Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Advertisement

Communications Physics
  • View all journals
  • Search
  • My Account Login
  • Content Explore content
  • About the journal
  • Publish with us
  • Sign up for alerts
  • RSS feed
  1. nature
  2. communications physics
  3. articles
  4. article
Generation of wave turbulence in dipolar gases driven across their phase transitions
Download PDF
Download PDF
  • Article
  • Open access
  • Published: 16 January 2026

Generation of wave turbulence in dipolar gases driven across their phase transitions

  • George A. Bougas  ORCID: orcid.org/0009-0002-5643-92931,
  • Koushik Mukherjee2 &
  • Simeon I. Mistakidis  ORCID: orcid.org/0000-0002-5118-57921 

Communications Physics , Article number:  (2026) Cite this article

  • 959 Accesses

  • Metrics details

We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors present which affect the content, and all legal disclaimers apply.

Subjects

  • Theoretical physics
  • Ultracold gases

Abstract

Ultracold quantum gases with long-range anisotropic interactions host novel exotic phases of matter, such as supersolids, exhibiting both rigid and superfluid characteristics. The impact of this interplay on the out-of-equilibrium dynamics of dipolar gases, and in particular its connection with universal turbulent behavior, remains highly unexplored. Here, upon considering a dipolar Bose-Einstein condensate of dysprosium atoms being dynamically driven across the supersolid-superfluid phase transition and vice versa, we unveil the emergence of a robust nonequilibrium quasi-steady state. This state displays self-similar momentum distributions exhibiting algebraic decay at large momenta, with scaling exponents supporting the existence of wave turbulence. We demonstrate that supersolidity sustaining higher-lying momenta, associated with the roton minimum, promotes the development of turbulence. Our results provide a stepping stone toward unraveling and exploiting turbulent and self-similar behavior in anisotropically long-range interacting quantum gases amenable in current experiments.

Similar content being viewed by others

Observation of vortices in a dipolar supersolid

Article 06 November 2024

Two-dimensional supersolidity in a dipolar quantum gas

Article 18 August 2021

Supersolidity in ultracold dipolar gases

Article 09 October 2023

Data availability

The data associated with this work are available from the corresponding author upon request.

Code availability

The code associated with this work are available from the corresponding author upon request.

References

  1. Kolmogorov, A. N. The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301 (1941).

    Google Scholar 

  2. Obukhov, A. On the distribution of energy in the spectrum of turbulent flow. Dokl. Akad. Nauk SSSR. 32, 22 (1941).

  3. Kraichnan, R. H. Inertial ranges in two-dimensional turbulene. Phys. Fluids 10, 1417 (1967).

    Google Scholar 

  4. Frisch, U. Turbulence: the legacy of AN Kolmogorov (Cambridge University Press, 1995).

  5. Hwang, P. A., Wang, D. W., Walsh, E. J., Krabill, W. B. & Swift, R. N. Airborne measurements of the wavenumber spectra of ocean surface waves. Part I: spectral slope and dimensionless spectral coefficient. J. Phys. Oceanogr. 30, 2753 (2000).

    Google Scholar 

  6. Cobelli, P. et al. Different regimes for water wave turbulence. Phys. Rev. Lett. 107, 214503 (2011).

    Google Scholar 

  7. Choi, H. J., Lim, S. T., Lai, P.-Y. & Chan, C. K. Turbulent drag reduction and degradation of DNA. Phys. Rev. Lett. 89, 088302 (2002).

    Google Scholar 

  8. Bakas, N. A. & Ioannou, P. J. Emergence of large scale structure in barotropic β-plane turbulence. Phys. Rev. Lett. 110, 224501 (2013).

    Google Scholar 

  9. Diamond, P. H., Itoh, S.-I., Itoh, K. & Hahm, T. S. Zonal flows in plasma-a review. Plasma Phys. Control. Fusion 47, R35 (2005).

    Google Scholar 

  10. Mösta, P. et al. A large-scale dynamo and magnetoturbulence in rapidly rotating core-collapse supernovae. Nature 528, 376 (2015).

    Google Scholar 

  11. Chorin, A. J. Vorticity and Turbulence (Springer, 1994).

  12. Yao, J. & Hussain, F. Vortex reconnection and turbulence cascade. Annu. Rev. Fluid Mech. 54, 317 (2022).

    Google Scholar 

  13. Zakharov, V. E., L’vov, V. S. & Falkovich, G. Kolmogorov spectra of turbulence I: Wave turbulence (Springer, 1992).

  14. Nazarenko, S. Wave turbulence, 825 (Springer, 2011).

  15. Dyachenko, S., Newell, A., Pushkarev, A. & Zakharov, V. Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear schrödinger equation. Phys. D: Nonlinear Phenom. 57, 96 (1992).

