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Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate

Abstract

Revealing universal behaviours is a hallmark of statistical physics. Phenomena such as the stochastic growth of crystalline surfaces1 and of interfaces in bacterial colonies2, and spin transport in quantum magnets3,4,5,6 all belong to the same universality class, despite the great plurality of physical mechanisms they involve at the microscopic level. More specifically, in all these systems, space–time correlations show power-law scalings characterized by universal critical exponents. This universality stems from a common underlying effective dynamics governed by the nonlinear stochastic Kardar–Parisi–Zhang (KPZ) equation7. Recent theoretical works have suggested that this dynamics also emerges in the phase of out-of-equilibrium systems showing macroscopic spontaneous coherence8,9,10,11,12,13,14,15,16,17. Here we experimentally demonstrate that the evolution of the phase in a driven-dissipative one-dimensional polariton condensate falls in the KPZ universality class. Our demonstration relies on a direct measurement of KPZ space–time scaling laws18,19, combined with a theoretical analysis that reveals other key signatures of this universality class. Our results highlight fundamental physical differences between out-of-equilibrium condensates and their equilibrium counterparts, and open a paradigm for exploring universal behaviours in driven open quantum systems.

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Fig. 1: KPZ physics in the phase dynamics of a 1D polariton condensate.
Fig. 2: Probing the coherence of 1D polariton condensates.
Fig. 3: KPZ scaling in the coherence decay of a 1D polariton condensate.
Fig. 4: Analysis of the simulated phase dynamics.

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Data availability

All datasets generated and analysed during this study are available upon request from the corresponding authors. Source data are provided with this paper.

Code availability

All codes generated during this study are available upon request from the corresponding authors.

References

  1. Krug, J. & Meakin, P. Universal finite-size effects in the rate of growth processes. J. Phys. A 23, L987 (1990).

    Article  ADS  Google Scholar 

  2. Wakita, J.-i, Itoh, H., Matsuyama, T. & Matsushita, M. Self-affinity for the growing interface of bacterial colonies. J. Phys. Soc. Jpn 66, 67–72 (1997).

    Article  ADS  CAS  Google Scholar 

  3. Ljubotina, M., Žnidarič, M. & Prosen, T. Spin diffusion from an inhomogeneous quench in an integrable system. Nat. Commun. 8, 16117 (2017).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  4. Ljubotina, M., Žnidarič, M. & Prosen, T. Kardar–Parisi–Zhang physics in the quantum Heisenberg magnet. Phys. Rev. Lett. 122, 210602 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Scheie, A. et al. Detection of Kardar–Parisi–Zhang hydrodynamics in a quantum Heisenberg spin-1/2 chain. Nat. Phys. 17, 726–730 (2021).

    Article  CAS  Google Scholar 

  6. Wei, D. et al. Quantum gas microscopy of Kardar–Parisi–Zhang superdiffusion. Science 376, 716–720 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Kardar, M., Parisi, G. & Zhang, Y.-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  8. Altman, E., Sieberer, L. M., Chen, L., Diehl, S. & Toner, J. Two-dimensional superfluidity of exciton polaritons requires strong anisotropy. Phys. Rev. X 5, 011017 (2015).

    CAS  Google Scholar 

  9. Ji, K., Gladilin, V. N. & Wouters, M. Temporal coherence of one-dimensional nonequilibrium quantum fluids. Phys. Rev. B 91, 045301 (2015).

    Article  ADS  Google Scholar 

  10. He, L., Sieberer, L. M., Altman, E. & Diehl, S. Scaling properties of one-dimensional driven-dissipative condensates. Phys. Rev. B 92, 155307 (2015).

    Article  ADS  Google Scholar 

  11. Zamora, A., Sieberer, L., Dunnett, K., Diehl, S. & Szymańska, M. Tuning across universalities with a driven open condensate. Phys. Rev. X 7, 041006 (2017).

