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.

  • Review Article
  • Published:

Non-equilibrium Bose–Einstein condensation in photonic systems

Nature Reviews Physics volume 4, pages 470–488 (2022)Cite this article

Subjects

Abstract

The study of Bose–Einstein condensation effects in photonic systems has revealed a rich phenomenology related to spontaneous coherence generation in driven-dissipative spatially extended systems and is providing a new platform for the study of non-equilibrium phase transitions and critical behaviours. In this Review, we give an interdisciplinary overview of condensation phenomena in photonic systems. We cover a wide range of systems, from lasers to photon condensates in dye-filled cavities, to excitons in semiconductor heterostructures, to microcavity polaritons, as well as emerging systems, such as mode-locked lasers and classical light waves. Our aim is to highlight novel universal phenomena that stem from the driven-dissipative, non-equilibrium nature of these systems and affect the static, dynamic, superfluid and coherence properties of the condensate. Finally, we provide our perspectives on the future of fundamental science and technological applications in this field.

Key points

  • Bose–Einstein condensation in photonic systems is an example of a non-equilibrium phase transition leading to spontaneous coherence generation in a spatially extended system.

  • The physical platforms can be classified in terms of the typical strength of the driven-dissipative condition — from quasi-equilibrium towards fully non-equilibrium regimes — as follows: excitons and classical light in waveguides, then photons in dye-filled cavities, exciton–polaritons and, finally, optical parametric oscillators and conventional lasers.

  • The non-equilibrium condition is responsible for a richer mean-field dynamics, such as condensation in excited states, topological lasing, outward flows in localized condensates, spiralling condensate phases around quantized vortices, diffusive Goldstone modes and generalized Landau criteria for superfluidity.

  • Classical and quantum fluctuations in the driven-dissipative regime display novel critical behaviours such as Kardar–Parisi–Zhang spatiotemporal scalings in the coherence of the emitted light.

  • Strongly interacting photon fluids in the presence of strong non-linearities have been anticipated to feature exotic strongly correlated and/or topological states of photonic matter, such as Mott insulators and fractional quantum Hall states.

  • All these developments are expected to lead to novel sources of coherent and/or quantum light, as well as to sophisticated schemes for optical computation.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: General features of condensation.
Fig. 2: Bose–Einstein condensation of photons.
Fig. 3: Bose–Einstein condensation of excitons.
Fig. 4: Bose–Einstein condensation of exciton–polaritons.
Fig. 5: New features of non-equilibrium condensates.
Fig. 6: Future perspectives.

Similar content being viewed by others

References

  1. Huang, K. Statistical Mechanics (Wiley, 1987).

  2. Pitaevskii, L. P. & Stringari, S. Bose–Einstein Condensation (Clarendon, 2003).

    MATH  Google Scholar 

  3. Graham, R. & Haken, H. Laserlight — first example of a second-order phase transition far away from thermal equilibrium. Z. Phys. 237, 31–46 (1970).

    Article  ADS  Google Scholar 

  4. DeGiorgio, V. & Scully, M. O. Analogy between the laser threshold region and a second-order phase transition. Phys. Rev. A 2, 1170–1177 (1970).

    Article  ADS  Google Scholar 

  5. Schmittmann, B. & Zia, R. K. P. Statistical mechanics of driven diffusive systems. Phase Transit. Crit. Phenom. 17, 3–214 (1995).

    Article  Google Scholar 

  6. Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Lundeberg, L., Lousberg, G., Boiko, D. & Kapon, E. Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers. Appl. Phys. Lett. 90, 021103 (2007).

    Article  ADS  Google Scholar 

  9. Staliūnas, K. & Sánchez-Morcillo, V. Transverse Patterns in Nonlinear Optical Resonators (Springer, 2003).

  10. Krupa, K. et al. Spatial beam self-cleaning in multimode fibres. Nat. Photonics 11, 237–241 (2017).

    Article  ADS  Google Scholar 

  11. Baudin, K. et al. Classical Rayleigh-Jeans condensation of light waves: observation and thermodynamic characterization. Phys. Rev. Lett. 125, 244101 (2020).

    Article  ADS  Google Scholar 

  12. Oren, G., Bekker, A. & Fischer, B. Classical condensation of light pulses in a loss trap in a laser cavity. Optica 1, 145–148 (2014).

    Article  ADS  Google Scholar 

  13. Klaers, J., Schmitt, J., Vewinger, F. & Weitz, M. Bose–Einstein condensation of photons in an optical microcavity. Nature 468, 545–548 (2010).

    Article  ADS  Google Scholar 

  14. High, A. et al. Spontaneous coherence in a cold exciton gas. Nature 483, 584–588 (2012).

    Article  ADS  Google Scholar 

  15. Alloing, M. et al. Evidence for a Bose-Einstein condensate of excitons. EPL 107, 10012 (2014).

    Article  ADS  Google Scholar 

  16. Stevenson, R. M. et al. Continuous wave observation of massive polariton redistribution by stimulated scattering in semiconductor microcavities. Phys. Rev. Lett. 85, 3680–3683 (2000).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. Sun, Y. et al. Bose-Einstein condensation of long-lifetime polaritons in thermal equilibrium. Phys. Rev. Lett. 118, 016602 (2017).

    Article  ADS  Google Scholar 

  20. Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation. Rev. Mod. Phys. 82, 1489–1537 (2010).

    Article  ADS  Google Scholar 

  21. Sanvitto, D. & Kéna-Cohen, S. The road towards polaritonic devices. Nat. Mater. 15, 1061–1073 (2016).

    Article  ADS  Google Scholar 

  22. Kavokin, A., Baumberg, J. J., Malpuech, G. & Laussy, F. P. Microcavities (Oxford Univ. Press, 2017).

  23. Sieberer, L. M., Buchhold, M. & Diehl, S. Keldysh field theory for driven open quantum systems. Rep. Prog. Phys. 79, 096001 (2016).

    Article  ADS  Google Scholar 

  24. Fraser, M. D., Höfling, S. & Yamamoto, Y. Physics and applications of exciton–polariton lasers. Nat. Mater. 15, 1049–1052 (2016).

    Article  ADS  Google Scholar 

  25. Hartmann, M. J. Quantum simulation with interacting photons. J. Opt. 18, 104005 (2016).

    Article  ADS  Google Scholar 

  26. Noh, C. & Angelakis, D. G. Quantum simulations and many-body physics with light. Rep. Prog. Phys. 80, 016401 (2017).

    Article  ADS  Google Scholar 

  27. Ballarini, D. & Liberato, S. D. Polaritonics: from microcavities to sub-wavelength confinement. Nanophotonics 8, 641–654 (2019).

    Article  Google Scholar 

  28. Carusotto, I. et al. Photonic materials in circuit quantum electrodynamics. Nat. Phys. 16, 268–279 (2020).

    Article  Google Scholar 

  29. Keeling, J. & Kéna-Cohen, S. Bose–Einstein condensation of exciton-polaritons in organic microcavities. Ann. Rev. Phys. Chem. 71, 435–459 (2020).

    Article  Google Scholar 

  30. Segev, M. & Bandres, M. A. Topological photonics: where do we go from here? Nanophotonics 10, 425–434 (2021).

    Article  Google Scholar 

  31. Colombelli, R. & Manceau, J.-M. Perspectives for intersubband polariton lasers. Phys. Rev. X 5, 011031 (2015).

    Google Scholar 

  32. Sanvitto, D. & Kéna-Cohen, S. The road towards polaritonic devices. Nat. Mater. 15, 1061–1073 (2016).

    Article  ADS  Google Scholar 

  33. Berloff, N. G. et al. Realizing the classical XY Hamiltonian in polariton simulators. Nat. Mater. 16, 1120–1126 (2017).

    Article  ADS  Google Scholar 

  34. Nixon, M., Ronen, E., Friesem, A. A. & Davidson, N. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett. 110, 184102 (2013).