    Google Scholar 

  16. Madeira, L., Caracanhas, M. A., dos Santos, F. & Bagnato, V. S. Quantum turbulence in quantum gases. Annu. Rev. Condens. Matter Phys. 11, 37 (2020).

    Google Scholar 

  17. Tsatsos, M. C. et al. Quantum turbulence in trapped atomic Bose–Einstein condensates. Phys. Rep. 622, 1 (2016).

    Google Scholar 

  18. Navon, N., Gaunt, A. L., Smith, R. P. & Hadzibabic, Z. Emergence of a turbulent cascade in a quantum gas. Nature 539, 72 (2016).

    Google Scholar 

  19. Vinen, W. & Niemela, J. Quantum turbulence. J. Low Temp. Phys. 128, 167 (2002).

    Google Scholar 

  20. Seman, J. A. et al. Route to turbulence in a trapped Bose-Einstein condensate. Laser Phys. Lett. 8, 691 (2011).

    Google Scholar 

  21. Yukalov, V. I., Yukalova, E. P. & Bagnato, V. S. Trapped Bose-Einstein condensates with nonlinear coherent modes. Laser Phys. 33, 123001 (2023).

    Google Scholar 

  22. Kwon, W. J., Moon, G., Choi, J. -y, Seo, S. W. & Shin, Y. -i Relaxation of superfluid turbulence in highly oblate bose-einstein condensates. Phys. Rev. A 90, 063627 (2014).

    Google Scholar 

  23. Henn, E. A. L., Seman, J. A., Roati, G., MagalhĂŁes, K. M. F. & Bagnato, V. S. Emergence of turbulence in an oscillating Bose-Einstein condensate. Phys. Rev. Lett. 103, 045301 (2009).

    Google Scholar 

  24. Neely, T. W. et al. Characteristics of two-dimensional quantum turbulence in a compressible superfluid. Phys. Rev. Lett. 111, 235301 (2013).

    Google Scholar 

  25. Gauthier, G. et al. Giant vortex clusters in a two-dimensional quantum fluid. Science 364, 1264 (2019).

    Google Scholar 

  26. Johnstone, S. P. et al. Evolution of large-scale flow from turbulence in a two-dimensional superfluid. Science 364, 1267 (2019).

    Google Scholar 

  27. Baggaley, A. W. & Barenghi, C. F. Spectrum of turbulent Kelvin-waves cascade in superfluid helium. Phys. Rev. B 83, 134509 (2011).

    Google Scholar 

  28. Mossman, M. E., Hoefer, M. A., Julien, K., Kevrekidis, P. G. & Engels, P. Dissipative shock waves generated by a quantum-mechanical piston. Nat. Commun. 9, 4665 (2018).

    Google Scholar 

  29. Ghosh Dastidar, M., Das, S., Mukherjee, K. & Majumder, S. Pattern formation and evidence of quantum turbulence in binary Bose-Einstein condensates interacting with a pair of Laguerre-Gaussian laser beams. Phys. Lett. A 421, 127776 (2022).

    Google Scholar 

  30. Das, S., Mukherjee, K. & Majumder, S. Vortex formation and quantum turbulence with rotating paddle potentials in a two-dimensional binary Bose-Einstein condensate. Phys. Rev. A 106, 023306 (2022).

    Google Scholar 

  31. Bulgac, A., Kafker, M., Abdurrahman, I. & Wlazłowski, G. Quantum turbulence, superfluidity, non-Markovian dynamics, and wave function thermalization. Phys. Rev. Res. 6, L042003 (2024).

    Google Scholar 

  32. Gałka, M. et al. Emergence of isotropy and dynamic scaling in 2d wave turbulence in a homogeneous Bose gas. Phys. Rev. Lett. 129, 190402 (2022).

    Google Scholar 

  33. Navon, N. et al. Synthetic dissipation and cascade fluxes in a turbulent quantum gas. Science 366, 382 (2019).

    Google Scholar 

  34. Dogra, L. H. et al. Universal equation of state for wave turbulence in a quantum gas. Nature 620, 521 (2023).

    Google Scholar 

  35. Martirosyan, G., Fujimoto, K. & Navon, N. An equation of state for turbulence in the gross-pitaevskii model. Preprint at https://doi.org/10.48550/arXiv.2407.08738 (2024).

