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Crystallization of bosonic quantum Hall states in a rotating quantum gas

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Abstract

The dominance of interactions over kinetic energy lies at the heart of strongly correlated quantum matter, from fractional quantum Hall liquids1, to atoms in optical lattices2 and twisted bilayer graphene3. Crystalline phases often compete with correlated quantum liquids, and transitions between them occur when the energy cost of forming a density wave approaches zero. A prime example occurs for electrons in high-strength magnetic fields, where the instability of quantum Hall liquids towards a Wigner crystal4,5,6,7,8,9 is heralded by a roton-like softening of density modulations at the magnetic length7,10,11,12. Remarkably, interacting bosons in a gauge field are also expected to form analogous liquid and crystalline states13,14,15,16,17,18,19,20,21. However, combining interactions with strong synthetic magnetic fields has been a challenge for experiments on bosonic quantum gases18,21. Here we study the purely interaction-driven dynamics of a Landau gauge Bose–Einstein condensate22 in and near the lowest Landau level. We observe a spontaneous crystallization driven by condensation of magneto-rotons7,10, excitations visible as density modulations at the magnetic length. Increasing the cloud density smoothly connects this behaviour to a quantum version of the Kelvin–Helmholtz hydrodynamic instability, driven by the sheared internal flow profile of the rapidly rotating condensate. At long times the condensate self-organizes into a persistent array of droplets separated by vortex streets, which are stabilized by a balance of interactions and effective magnetic forces.

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Fig. 1: Spontaneous crystallization of an interacting Bose–Einstein condensate in an artificial magnetic field.
Fig. 2: Structure factor and lengthscale of the emergent crystal.
Fig. 3: Instability growth dynamics.
Fig. 4: Spontaneous breaking of translational symmetry.

Data availability

All data files are available from the corresponding author upon request. Accompanying data, including those for figures, are available from Zenodo (https://doi.org/10.5281/zenodo.5533142).

Code availability

The simulation and analysis code are available from the corresponding author upon reasonable request.

Change history

  • 07 January 2022

    This Article was amended to correct the Peer review information.

References

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

    MathSciNet  CAS  Google Scholar 

  2. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    ADS  CAS  Google Scholar 

  3. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    ADS  CAS  PubMed  Google Scholar 

  4. Wigner, E. On the interaction of electrons in metals. Phys. Rev. 46, 1002–1011 (1934).

    ADS  CAS  MATH  Google Scholar 

  5. Yoshioka, D. & Fukuyama, H. Charge density wave state of two-dimensional electrons in strong magnetic fields. J. Phys. Soc. Jpn. 47, 394–402 (1979).

    ADS  CAS  Google Scholar 

  6. Lam, P. K. & Girvin, S. M. Liquid–solid transition and the fractional quantum-Hall effect. Phys. Rev. B 30, 473–475 (1984).

    ADS  CAS  Google Scholar 

  7. Girvin, S. M., MacDonald, A. H. & Platzman, P. M. Magneto-roton theory of collective excitations in the fractional quantum Hall effect. Phys. Rev. B 33, 2481–2494 (1986).

    ADS  CAS  Google Scholar 

  8. Jiang, H. W. et al. Quantum liquid versus electron solid around ν = 1/5 Landau-level filling. Phys. Rev. Lett. 65, 633–636 (1990).

    ADS  CAS  PubMed  Google Scholar 

  9. Jang, J., Hunt, B. M., Pfeiffer, L. N., West, K. W. & Ashoori, R. C. Sharp tunnelling resonance from the vibrations of an electronic Wigner crystal. Nat. Phys. 13, 340–344 (2017).

    CAS  Google Scholar 

  10. Haldane, F. D. M. & Rezayi, E. H. Finite-size studies of the incompressible state of the fractionally quantized Hall effect and its excitations. Phys. Rev. Lett. 54, 237–240 (1985).

