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Onset of vortex clustering and inverse energy cascade in dissipative quantum fluids

Nature Photonics volume 17, pages 451–456 (2023)Cite this article

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An Author Correction to this article was published on 28 August 2024

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Abstract

Turbulent phenomena are among the most striking effects that both classical and quantum fluids can exhibit. Although classical turbulence is ubiquitous in nature, the observation of quantum turbulence requires the precise manipulation of quantum fluids such as superfluid helium or atomic Bose–Einstein condensates. Here we demonstrate the turbulent dynamics of a two-dimensional quantum fluid of exciton–polaritons, hybrid light–matter quasiparticles, both by measuring the kinetic energy spectrum and showing the onset of vortex clustering. We demonstrate that the formation of clusters of quantum vortices is triggered by the increase of the incompressible kinetic energy per vortex, showing the tendency of the vortex-gas towards highly excited configurations despite the dissipative nature of our system. These results lay the basis for investigations of quantum turbulence in two-dimensional fluids of light.

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Fig. 1: Injection and trapping of a polariton quantum fluid.
Fig. 2: Vortex classification and velocity decomposition.
Fig. 3: Energy transfer and clusterization.
Fig. 4: Inverse energy cascade.

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Data availability

The data that support the findings of this study are available from the authors upon reasonable request.

Code availability

The codes used in this study will be provided upon reasonable request.

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References

  1. Frisch, U. Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1995).

  2. Hall, H. E., Vinen, W. F. & Shoenberg, D. The rotation of liquid helium II II. The theory of mutual friction in uniformly rotating helium II. Proc. R. Soc. Lond. A Math. Phys. Sci. 238, 215–234 (1956).

    Article  ADS  Google Scholar 

  3. Barenghi, C. F., Skrbek, L. & Sreenivasan, K. R. Introduction to quantum turbulence. Proc. Natl Acad. Sci. USA 111, 4647–4652 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  4. Feynman, R. P. in Progress in Low Temperature Physics Vol. 1 (ed Gorter, C. J.) Ch. 2, 17–53 (Elsevier, 1995).

  5. Barenghi, C. F. & Parker, N. G. A Primer on Quantum Fluids (Springer, 2016).

  6. Tsubota, M. Quantum turbulence. J. Phys. Soc. Jpn 77, 111006 (2008).

    Article  ADS  Google Scholar 

  7. Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995).

    Article  ADS  Google Scholar 

  8. Davis, K. B. et al. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. White, A. C., Anderson, B. P. & Bagnato, V. S. Vortices and turbulence in trapped atomic condensates. Proc. Natl Acad. Sci. USA 111, 4719–4726 (2014).

    Article  ADS  Google Scholar 

  11. Bradley, A. S. & Anderson, B. P. Energy spectra of vortex distributions in two-dimensional quantum turbulence. Phys. Rev. X 2, 041001 (2012).

    Google Scholar 

  12. 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).

    Article  ADS  Google Scholar 

  13. Billam, T. P., Reeves, M. T., Anderson, B. P. & Bradley, A. S. Onsager-Kraichnan condensation in decaying two-dimensional quantum turbulence. Phys. Rev. Lett. 112, 145301 (2014).

    Article  ADS  Google Scholar 

  14. Simula, T., Davis, M. J. & Helmerson, K. Emergence of order from turbulence in an isolated planar superfluid. Phys. Rev. Lett. 113, 165302 (2014).

    Article  ADS  Google Scholar 

  15. Groszek, A. J., Davis, M. J., Paganin, D. M., Helmerson, K. & Simula, T. P. Vortex thermometry for turbulent two-dimensional fluids. Phys. Rev. Lett. 120, 034504 (2018).

    Article  ADS  Google Scholar 

  16. Han, J. & Tsubota, M. Onsager vortex formation in two-component Bose-Einstein condensates. J. Phys. Soc. Jpn 87, 063601 (2018).

    Article  ADS  Google Scholar 

  17. Boffetta, G. & Ecke, R. E. Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451 (2012).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Onsager, L. Statistical hydrodynamics. Il Nuovo Cimento 6, 279 (1949).

    Article  ADS  MathSciNet  Google Scholar 

  20. Johnstone, S. P. et al. Evolution of large-scale flow from turbulence in a two-dimensional superfluid. Science 364, 1267–1271 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  21. Gauthier, G. et al. Giant vortex clusters in a two-dimensional quantum fluid. Science 364, 1264–1267 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  22. Arecchi, F. T., Giacomelli, G., Ramazza, P. L. & Residori, S. Vortices and defect statistics in two-dimensional optical chaos. Phys. Rev. Lett. 67, 3749–3752 (1991).

    Article  ADS  Google Scholar 

  23. Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).

    Article  ADS  Google Scholar 

  24. Fontaine, Q. et al. Observation of the Bogoliubov dispersion in a fluid of light. Phys. Rev. Lett. 121, 183604 (2018).