  36. Zhu, Y., Krstulovic, G. & Nazarenko, S. Turbulence and far-from-equilibrium equation of state of Bogoliubov waves in Bose-Einstein condensates. Preprint at https://doi.org/10.48550/arXiv.2408.15163 (2024).

  37. Karailiev, A. et al. Observation of an inverse turbulent-wave cascade in a driven quantum gas. Phys. Rev. Lett. 133, 243402 (2024).

    Google Scholar 

  38. Bland, T., Stagg, G. W., Galantucci, L., Baggaley, A. & Parker, N. G. Quantum ferrofluid turbulence. Phys. Rev. Lett. 121, 174501 (2018).

    Google Scholar 

  39. Chomaz, L. et al. Dipolar physics: a review of experiments with magnetic quantum gases. Rep. Prog. Phys. 86, 026401 (2022).

    Google Scholar 

  40. Lahaye, T., Menotti, C., Santos, L., Lewenstein, M. & Pfau, T. The physics of dipolar bosonic quantum gases. Rep. Prog. Phys. 72, 126401 (2009).

    Google Scholar 

  41. Sabari, S., Kishor Kumar, R. & Tomio, L. Vortex dynamics and turbulence in dipolar Bose-Einstein condensates. Phys. Rev. A 109, 023313 (2024).

    Google Scholar 

  42. Prasad, S. B., Parker, N. G. & Baggaley, A. W. Crow instability of vortex lines in dipolar superfluids. Sci. Rep. 15, 33364 (2025).

    Google Scholar 

  43. Altmeyer, S., Do, Y. & Lai, Y.-C. Transition to turbulence in Taylor-Couette ferrofluidic flow. Sci. Rep. 5, 10781 (2015).

    Google Scholar 

  44. Mouraya, S., Pan, N. & Banerjee, S. Stationary and nonstationary energy cascades in homogeneous ferrofluid turbulence. Phys. Rev. Fluids 9, 094604 (2024).

    Google Scholar 

  45. Altmeyer, S., Do, Y. & Lai, Y.-C. Magnetic field induced flow pattern reversal in a ferrofluidic Taylor-Couette system. Sci. Rep. 5, 18589 (2015).

    Google Scholar 

  46. Lu, M., Burdick, N. Q., Youn, S. H. & Lev, B. L. Strongly dipolar Bose-Einstein condensate of dysprosium. Phys. Rev. Lett. 107, 190401 (2011).

    Google Scholar 

  47. Aikawa, K. et al. Bose-Einstein condensation of erbium. Phys. Rev. Lett. 108, 210401 (2012).

    Google Scholar 

  48. Miyazawa, Y., Inoue, R., Matsui, H., Nomura, G. & Kozuma, M. Bose-Einstein condensation of europium. Phys. Rev. Lett. 129, 223401 (2022).

    Google Scholar 

  49. Kadau, H. et al. Observing the Rosensweig instability of a quantum ferrofluid. Nature 530, 194 (2016).

    Google Scholar 

  50. Schmitt, M., Wenzel, M., Böttcher, F., Ferrier-Barbut, I. & Pfau, T. Self-bound droplets of a dilute magnetic quantum liquid. Nature 539, 259 (2016).

    Google Scholar 

  51. Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau, T. Observation of quantum droplets in a strongly dipolar Bose gas. Phys. Rev. Lett. 116, 215301 (2016).

    Google Scholar 

  52. Chomaz, L. et al. Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid. Phys. Rev. X 6, 041039 (2016).

    Google Scholar 

  53. Wächtler, F. & Santos, L. Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einstein condensates. Phys. Rev. A 94, 043618 (2016).

    Google Scholar 

  54. Wächtler, F. & Santos, L. Quantum filaments in dipolar Bose-Einstein condensates. Phys. Rev. A 93, 061603 (2016).

    Google Scholar 

  55. Chester, G. V. Speculations on Bose-Einstein condensation and quantum crystals. Phys. Rev. A 2, 256 (1970).

    Google Scholar 

  56. Leggett, A. J. Can a Solid Be “Superfluid"?. Phys. Rev. Lett. 25, 1543 (1970).

    Google Scholar 

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

    Google Scholar 

  58. Böttcher, F. et al. Transient supersolid properties in an array of dipolar quantum droplets. Phys. Rev. X 9, 011051 (2019).

