupnik
    •  & Jonathan Simon
  2. Initiative for the Theoretical Sciences, The Graduate Center, CUNY, New York, NY, USA

    • Tankut Can
  3. Kadanoff Center for Theoretical Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL, USA

    • Andrey Gromov

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Contributions

N.S., M.C. and J.S. designed and built the experiment. N.S. and M.C. collected and analysed the data. T.C. and A.G. developed the theory concerning \(\bar{s}\) and c. All authors contributed to the manuscript.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Jonathan Simon.

Extended data figures and tables

  1. Extended Data Fig. 1 Resonator imaging comparison.

    The LDOS in the second excited Landau level with an effective magnetic flux of ΦB/(2π) = −2/3 threading the cone tip highlights the improvements in the resonator design and the imaging system. The previous resonator28 (top left) exhibits substantial diagonal astigmatism, which has been removed in the resonator used here (top right). Images of modes in the lowest Landau level provide estimates of the expectation value of r2 (bottom), errors in which directly cause systematic errors in measurements of the shifted second moment. The substantial reduction in deviations from the ideal system enables measurements of the central charge and extensions to higher Landau levels. Error bars are calculated from the uncertainty in the centre location and waist size of the modes and are all smaller than the symbol size.

Supplementary information

  1. Supplementary Information

    This 12-page document contains 8 sections and 9 figures. These provide additional details about the experimental methods, theoretical background for numerical calculations and the theoretical results connecting LDOS measurements to topological invariants.

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DOI

https://doi.org/10.1038/s41586-018-0817-4

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