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Crystallization of bosonic quantum Hall states in a rotating quantum gas

Nature volume 601, pages 58–62 (2022)Cite this article

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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.

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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.

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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.

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Authors and Affiliations

  1. MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Biswaroop Mukherjee, Airlia Shaffer, Parth B. Patel, Zhenjie Yan, Cedric C. Wilson, Valentin Crépel, Richard J. Fletcher & Martin Zwierlein

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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.

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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.

a–c, 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. d–e, 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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