    ADS  CAS  PubMed  Google Scholar 

  11. Pinczuk, A., Dennis, B. S., Pfeiffer, L. N. & West, K. Observation of collective excitations in the fractional quantum Hall effect. Phys. Rev. Lett. 70, 3983–3986 (1993).

    ADS  CAS  PubMed  Google Scholar 

  12. Kukushkin, I. V., Smet, J. H., Scarola, V. W., Umansky, V. & von Klitzing, K. Dispersion of the excitations of fractional quantum Hall states. Science 324, 1044–1047 (2009).

    ADS  CAS  PubMed  Google Scholar 

  13. Ho, T.-L. Bose–Einstein condensates with large number of vortices. Phys. Rev. Lett. 87, 060403 (2001).

    ADS  CAS  PubMed  Google Scholar 

  14. Oktel, M. Ö. Vortex lattice of a Bose–Einstein condensate in a rotating anisotropic trap. Phys. Rev. A 69, 023618 (2004).

    ADS  Google Scholar 

  15. Sinha, S. & Shlyapnikov, G. V. Two-dimensional Bose–Einstein condensate under extreme rotation. Phys. Rev. Lett. 94, 150401 (2005).

    ADS  CAS  PubMed  Google Scholar 

  16. Cooper, N. R. Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008).

    ADS  CAS  Google Scholar 

  17. Aftalion, A., Blanc, X. & Lerner, N. Fast rotating condensates in an asymmetric harmonic trap. Phys. Rev. A 79, 011603 (2009).

    ADS  MATH  Google Scholar 

  18. Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).

    ADS  CAS  Google Scholar 

  19. Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic systems. Science 338, 1604–1606 (2012).

    ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  20. Senthil, T. & Levin, M. Integer quantum Hall effect for bosons. Phys. Rev. Lett. 110, 046801 (2013).

    ADS  CAS  PubMed  Google Scholar 

  21. Goldman, N., Juzeliūnas, G., Öhberg, P. & Spielman, I. B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 77, 126401 (2014).

    ADS  CAS  PubMed  Google Scholar 

  22. Fletcher, R. J. et al. Geometric squeezing into the lowest Landau level. Science 372, 1318–1322 (2021).

    ADS  CAS  PubMed  Google Scholar 

  23. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    ADS  CAS  Google Scholar 

  24. Nozières, P. Is the roton in superfluid 4He the ghost of a Bragg spot? J. Low Temp. Phys. 137, 45–67 (2004).

    ADS  Google Scholar 

  25. Graß, T. et al. Fractional quantum Hall phases of bosons with tunable interactions: from the Laughlin liquid to a fractional Wigner crystal. Phys. Rev. Lett. 121, 253403 (2018).

    ADS  PubMed  PubMed Central  Google Scholar 

  26. Galitski, V. & Spielman, I. B. Spin–orbit coupling in quantum gases. Nature 494, 49–54 (2013).

    ADS  CAS  PubMed  Google Scholar 

  27. Chalopin, T. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. 16, 1017–1021 (2020).

    CAS  Google Scholar 

  28. Struck, J. et al. Tunable gauge potential for neutral and spinless particles in driven optical lattices. Phys. Rev. Lett. 108, 225304 (2012).

    ADS  CAS  PubMed  Google Scholar 

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

    ADS  CAS  PubMed  Google Scholar 

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

    CAS  Google Scholar 

  31. 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  CAS  PubMed  MATH  Google Scholar 

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

    ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  33. 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  CAS  PubMed  Google Scholar 

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

    ADS  PubMed  Google Scholar 

  35. Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).

    ADS  Google Scholar 

  36. Iordanskii, S. V. & Pitaevskii, L. P. Bose condensation of moving rotons. Sov. Phys. Usp. 23, 317–318 (1980).

    ADS  Google Scholar 

  37. Pitaevskii, L. P. Layered structure of superfluid 4He with super-critical motion. JETP Lett. 39, 511–514 (1984).

    ADS  Google Scholar 

  38. Martone, G. I., Recati, A. & Pavloff, N. Supersolidity of cnoidal waves in an ultracold Bose gas. Phys. Rev. Res. 3, 013143 (2021).