    Google Scholar 

  59. Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 021012 (2019).

    Google Scholar 

  60. Norcia, M. A. et al. Two-dimensional supersolidity in a dipolar quantum gas. Nature 596, 357 (2021).

    Google Scholar 

  61. Biagioni, G. et al. Dimensional crossover in the superfluid-supersolid quantum phase transition. Phys. Rev. X 12, 021019 (2022).

    Google Scholar 

  62. Bland, T. et al. Two-dimensional supersolid formation in dipolar condensates. Phys. Rev. Lett. 128, 195302 (2022).

    Google Scholar 

  63. Petter, D. et al. Probing the Roton excitation spectrum of a stable dipolar Bose Gas. Phys. Rev. Lett. 122, 183401 (2019).

    Google Scholar 

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

    Google Scholar 

  65. Boninsegni, M. & Prokof’ev, N. V. Colloquium: supersolids: what and where are they?. Rev. Mod. Phys. 84, 759 (2012).

    Google Scholar 

  66. Poli, E., Baillie, D., Ferlaino, F. & Blakie, P. B. Excitations of a two-dimensional supersolid. Phys. Rev. A 110, 053301 (2024).

    Google Scholar 

  67. Lee, T. D., Huang, K. & Yang, C. N. Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, 1135 (1957).

    Google Scholar 

  68. Zhang, F. & Yin, L. Phonon stability of quantum droplets in dipolar Bose gases. Chinese Phys. Lett. 39, 060301 (2022).

    Google Scholar 

  69. Lima, A. R. P. & Pelster, A. Quantum fluctuations in dipolar Bose gases. Phys. Rev. A 84, 041604 (2011).

    Google Scholar 

  70. Baillie, D. & Blakie, P. B. Droplet crystal ground states of a dipolar Bose gas. Phys. Rev. Lett. 121, 195301 (2018).

    Google Scholar 

  71. Casotti, E. et al. Observation of vortices in a dipolar supersolid. Nature 635, 327 (2024).

    Google Scholar 

  72. Ronen, S., Bortolotti, D. C. E., Blume, D. & Bohn, J. L. Dipolar Bose-Einstein condensates with dipole-dependent scattering length. Phys. Rev. A 74, 033611 (2006).

    Google Scholar 

  73. Yi, S. & You, L. Trapped condensates of atoms with dipole interactions. Phys. Rev. A 63, 053607 (2001).

    Google Scholar 

  74. Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225 (2010).

    Google Scholar 

  75. Maier, T. et al. Broad universal feshbach resonances in the chaotic spectrum of dysprosium atoms. Phys. Rev. A 92, 060702 (2015).

    Google Scholar 

  76. Schützhold, R., Uhlmann, M., Xu, Y. & Fischer, U. R. Mean-field expansion in Bose–Einstein condensates with finite-range interactions. Int. J. Mod. Phys. B 20, 3555 (2006).

    Google Scholar 

  77. Zhu, Y., Semisalov, B., Krstulovic, G. & Nazarenko, S. Direct and inverse cascades in turbulent Bose-Einstein condensates. Phys. Rev. Lett. 130, 133001 (2023).

    Google Scholar 

  78. Kolmakov, G. V., McClintock, P. V. E. & Nazarenko, S. V. Wave turbulence in quantum fluids. Proc. Natl. Acad. Sci. USA 111, 4727 (2014).

    Google Scholar 

  79. Ripley, B. T. E., Baillie, D. & Blakie, P. B. Two-dimensional supersolidity in a planar dipolar Bose gas. Phys. Rev. A 108, 053321 (2023).

    Google Scholar 

  80. Hertkorn, J. et al. Pattern formation in quantum ferrofluids: from supersolids to superglasses. Phys. Rev. Res. 3, 033125 (2021).

    Google Scholar 

  81. Reeves, M. T., Billam, T. P., Anderson, B. P. & Bradley, A. S. Inverse energy cascade in forced two-dimensional quantum turbulence. Phys. Rev. Lett. 110, 104501 (2013).

    Google Scholar 

  82. Halder, S. et al. Control of 164Dy Bose-Einstein condensate phases and dynamics with dipolar anisotropy. Phys. Rev. Res. 4, 043124 (2022).

    Google Scholar 

  83. Mukherjee, K., Tengstrand, M. N., Cardinale, T. A. & Reimann, S. M. Supersolid stacks in antidipolar Bose-Einstein condensates. Phys. Rev. A 108, 023302 (2023).