    Article  ADS  Google Scholar 

  35. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  36. Ma, R. et al. A dissipatively stabilized Mott insulator of photons. Nature 566, 51–57 (2019).

    Article  ADS  Google Scholar 

  37. Roushan, P. et al. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nat. Phys. 13, 146–151 (2017).

    Article  Google Scholar 

  38. 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 

  39. 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 

  40. 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 

  41. Gunton, J. D. & Buckingham, M. J. Condensation of the ideal Bose gas as a cooperative transition. Phys. Rev. 166, 152–158 (1968).

    Article  ADS  Google Scholar 

  42. Cohen-Tannoudji, C. Lectures at Collège de France: Coherence properties of Bose-Einstein condensates [French]. ENS http://www.phys.ens.fr/~cct/college-de-france/1999-00/1999-00.htm (1999).

  43. Aharoni, A. Introduction to the Theory of Ferromagnetism 2nd edn (Clarendon, 2001).

  44. Andrews, M. et al. Observation of interference between two Bose condensates. Science 275, 637–641 (1997).

    Article  Google Scholar 

  45. Wilmsen, C. W., Temkin, H. & Coldren, L. A. Vertical-Cavity Surface-Emitting Lasers: Design, Fabrication, Characterization, and Applications Vol. 24 (Cambridge Univ. Press, 2001).

  46. Iga, K. Vertical-cavity surface-emitting laser: its conception and evolution. Jpn. J. Appl. Phys. 47, 1 (2008).

    Article  ADS  Google Scholar 

  47. Hegarty, S. P., Huyet, G., McInerney, J. G. & Choquette, K. D. Pattern formation in the transverse section of a laser with a large Fresnel number. Phys. Rev. Lett. 82, 1434–1437 (1999).

    Article  ADS  Google Scholar 

  48. Lugiato, L., Prati, F. & Brambilla, M. Nonlinear Optical Systems (Cambridge Univ. Press, 2015).

  49. Longhi, S. & Feng, L. Invited article: Mitigation of dynamical instabilities in laser arrays via non-Hermitian coupling. APL Photonics 3, 060802 (2018).

    Article  ADS  Google Scholar 

  50. Greveling, S., Perrier, K. & van Oosten, D. Density distribution of a Bose-Einstein condensate of photons in a dye-filled microcavity. Phys. Rev. A 98, 013810 (2018).

    Article  ADS  Google Scholar 

  51. Strinati, M. C. & Conti, C. Bose-Einstein condensation of photons with nonlocal nonlinearity in a dye-doped graded-index microcavity. Phys. Rev. A 90, 043853 (2014).

    Article  ADS  Google Scholar 

  52. Alaeian, H., Schedensack, M., Bartels, C., Peterseim, D. & Weitz, M. Thermo-optical interactions in a dye-microcavity photon Bose–Einstein condensate. New J. Phys. 19, 115009 (2017).

    Article  ADS  Google Scholar 

  53. Kennard, E. H. On the thermodynamics of fluorescence. Phys. Rev. 11, 29–38 (1918).

    Article  ADS  Google Scholar 

  54. Stepanov, B. A universal relation between the absorption and luminescence spectra of complex molecules. Sov. Phys. Dokl. 2, 81 (1957).

    ADS  Google Scholar 

  55. Moroshkin, P., Weller, L., Saß, A., Klaers, J. & Weitz, M. Kennard-Stepanov relation connecting absorption and emission spectra in an atomic gas. Phys. Rev. Lett. 113, 063002 (2014).

    Article  ADS  Google Scholar 

  56. Schmitt, J. et al. Thermalization kinetics of light: from laser dynamics to equilibrium condensation of photons. Phys. Rev. A 92, 011602 (2015).

    Article  ADS  Google Scholar 

  57. Walker, B. T. et al. Driven-dissipative non-equilibrium Bose–Einstein condensation of less than ten photons. Nat. Phys. 14, 1173–1177 (2018).

    Article  Google Scholar 

  58. Hesten, H. J., Nyman, R. A. & Mintert, F. Decondensation in nonequilibrium photonic condensates: when less is more. Phys. Rev. Lett. 120, 040601 (2018).

    Article  ADS  Google Scholar 

  59. Bajoni, D., Senellart, P., Lemaître, A. & Bloch, J. Photon lasing in GaAs microcavity: similarities with a polariton condensate. Phys. Rev. B 76, 201305 (2007).

    Article  ADS  Google Scholar 

  60. Chuang, S. L. Physics of Photonic Devices Vol. 80 (Wiley, 2012).

  61. Barland, S., Azam, P., Lippi, G. L., Nyman, R. A. & Kaiser, R. Photon thermalization and a condensation phase transition in an electrically pumped semiconductor microresonator. Opt. Express 29, 8368–8375 (2021).

    Article  ADS  Google Scholar 

  62. van Roosbroeck, W. & Shockley, W. Photon-radiative recombination of electrons and holes in germanium. Phys. Rev. 94, 1558–1560 (1954).

    Article  ADS  Google Scholar 

  63. Hakala, T. K. et al. Bose–Einstein condensation in a plasmonic lattice. Nat. Phys. 14, 739–744 (2018).

    Article  Google Scholar 

  64. Väkeväinen, A. I. et al. Sub-picosecond thermalization dynamics in condensation of strongly coupled lattice plasmons. Nat. Commun. 11, 3139 (2020).

    Article  ADS  Google Scholar 

  65. Schmitt, J. Dynamics and correlations of a Bose–Einstein condensate of photons. J. Phys. B 51, 173001 (2018).

    Article  ADS  Google Scholar 

  66. Kirton, P. & Keeling, J. Nonequilibrium model of photon condensation. Phys. Rev. Lett. 111, 100404 (2013).

    Article  ADS  Google Scholar 

  67. Gladilin, V. N. & Wouters, M. Classical field model for arrays of photon condensates. Phys. Rev. A 101, 043814 (2020).

    Article  ADS  Google Scholar 

  68. Blatt, J. M., Böer, K. & Brandt, W. Bose-Einstein condensation of excitons. Phys. Rev. 126, 1691 (1962).

    Article  ADS  Google Scholar 

  69. Moskalenko, S. Inverse optical-hydrodynamic phenomena in a non-ideal excitonic gase. Fiz. Tverd. Tela 4, 276 (1962).

    ADS  Google Scholar 

  70. Keldysh, L. & Kopaev, Y. V. Possible instability of semimetallic state toward Coulomb interaction. Sov. Phys. Solid State 6, 2219–2224 (1965).

    Google Scholar 

  71. Mysyrowicz, A., Hulin, D. & Antonetti, A. Long exciton lifetime in Cu2O. Phys. Rev. Lett. 43, 1123–1126 (1979).

    Article  ADS  Google Scholar 

  72. Snoke, D. & Kavoulakis, G. M. Bose–Einstein condensation of excitons in Cu2O: progress over 30 years. Rep. Prog. Phys. 77, 116501 (2014).

    Article  ADS  Google Scholar 

  73. Ivanov, A. L., Littlewood, P. B. & Haug, H. Bose-Einstein statistics in thermalization and photoluminescence of quantum-well excitons. Phys. Rev. B 59, 5032–5048 (1999).

    Article  ADS  Google Scholar 

  74. Fukuzawa, T., Mendez, E. E. & Hong, J. M. Phase transition of an exciton system in GaAs coupled quantum wells. Phys. Rev. Lett. 64, 3066–3069 (1990).

    Article  ADS  Google Scholar 

  75. Kash, J. A., Zachau, M., Mendez, E. E., Hong, J. M. & Fukuzawa, T. Fermi-Dirac distribution of excitons in coupled quantum wells. Phys. Rev. Lett. 66, 2247–2250 (1991).

    Article  ADS  Google Scholar 

  76. Butov, L. V., Zrenner, A., Abstreiter, G., Böhm, G. & Weimann, G. Condensation of indirect excitons in coupled AlAs/GaAs quantum wells. Phys. Rev. Lett. 73, 304–307 (1994).

    Article  ADS  Google Scholar 

  77. Butov, L. V. & Filin, A. I. Anomalous transport and luminescence of indirect excitons in AlAs/GaAs coupled quantum wells as evidence for exciton condensation. Phys. Rev. B 58, 1980–2000 (1998).

    Article  ADS  Google Scholar 

  78. Butov, L. V., Gossard, A. C. & Chemla, D. S. Macroscopically ordered state in an exciton system. Nature 418, 751–754 (2002).

    Article  ADS  Google Scholar 

  79. Snoke, D., Denev, S., Liu, Y., Pfeiffer, L. & West, K. Long-range transport in excitonic dark states in coupled quantum wells. Nature 418, 754–757 (2002).

    Article  ADS  Google Scholar 

  80. Butov, L. V. et al. Formation mechanism and low-temperature instability of exciton rings. Phys. Rev. Lett. 92, 117404 (2004).

    Article  ADS  Google Scholar 

  81. Rapaport, R. et al. Charge separation of dense two-dimensional electron-hole gases: mechanism for exciton ring pattern formation. Phys. Rev. Lett. 92, 117405 (2004).

    Article  ADS  Google Scholar 

  82. Ivanov, A. L. et al. Origin of the inner ring in photoluminescence patterns of quantum well excitons. Europhys. Lett. 73, 920–926 (2006).

    Article  ADS  Google Scholar 

  83. High, A. A. et al. Spin currents in a coherent exciton gas. Phys. Rev. Lett. 110, 246403 (2013).

    Article  ADS  Google Scholar 

  84. Vishnevsky, D. et al. Skyrmion formation and optical spin-Hall effect in an expanding coherent cloud of indirect excitons. Phys. Rev Lett. 110, 246404 (2013).

    Article  ADS  Google Scholar 

  85. Leonard, J. R. et al. Pancharatnam–Berry phase in condensate of indirect excitons. Nat. Commun. 9, 2158 (2018).

    Article  ADS  Google Scholar 

  86. Leonard, J. et al. Moiré pattern of interference dislocations in condensate of indirect excitons. Nat. Commun. 12, 1175 (2021).

    Article  ADS  Google Scholar 

  87. Combescot, M., Betbeder-Matibet, O. & Combescot, R. Bose-Einstein condensation in semiconductors: the key role of dark excitons. Phys. Rev. Lett. 99, 176403 (2007).

    Article  ADS  Google Scholar 

  88. Combescot, R. & Combescot, M. “Gray” BCS condensate of excitons and internal Josephson effect. Phys. Rev. Lett. 109, 26401 (2012).

    Article  ADS  Google Scholar 

  89. Shiau, S.-Y. & Combescot, M. Optical signature of quantum coherence in fully dark exciton condensates. Phys. Rev. Lett. 123, 097401 (2019).

    Article  ADS  Google Scholar 

  90. Remeika, M. et al. Localization-delocalization transition of indirect excitons in lateral electrostatic lattices. Phys. Rev. Lett. 102, 186803 (2009).

    Article  ADS  Google Scholar 

  91. Vörös, Z., Snoke, D. W., Pfeiffer, L. & West, K. Direct measurement of exciton-exciton interaction energy. Phys. Rev. Lett. 103, 016403 (2009).

    Article  ADS  Google Scholar 

  92. Shilo, Y. et al. Particle correlations and evidence for dark state condensation in a cold dipolar exciton fluid. Nat. Commun. 4, 2335 (2013).

    Article  ADS  Google Scholar 

  93. Combescot, M., Combescot, R., Alloing, M. & Dubin, F. Optical signatures of a fully dark exciton condensate. EPL 105, 47011 (2014).

    Article  ADS  Google Scholar 

  94. Combescot, M., Combescot, R. & Dubin, F. Bose–Einstein condensation and indirect excitons: a review. Rep. Prog. Phys. 80, 066501 (2017).

    Article  ADS  Google Scholar 

  95. Leonard, J. R. et al. Spin transport of excitons. Nano Lett. 9, 4204–4208 (2009).

    Article  ADS  Google Scholar 

  96. Mazuz-Harpaz, Y. et al. Dynamical formation of a strongly correlated dark condensate of dipolar excitons. Proc. Natl Acad. Sci. USA 116, 18328–18333 (2019).

    Article  ADS  Google Scholar 

  97. Hammack, A. T. et al. Excitons in electrostatic traps. J. Appl. Phys. 99, 066104 (2006).

    Article  ADS  Google Scholar 

  98. High, A. et al. Condensation of excitons in a trap. Nano Lett. 12, 2605–2609 (2012).

    Article  ADS  Google Scholar 

  99. Anankine, R. et al. Quantized vortices and four-component superfluidity of semiconductor excitons. Phys. Rev. Lett. 118, 127402 (2017).

    Article  ADS  Google Scholar 

  100. Dang, S. et al. Defect proliferation at the quasicondensate crossover of two-dimensional dipolar excitons trapped in coupled GaAs quantum wells. Phys. Rev. Lett. 122, 117402 (2019).

    Article  ADS  Google Scholar 

  101. Dang, S. et al. Observation of algebraic time order for two-dimensional dipolar excitons. Phys. Rev. Lett. 2, 032013 (2020).

    Google Scholar 

  102. Remeika, M., Fogler, M. M., Butov, L. V., Hanson, M. & Gossard, A. C. Two-dimensional electrostatic lattices for indirect excitons. Appl. Phys. Lett. 100, 061103 (2012).

    Article  ADS  Google Scholar 

  103. Lagoin, C. et al. Quasicondensation of bilayer excitons in a periodic potential. Phys. Rev. Lett. 126, 067404 (2021).

    Article  ADS  Google Scholar 

  104. Lagoin, C., Suffit, S., Baldwin, K., Pfeiffer, L. & Dubin, F. Mott insulator of strongly interacting two-dimensional semiconductor excitons. Nat. Phys. 18, 149–153 (2022).

    Article  Google Scholar 

  105. Lozovik, Y. E., Kurbakov, I., Astrakharchik, G. & Willander, M. Bose condensation of two-dimensional dipolar excitons: simulation by the quantum Monte Carlo method. J. Exp. Theor. Phys. 106, 296–315 (2008).

    Article  ADS  Google Scholar 

  106. Laikhtman, B. & Rapaport, R. Correlations in a two-dimensional Bose gas with long-range interaction. EPL 87, 27010 (2009).

    Article  ADS  Google Scholar 

  107. Wang, Z. et al. Evidence of high-temperature exciton condensation in two-dimensional atomic double layers. Nature 574, 76–80 (2019).

    Article  ADS  Google Scholar 

  108. Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).

    Article  ADS  Google Scholar 

  109. Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).

    Article  ADS  Google Scholar 

  110. Jin, C. et al. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature 567, 76–80 (2019).

    Article  ADS  Google Scholar 

  111. Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures. Nature 567, 81–86 (2019).

    Article  ADS  Google Scholar 

  112. Lagoin, C. & Dubin, Fmc Key role of the moiré potential for the quasicondensation of interlayer excitons in van der Waals heterostructures. Phys. Rev. B 103, L041406 (2021).

    Article  ADS  Google Scholar 

  113. Eisenstein, J. Exciton condensation in bilayer quantum Hall systems. Annu. Rev. Condens. Matter Phys. 5, 159–181 (2014).

    Article  ADS  Google Scholar 

  114. Li, J., Taniguchi, T., Watanabe, K., Hone, J. & Dean, C. Excitonic superfluid phase in double bilayer graphene. Nat. Phys. 13, 751–755 (2017).

    Article  Google Scholar 

  115. Liu, X., Watanabe, K., Taniguchi, T., Halperin, B. I. & Kim, P. Quantum Hall drag of exciton condensate in graphene. Nat. Phys. 13, 746–750 (2017).

    Article  Google Scholar 

  116. Giamarchi, T., Rüegg, C. & Tchernyshyov, O. Bose–Einstein condensation in magnetic insulators. Nat. Phys. 4, 198–204 (2008).

    Article  Google Scholar 

  117. Demokritov, S. et al. Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443, 430–433 (2006).

    Article  ADS  Google Scholar 

  118. Kavokin, A. V., Baumberg, J. J., Malpuech, G. & Laussy, F. P. Microcavities Vol. 21 (Oxford Univ. Press, 2017).

  119. Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992).

    Article  ADS  Google Scholar 

  120. Houdré, R. et al. Measurement of cavity-polariton dispersion curve from angle-resolved photoluminescence experiments. Phys. Rev. Lett. 73, 2043–2046 (1994).

    Article  ADS  Google Scholar 

  121. Imamğlu, A., Ram, R. J., Pau, S. & Yamamoto, Y. Nonequilibrium condensates and lasers without inversion: exciton-polariton lasers. Phys. Rev. A 53, 4250–4253 (1996).

    Article  ADS  Google Scholar 

  122. Tassone, F., Piermarocchi, C., Savona, V., Quattropani, A. & Schwendimann, P. Bottleneck effects in the relaxation and photoluminescence of microcavity polaritons. Phys. Rev. B 56, 7554–7563 (1997).

    Article  ADS  Google Scholar 

  123. Tassone, F. & Yamamoto, Y. Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons. Phys. Rev. B 59, 10830–10842 (1999).

    Article  ADS  Google Scholar 

  124. Dang, L., Heger, D., Andre, R., Boeuf, F. & Romestain, R. Stimulation of polariton photoluminescence in semiconductor microcavity. Phys. Rev. Lett. 81, 3920–3923 (1998).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  126. Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose-Einstein condensation of microcavity polaritons in a trap. Science 316, 1007–1010 (2007).

    Article  ADS  Google Scholar 

  127. Byrnes, T., Kim, N. Y. & Yamamoto, Y. Exciton–polariton condensates. Nat. Phys. 10, 803–813 (2014).

    Article  Google Scholar 

  128. Kira, M. et al. Quantum theory of nonlinear semiconductor microcavity luminescence explaining “boser” experiments. Phys. Rev. Lett. 79, 5170–5173 (1997).

    Article  ADS  Google Scholar 

  129. Butté, R., Skolnick, M. S., Whittaker, D. M., Bajoni, D. & Roberts, J. S. Dependence of stimulated scattering in semiconductor microcavities on pump power, angle, and energy. Phys. Rev. B 68, 115325 (2003).

    Article  ADS  Google Scholar 

  130. Deng, H., Weihs, G., Snoke, D., Bloch, J. & Yamamoto, Y. Polariton lasing vs. photon lasing in a semiconductor microcavity. Proc. Natl Acad. Sci. USA 100, 15318–15323 (2003).

    Article  ADS  Google Scholar 

  131. Bajoni, D. et al. Polariton laser using single micropillar GaAs–GaAlAs semiconductor cavities. Phys. Rev. Lett. 100, 047401 (2008).

    Article  ADS  Google Scholar 

  132. Hu, J. et al. Polariton laser in the Bardeen-Cooper-Schrieffer regime. Phys. Rev. X 11, 011018 (2021).

    Google Scholar 

  133. Schneider, C. et al. An electrically pumped polariton laser. Nature 497, 348–352 (2013).

    Article  ADS  Google Scholar 

  134. Fischer, J. et al. Anomalies of a nonequilibrium spinor polariton condensate in a magnetic field. Phys. Rev. Lett. 112, 093902 (2014).

    Article  ADS  Google Scholar 

  135. Ménard, J.-M. et al. Revealing the dark side of a bright exciton–polariton condensate. Nat. Commun. 5, 4648 (2014).

    Article  ADS  Google Scholar 

  136. Baumberg, J. J. et al. Parametric oscillation in a vertical microcavity: a polariton condensate or micro-optical parametric oscillation. Phys. Rev. B 62, R16247–R16250 (2000).

    Article  ADS  Google Scholar 

  137. Krizhanovskii, D. N. et al. Dominant effect of polariton-polariton interactions on the coherence of the microcavity optical parametric oscillator. Phys. Rev. Lett. 97, 097402 (2006).

    Article  ADS  Google Scholar 

  138. Carusotto, I. & Ciuti, C. Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies. Phys. Rev. B 72, 125335 (2005).

    Article  ADS  Google Scholar 

  139. del Valle, E. et al. Dynamics of the formation and decay of coherence in a polariton condensate. Phys. Rev. Lett. 103, 096404 (2009).

    Article  ADS  Google Scholar 

  140. Dagvadorj, G. et al. Nonequilibrium phase transition in a two-dimensional driven open quantum system. Phys. Rev. X 5, 041028 (2015).

    Google Scholar 

  141. Kasprzak, J., Solnyshkov, D. D., André, R., Dang, L. S. & Malpuech, G. Formation of an exciton polariton condensate: thermodynamic versus kinetic regimes. Phys. Rev. Lett. 101, 146404 (2008).

    Article  ADS  Google Scholar 

  142. Chiocchetta, A., Gambassi, A. & Carusotto, I. in Universal Themes of Bose-Einstein Condensation (eds Proukakis, N. P., Snoke, D. W. & Littlewood, P. B.) 409–423 (Cambridge Univ. Press, 2017).

  143. Amo, A. et al. Light engineering of the polariton landscape in semiconductor microcavities. Phys. Rev. B 82, 081301 (2010).

    Article  ADS  Google Scholar 

  144. Tosi, G. et al. Sculpting oscillators with light within a nonlinear quantum fluid. Nat. Phys. 8, 190–194 (2012).

    Article  Google Scholar 

  145. Gao, T. et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015).

    Article  ADS  Google Scholar 

  146. Gao, T. et al. Polariton condensate transistor switch. Phys. Rev. B 85, 235102 (2012).

    Article  ADS  Google Scholar 

  147. Ballarini, D. et al. All-optical polariton transistor. Nat. Commun. 4, 1778 (2013).

    Article  ADS  Google Scholar 

  148. Sturm, C. et al. All-optical phase modulation in a cavity-polariton Mach–Zehnder interferometer. Nat. Commun. 5, 3278 (2014).

    Article  ADS  Google Scholar 

  149. Nguyen, H. S. et al. Realization of a double-barrier resonant tunneling diode for cavity polaritons. Phys. Rev. Lett. 110, 236601 (2013).

    Article  ADS  Google Scholar 

  150. Shelykh, I. A., Kavokin, A. V. & Malpuech, G. Spin dynamics of exciton polaritons in microcavities. Phys. Status Solidi B 242, 2271–2289 (2005).

    Article  ADS  MATH  Google Scholar 

  151. Kavokin, A., Malpuech, G. & Glazov, M. Optical spin Hall effect. Phys. Rev. Lett. 95, 136601 (2005).

    Article  ADS  Google Scholar 

  152. Leyder, C. et al. Observation of the optical spin Hall effect. Nat. Phys. 3, 628–631 (2007).

    Article  Google Scholar 

  153. Kammann, E. et al. Nonlinear optical spin Hall effect and long-range spin transport in polariton lasers. Phys. Rev. Lett. 109, 036404 (2012).

    Article  ADS  Google Scholar 

  154. Sala, V. et al. Spin-orbit coupling for photons and polaritons in microstructures. Phys. Rev. X 5, 011034 (2015).

    Google Scholar 

  155. Nalitov, A. V., Solnyshkov, D. D. & Malpuech, G. Polariton \({\mathbb{Z}}\) topological insulator. Phys. Rev. Lett. 114, 116401 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  156. Karzig, T., Bardyn, C.-E., Lindner, N. H. & Refael, G. Topological polaritons. Phys. Rev. X 5, 031001 (2015).

    Google Scholar 

  157. Klembt, S. et al. Exciton-polariton topological insulator. Nature 562, 552–556 (2018).

    Article  ADS  Google Scholar 

  158. Taskinen, J. M., Kliuiev, P., Moilanen, A. J. & Törmä, P. Polarization and phase textures in lattice plasmon condensates. Nano Lett. 12, 5262–5268 (2021).

    Article  ADS  Google Scholar 

  159. Ciuti, C., Savona, V., Piermarocchi, C., Quattropani, A. & Schwendimann, P. Role of the exchange of carriers in elastic exciton-exciton scattering in quantum wells. Phys. Rev. B 58, 7926–7933 (1998).

    Article  ADS  Google Scholar 

  160. Vladimirova, M. et al. Polariton-polariton interaction constants in microcavities. Phys. Rev. B 82, 075301 (2010).

    Article  ADS  Google Scholar 

  161. Carusotto, I., Volz, T. & Imamoglu, A. Feshbach blockade: single-photon nonlinear optics using resonantly enhanced cavity polariton scattering from biexciton states. EPL 90, 37001 (2010).

    Article  ADS  Google Scholar 

  162. Takemura, N., Trebaol, S., Wouters, M., Portella-Oberli, M. T. & Deveaud, B. Polaritonic Feshbach resonance. Nat. Phys. 10, 500–504 (2014).

    Article  Google Scholar 

  163. Rubo, Y. G., Kavokin, A. & Shelykh, I. Suppression of superfluidity of exciton-polaritons by magnetic field. Phys. Lett. A 358, 227–230 (2006).

    Article  ADS  MATH  Google Scholar 

  164. Read, D., Liew, T. C. H., Rubo, Y. G. & Kavokin, A. V. Stochastic polarization formation in exciton-polariton Bose-Einstein condensates. Phys. Rev. B 80, 195309 (2009).

    Article  ADS  Google Scholar 

  165. Baumberg, J. J. et al. Spontaneous polarization buildup in a room-temperature polariton laser. Phys. Rev. Lett. 101, 136409 (2008).

    Article  ADS  Google Scholar 

  166. Ohadi, H. et al. Spontaneous symmetry breaking in a polariton and photon laser. Phys. Rev. Lett. 109, 016404 (2012).

    Article  ADS  Google Scholar 

  167. Ohadi, H. et al. Spontaneous spin bifurcations and ferromagnetic phase transitions in a spinor exciton-polariton condensate. Phys. Rev. X 5, 031002 (2015).

    Google Scholar 

  168. Sala, V. G. et al. Stochastic precession of the polarization in a polariton laser. Phys. Rev. B 93, 115313 (2016).

    Article  ADS  Google Scholar 

  169. Larionov, A. V. et al. Polarized nonequilibrium Bose-Einstein condensates of spinor exciton polaritons in a magnetic field. Phys. Rev. Lett. 105, 256401 (2010).

    Article  ADS  Google Scholar 

  170. Walker, P. et al. Suppression of Zeeman splitting of the energy levels of exciton-polariton condensates in semiconductor microcavities in an external magnetic field. Phys. Rev. Lett. 106, 257401 (2011).

    Article  ADS  Google Scholar 

  171. Król, M. et al. Spin polarized semimagnetic exciton-polariton condensate in magnetic field. Sci. Rep. 8, 6694 (2018).

    Article  ADS  Google Scholar 

  172. Król, M. et al. Giant spin Meissner effect in a nonequilibrium exciton-polariton gas. Phys. Rev. B 99, 115318 (2019).

    Article  ADS  Google Scholar 

  173. Shelykh, I. A., Liew, T. C. H. & Kavokin, A. V. Interplay between superfluidity and magnetic self-trapping of exciton polaritons. Phys. Rev. B 80, 201306 (2009).

    Article  ADS  Google Scholar 

  174. Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities. Phys. Rev. Lett. 98, 126405 (2007).

    Article  ADS  Google Scholar 

  175. Christmann, G., Butté, R., Feltin, E., Carlin, J.-F. & Grandjean, N. Room temperature polariton lasing in a GaN/AlGaN multiple quantum well microcavity. Appl. Phys. Lett. 93, 051102 (2008).

    Article  ADS  Google Scholar 

  176. Lu, T.-C. et al. Room temperature polariton lasing vs. photon lasing in a ZnO-based hybrid microcavity. Opt. Express 20, 5530–5537 (2012).

    Article  ADS  Google Scholar 

  177. Li, F. et al. From excitonic to photonic polariton condensate in a ZnO-based microcavity. Phys. Rev. Lett. 110, 196406 (2013).

    Article  ADS  Google Scholar 

  178. Kéna-Cohen, S. & Forrest, S. Room-temperature polariton lasing in an organic single-crystal microcavity. Nat. Photonics 4, 371–375 (2010).

    Article  ADS  Google Scholar 

  179. Plumhof, J. D., Stöferle, T., Mai, L., Scherf, U. & Mahrt, R. F. Room-temperature Bose–Einstein condensation of cavity exciton–polaritons in a polymer. Nat. Mater. 13, 247–252 (2014).

    Article  ADS  Google Scholar 

  180. Dietrich, C. P. et al. An exciton-polariton laser based on biologically produced fluorescent protein. Sci. Adv. 2, e1600666 (2016).

    Article  ADS  Google Scholar 

  181. Su, R. et al. Room-temperature polariton lasing in all-inorganic perovskite nanoplatelets. Nano Lett. 17, 3982–3988 (2017).

    Article  ADS  Google Scholar 

  182. Su, R. et al. Observation of exciton polariton condensation in a perovskite lattice at room temperature. Nat. Phys. 16, 301–306 (2020).

    Article  Google Scholar 

  183. Su, R. et al. Perovskite semiconductors for room-temperature exciton-polaritonics. Nat. Mater. 20, 1315–1324 (2021).

    Article  ADS  Google Scholar 

  184. Anton-Solanas, C. et al. Bosonic condensation of exciton–polaritons in an atomically thin crystal. Nat. Mater. 20, 1233–1239 (2021).

    Article  ADS  Google Scholar 

  185. Zhao, J. et al. Ultralow threshold polariton condensate in a monolayer semiconductor microcavity at room temperature. Nano Lett. 21, 3331–3339 (2021).

    Article  ADS  Google Scholar 

  186. Ramezani, M. et al. Plasmon-exciton-polariton lasing. Optica 4, 31–37 (2017).

    Article  ADS  Google Scholar 

  187. Moilanen, A. J., Daskalakis, K. S., Taskinen, J. M. & Törmä, P. Spatial and temporal coherence in strongly coupled plasmonic Bose-Einstein condensates. Phys. Rev. Lett. 127, 255301 (2021).

    Article  ADS  Google Scholar 

  188. Bhattacharya, P., Xiao, B., Das, A., Bhowmick, S. & Heo, J. Solid state electrically injected exciton-polariton laser. Phys. Rev. Lett. 110, 206403 (2013).

    Article  ADS  Google Scholar 

  189. Bhattacharya, P. et al. Room temperature electrically injected polariton laser. Phys. Rev. Lett. 112, 236802 (2014).

    Article  ADS  Google Scholar 

  190. Deveaud, B. Comment on “Room temperature electrically injected polariton laser”. Phys. Rev. Lett. 117, 029701 (2016).

    Article  ADS  Google Scholar 

  191. Suchomel, H. et al. Platform for electrically pumped polariton simulators and topological lasers. Phys. Rev. Lett. 121, 257402 (2018).

    Article  ADS  Google Scholar 

  192. Suchomel, H. et al. Spatio-temporal coherence in vertically emitting GaAs-based electrically driven polariton lasers. Appl. Phys. Lett. 116, 171103 (2020).

    Article  ADS  Google Scholar 

  193. Dini, D., Köhler, R., Tredicucci, A., Biasiol, G. & Sorba, L. Microcavity polariton splitting of intersubband transitions. Phys. Rev. Lett. 90, 116401 (2003).

    Article  ADS  Google Scholar 

  194. Anappara, A. A. et al. Signatures of the ultrastrong light-matter coupling regime. Phys. Rev. B 79, 201303 (2009).

    Article  ADS  Google Scholar 

  195. Ciuti, C., Bastard, G. & Carusotto, I. Quantum vacuum properties of the intersubband cavity polariton field. Phys. Rev. B 72, 115303 (2005).

    Article  ADS  Google Scholar 

  196. Ohtani, K. et al. An electrically pumped phonon-polariton laser. Sci. Adv. 5, eaau1632 (2019).

    Article  ADS  Google Scholar 

  197. Agrawal, G. Nonlinear Fiber Optics (Elsevier, 2007).

  198. Larré, P.-É. & Carusotto, I. Propagation of a quantum fluid of light in a cavityless nonlinear optical medium: general theory and response to quantum quenches. Phys. Rev. A 92, 043802 (2015).

    Article  ADS  Google Scholar 

  199. Sun, C. et al. Observation of the kinetic condensation of classical waves. Nat. Phys. 8, 471–475 (2012).

    Article  Google Scholar 

  200. Šantić, N. et al. Nonequilibrium precondensation of classical waves in two dimensions propagating through atomic vapors. Phys. Rev. Lett. 120, 055301 (2018).

    Article  ADS  Google Scholar 

  201. Krupa, K. et al. Multimode nonlinear fiber optics, a spatiotemporal avenue. APL Photonics 4, 110901 (2019).

    Article  ADS  Google Scholar 

  202. Connaughton, C., Josserand, C., Picozzi, A., Pomeau, Y. & Rica, S. Condensation of classical nonlinear waves. Phys. Rev. Lett. 95, 263901 (2005).

    Article  ADS  Google Scholar 

  203. Aschieri, P., Garnier, J., Michel, C., Doya, V. & Picozzi, A. Condensation and thermalization of classsical optical waves in a waveguide. Phys. Rev. A 83, 033838 (2011).

    Article  ADS  Google Scholar 

  204. Picozzi, A. et al. Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep. 542, 1–132 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  205. Wu, F. O., Hassan, A. U. & Christodoulides, D. N. Thermodynamic theory of highly multimoded nonlinear optical systems. Nat. Photonics 13, 776–782 (2019).

    Article  ADS  Google Scholar 

  206. Wu, F. O. et al. Thermalization of light’s orbital angular momentum in nonlinear multimode waveguide systems. Phys. Rev. Lett. 128, 123901 (2022).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  208. Chiocchetta, A., Larré, P.-É. & Carusotto, I. Thermalization and Bose-Einstein condensation of quantum light in bulk nonlinear media. EPL 115, 24002 (2016).

    Article  ADS  Google Scholar 

  209. Peyronel, T. et al. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature 488, 57–60 (2012).

    Article  ADS  Google Scholar 

  210. Steinhauer, J. et al. Analogue cosmological particle creation in an ultracold quantum fluid of light. Nat. Commun. 13, 2890 (2022).

    Article  ADS  Google Scholar 

  211. Richard, M., Kasprzak, J., Romestain, R., Andre, R. & Dang, L. Spontaneous coherent phase transition of polaritons in CdTe microcavities. Phys. Rev. Lett. 94, 187401 (2005).

    Article  ADS  Google Scholar 

  212. Wouters, M., Carusotto, I. & Ciuti, C. Spatial and spectral shape of inhomogeneous nonequilibrium exciton-polariton condensates. Phys. Rev. B 77, 115340 (2008).

    Article  ADS  Google Scholar 

  213. Coullet, P., Gil, L. & Rocca, F. Optical vortices. Opt. Commun. 73, 403–408 (1989).

    Article  ADS  Google Scholar 

  214. Lugiato, L., Oppo, G., Tredicce, J., Narducci, L. & Pernigo, M. Instabilities and spatial complexity in a laser. JOSA B 7, 1019–1033 (1990).

    Article  ADS  Google Scholar 

  215. 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 

  216. Mamaev, A. V. & Saffman, M. Optical vortex patterns in a unidirectional ring oscillator. Phys. Scr. T67, 21–25 (1996).

    Article  ADS  Google Scholar 

  217. Scheuer, J. & Orenstein, M. Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities. Science 285, 230–233 (1999).

    Article  Google Scholar 

  218. Chen, Y. F. & Lan, Y. P. Transverse pattern formation of optical vortices in a microchip laser with a large Fresnel number. Phys. Rev. A 65, 013802 (2001).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  220. Lagoudakis, K. G. et al. Observation of half-quantum vortices in an exciton-polariton condensate. Science 326, 974–976 (2009).

    Article  ADS  Google Scholar 

  221. Jimenez-Garcia, J., Rodriguez, P., Guillet, T. & Ackemann, T. Spontaneous formation of vector vortex beams in vertical-cavity surface-emitting lasers with feedback. Phys. Rev. Lett. 119, 113902 (2017).

    Article  ADS  Google Scholar 

  222. Manni, F., Lagoudakis, K. G., Liew, T. C. H., André, R. & Deveaud-Plédran, B. Spontaneous pattern formation in a polariton condensate. Phys. Rev. Lett. 107, 106401 (2011).

    Article  ADS  Google Scholar 

  223. Tosi, G. et al. Geometrically locked vortex lattices in semiconductor quantum fluids. Nat. Commun. 3, 1243 (2012).

    Article  ADS  Google Scholar 

  224. Richard, M. et al. Experimental evidence for nonequilibrium Bose condensation of exciton polaritons. Phys. Rev. B 72, 201301 (2005).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  226. Lai, C. et al. Coherent zero-state and π-state in an exciton–polariton condensate array. Nature 450, 529–532 (2007).

    Article  ADS  Google Scholar 

  227. Zhang, L. et al. Weak lasing in one-dimensional polariton superlattices. Proc. Natl Acad. Sci. USA 112, E1516–E1519 (2015).

    Google Scholar 

  228. Tanese, D. et al. Polariton condensation in solitonic gap states in a one-dimensional periodic potential. Nat. Commun. 4, 1749 (2013).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  230. Jacqmin, T. et al. Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons. Phys. Rev. Lett. 112, 116402 (2014).

    Article  ADS  Google Scholar 

  231. Kim, N. Y. et al. Dynamical d-wave condensation of exciton–polaritons in a two-dimensional square-lattice potential. Nat. Phys. 7, 681–686 (2011).

    Article  Google Scholar 

  232. Baboux, F. et al. Bosonic condensation and disorder-induced localization in a flat band. Phys. Rev. Lett. 116, 066402 (2016).

    Article  ADS  Google Scholar 

  233. Lledó, C., Carusotto, I. & Szymańska, M. H. Spontaneous vortex lattices in polariton condensates in the lowest Landau level of a strained honeycomb lattice. SciPost Phys. 12, 068 (2022).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  235. Meystre, P. & Sargent, M. Elements of Quantum Optics (Springer, 2007).

  236. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence (Springer, 1984).

  237. Baas, A. et al. Synchronized and desynchronized phases of exciton-polariton condensates in the presence of disorder. Phys. Rev. Lett. 100, 170401 (2008).

    Article  ADS  Google Scholar 

  238. Wouters, M. Synchronized and desynchronized phases of coupled nonequilibrium exciton-polariton condensates. Phys. Rev. B 77, 121302 (2008).

    Article  ADS  Google Scholar 

  239. Eastham, P. Mode locking and mode competition in a nonequilibrium solid-state condensate. Phys. Rev. B 78, 035319 (2008).

    Article  ADS  Google Scholar 

  240. Ohadi, H. et al. Synchronization crossover of polariton condensates in weakly disordered lattices. Phys. Rev. B 97, 195109 (2018).

    Article  ADS  Google Scholar 

  241. Aleiner, I. L., Altshuler, B. L. & Rubo, Y. G. Radiative coupling and weak lasing of exciton-polariton condensates. Phys. Rev. B 85, 121301 (2012).

    Article  ADS  Google Scholar 

  242. Harrison, S. L., Sigurdsson, H. & Lagoudakis, P. G. Synchronization in optically trapped polariton Stuart-Landau networks. Phys. Rev. B 101, 155402 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  244. Lagoudakis, P. G. & Berloff, N. G. A polariton graph simulator. New J. Phys. 19, 125008 (2017).

    Article  ADS  Google Scholar 

  245. 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 

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

    Article  Google Scholar 

  247. Kodigala, A. et al. Lasing action from photonic bound states in continuum. Nature 541, 196–199 (2017).

    Article  ADS  Google Scholar 

  248. Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–640 (2017).

    Article  ADS  Google Scholar 

  249. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  251. Dikopoltsev, A. et al. Topological insulator vertical-cavity laser array. Science 373, 1514–1517 (2021).

    Article  ADS  Google Scholar 

  252. Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).

    Article  Google Scholar 

  253. Seclì, M., Capone, M. & Carusotto, I. Theory of chiral edge state lasing in a two-dimensional topological system. Phys. Rev. Res. 1, 033148 (2019).

    Article  Google Scholar 

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

    Google Scholar 

  255. Wouters, M., Liew, T. C. H. & Savona, V. Energy relaxation in one-dimensional polariton condensates. Phys. Rev. B 82, 245315 (2010).

    Article  ADS  Google Scholar 

  256. Askitopoulos, A. et al. Polariton condensation in an optically induced two-dimensional potential. Phys. Rev. B 88, 041308 (2013).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  258. Zambon, N. C. et al. Orbital angular momentum bistability in a microlaser. Opt. Lett. 44, 4531–4534 (2019).

    Article  ADS  Google Scholar 

  259. Aranson, I. S. & Kramer, L. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99–143 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  260. Weiss, C., Vaupel, M., Staliunas, K., Slekys, G. & Taranenko, V. Solitons and vortices in lasers. Appl. Phys. B 68, 151–168 (1999).

    Article  ADS  MATH  Google Scholar 

  261. Gladilin, V. N. & Wouters, M. Interaction and motion of vortices in nonequilibrium quantum fluids. New J. Phys. 19, 105005 (2017).

    Article  ADS  Google Scholar 

  262. Wachtel, G., Sieberer, L., Diehl, S. & Altman, E. Electrodynamic duality and vortex unbinding in driven-dissipative condensates. Phys. Rev. B 94, 104520 (2016).

    Article  ADS  Google Scholar 

  263. Gladilin, V. N. & Wouters, M. Noise-induced transition from superfluid to vortex state in two-dimensional nonequilibrium polariton condensates. Phys. Rev. B 100, 214506 (2019).

    Article  ADS  Google Scholar 

  264. Gladilin, V. N. & Wouters, M. Multivortex states and dynamics in nonequilibrium polariton condensates. J. Phys. A 52, 395303 (2019).

    Article  MathSciNet  Google Scholar 

  265. Szymańska, M. H., Keeling, J. & Littlewood, P. B. Nonequilibrium quantum condensation in an incoherently pumped dissipative system. Phys. Rev. Lett. 96, 230602 (2006).

    Article  ADS  Google Scholar 

  266. Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007).

    Article  ADS  Google Scholar 

  267. Hidaka, Y. & Minami, Y. Spontaneous symmetry breaking and Nambu–Goldstone modes in open classical and quantum systems. Prog. Theor. Exp. Phys 2020, 033A01 (2020).

    Article  MATH  Google Scholar 

  268. Wouters, M. & Carusotto, I. Goldstone mode of optical parametric oscillators in planar semiconductor microcavities in the strong-coupling regime. Phys. Rev. A 76, 043807 (2007).

    Article  ADS  Google Scholar 

  269. Ballarini, D. et al. Observation of long-lived polariton states in semiconductor microcavities across the parametric threshold. Phys. Rev. Lett. 102, 056402 (2009).

    Article  ADS  Google Scholar 

  270. Utsunomiya, S. et al. Observation of Bogoliubov excitations in exciton-polariton condensates. Nat. Phys. 4, 700–705 (2008).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  272. Leggett, A. Superfluidity. Rev. Mod. Phys. 71, S318–S323 (1999).

    Article  Google Scholar 

  273. Marchetti, F. M., Szymańska, M. H., Tejedor, C. & Whittaker, D. M. Spontaneous and triggered vortices in polariton optical-parametric-oscillator superfluids. Phys. Rev. Lett. 105, 063902 (2010).

    Article  ADS  Google Scholar 

  274. Sanvitto, D. et al. Persistent currents and quantized vortices in a polariton superfluid. Nat. Phys. 6, 527–533 (2010).

    Article  Google Scholar 

  275. Tosi, G. et al. Onset and dynamics of vortex-antivortex pairs in polariton optical parametric oscillator superfluids. Phys. Rev. Lett. 107, 036401 (2011).

    Article  ADS  Google Scholar 

  276. Dalfovo, F. & Stringari, S. Static response function for longitudinal and transverse excitations in superfluid helium. Phys. Rev. B 46, 13991–13996 (1992).

    Article  ADS  Google Scholar 

  277. Carusotto, I. & Castin, Y. Nonequilibrium and local detection of the normal fraction of a trapped two-dimensional Bose gas. Phys. Rev. A 84, 053637 (2011).

    Article  ADS  Google Scholar 

  278. Keeling, J. Superfluid density of an open dissipative condensate. Phys. Rev. Lett. 107, 080402 (2011).

    Article  ADS  Google Scholar 

  279. Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities. Nat. Phys. 5, 805–810 (2009).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  281. Wouters, M. & Carusotto, I. Superfluidity and critical velocities in nonequilibrium Bose-Einstein condensates. Phys. Rev. Lett. 105, 020602 (2010).

    Article  ADS  Google Scholar 

  282. Wouters, M. & Carusotto, I. Absence of long-range coherence in the parametric emission of photonic wires. Phys. Rev. B 74, 245316 (2006).

    Article  ADS  Google Scholar 

  283. Chiocchetta, A. & Carusotto, I. Non-equilibrium quasi-condensates in reduced dimensions. EPL 102, 67007 (2013).

    Article  ADS  Google Scholar 

  284. Kulkarni, M. & Lamacraft, A. Finite-temperature dynamical structure factor of the one-dimensional Bose gas: from the Gross-Pitaevskii equation to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena. Phys. Rev. A 88, 021603 (2013).

    Article  ADS  Google Scholar 

  285. 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).

    Google Scholar 

  286. Gladilin, V. N., Ji, K. & Wouters, M. Spatial coherence of weakly interacting one-dimensional nonequilibrium bosonic quantum fluids. Phys. Rev. A 90, 023615 (2014).

    Article  ADS  Google Scholar 

  287. 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 

  288. 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 

  289. 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  Google Scholar 

  290. 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 

  291. 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  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  293. 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 

  294. Fontaine, Q. Observation of KPZ universal scaling in a one-dimensional driven dissipative condensate. (In the press).

  295. Malpuech, G., Rubo, Y., Laussy, F., Bigenwald, P. & Kavokin, A. Polariton laser: thermodynamics and quantum kinetic theory. Semicond. Sci. Technol. 18, S395 (2003).