    CAS  Google Scholar 

  39. Mottl, R. et al. Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions. Science 336, 1570–1573 (2012).

    ADS  CAS  PubMed  Google Scholar 

  40. Léonard, J., Morales, A., Zupancic, P., Esslinger, T. & Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87–90 (2017).

    ADS  PubMed  Google Scholar 

  41. Ji, S.-C. et al. Softening of roton and phonon modes in a Bose–Einstein condensate with spin–orbit coupling. Phys. Rev. Lett. 114, 105301 (2015).

    ADS  PubMed  Google Scholar 

  42. Li, J.-R. et al. A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates. Nature 543, 91–94 (2017).

    ADS  CAS  PubMed  Google Scholar 

  43. Ha, L.-C., Clark, L. W., Parker, C. V., Anderson, B. M. & Chin, C. Roton-maxon excitation spectrum of Bose condensates in a shaken optical lattice. Phys. Rev. Lett. 114, 055301 (2015).

    ADS  CAS  PubMed  Google Scholar 

  44. Feng, L., Clark, L. W., Gaj, A. & Chin, C. Coherent inflationary dynamics for Bose–Einstein condensates crossing a quantum critical point. Nat. Phys. 14, 269–272 (2018).

    CAS  Google Scholar 

  45. Zhang, Z., Yao, K.-X., Feng, L., Hu, J. & Chin, C. Pattern formation in a driven Bose–Einstein condensate. Nat. Phys. 16, 652–656 (2020).

    CAS  Google Scholar 

  46. Petter, D. et al. Probing the roton excitation spectrum of a stable dipolar Bose gas. Phys. Rev. Lett. 122, 183401 (2019).

    ADS  CAS  PubMed  Google Scholar 

  47. Hertkorn, J. et al. Density fluctuations across the superfluid–supersolid phase transition in a dipolar quantum gas. Phys. Rev. X 11, 011037 (2021).

    CAS  Google Scholar 

  48. Schmidt, J.-N. et al. Roton excitations in an oblate dipolar quantum gas. Phys. Rev. Lett. 126, 193002 (2021).

    ADS  CAS  PubMed  Google Scholar 

  49. Guo, M. et al. The low-energy Goldstone mode in a trapped dipolar supersolid. Nature 574, 386–389 (2019).

    ADS  CAS  PubMed  Google Scholar 

  50. Tanzi, L. et al. Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas. Nature 574, 382–385 (2019).

    ADS  CAS  PubMed  Google Scholar 

  51. Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 21012 (2019).

    CAS  Google Scholar 

  52. Davidson, R. C., Chan, H.-W., Chen, C. & Lund, S. Equilibrium and stability properties of intense non-neutral electron flow. Rev. Mod. Phys. 63, 341–374 (1991).

    ADS  Google Scholar 

  53. Cerfon, A. J. Vortex dynamics and shear-layer instability in high-intensity cyclotrons. Phys. Rev. Lett. 116, 174801 (2016).

    ADS  PubMed  Google Scholar 

  54. Chandrasekhar, S. C. Hydrodynamic and Hydromagnetic Stability (Clarendon Press, 1961).

  55. Landau, L. D. & Lifshitz, E. M. Fluid Mechanics (Pergamon Press, 1987).

  56. Finne, A. P. et al. Dynamics of vortices and interfaces in superfluid 3He. Rep. Prog. Phys. 69, 3157–3230 (2006).

    ADS  CAS  Google Scholar 

  57. Baggaley, A. W. & Parker, N. G. Kelvin–Helmholtz instability in a single-component atomic superfluid. Phys. Rev. A 97, 053608 (2018).

    ADS  CAS  Google Scholar 

  58. Fetter, A. L. & Walecka, J. D. Quantum Theory of Many-particle Systems (McGraw-Hill, 1971).

  59. Recati, A., Zambelli, F. & Stringari, S. Overcritical rotation of a trapped Bose–Einstein condensate. Phys. Rev. Lett. 86, 377–380 (2001).

    ADS  CAS  PubMed  Google Scholar 

  60. Petrich, W., Anderson, M. H., Ensher, J. R. & Cornell, E. A. Stable, tightly confining magnetic trap for evaporative cooling of neutral atoms. Phys. Rev. Lett. 74, 3352–3355 (1995).

    ADS  CAS  PubMed  Google Scholar 

  61. Sinha, S. & Castin, Y. Dynamic instability of a rotating Bose–Einstein condensate. Phys. Rev. Lett. 87, 190402 (2001).

    ADS  CAS  PubMed  Google Scholar 

  62. Ronveaux, A. (ed.) Heun’s Differential Equations (Oxford Univ. Press, 1995).

  63. Bao, W. & Wang, H. An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose–Einstein condensates. J. Comput. Phys. 217, 612–626 (2006).

    ADS  MathSciNet  CAS  MATH  Google Scholar 

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Acknowledgements

We thank T. Pfau and his research group, J. Dunkel, A. Fetter, T. Senthil, T. Simula and W. Zwerger for discussions. This work was supported by the National Science Foundation (Center for Ultracold Atoms and grant no. PHY-2012110), Air Force Office of Scientific Research (FA9550-16-1-0324 and MURI Quantum Phases of Matter FA9550-14-1-0035), Office of Naval Research (N00014-17-1-2257), the DARPA A-PhI program through ARO grant W911NF-19-1-0511, and the Vannevar Bush Faculty Fellowship. A.S. acknowledges support from the NSF GRFP. M.Z. acknowledges funding from the Alexander von Humboldt Foundation.

Author information

Authors and Affiliations

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Contributions

B.M., A.S., C.C.W., P.B.P., Z.Y. and R.J.F. constructed the apparatus. B.M., A.S., C.C.W. and R.J.F. performed the measurements. B.M. and A.S. analysed the data. B.M. contributed the GP numerical simulations. V.C., R.J.F. and M.Z. developed the theoretical description. R.J.F. and M.Z. supervised the project. All authors contributed to interpretation of the results and preparation of the manuscript.

Corresponding author

Correspondence to Martin Zwierlein.

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Nature thanks Alessio Recati 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

Extended Data Fig. 1 Numerical GP simulation of the condensate evolution in the rotating frame.

ac, Time evolution of the condensate density without the addition of noise (top), with added phase noise (middle), and in the experiment (bottom). The frames correspond to times ωct/(2π) = 0, 4 and 6. de, Evolution of the structure factor Sk(t) extracted from the simulation (d) and the experiment (e) which show good agreement. f, The extracted instability growth rate as a function of wavevector k. The experimental measurements are shown by red points, and the Bogoliubov prediction by the black line. The blue line shows the result of the GP simulation. Here, the blue shading and the red error bars indicate 1σ standard error. This model captures the experimentally measured growth at wavevectors above the instability region provided by the linear Bogoliubov description.

Extended Data Fig. 2 Phase profile of the crystal.

a, b, The density profiles of the crystals in the experiment (a) and GP simulation (b) appear to contain vortices, which are marked in c and d. e, The phase of the macroscopic wavefunction can be inferred from the locations of the vortices in the experimental image. Note that additional contributions from undetected vortices may exist. f, The simulated phase profile from a GP simulation shows a similar structure of irrotational flow within each segment of the crystal. In both e and f, the phase shown is in the rotating frame.

Supplementary information

Supplementary Information

This Supplementary Information file contains details on our Bogoliubov stability analysis, as well as on superfluid hydrodynamics in the Landau gauge, and includes Supplementary Figures 1–3 and additional references.

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Mukherjee, B., Shaffer, A., Patel, P.B. et al. Crystallization of bosonic quantum Hall states in a rotating quantum gas. Nature 601, 58–62 (2022). https://doi.org/10.1038/s41586-021-04170-2

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