    Google Scholar 

  84. Alexakis, A. Quasi-two-dimensional turbulence. Rev. Mod. Plasma Phys. 7, 31 (2023).

    Google Scholar 

  85. Danilov, S. D. & Gurarie, D. Quasi-two-dimensional turbulence. Phys. Usp. 43, 863 (2000).

    Google Scholar 

  86. Crank, J. & Nicolson, P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Math. Proc. Camb. Philos. Soc. 43, 50–67 (1947).

  87. Antoine, X., Bao, W. & Besse, C. Computational methods for the dynamics of the nonlinear schrödinger/gross–pitaevskii equations. Comput. Phys. Commun. 184, 2621 (2013).

    Google Scholar 

Download references

Acknowledgements

S. I. M. acknowledges support from the Missouri University of Science and Technology, Department of Physics, in the framework of a Startup fund. Financial support by the Knut and Alice Wallenberg Foundation and the Swedish Research Council are also acknowledged (K. M.). S. I. M. acknowledges extensive discussions with H. R. Sadeghpour in the context of universal dynamics and supersolid character. K.M. gratefully acknowledges many discussions with Stephanie M. Reimann on the topic of supersolidity. The authors acknowledge the anonymous referees for their insightful comments.

Author information

Authors and Affiliations

  1. Department of Physics and LAMOR, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    George A. Bougas & Simeon I. Mistakidis

  2. Mathematical Physics and NanoLund, LTH, Lund University, Box 118, 22100, Lund, Sweden

    Koushik Mukherjee

Authors
  1. George A. Bougas
    View author publications

    Search author on:PubMed Google Scholar

  2. Koushik Mukherjee
    View author publications

    Search author on:PubMed Google Scholar

  3. Simeon I. Mistakidis
    View author publications

    Search author on:PubMed Google Scholar

Contributions

G.A.B. performed the numerical simulations and carried out the associated analysis. K.M. benchmarked part of the simulations and developed numerical scripts for portions of the analysis. S.I.M. conceived the idea of the work, supervised and funded the project. All authors contributed to the interpretation of the results and the writing of the manuscript.

Corresponding author

Correspondence to George A. Bougas.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Communications Physics thanks Thomas Bland and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Material

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bougas, G.A., Mukherjee, K. & Mistakidis, S.I. Generation of wave turbulence in dipolar gases driven across their phase transitions. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02487-w

Download citation

  • Received: 16 November 2024

  • Accepted: 30 December 2025

  • Published: 16 January 2026

  • DOI: https://doi.org/10.1038/s42005-026-02487-w

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Download PDF

Advertisement

Explore content

  • Research articles
  • Reviews & Analysis
  • News & Comment
  • Collections
  • Follow us on Twitter
  • Sign up for alerts
  • RSS feed

About the journal

  • Aims & Scope
  • Journal Information
  • Open Access Fees and Funding
  • Journal Metrics
  • Editors
  • Editorial Board
  • Calls for Papers
  • Editorial Values Statement
  • Editorial policies
  • Referees
  • Conferences
  • Contact

Publish with us

  • For authors
  • Language editing services
  • Open access funding
  • Submit manuscript

Search

Advanced search

Quick links

  • Explore articles by subject
  • Find a job
  • Guide to authors
  • Editorial policies

Communications Physics (Commun Phys)

ISSN 2399-3650 (online)

nature.com sitemap

About Nature Portfolio

  • About us
  • Press releases
  • Press office
  • Contact us

Discover content

  • Journals A-Z
  • Articles by subject
  • protocols.io
  • Nature Index

Publishing policies

  • Nature portfolio policies
  • Open access

Author & Researcher services

  • Reprints & permissions
  • Research data
  • Language editing
  • Scientific editing
  • Nature Masterclasses
  • Research Solutions

Libraries & institutions

  • Librarian service & tools
  • Librarian portal
  • Open research
  • Recommend to library

Advertising & partnerships

  • Advertising
  • Partnerships & Services
  • Media kits
  • Branded content

Professional development

  • Nature Awards
  • Nature Careers
  • Nature Conferences

Regional websites

  • Nature Africa
  • Nature China
  • Nature India
  • Nature Japan
  • Nature Middle East
  • Privacy Policy
  • Use of cookies
  • Legal notice
  • Accessibility statement
  • Terms & Conditions
  • Your US state privacy rights
Springer Nature

© 2026 Springer Nature Limited

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing