Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate

V. Yu. Shishkov1,2,3,4, E. S. Andrianov1,2,3,4, and Yu. E. Lozovik1,5,6

1Dukhov Research Institute of Automatics (VNIIA), 22 Sushchevskaya, Moscow 127055, Russia;
2Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, Dolgoprudny 141700, Moscow region, Russia;
3Center for Photonics and Quantum Materials, Skolkovo Institute of Science and Technology, Moscow, Russian Federation
4Laboratories for Hybrid Photonics, Skolkovo Institute of Science and Technology, Moscow, Russian Federation
5Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia;
6Institute for Spectroscopy RAS, 5 Fizicheskaya, Troitsk 142190, Russia;

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


The theoretical description of non-equilibrium Bose–Einstein condensate (BEC) is one of the main challenges in modern statistical physics and kinetics. The non-equilibrium nature of BEC makes it impossible to employ the well-established formalism of statistical mechanics. We develop a framework for the analytical description of a non-equilibrium phase transition to BEC that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states. We consider the limit of fast thermalization and obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC which also covers the equilibrium case. For the particular cases of 2D and 3D, we investigate the non-equilibrium formation of BEC by finding the temperature dependence of the ground state occupation and second-order coherence function. We show that for a given pumping rate, the macroscopic occupation of the ground state and buildup of coherence may occur at different temperatures. Moreover, the buildup of coherence strongly depends on the pumping scheme. We also investigate the condensate linewidth and show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

🇺🇦 Quantum strongly condemns the 2022 invasion of Ukraine, the loss of life and war crimes inflicted by Russian forces. For more information on our policy on publishing articles by authors based in Russian institutions, see this post

We develop a framework for the analytical description of a non-equilibrium phase transition to Bose–Einstein condensate (BEC) that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states.
We obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC in the fast thermalization limit.
We find the temperature dependence of the ground state occupation and second-order coherence function.
We show that the buildup of coherence of the ground state strongly depends on the pumping scheme.
We also show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

► BibTeX data

► References

[1] J.D. Plumhof, T. Stöferle, L. Mai, U. Scherf, and R.F. Mahrt. Room-temperature Bose–Einstein condensation of cavity exciton–polaritons in a polymer. Nature Materials, 13 (3): 247–252, 2014. https:/​/​​10.1038/​nmat3825.

[2] A.V. Zasedatelev, A.V. Baranikov, D. Urbonas, F. Scafirimuto, U. Scherf, T. Stöferle, R.F. Mahrt, and P.G. Lagoudakis. A room-temperature organic polariton transistor. Nature Photonics, 13 (6): 378–383, 2019. https:/​/​​10.1038/​s41566-019-0392-8.

[3] A.V. Zasedatelev, A.V. Baranikov, D. Sannikov, D. Urbonas, F. Scafirimuto, V.Yu. Shishkov, E.S. Andrianov, Y.E. Lozovik, U. Scherf, T. Stöferle, R.F. Mahrt, and P.G. Lagoudakis. Single-photon nonlinearity at room temperature. Nature, 597: 493–497, 2021. https:/​/​​10.1038/​s41586-021-03866-9.

[4] D. Sanvitto and S. Kéna-Cohen. The road towards polaritonic devices. Nature Materials, 15 (10): 1061–1073, 2016. https:/​/​​10.1038/​nmat4668.

[5] J. Keeling and S. Kéna-Cohen. Bose–Einstein condensation of exciton-polaritons in organic microcavities. Annual Review of Physical Chemistry, 71: 435–459, 2020. https:/​/​​10.1146/​annurev-physchem-010920-102509.

[6] H. Deng, G. Weihs, D. Snoke, J. Bloch, and Y. Yamamoto. Polariton lasing vs. photon lasing in a semiconductor microcavity. Proceedings of the National Academy of Sciences, 100 (26): 15318–15323, 2003. https:/​/​​10.1073/​pnas.2634328100.

[7] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M. Keeling, F.M. Marchetti, M.H. Szymańska, R. André, J.L. Staehli, and V. Savona. Bose–Einstein condensation of exciton polaritons. Nature, 443 (7110): 409–414, 2006. https:/​/​​10.1038/​nature05131.

[8] M. Combescot and S.-Y. Shiau. Excitons and Cooper pairs: two composite bosons in many-body physics. Oxford University Press, 2015.

[9] T. Byrnes, N. Y. Kim, and Y. Yamamoto. Exciton–polariton condensates. Nature Physics, 10 (11): 803–813, 2014. https:/​/​​10.1038/​nphys3143.

[10] E. Wertz, L. Ferrier, D.D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaı̂tre, I. Sagnes, R. Grousson, A.V. Kavokin, P. Senellart, and G. Malpuech. Spontaneous formation and optical manipulation of extended polariton condensates. Nature Physics, 6 (11): 860–864, 2010. https:/​/​​10.1038/​nphys1750.

[11] R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West. Bose–Einstein condensation of microcavity polaritons in a trap. Science, 316 (5827): 1007–1010, 2007. https:/​/​​10.1126/​science.1140990.

[12] E. Estrecho, T. Gao, N. Bobrovska, M.D. Fraser, M. Steger, L. Pfeiffer, K. West, T.C.H. Liew, M. Matuszewski, D.W. Snoke, and A.G. Truscott. Single-shot condensation of exciton polaritons and the hole burning effect. Nature Communications, 9 (1): 1–9, 2018. https:/​/​​10.1038/​s41467-018-05349-4.

[13] Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L.N. Pfeiffer, K. West, D.W. Snoke, and K.A. Nelson. Bose–Einstein condensation of long-lifetime polaritons in thermal equilibrium. Physical Review Letters, 118 (1): 016602, 2017. https:/​/​​10.1103/​PhysRevLett.118.016602.

[14] H. Deng, H. Haug, and Y. Yamamoto. Exciton-polariton Bose-Einstein condensation. Reviews of Modern Physics, 82 (2): 1489, 2010. https:/​/​​10.1103/​RevModPhys.82.1489.

[15] M. Klaas, E. Schlottmann, H. Flayac, F.P. Laussy, F. Gericke, M. Schmidt, M. V. Helversen, J. Beyer, S. Brodbeck, H. Suchomel, et al. Photon-number-resolved measurement of an exciton-polariton condensate. Physical Review Letters, 121 (4): 047401, 2018. https:/​/​​10.1103/​PhysRevLett.121.047401.

[16] A. Imamoglu, R.J. Ram, S. Pau, and Y. Yamamoto. Nonequilibrium condensates and lasers without inversion: Exciton-polariton lasers. Physical Review A, 53 (6): 4250, 1996. https:/​/​​10.1103/​PhysRevA.53.4250.

[17] M. Wei, S.K. Rajendran, H. Ohadi, L. Tropf, M.C. Gather, G.A. Turnbull, and I.D.W. Samuel. Low-threshold polariton lasing in a highly disordered conjugated polymer. Optica, 6 (9): 1124–1129, 2019. https:/​/​​10.1364/​OPTICA.6.001124.

[18] R. Weill, A. Bekker, B. Levit, and B. Fischer. Bose–Einstein condensation of photons in an erbium–ytterbium co-doped fiber cavity. Nature Communications, 10 (1): 1–6, 2019. https:/​/​​10.1038/​s41467-019-08527-0.

[19] T.K. Hakala, A.J. Moilanen, A.I. Väkeväinen, R. Guo, J.-P. Martikainen, K.S. Daskalakis, H.T. Rekola, A. Julku, and P. Törmä. Bose–Einstein condensation in a plasmonic lattice. Nature Physics, 14 (7): 739–744, 2018. https:/​/​​10.1038/​s41567-018-0109-9.

[20] A.I. Väkeväinen, A.J. Moilanen, M. Nečada, T.K. Hakala, K.S. Daskalakis, and P. Törmä. Sub-picosecond thermalization dynamics in condensation of strongly coupled lattice plasmons. Nature Communications, 11 (1): 1–12, 2020. https:/​/​​10.1038/​s41467-020-16906-1.

[21] G. Malpuech, A. Di Carlo, A. Kavokin, J.J. Baumberg, M. Zamfirescu, and P. Lugli. Room-temperature polariton lasers based on gan microcavities. Applied Physics Letters, 81 (3): 412–414, 2002. https:/​/​​10.1063/​1.1494126.

[22] L. Banyai and P. Gartner. Real-time Bose–Einstein condensation in a finite volume with a discrete spectrum. Physical Review Letters, 88 (21): 210404, 2002. https:/​/​​10.1103/​PhysRevLett.88.210404.

[23] H.T. Cao, T.D. Doan, D.B.T. Thoai, and H. Haug. Condensation kinetics of cavity polaritons interacting with a thermal phonon bath. Physical Review B, 69 (24): 245325, 2004. https:/​/​​10.1103/​PhysRevB.69.245325.

[24] T.D. Doan, H. Thien Cao, D.B.T. Thoai, and H. Haug. Coherence of condensed microcavity polaritons calculated within boltzmann-master equations. Physical Review B, 78 (20): 205306, 2008. https:/​/​​10.1103/​PhysRevB.78.205306.

[25] F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, and P. Schwendimann. Bottleneck effects in the relaxation and photoluminescence of microcavity polaritons. Physical Review B, 56 (12): 7554, 1997. https:/​/​​10.1103/​PhysRevB.56.7554.

[26] P. Kirton and J. Keeling. Nonequilibrium model of photon condensation. Physical Review Letters, 111 (10): 100404, 2013. https:/​/​​10.1103/​PhysRevLett.111.100404.

[27] P. Kirton and J. Keeling. Thermalization and breakdown of thermalization in photon condensates. Physical Review A, 91 (3): 033826, 2015. https:/​/​​10.1103/​PhysRevA.91.033826.

[28] A. Strashko, P. Kirton, and J. Keeling. Organic polariton lasing and the weak to strong coupling crossover. Physical Review Letters, 121 (19): 193601, 2018. https:/​/​​10.1103/​PhysRevLett.121.193601.

[29] K.B. Arnardottir, A.J. Moilanen, A. Strashko, P. Törmä, and J. Keeling. Multimode organic polariton lasing. Physical Review Letters, 125 (23): 233603, 2020. https:/​/​​10.1103/​PhysRevLett.125.233603.

[30] I. Carusotto and C. Ciuti. Quantum fluids of light. Reviews of Modern Physics, 85 (1): 299, 2013. https:/​/​​10.1103/​RevModPhys.85.299.

[31] A. Kavokin, J.J. Baumberg, G. Malpuech, and F.P. Laussy. Microcavities. Oxford university press, 2017.

[32] F.P. Laussy. Exciton polaritons in microcavities: New frontiers. In Daniele Sanvitto and Vladislav Timofeev, editors, Springer Series in solid–state sciences, volume 172, chapter 1, pages 1–42. Springer Science & Business Media, 2012.

[33] F.P. Laussy, G. Malpuech, and A. Kavokin. Spontaneous coherence buildup in a polariton laser. Physica Status Solidi (c), 1 (6): 1339–1350, 2004a. https:/​/​​10.1002/​pssc.200304064.

[34] F.P. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald. Spontaneous coherence buildup in a polariton laser. Physical Review Letters, 93: 016402, Jun 2004b. https:/​/​​10.1103/​PhysRevLett.93.016402.

[35] V.Yu. Shishkov, E.S. Andrianov, A.V. Zasedatelev, P.G. Lagoudakis, and Y.E. Lozovik. Exact analytical solution for the density matrix of a nonequilibrium polariton Bose–Einstein condensate. Physical Review Letters, 128: 065301, Feb 2022. https:/​/​​10.1103/​PhysRevLett.128.065301.

[36] P.G. Lagoudakis, M.D. Martin, J.J. Baumberg, A. Qarry, E. Cohen, and L.N. Pfeiffer. Electron-polariton scattering in semiconductor microcavities. Physical Review Letters, 90 (20): 206401, 2003. https:/​/​​10.1103/​PhysRevLett.90.206401.

[37] M. Maragkou, A.J.D. Grundy, T. Ostatnickỳ, and P.G. Lagoudakis. Longitudinal optical phonon assisted polariton laser. Applied Physics Letters, 97 (11): 111110, 2010. https:/​/​​10.1063/​1.3488012.

[38] D. M. Coles, P. Michetti, C. Clark, W. C. Tsoi, A. M. Adawi, J.-S. Kim, and D. G. Lidzey. Vibrationally assisted polariton-relaxation processes in strongly coupled organic-semiconductor microcavities. Advanced Functional Materials, 21 (19): 3691–3696, 2011. https:/​/​​10.1002/​adfm.201100756.

[39] H.-P. Breuer and F. Petruccione. The theory of open quantum systems. Oxford University Press on Demand, 2002.

[40] M. Litinskaya, P. Reineker, and V.M. Agranovich. Fast polariton relaxation in strongly coupled organic microcavities. Journal of Luminescence, 110 (4): 364–372, 2004. https:/​/​​10.1016/​j.jlumin.2004.08.033.

[41] L. Mazza, L. Fontanesi, and G.C. La Rocca. Organic-based microcavities with vibronic progressions: Photoluminescence. Physical Review B, 80 (23): 235314, 2009. https:/​/​​10.1103/​PhysRevB.80.235314.

[42] E.R. Bittner and C. Silva. Estimating the conditions for polariton condensation in organic thin-film microcavities. The Journal of Chemical Physics, 136 (3): 034510, 2012. https:/​/​​10.1063/​1.3678015.

[43] J.A. Ć wik, S. Reja, P.B. Littlewood, and J. Keeling. Polariton condensation with saturable molecules dressed by vibrational modes. Europhysics Letters, 105 (4): 47009, 2014. https:/​/​​10.1209/​0295-5075/​105/​47009.

[44] N. Somaschi, L. Mouchliadis, D. Coles, I.E. Perakis, D.G. Lidzey, P.G. Lagoudakis, and P.G. Savvidis. Ultrafast polariton population build-up mediated by molecular phonons in organic microcavities. Applied Physics Letters, 99 (14): 209, 2011. https:/​/​​10.1063/​1.3645633.

[45] M. Ramezani, Q. Le-Van, A. Halpin, and J.G. Rivas. Nonlinear emission of molecular ensembles strongly coupled to plasmonic lattices with structural imperfections. Physical Review Letters, 121 (24): 243904, 2018. https:/​/​​10.1103/​PhysRevLett.121.243904.

[46] P.G. Savvidis, J.J. Baumberg, R.M. Stevenson, M.S. Skolnick, D.M. Whittaker, and J.S. Roberts. Angle-resonant stimulated polariton amplifier. Physical Review Letters, 84 (7): 1547, 2000. https:/​/​​10.1103/​PhysRevLett.84.1547.

[47] R. Kosloff. Quantum thermodynamics: A dynamical viewpoint. Entropy, 15 (6): 2100–2128, 2013. https:/​/​​10.3390/​e15062100.

[48] V.Yu. Shishkov, E.S. Andrianov, A.A. Pukhov, A.P. Vinogradov, and A.A. Lisyansky. Zeroth law of thermodynamics for thermalized open quantum systems having constants of motion. Physical Review E, 98 (2): 022132, 2018. https:/​/​​10.1103/​PhysRevE.98.022132.

[49] O.L. Berman, Yu.E. Lozovik, and D.W. Snoke. Theory of Bose–Einstein condensation and superfluidity of two-dimensional polaritons in an in-plane harmonic potential. Physical Review B, 77 (15): 155317, 2008. https:/​/​​10.1103/​PhysRevB.77.155317.

[50] M. Toda, R. Kubo, and N. Saito. Statistical Physics I: Equilibrium Statistical Mechanics. Springer Series in Solid-State Sciences №30. Springer Berlin Heidelberg, 1983.

[51] L.D. Landau and E.M. Lifshitz. Course of theoretical physics. Elsevier, 2013.

[52] A.R. Fraser. XV. The condensation of a perfect Bose–Einstein gas.–II. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 42 (325): 165–175, 1951a. https:/​/​​10.1080/​14786445108560984.

[53] P. Borrmann and G. Franke. Recursion formulas for quantum statistical partition functions. The Journal of Chemical Physics, 98 (3): 2484, 1993. https:/​/​​10.1063/​1.464180.

[54] F. Brosens, J.T. Devreese, and L.F. Lemmens. Thermodynamics of coupled identical oscillators within the path-integral formalism. Physical Review E, 55 (1): 227, 1997. https:/​/​​10.1103/​PhysRevE.55.227.

[55] C. Weiss and M. Wilkens. Particle number counting statistics in ideal Bose gases. Optics Express, 1 (10): 272–283, 1997. https:/​/​​10.1364/​OE.1.000272.

[56] K.C. Chase, A.Z. Mekjian, and L. Zamick. Canonical and microcanonical ensemble approaches to Bose–Einstein condensation: The thermodynamics of particles in harmonic traps. The European Physical Journal B-Condensed Matter and Complex Systems, 8 (2): 281–285, 1999. https:/​/​​10.1007/​s100510050691.