    Google Scholar 

  12. Comaron, P. et al. Dynamical critical exponents in driven-dissipative quantum systems. Phys. Rev. Lett. 121, 095302 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  13. Squizzato, D., Canet, L. & Minguzzi, A. Kardar–Parisi–Zhang universality in the phase distributions of one-dimensional exciton–polaritons. Phys. Rev. B 97, 195453 (2018).

    Article  ADS  CAS  Google Scholar 

  14. Amelio, I. & Carusotto, I. Theory of the coherence of topological lasers. Phys. Rev. X 10, 041060 (2020).

    CAS  Google Scholar 

  15. Ferrier, A., Zamora, A., Dagvadorj, G. & Szymańska, M. Searching for the Kardar–Parisi–Zhang phase in microcavity polaritons. Phys. Rev. B 105, 205301 (2022)

    Article  ADS  CAS  Google Scholar 

  16. Deligiannis, K., Squizzato, D., Minguzzi, A. & Canet, L. Accessing Kardar–Parisi–Zhang universality sub-classes with exciton polaritons. Europhys. Lett. 132, 67004 (2021).

    Article  ADS  Google Scholar 

  17. Mei, Q., Ji, K. & Wouters, M. Spatiotemporal scaling of two-dimensional nonequilibrium exciton–polariton systems with weak interactions. Phys. Rev. B 103, 045302 (2021).

    Article  ADS  CAS  Google Scholar 

  18. Family, F. & Vicsek, T. Scaling of the active zone in the eden process on percolation networks and the ballistic deposition model. J. Phys. A 18, L75 (1985).

    Article  ADS  Google Scholar 

  19. Halpin-Healy, T. & Zhang, Y.-C. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Phys. Rep. 254, 215–414 (1995).

    Article  ADS  Google Scholar 

  20. Krug, J. Origins of scale invariance in growth processes. Adv. Phys. 46, 139–282 (1997).

    Article  ADS  CAS  Google Scholar 

  21. Takeuchi, K. A. An appetizer to modern developments on the Kardar–Parisi–Zhang universality class. Physica A 504, 77–105 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Lauter, R., Mitra, A. & Marquardt, F. From Kardar–Parisi–Zhang scaling to explosive desynchronization in arrays of limit-cycle oscillators. Phys. Rev. E 96, 012220 (2017).

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  23. Chen, L. & Toner, J. et al. Universality for moving stripes: a hydrodynamic theory of polar active smectics. Phys. Rev. Lett. 111, 088701 (2013).

    Article  ADS  PubMed  Google Scholar 

  24. He, L., Sieberer, L. M. & Diehl, S. Space–time vortex driven crossover and vortex turbulence phase transition in one-dimensional driven open condensates. Phys. Rev. Lett. 118, 085301 (2017).

    Article  ADS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. Schneider, C. et al. Exciton–polariton trapping and potential landscape engineering. Rep. Prog. Phys. 80, 016503 (2016).

    Article  ADS  PubMed  Google Scholar 

  27. Deng, H., Weihs, G., Santori, C., Bloch, J. & Yamamoto, Y. Condensation of semiconductor microcavity exciton polaritons. Science 298, 199–202 (2002).

    Article  ADS  CAS  PubMed  Google Scholar 

  28. Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006).

    Article  ADS  CAS  PubMed  Google Scholar 

  29. Love, A. et al. Intrinsic decoherence mechanisms in the microcavity polariton condensate. Phys. Rev. Lett. 101, 067404 (2008).

    Article  ADS  CAS  PubMed  Google Scholar 

  30. Roumpos, G. et al. Power-law decay of the spatial correlation function in exciton-polariton condensates. Proc. Natl Acad. Sci. 109, 6467–6472 (2012).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  31. Fischer, J. et al. Spatial coherence properties of one dimensional exciton–polariton condensates. Phys. Rev. Lett. 113, 203902 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Bobrovska, N., Ostrovskaya, E. A. & Matuszewski, M. Stability and spatial coherence of nonresonantly pumped exciton-polariton condensates. Phys. Rev. B 90, 205304 (2014).

    Article  ADS  Google Scholar 

  33. Daskalakis, K. S., Maier, S. A. & Kéna-Cohen, S. Spatial coherence and stability in a disordered organic polariton condensate. Phys. Rev. Lett. 115, 035301 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  34. Estrecho, E. et al. Single-shot condensation of exciton polaritons and the hole burning effect. Nat. Commun. 9, 2944 (2018).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  35. Bobrovska, N., Matuszewski, M., Daskalakis, K. S., Maier, S. A. & Kéna-Cohen, S. Dynamical instability of a nonequilibrium exciton–polariton condensate. ACS Photon. 5, 111–118 (2018).

    Article  CAS  Google Scholar 

  36. Smirnov, L. A., Smirnova, D. A., Ostrovskaya, E. A. & Kivshar, Y. S. Dynamics and stability of dark solitons in exciton–polariton condensates. Phys. Rev. B 89, 235310 (2014).

    Article  ADS  Google Scholar 

  37. Liew, T. C. H. et al. Instability-induced formation and nonequilibrium dynamics of phase defects in polariton condensates. Phys. Rev. B 91, 085413 (2015).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  39. Baboux, F. et al. Unstable and stable regimes of polariton condensation. Optica 5, 1163–1170 (2018).

    Article  ADS  Google Scholar 

  40. Edwards, S. F. & Wilkinson, D. The surface statistics of a granular aggregate. Proc. R. Soc. Lond. A 381, 17–31 (1982).

    Article  ADS  Google Scholar 

  41. Prähofer, M. & Spohn, H. Exact scaling functions for one-dimensional stationary kpz growth. J. Stat. Phys. 115, 255–279 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Dennis, G. R., Hope, J. J. & Johnsson, M. T. Xmds2: fast, scalable simulation of coupled stochastic partial differential equations. Comput. Phys. Commun. 184, 201–208 (2013).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  43. Werner, M. & Drummond, P. Robust algorithms for solving stochastic partial differential equations. J. Comput. Phys. 132, 312–326 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank V. Goblot, D. Vajner and A. Toor for their assistance in the early development of the experiment. This work was supported by the Paris Ile-de-France Région in the framework of DIM SIRTEQ, the French RENATECH network, the H2020-FETFLAG project PhoQus (820392), the QUANTERA project Interpol (ANR-QUAN-0003-05), the European Research Council via the project ARQADIA (949730), EmergenTopo (865151) and RG.BIO (785932), the French government through the Programme Investissement d’Avenir (I-SITE ULNE / ANR-16-IDEX-0004 ULNE) managed by the Agence Nationale de la Recherche, and the Labex CEMPI (ANR-11-LABX-0007). L.C. acknowledges support from ANR (grant ANR-18-CE92-0019) and from Institut Universitaire de France.

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Contributions

Q.F. built the experimental set-up, performed the experiments and analysed the data. D.S. realized the theoretical calculations and numerical simulations. F.B. contributed to the design of the sample structure and initial characterization of the sample. A.L. and M.M. grew the sample by molecular beam epitaxy. I.S., L.L.G. and A.H. fabricated the polariton lattices. Q.F., D.S., I.A., M.W., I.C., A.A., M.R., A.M., L.C., S.R. and J.B. participated in the scientific discussions about all aspects of the work. Q.F., A.M., L.C., S.R. and J.B. wrote the original draft of the paper. Q.F., D.S., I.A., M.W., I.C., A.A., M.R., A.M., L.C., S.R. and J.B. reviewed and edited the paper into its current form. A.M., L.C., S.R. and J.B. supervised the work.

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Correspondence to Léonie Canet or Jacqueline Bloch.

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Nature thanks Sebastian Diehl, Michael Fraser and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Supplementary Information

This Supplementary Information file contains four sections with 16 figures: (I) Overview; (II) The theoretical model: emergence of KPZ dynamics in incoherently pumped polaritons; (III) Experiments: additional information and data; and (IV) Numerical simulations: discussion.

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Fontaine, Q., Squizzato, D., Baboux, F. et al. Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate. Nature 608, 687–691 (2022). https://doi.org/10.1038/s41586-022-05001-8

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