Huang, K. Introduction to Statistical Physics 2nd edn (CRC, 2010).
Sachdev, S. Quantum Phase Transitions 2nd edn (Cambridge Univ. Press, 2011).
Nambu, Y. & Jona-Lasinio, G. Dynamical model of elementary particles based on an analogy with superconductivity. I. Phys. Rev. 122, 345–358 (1961).
Goldstone, J. Field theories with ‘superconductor’ solutions. Il Nuovo Cimento (1955-1965) 19, 154–164 (1961).
Goldstone, J., Salam, A. & Weinberg, S. Broken symmetries. Phys. Rev. 127, 965–970 (1962).
Lange, R. V. Nonrelativistic theorem analogous to the Goldstone theorem. Phys. Rev. 146, 301–303 (1966).
Takahashi, D. A. & Nitta, M. Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach. Ann. Phys. 354, 101–156 (2015).
Gunton, J. D. & Buckingham, M. J. Condensation of the ideal Bose gas as a cooperative transition. Phys. Rev. 166, 152–158 (1968).
Pitaevskii, L. & Stringari, S. Bose-Einstein Condensation and Superfluidity (Oxford Univ. Press, 2016).
Steinhauer, J., Ozeri, R., Katz, N. & Davidson, N. Excitation spectrum of a Bose-Einstein condensate. Phys. Rev. Lett. 88, 120407 (2002).
Schmittmann, B. & Zia, R. Statistical Mechanics of Driven Diffusive Systems (Elsevier, 1995).
Hidaka, Y. & Minami, Y. Spontaneous symmetry breaking and Nambu-Goldstone modes in open classical and quantum systems. Prog. Theor. Exp. Phys. 2020, 033A01 (2020).
Bloch, J., Carusotto, I. & Wouters, M. Non-equilibrium Bose-Einstein condensation in photonic systems. Nat. Rev. Phys. 4, 470–488 (2022).
Graham, R. & Haken, H. The quantum-fluctuations of the optical parametric oscillator. I. Z. Phys. A: Hadrons Nucl. 210, 276–302 (1968).
DeGiorgio, V. & Scully, M. O. Analogy between the laser threshold region and a second-order phase transition. Phys. Rev. A 2, 1170–1177 (1970).
Graham, R. & Haken, H. Laserlight—first example of a second-order phase transition far away from thermal equilibrium. Z. Phys. 237, 31–46 (1970).
Grossmann, S. & Richter, P. H. Laser threshold and nonlinear Landau fluctuation theory of phase transitions. Z. Phys. A: Hadrons Nucl. 242, 458–475 (1971).
Baumberg, J. et al. Parametric oscillation in a vertical microcavity: a polariton condensate or micro-optical parametric oscillation. Phys. Rev. B 62, R16247 (2000).
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).
Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006).
Richard, M. et al. Experimental evidence for nonequilibrium Bose condensation of exciton polaritons. Phys. Rev. B 72, 201301 (2005).
Szymańska, M. H., Keeling, J. & Littlewood, P. B. Nonequilibrium quantum condensation in an incoherently pumped dissipative system. Phys. Rev. Lett. 96, 230602 (2006).
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).
Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007).
Keeling, J., Szymańska, M. H. & Littlewood, P. B. in Optical Generation and Control of Quantum Coherence in Semiconductor Nanostructures (eds Slavcheva, G. & Roussignol, P.) 293–329 (Springer, 2010).
Dunnett, K. & Szymańska, M. H. Keldysh field theory for nonequilibrium condensation in a parametrically pumped polariton system. Phys. Rev. B 93, 195306 (2016).
Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).
Carusotto, I. & Ciuti, C. Spontaneous microcavity-polariton coherence across the parametric threshold: quantum Monte Carlo studies. Phys. Rev. B 72, 125335 (2005).
Stepanov, P. et al. Dispersion relation of the collective excitations in a resonantly driven polariton fluid. Nat. Commun. 10, 3869 (2019).
Claude, F. et al. High-resolution coherent probe spectroscopy of a polariton quantum fluid. Phys. Rev. Lett. 129, 103601 (2022).
Chow, W. W., Scully, M. O. & Van Stryland, E. W. Line narrowing in a symmetry broken laser. Opt. Commun. 15, 6–9 (1975).
Courtois, J., Smith, A., Fabre, C. & Reynaud, S. Phase diffusion and quantum noise in the optical parametric oscillator: a semiclassical approach. J. Mod. Opt. 38, 177–191 (1991).
Savvidis, P. G. et al. Angle-resonant stimulated polariton amplifier. Phys. Rev. Lett. 84, 1547–1550 (2000).
Baas, A., Karr, J.-P., Romanelli, M., Bramati, A. & Giacobino, E. Quantum degeneracy of microcavity polaritons. Phys. Rev. Lett. 96, 176401 (2006).
Aßmann, M. et al. From polariton condensates to highly photonic quantum degenerate states of bosonic matter. Proc. Natl Acad. Sci. USA 108, 1804–1809 (2011).
Nakayama, M. & Ueda, M. Observation of diffusive and dispersive profiles of the nonequilibrium polariton-condensate dispersion relation in a CuBr microcavity. Phys. Rev. B 95, 125315 (2017).
Ballarini, D. et al. Observation of long-lived polariton states in semiconductor microcavities across the parametric threshold. Phys. Rev. Lett. 102, 056402 (2009).
Pieczarka, M. et al. Observation of quantum depletion in a non-equilibrium exciton-polariton condensate. Nat. Commun. 11, 429 (2020).
Estrecho, E. et al. Low-energy collective oscillations and Bogoliubov sound in an exciton-polariton condensate. Phys. Rev. Lett. 126, 075301 (2021).
Ballarini, D. et al. Directional Goldstone waves in polariton condensates close to equilibrium. Nat. Commun. 11, 217 (2020).
Claude, F. et al. Spectrum of collective excitations of a quantum fluid of polaritons. Phys. Rev. B 107, 174507 (2023).
Wouters, M. & Carusotto, I. Absence of long-range coherence in the parametric emission of photonic wires. Phys. Rev. B 74, 245316 (2006).
Porras, D. & Tejedor, C. Linewidth of a polariton laser: theoretical analysis of self-interaction effects. Phys. Rev. B 67, 161310 (2003).
Love, A. P. D. et al. Intrinsic decoherence mechanisms in the microcavity polariton condensate. Phys. Rev. Lett. 101, 067404 (2008).
Amelio, I. & Carusotto, I. Bogoliubov theory of the laser linewidth and application to polariton condensates. Phys. Rev. A 105, 023527 (2022).
Alaeian, H., Giedke, G., Carusotto, I., Löw, R. & Pfau, T. Limit cycle phase and goldstone mode in driven dissipative systems. Phys. Rev. A 103, 013712 (2021).
Ji, K., Gladilin, V. N. & Wouters, M. Temporal coherence of one-dimensional nonequilibrium quantum fluids. Phys. Rev. B 91, 045301 (2015).
Fontaine, Q. et al. Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate. Nature 608, 687–691 (2022).
Ferrier, A., Zamora, A., Dagvadorj, G. & Szymańska, M. H. Searching for the Kardar-Parisi-Zhang phase in microcavity polaritons. Phys. Rev. B 105, 205301 (2022).
Léonard, J., Morales, A., Zupancic, P., Donner, T. & Esslinger, T. Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas. Science 358, 1415–1418 (2017).
Tanzi, L. et al. Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas. Nature 574, 382–385 (2019).
Trypogeorgos, D. et al. Emerging supersolidity in photonic-crystal polariton condensates. Nature 639, 337–341 (2025).
Nigro, D. et al. Supersolidity of polariton condensates in photonic crystal waveguides. Phys. Rev. Lett. 134, 056002 (2025).
Ma, R. et al. A dissipatively stabilized Mott insulator of photons. Nature 566, 51–57 (2019).
Clark, L. W., Schine, N., Baum, C., Jia, N. & Simon, J. Observation of Laughlin states made of light. Nature 582, 41–45 (2020).
Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).
Claude, F. et al. Observation of the diffusive Nambu-Goldstone mode of a non-equilibrium phase transition. Zenodo https://doi.org/10.5281/zenodo.15079205 (2025).