[57] V.V. Kocharovsky, V.V. Kocharovsky, M. Holthaus, C.H.R. Ooi, A. Svidzinsky, W. Ketterle, and M.O. Scully. Fluctuations in ideal and interacting Bose–Einstein condensates: From the laser phase transition analogy to squeezed states and bogoliubov quasiparticles. Advances in Atomic, Molecular, and Optical Physics, 53: 291–411, 2006. https:/​/​​10.1016/​S1049-250X(06)53010-1.

[58] A.R. Fraser. XIV. The condensation of a perfect Bose–Einstein gas.–I. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 42 (325): 156–164, 1951b. https:/​/​​10.1080/​14786445108560983.

[59] G. Schubert. Zur Bose–Statistik. Zeitschrift für Naturforschung A, 1 (3): 113–120, 1946. https:/​/​​10.1515/​zna-1946-0301.

[60] G. Schubert. Zur Bose-Statistik (Nachtrag). Zeitschrift für Naturforschung A, 2 (5): 250–251, 1947. https:/​/​​10.1515/​zna-1947-0503.

[61] M. Holthaus and E. Kalinowski. The saddle-point method for condensed Bose gases. Annals of Physics, 276 (2): 321–360, 1999. https:/​/​​10.1006/​aphy.1999.5950.

[62] L. Pitaevskii and S. Stringari. Bose–Einstein Condensation. International series of monographs on physics 116 Oxford science publications. Clarendon Press, 2003.

[63] W. Ketterle and N.J. Van Druten. Bose–Einstein condensation of a finite number of particles trapped in one or three dimensions. Physical Review A, 54 (1): 656, 1996. https:/​/​​10.1103/​PhysRevA.54.656.

[64] M.O. Scully and M.S. Zubairy. Quantum optics, 1999.

[65] N.D. Mermin and H. Wagner. Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic heisenberg models. Physical Review Letters, 17 (22): 1133, 1966. https:/​/​​10.1103/​PhysRevLett.17.1133.

[66] L. Mouchliadis and A.L. Ivanov. First-order spatial coherence of excitons in planar nanostructures: A k-filtering effect. Physical Review B, 78 (3): 033306, 2008. https:/​/​​10.1103/​PhysRevB.78.033306.

[67] M. Wilkens and C. Weiss. Particle number fluctuations in an ideal Bose gas. Journal of Modern Optics, 44 (10): 1801–1814, 1997. https:/​/​​10.1080/​09500349708231847.

[68] J. Mossel and J.-S. Caux. Exact time evolution of space-and time-dependent correlation functions after an interaction quench in the one-dimensional Bose gas. New Journal of Physics, 14 (7): 075006, 2012. https:/​/​​10.1088/​1367-2630/​14/​7/​075006.

[69] L. Comtet. Advanced Combinatorics: The art of finite and infinite expansions. Springer Science & Business Media, 2012.

Cited by

[1] Vladislav Yu. Shishkov and Evgeny S. Andrianov, "Negative compressibility of a nonequilibrium nonideal Bose-Einstein condensate", Physical Review E 106 6, 064108 (2022).

[2] Ivan Amelio, Alessio Chiocchetta, and Iacopo Carusotto, "Kardar-Parisi-Zhang universality in the coherence time of nonequilibrium one-dimensional quasicondensates", Physical Review E 109 1, 014104 (2024).

[3] Evgeny A. Tereshchenkov, Ivan V. Panyukov, Mikhail Misko, Vladislav Y. Shishkov, Evgeny S. Andrianov, and Anton V. Zasedatelev, "Thermalization rate of polaritons in strongly-coupled molecular systems", Nanophotonics 13 14, 2635 (2024).

[4] Andrey S. Plyashechnik, Alexey A. Sokolik, and Yurii E. Lozovik, "Bose-Einstein condensation in a canonical ensemble with fixed total momentum", Physical Review A 110 1, 013301 (2024).

[5] Fabrice P. Laussy, "A Quantum Theory for Bose--Einstein Condensation of the Ideal Gas", Quantum Views 6, 67 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-07-15 21:12:46) and SAO/NASA ADS (last updated successfully 2024-07-15 21:12:47). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate

  1. Pingback: Perspective in Quantum Views by Fabrice P. Laussy "A Quantum Theory for Bose--Einstein Condensation of the Ideal Gas"