    Article  ADS  Google Scholar 

  25. Ballarini, D. et al. Directional Goldstone waves in polariton condensates close to equilibrium. Nat. Commun. 11, 217 (2020).

    Article  ADS  Google Scholar 

  26. Öztürk, F. E. et al. Observation of a non-Hermitian phase transition in an optical quantum gas. Science 372, 88–91 (2021).

    Article  ADS  Google Scholar 

  27. Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity. Nature 457, 291–295 (2009).

    Article  ADS  Google Scholar 

  28. Lagoudakis, K. G. et al. Quantized vortices in an exciton–polariton condensate. Nat. Phys. 4, 706–710 (2008).

    Article  Google Scholar 

  29. Sanvitto, D. et al. All-optical control of the quantum flow of a polariton condensate. Nat. Photon. 5, 610–614 (2011).

    Article  ADS  Google Scholar 

  30. Nardin, G. et al. Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid. Nat. Phys. 7, 635–641 (2011).

    Article  Google Scholar 

  31. Panico, R. et al. Dynamics of a vortex lattice in an expanding polariton quantum fluid. Phys. Rev. Lett. 127, 047401 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  33. Alyatkin, S., Töpfer, J. D., Askitopoulos, A., Sigurdsson, H. & Lagoudakis, P. G. Optical control of couplings in polariton condensate lattices. Phys. Rev. Lett. 124, 207402 (2020).

    Article  ADS  Google Scholar 

  34. Pieczarka, M. et al. Topological phase transition in an all-optical exciton-polariton lattice. Optica 8, 1084–1091 (2021).

    Article  ADS  Google Scholar 

  35. Caputo, D. et al. Topological order and thermal equilibrium in polariton condensates. Nat. Mater. 17, 145–151 (2018).

    Article  ADS  Google Scholar 

  36. Galantucci, L., Baggaley, A. W., Parker, N. G. & Barenghi, C. F. Crossover from interaction to driven regimes in quantum vortex reconnections. Proc. Natl Acad. Sci. USA 116, 12204–12211 (2019).

    Article  ADS  Google Scholar 

  37. Zamora, A. et al. Kibble-Zurek mechanism in driven dissipative systems crossing a nonequilibrium phase transition. Phys. Rev. Lett. 125, 095301 (2020).

    Article  ADS  Google Scholar 

  38. Berloff, N. G. Turbulence in exciton-polariton condensates. Preprint at https://arxiv.org/abs/1010.5225 (2010).

  39. Koniakhin, S., Bleu, O., Malpuech, G. & Solnyshkov, D. 2D quantum turbulence in a polariton quantum fluid. Chaos Solitons Fractals 132, 109574 (2020).

    Article  MathSciNet  Google Scholar 

  40. Skaugen, A. & Angheluta, L. Vortex clustering and universal scaling laws in two-dimensional quantum turbulence. Phys. Rev. E 93, 032106 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  41. Garcia-Orozco, A. D., Madeira, L., Galantucci, L., Barenghi, C. F. & Bagnato, V. S. Intra-scales energy transfer during the evolution of turbulence in a trapped Bose-Einstein condensate. Europhys. Lett. 130, 46001 (2020).

    Article  ADS  Google Scholar 

  42. Steger, M. et al. Long-range ballistic motion and coherent flow of long-lifetime polaritons. Phys. Rev. B 88, 235314 (2013).

    Article  ADS  Google Scholar 

  43. Alyatkin, S., Sigurdsson, H., Askitopoulos, A., Töpfer, J. D. & Lagoudakis, P. G. Quantum fluids of light in all-optical scatterer lattices. Nat. Commun. 12, 5571 (2021).

    Article  ADS  Google Scholar 

  44. Wertz, E. et al. Spontaneous formation and optical manipulation of extended polariton condensates. Nat. Phys. 6, 860–864 (2010).

    Article  Google Scholar 

  45. Donati, S. et al. Twist of generalized skyrmions and spin vortices in a polariton superfluid. Proc. Natl Acad. Sci. USA 113, 14926–14931 (2016).

    Article  ADS  Google Scholar 

  46. Valani, R. N., Groszek, A. J. & Simula, T. P. Einstein-Bose condensation of Onsager vortices. New J. Phys. 20, 053038 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  47. Kolmakov, G. V., McClintock, P. V. E. & Nazarenko, S. V. Wave turbulence in quantum fluids. Proc. Natl Acad. Sci. USA 111, 4727–4734 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  48. Kanai, T. & Guo, W. True mechanism of spontaneous order from turbulence in two-dimensional superfluid manifolds. Phys. Rev. Lett. 127, 095301 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  49. Bradley, A. S., Kumar, R. K., Pal, S. & Yu, X. Spectral analysis for compressible quantum fluids. Phys. Rev. A 106, 043322 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  50. Kolmogorov, A. N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS 30, 301 (1941).