    Article  ADS  Google Scholar 

  296. Liew, T., Shelykh, I. & Malpuech, G. Polaritonic devices. Phys. E 43, 1543–1568 (2011).

    Article  Google Scholar 

  297. Khalatpour, A., Paulsen, A. K., Deimert, C., Wasilewski, Z. R. & Hu, Q. High-power portable terahertz laser systems. Nat. Photonics 15, 16–20 (2021).

    Article  ADS  Google Scholar 

  298. Sirtori, C. Terahertz race heats up. Nat. Photonics 15, 1–2 (2021).

    Article  ADS  Google Scholar 

  299. Jamadi, O. et al. Edge-emitting polariton laser and amplifier based on a ZnO waveguide. Light Sci Appl. 7, 82 (2018).

    Article  ADS  Google Scholar 

  300. Espinosa-Ortega, T. & Liew, T. C. H. Complete architecture of integrated photonic circuits based on AND and NOT logic gates of exciton polaritons in semiconductor microcavities. Phys. Rev. B 87, 195305 (2013).

    Article  ADS  Google Scholar 

  301. Rozas, E. et al. Impact of the energetic landscape on polariton condensates’ propagation along a coupler. Adv. Opt. Mater. 8, 2070072 (2020).

    Article  Google Scholar 

  302. Beierlein, J. Propagative oscillations in codirectional polariton waveguide couplers. Phys. Rev. Lett. 126, 075302 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  304. Matuszewski, M. & Witkowska, E. Universality in nonequilibrium condensation of exciton-polaritons. Phys. Rev. B 89, 155318 (2014).

    Article  ADS  Google Scholar 

  305. Solnyshkov, D. D., Nalitov, A. V. & Malpuech, G. Kibble-Zurek mechanism in topologically nontrivial zigzag chains of polariton micropillars. Phys. Rev. Lett. 116, 046402 (2016).

    Article  ADS  Google Scholar 

  306. Kulczykowski, M. & Matuszewski, M. Phase ordering kinetics of a nonequilibrium exciton-polariton condensate. Phys. Rev. B 95, 075306 (2017).

    Article  ADS  Google Scholar 

  307. 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 

  308. Calvanese Strinati, M., Pierangeli, D. & Conti, C. All-optical scalable spatial coherent Ising machine. Phys. Rev. Appl. 16, 054022 (2021).

    Article  ADS  Google Scholar 

  309. Imamoḡlu, A., Schmidt, H., Woods, G. & Deutsch, M. Strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett. 79, 1467 (1997).

    Article  ADS  Google Scholar 

  310. Verger, A., Ciuti, C. & Carusotto, I. Polariton quantum blockade in a photonic dot. Phys. Rev. B 73, 193306 (2006).

    Article  ADS  Google Scholar 

  311. Chang, D. E., Vuletić V. & Lukin, M. D. Quantum nonlinear optics — photon by photon. Nat. Photonics 8, 685–694 (2014).

    Article  ADS  Google Scholar 

  312. Delteil, A. et al. Towards polariton blockade of confined exciton–polaritons. Nat. Mater. 18, 219–222 (2019).

    Article  Google Scholar 

  313. Muñoz-Matutano, G. et al. Emergence of quantum correlations from interacting fibre-cavity polaritons. Nat. Mater. 18, 213–218 (2019).

    Article  Google Scholar 

  314. Hartmann, M. J., Brandão, F. G. S. L. & Plenio, M. B. Strongly interacting polaritons in coupled arrays of cavities. Nat. Phys. 2, 849–855 (2006).

    Article  Google Scholar 

  315. Greentree, A. D., Tahan, C., Cole, J. H. & Hollenberg, L. C. L. Quantum phase transitions of light. Nat. Phys. 2, 856–861 (2006).

    Article  Google Scholar 

  316. Angelakis, D. G., Santos, M. F. & Bose, S. Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays. Phys. Rev. A 76, 031805 (2007).

    Article  ADS  Google Scholar 

  317. Biella, A. et al. Phase diagram of incoherently driven strongly correlated photonic lattices. Phys. Rev. A 96, 023839 (2017).

    Article  ADS  Google Scholar 

  318. Lebreuilly, J. et al. Stabilizing strongly correlated photon fluids with non-Markovian reservoirs. Phys. Rev. A 96, 033828 (2017).

    Article  ADS  Google Scholar 

  319. Carusotto, I. et al. Fermionized photons in an array of driven dissipative nonlinear cavities. Phys. Rev. Lett. 103, 033601 (2009).

    Article  ADS  Google Scholar 

  320. Rota, R., Storme, F., Bartolo, N., Fazio, R. & Ciuti, C. Critical behavior of dissipative two-dimensional spin lattices. Phys. Rev. B 95, 134431 (2017).

    Article  ADS  Google Scholar 

  321. Rota, R., Minganti, F., Ciuti, C. & Savona, V. Quantum critical regime in a quadratically driven nonlinear photonic lattice. Phys. Rev. Lett. 122, 110405 (2019).

    Article  ADS  Google Scholar 

  322. Umucalılar, R. O. & Carusotto, I. Fractional quantum Hall states of photons in an array of dissipative coupled cavities. Phys. Rev. Lett. 108, 206809 (2012).

    Article  ADS  Google Scholar 

  323. Kapit, E., Hafezi, M. & Simon, S. H. Induced self-stabilization in fractional quantum Hall states of light. Phys. Rev. X 4, 031039 (2014).

    Google Scholar 

  324. Zasedatelev, A. V. et al. Single-photon nonlinearity at room temperature. Nature 597, 493–497 (2021).

    Article  ADS  Google Scholar 

  325. Moxley, F. I., Ilo-Okeke, E. O., Mudaliar, S. & Byrnes, T. Emergent Materials. Emergent Mater. 4, 971–988 (2021).

    Article  Google Scholar 

  326. Scheel, S. & Szameit, A. PT-symmetric photonic quantum systems with gain and loss do not exist. EPL 122, 34001 (2018).

    Article  ADS  Google Scholar 

  327. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  328. Lahtinen, V. & Pachos, J. A short introduction to topological quantum computation. SciPost Phys. 3, 021 (2017).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

J.B. acknowledges financial support from the Paris Île-de-France Région in the framework of DIM SIRTEQ, the QuantERA project ‘InterPol’ (ANR-QUAN-0003-05), the French National Research Agency project ‘Quantum Fluids of Light’ (ANR-16-CE30-0021) and the French RENATECH network. I.C. acknowledges financial support from the European Union FET-Open grant ‘MIR-BOSE’ (no. 737017), from the Provincia Autonoma di Trento, from the Q@TN initiative and from Google via the quantum NISQ award. J.B. and I.C. acknowledge support from the European Union H2020-FETFLAG-2018-2020 project ‘PhoQuS’ (no. 820392). M.W. acknowledges financial support from the FWO-Vlaanderen (grant no. G016219N).

Author information

Authors and Affiliations

  1. Centre de Nanosciences et de Nanotechnologies, Université Paris-Saclay, CNRS, Palaiseau, France

    Jacqueline Bloch

  2. INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, Trento, Italy

    Iacopo Carusotto

  3. TQC, Universiteit Antwerpen, Antwerpen, Belgium

    Michiel Wouters

Authors
  1. Jacqueline Bloch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iacopo Carusotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michiel Wouters
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed to all aspects of the article.

Corresponding authors

Correspondence to Jacqueline Bloch, Iacopo Carusotto or Michiel Wouters.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Physics thanks Alexey Kavokin, Na Young Kim and the other, anonymous, reviewers 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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bloch, J., Carusotto, I. & Wouters, M. Non-equilibrium Bose–Einstein condensation in photonic systems. Nat Rev Phys 4, 470–488 (2022). https://doi.org/10.1038/s42254-022-00464-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-022-00464-0

This article is cited by

Search

Advanced search

Quick links

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