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.
This is a preview of subscription content, access via your institution
Subscribe to Nature+
Get immediate online access to the entire Nature family of 50+ journals
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
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).
The simulation and analysis code are available from the corresponding author upon reasonable request.
Stormer, H. L., Tsui, D. C. & Gossard, A. C. The fractional quantum Hall effect. Rev. Mod. Phys. 71, S298–S305 (1999).
Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Wigner, E. On the interaction of electrons in metals. Phys. Rev. 46, 1002–1011 (1934).
Yoshioka, D. & Fukuyama, H. Charge density wave state of two-dimensional electrons in strong magnetic fields. J. Phys. Soc. Jpn. 47, 394–402 (1979).
Lam, P. K. & Girvin, S. M. Liquid–solid transition and the fractional quantum-Hall effect. Phys. Rev. B 30, 473–475 (1984).
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).
Jiang, H. W. et al. Quantum liquid versus electron solid around ν = 1/5 Landau-level filling. Phys. Rev. Lett. 65, 633–636 (1990).
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).
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).
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).
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).
Ho, T.-L. Bose–Einstein condensates with large number of vortices. Phys. Rev. Lett. 87, 060403 (2001).
Oktel, M. Ö. Vortex lattice of a Bose–Einstein condensate in a rotating anisotropic trap. Phys. Rev. A 69, 023618 (2004).
Sinha, S. & Shlyapnikov, G. V. Two-dimensional Bose–Einstein condensate under extreme rotation. Phys. Rev. Lett. 94, 150401 (2005).
Cooper, N. R. Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008).
Aftalion, A., Blanc, X. & Lerner, N. Fast rotating condensates in an asymmetric harmonic trap. Phys. Rev. A 79, 011603 (2009).
Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).
Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic systems. Science 338, 1604–1606 (2012).
Senthil, T. & Levin, M. Integer quantum Hall effect for bosons. Phys. Rev. Lett. 110, 046801 (2013).
Goldman, N., Juzeliūnas, G., Öhberg, P. & Spielman, I. B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 77, 126401 (2014).
Fletcher, R. J. et al. Geometric squeezing into the lowest Landau level. Science 372, 1318–1322 (2021).
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).
Nozières, P. Is the roton in superfluid 4He the ghost of a Bragg spot? J. Low Temp. Phys. 137, 45–67 (2004).
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).
Galitski, V. & Spielman, I. B. Spin–orbit coupling in quantum gases. Nature 494, 49–54 (2013).
Chalopin, T. et al. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nat. Phys. 16, 1017–1021 (2020).
Struck, J. et al. Tunable gauge potential for neutral and spinless particles in driven optical lattices. Phys. Rev. Lett. 108, 225304 (2012).
Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).
Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).
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).
Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).
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).
Bretin, V., Stock, S., Seurin, Y. & Dalibard, J. Fast rotation of a Bose–Einstein condensate. Phys. Rev. Lett. 92, 050403 (2004).
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).
Iordanskii, S. V. & Pitaevskii, L. P. Bose condensation of moving rotons. Sov. Phys. Usp. 23, 317–318 (1980).
Pitaevskii, L. P. Layered structure of superfluid 4He with super-critical motion. JETP Lett. 39, 511–514 (1984).
Martone, G. I., Recati, A. & Pavloff, N. Supersolidity of cnoidal waves in an ultracold Bose gas. Phys. Rev. Res. 3, 013143 (2021).
Mottl, R. et al. Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions. Science 336, 1570–1573 (2012).
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).
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).
Li, J.-R. et al. A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates. Nature 543, 91–94 (2017).
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).
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).
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).
Petter, D. et al. Probing the roton excitation spectrum of a stable dipolar Bose gas. Phys. Rev. Lett. 122, 183401 (2019).
Hertkorn, J. et al. Density fluctuations across the superfluid–supersolid phase transition in a dipolar quantum gas. Phys. Rev. X 11, 011037 (2021).
Schmidt, J.-N. et al. Roton excitations in an oblate dipolar quantum gas. Phys. Rev. Lett. 126, 193002 (2021).
Guo, M. et al. The low-energy Goldstone mode in a trapped dipolar supersolid. Nature 574, 386–389 (2019).
Tanzi, L. et al. Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas. Nature 574, 382–385 (2019).
Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 21012 (2019).
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).
Cerfon, A. J. Vortex dynamics and shear-layer instability in high-intensity cyclotrons. Phys. Rev. Lett. 116, 174801 (2016).
Chandrasekhar, S. C. Hydrodynamic and Hydromagnetic Stability (Clarendon Press, 1961).
Landau, L. D. & Lifshitz, E. M. Fluid Mechanics (Pergamon Press, 1987).
Finne, A. P. et al. Dynamics of vortices and interfaces in superfluid 3He. Rep. Prog. Phys. 69, 3157–3230 (2006).
Baggaley, A. W. & Parker, N. G. Kelvin–Helmholtz instability in a single-component atomic superfluid. Phys. Rev. A 97, 053608 (2018).
Fetter, A. L. & Walecka, J. D. Quantum Theory of Many-particle Systems (McGraw-Hill, 1971).
Recati, A., Zambelli, F. & Stringari, S. Overcritical rotation of a trapped Bose–Einstein condensate. Phys. Rev. Lett. 86, 377–380 (2001).
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).
Sinha, S. & Castin, Y. Dynamic instability of a rotating Bose–Einstein condensate. Phys. Rev. Lett. 87, 190402 (2001).
Ronveaux, A. (ed.) Heun’s Differential Equations (Oxford Univ. Press, 1995).
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).
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.
The authors declare no competing interests.
Peer review information
Nature thanks Alessio Recati 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
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.
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.
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.
About this article
Cite this article
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