    MathSciNet  Google Scholar 

  51. Comaron, P., Carusotto, I., Szymańska, M. H. & Proukakis, N. P. Non-equilibrium Berezinskii-Kosterlitz-Thouless transition in driven-dissipative condensates. Europhys. Lett. 133, 17002 (2021).

    Article  ADS  Google Scholar 

  52. Lagoudakis, K. et al. Probing the dynamics of spontaneous quantum vortices in polariton superfluids. Phys. Rev. Lett. 106, 115301 (2011).

    Article  ADS  Google Scholar 

  53. Caputo, D. et al. Josephson vortices induced by phase twisting a polariton superfluid. Nat. Photon. 13, 488–493 (2019).

    Article  ADS  Google Scholar 

  54. Michel, C., Boughdad, O., Albert, M., Larré, P.-E. & Bellec, M. Superfluid motion and drag-force cancellation in a fluid of light. Nat. Commun. 9, 2108 (2018).

    Article  ADS  Google Scholar 

  55. Situ, G. & Fleischer, J. W. Dynamics of the Berezinskii-Kosterlitz-Thouless transition in a photon fluid. Nat. Photon. 14, 517–522 (2020).

    Article  Google Scholar 

  56. Piekarski, C. et al. Measurement of the static structure factor in a paraxial fluid of light using Bragg-like spectroscopy. Phys. Rev. Lett. 127, 023401 (2021).

    Article  ADS  Google Scholar 

  57. Clark, L. W., Schine, N., Baum, C., Jia, N. & Simon, J. Observation of Laughlin states made of light. Nature 582, 41–45 (2020).

    Article  ADS  Google Scholar 

  58. Shelykh, I., Malpuech, G., Kavokin, K. V., Kavokin, A. V. & Bigenwald, P. Spin dynamics of interacting exciton polaritons in microcavities. Phys. Rev. B 70, 115301 (2004).

    Article  ADS  Google Scholar 

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Acknowledgements

R.P., A.S.L., D.T., G.G., M.D.G., V.A., D.S. and D.B. acknowledge the following projects: Italian Ministry of University (MUR) PRIN project ‘Interacting Photons in Polariton Circuits’ – INPhoPOL (grant no. 2017P9FJBS); the project ‘Hardware implementation of a polariton neural network for neuromorphic computing’ – Joint Bilateral Agreement CNR-RFBR (Russian Foundation for Basic Research) – Triennal Program 2021–2023; the MAECI project ‘Novel photonic platform for neuromorphic computing’, Joint Bilateral Project Italia-Polonia 2022–2023; PNRR MUR project ‘National Quantum Science and Technology Institute’ – NQSTI (PE0000023); PNRR MUR project ‘Integrated Infrastructure Initiative in Photonic and Quantum Sciences’ – I-PHOQS (IR0000016); Apulia Region, project ‘Progetto Tecnopolo per la Medicina di precisione’, Tecnomed 2 (grant no. Deliberazione della Giunta Regionale n. 2117 del 21/11/2018). M.M. and P.C. acknowledge funding from National Science Centre, Poland, grant no. 2016/22/E/ST3/00045. We are grateful to P. Cazzato for valuable technical support during the experiments. D.B. is grateful to B. Alegria for logistics and inspiration.

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Authors and Affiliations

  1. CNR NANOTEC, Institute of Nanotechnology, Lecce, Italy

    R. Panico, A. S. Lanotte, D. Trypogeorgos, G. Gigli, M. De Giorgi, V. Ardizzone, D. Sanvitto & D. Ballarini

  2. Dipartimento di Matematica e Fisica E. De Giorgi, Università del Salento, Lecce, Italy

    R. Panico, G. Gigli & V. Ardizzone

  3. Institute of Physics, Polish Academy of Sciences, Warsaw, Poland

    P. Comaron & M. Matuszewski

  4. Department of Physics and Astronomy, University College London, London, UK

    P. Comaron

  5. INFN, Sez. Lecce, Lecce, Italy

    A. S. Lanotte

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Contributions

R.P., P.C., M.M., D.B. and D.S. contributed to the formulation of the project. M.M., D.B. and D.S. supervised the project. R.P. and P.C. contributed to the data curation and, together with A.S.L. and D.B., to the formal analysis and development of the methodology. R.P., P.C. and D.T. cured the codes. Funding acquisition was managed by M.M., D.S. and G.G. All authors contributed to discussions and reviewing the manuscript.

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Correspondence to D. Sanvitto or D. Ballarini.

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Panico, R., Comaron, P., Matuszewski, M. et al. Onset of vortex clustering and inverse energy cascade in dissipative quantum fluids. Nat. Photon. 17, 451–456 (2023). https://doi.org/10.1038/s41566-023-01174-4

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