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Probing chiral edge dynamics and bulk topology of a synthetic Hall system

Nature Physics volume 16, pages 1017–1021 (2020)Cite this article

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Abstract

Quantum Hall systems are characterized by quantization of the Hall conductance—a bulk property rooted in the topological structure of the underlying quantum states1. In condensed matter devices, material imperfections hinder a direct connection to simple topological models2,3. Artificial systems, such as photonic platforms4 or cold atomic gases5, open novel possibilities by enabling specific probes of topology6,7,8,9,10,11,12,13 or flexible manipulation, for example using synthetic dimensions14,15,16,17,18,19,20,21. However, the relevance of topological properties requires the notion of a bulk, which was missing in previous works using synthetic dimensions of limited sizes. Here, we realize a quantum Hall system using ultracold dysprosium atoms in a two-dimensional geometry formed by one spatial dimension and one synthetic dimension encoded in the atomic spin J = 8. We demonstrate that the large number of magnetic sublevels leads to distinct bulk and edge behaviours. Furthermore, we measure the Hall drift and reconstruct the local Chern marker, an observable that has remained, so far, experimentally inaccessible22. In the centre of the synthetic dimension—a bulk of 11 states out of 17—the Chern marker reaches 98(5)% of the quantized value expected for a topological system. Our findings pave the way towards the realization of topological many-body phases.

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Fig. 1: Synthetic Hall system.
Fig. 2: Ground band characterization.
Fig. 3: Cyclotron and skipping orbits.
Fig. 4: The Hall response.
Fig. 5: Simulations of topological many-body systems.

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Data availability

Source data, as well as other datasets generated and analysed during the current study, are available from the corresponding author upon request. Source data are provided with this paper.

Code availability

The source code for the numerical simulations of the Abrikosov vortex lattices and the Laughlin states are available from the corresponding author upon request. Source data are provided with this paper.

References

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

    ADS  Google Scholar 

  2. Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632–5633 (1981).

    ADS  Google Scholar 

  3. Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982).

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  5. Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016).

    Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

  8. Hu, W. et al. Measurement of a topological edge invariant in a microwave network. Phys. Rev. X 5, 011012 (2015).

    Google Scholar 

  9. 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).

    ADS  Google Scholar 

  10. Wu, Z. et al. Realization of two-dimensional spin–orbit coupling for Bose–Einstein condensates. Science 354, 83–88 (2016).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  12. Ravets, S. et al. Polaron polaritons in the integer and fractional quantum Hall regimes. Phys. Rev. Lett. 120, 057401 (2018).

    ADS  Google Scholar 

  13. Schine, N., Chalupnik, M., Can, T., Gromov, A. & Simon, J. Electromagnetic and gravitational responses of photonic Landau levels. Nature 565, 173–179 (2019).

    Google Scholar 

  14. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    ADS  Google Scholar 

  15. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  16. 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).

    ADS  MathSciNet  MATH  Google Scholar 

  17. Livi, L. F. et al. Synthetic dimensions and spin–orbit coupling with an optical clock transition. Phys. Rev. Lett. 117, 220401 (2016).

    ADS  Google Scholar 

  18. Kolkowitz, S. et al. Spin–orbit-coupled fermions in an optical lattice clock. Nature 542, 66–70 (2017).

    ADS  Google Scholar 

  19. An, F. A., Meier, E. J. & Gadway, B. Direct observation of chiral currents and magnetic reflection in atomic flux lattices. Sci. Adv. 3, e1602685 (2017).

    ADS  Google Scholar 

  20. Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  22. Bianco, R. & Resta, R. Mapping topological order in coordinate space. Phys. Rev. B 84, 241106 (2011).

    ADS  Google Scholar 

  23. Stormer, H. L., Tsui, D. C. & Gossard, A. C. The fractional quantum Hall effect. Rev. Mod. Phys. 71, S298–S305 (1999).

    MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  25. Pesin, D. & MacDonald, A. H. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater. 11, 409–416 (2012).

    ADS  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  27. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  29. Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).

    ADS  Google Scholar 

  30. Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).

    ADS  Google Scholar 

  31. Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018).

    ADS  Google Scholar 

  32. Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018).

    ADS  Google Scholar 

  33. Lin, Y.-J., Jiménez-García, K. & Spielman, I. B. Spin–orbit-coupled Bose–Einstein condensates. Nature 471, 83–86 (2011).

    ADS  Google Scholar 

  34. Cui, X., Lian, B., Ho, T.-L., Lev, B. L. & Zhai, H. Synthetic gauge field with highly magnetic lanthanide atoms. Phys. Rev. A 88, 011601 (2013).

    ADS  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  36. Barbarino, S., Taddia, L., Rossini, D., Mazza, L. & Fazio, R. Magnetic crystals and helical liquids in alkaline-earth fermionic gases. Nat. Commun. 6, 8134 (2015).

    ADS  Google Scholar 

  37. Schweikhard, V., Coddington, I., Engels, P., Mogendorff, V. P. & Cornell, E. A. Rapidly rotating Bose–Einstein condensates in and near the lowest Landau level. Phys. Rev. Lett. 92, 040404 (2004).

    ADS  Google Scholar 

  38. Bretin, V., Stock, S., Seurin, Y. & Dalibard, J. Fast rotation of a Bose–Einstein condensate. Phys. Rev. Lett. 92, 050403 (2004).

    ADS  Google Scholar 

  39. Abrikosov, A. A. On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5, 1174–1182 (1957).

    Google Scholar 

  40. Kane, C. L., Mukhopadhyay, R. & Lubensky, T. C. Fractional quantum Hall effect in an array of quantum wires. Phys. Rev. Lett. 88, 036401 (2002).

    ADS  Google Scholar 

  41. Goldman, V. J., Su, B. & Jain, J. K. Detection of composite fermions by magnetic focusing. Phys. Rev. Lett. 72, 2065–2068 (1994).

    ADS  Google Scholar 

  42. Zeng, T.-S., Wang, C. & Zhai, H. Charge pumping of interacting fermion atoms in the synthetic dimension. Phys. Rev. Lett. 115, 095302 (2015).

    ADS  Google Scholar 

  43. Taddia, L. et al. Topological fractional pumping with alkaline-earth-like atoms in synthetic lattices. Phys. Rev. Lett. 118, 230402 (2017).

    ADS  Google Scholar 

  44. De Bièvre, S. & Pulé, J. V. Propagating edge states for a magnetic Hamiltonian. Math. Phys. Electron. J 2002, 39–55 (2002).

    MATH  Google Scholar 

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Acknowledgements

We thank J. Beugnon, N. Cooper, P. Delpace, N. Goldman, L. Mazza and H. Price for stimulating discussions. We acknowledge funding from the EU under ERC projects ‘UQUAM’ and ‘TOPODY’, and PSL research university under the project ‘MAFAG’.

Author information

Author notes

  1. Vasiliy Makhalov

    Present address: ICFO-The Institute of Photonic Sciences, Barcelona, Spain

  2. These authors contributed equally: Thomas Chalopin, Tanish Satoor.

Authors and Affiliations

  1. Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, Paris, France

    Thomas Chalopin, Tanish Satoor, Alexandre Evrard, Vasiliy Makhalov, Jean Dalibard, Raphael Lopes & Sylvain Nascimbene

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All authors contributed to the set-up of the experiment, data acquisition, data analysis and the writing of the manuscript.

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Correspondence to Sylvain Nascimbene.

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Extended data

Extended Data Fig. 1 Conservation of canonical momentum.

Difference between the measured canonical momentum \({p}_{{\rm{rec}}}\langle {\hat{J}}_{z}\rangle +M\langle \hat{v}\rangle\) and the targeted value p defined by the state preparation protocol. All error bars are the 1-σ standard deviation of typically 5 measurement repetitions.

Extended Data Fig. 2 Cyclotron orbits measurements.

a, b, c, Magnetization, velocity and position response as a function of time after application of a velocity kick vkick ≈ vrec. d, Bulk excitations corresponding to different velocity kicks, v = 0.23, 0.48, 1.02, 2.06, and 3.15 vrec, from left to right. The orbits are off-centred in real space for visual clarity. e, Skipping orbit for the momentum state p = − J prec following a sudden jump of the coupling strength Ω. f, Velocity drift of the orbits as a function of the amplitude kick. The solid line corresponds to the expected drift due to the non-harmonic spectrum of the energy bands. All error bars are the 1-σ standard deviation of typically 5 measurement repetitions.

Extended Data Fig. 3 Hall system in real dimensions.

a, Variation of the Hall mobility for classical skipping orbits, depending on the angle of rebound on a hard wall. The case of closed cyclotron orbits corresponds to θ = π. b, Dispersion relation of a quantum Hall system in a semi-infinite geometry y > 0. The blue line indicates the energy branch used for the Chern marker calculation, defined by E0(p) < ℏωc. c, Hall mobility μ as a function of momentum p. d, Local density of state in the (v, y) plane. e, Local Chern marker C(y) for the energy branch defined in b.

Extended Data Fig. 4 Hall mobility and local Chern markers.

a, Predicted dispersion relation for ℏΩ = Erec. The branch pictured in blue, chosen as E(p) < E* with E* at half the gap, is used for the computation of the local Chern marker. b, Measured mobility in x resulting from the application of a force along m, as presented in the main text. The points in blue, corresponding to ∣p∣ < p* (white area), are the ones considered for the Chern marker presented in the main text (see Fig. 4). c, Measured mobility in m resulting from the application of a force along x. As for b, the points in red are associated to momentum states lying below E*. d, Chern marker obtained from the measured mobility, using the whole energy branch (− ∞ < p < ∞, gray squares, using data in b), or using the branch defined in a (− p* < p < p*). For the latter, the blue dots correspond to the data in b, and are identical to Fig. 4. The red diamonds correspond to the data in c. Solid lines are theoretical values. The error bars are the 1-σ statistical uncertainty calculated from a bootstrap sampling analysis over typically 100 pictures (b,c) and 1000 pictures (d).

Extended Data Fig. 5 Effect of disorder.

a, Example of Chern marker distribution in the presence of disorder of strength Δ = Erec. b, Chern marker \(\bar{C}(m=0)\) averaged over the region ∣x∣ < λ/4 as a function of the disorder strength Δ. Each point is the average of 100 disorder realizations, the error bar showing the standard deviation of the mean.

Extended Data Fig. 6 Abrikosov vortex lattices.

a, Ground state density profile and b, associated phase, for ℏΩ = 3Erec and for μchem ≈ 4Erec. The local minima of the density exhibit a phase winding around them, and thus correspond to quantum vortices. c, Number of vortex lines as a function of the Raman coupling Ω and the chemical potential μchem. The dots identify the configurations for which a simulation was realized. The color encodes the number of vortex lines that characterizes the low-energy vortex lattice configuration. The phase separation lines are guides to the eye. The dashed line identifies the gap to the first excited band above which the atoms significantly occupy higher Landau levels. d, Momentum p0 associated to the spontaneous breaking of the translational invariance resulting from the appearance of a vortex lattice, as a function of Ω. The points were taken at a chemical potential corresponding to half the gap.

Supplementary information

Supplementary Information

Supplementary Discussion and Figs. 1–5.

Source data

Source Data Fig. 2

Source data for panels a and b.

Source Data Fig. 3

Source data.

Source Data Fig. 4

Source data for panels b and c.

Source Data Fig. 5

Source data for panel c.

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Chalopin, T., Satoor, T., Evrard, A. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. 16, 1017–1021 (2020). https://doi.org/10.1038/s41567-020-0942-5

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