Truncated Wigner Approximation for Nonequilibrium Polariton Quantum Fluids

Chapter
First Online:
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 172)

Abstract

In this chapter we review the stochastic approach that we recently developed to model the kinetics of polariton Bose–Einstein condensation, based on a truncated Wigner approximation. The approach consists in neglecting the third-order term appearing in the master equations for the Wigner distribution of the quantum field. The resulting Fokker–Planck equation can be modeled by numerically solving the corresponding stochastic Langevin equation, coupled to a phenomenological diffusion equation for the excitonic reservoir that provides the gain-loss mechanism. This approach is particularly well suited for polaritons, in which the neglected term is often negligible compared to the intrinsic loss rates of the polariton field. We apply our model to typical experimental situations and discuss the results, with particular focus on the dynamics of phase fluctuations and the possibility to observe a Berezinski-Kosterlitz-Thouless crossover in the polariton superfluid.

Keywords

Phase Fluctuation Quantum Degenerate Nonresonant Excitation Josephson Oscillation Lower Polariton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

References

  1. 1.
    C. Adrados, A. Amo, T.C.H. Liew, R. Hivet, R. Houdre, E. Giacobino, A.V. Kavokin, A. Bramati, Spin rings in bistable planar semiconductor microcavities. Phys. Rev. Lett. 105(21), 116402 (2010)Google Scholar
  2. 2.
    U. Al Khawaja, J.O. Andersen, N.P. Proukakis, H.T.C Stoof, Low dimensional bose gases. Phys. Rev. A, 66(1), 013615 (2002)Google Scholar
  3. 3.
    A. Amo, T.C.H. Liew, C. Adrados, R. Houdre, E. Giacobino, A.V. Kavokin, A. Bramati, Exciton-polariton spin switches. Nat. Photon. 4(6), 361–366 (2010)Google Scholar
  4. 4.
    A. Amo, S. Pigeon, D. Sanvitto, V.G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre, E. Giacobino, C. Ciuti, A. Bramati, Hydrodynamic solitons in polariton superfluids. Science, 332, 1167 (2011)Google Scholar
  5. 5.
    A. Amo, D. Sanvitto, F.P. Laussy, D. Ballarini, E. del Valle, M.D. Martin, A. Lemaitre, J. Bloch, D.N. Krizhanovskii, M.S. Skolnick, C. Tejedor, L. Vina, Collective fluid dynamics of a polariton condensate in a semiconductor microcavity. Nature, 457(7227), 291–U3 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, A. Bramati, Superfluidity of polaritons in semiconductor microcavities. Nat. Phys. 5(11), 805–810 (2009)CrossRefGoogle Scholar
  7. 7.
    A. Baas, J.P. Karr, H. Eleuch, E. Giacobino, Optical bistability in semiconductor microcavities. Phys. Rev. A 69(2) 023809 (2004)Google Scholar
  8. 8.
    D. Bajoni, E. Semenova, A. Lemaitre, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, J. Bloch, Optical bistability in a gaas-based polariton diode. Phys. Rev. Lett. 101(26), 267404 (2008)Google Scholar
  9. 9.
    D. Bajoni, P. Senellart, E. Wertz, I. Sagnes, A. Miard, A. Lemaitre, J. Bloch, Polariton laser using single micropillar gaas-gaalas semiconductor cavities. Phys. Rev. Lett. 100(4), 047401 (2008)Google Scholar
  10. 10.
    R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, K. West, Bose–einstein condensation of microcavity polaritons in a trap. Science 316(5827), 1007–1010 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    J. Bloch, R. Planel, V. Thierry-Mieg, J.M. Gerard, D. Barrier, J.Y. Marzin, E. Costard, Strong-coupling regime in pillar semiconductor microcavities. Superlattices Microst. 22(3), 371–374 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    H. Thien Cao, T.D. Doan, D.B. Tran Thoai, H. Haug, Condensation kinetics of cavity polaritons interacting with a thermal phonon bath. Phys. Rev. B 69(24), 245325 (2004)Google Scholar
  13. 13.
    G. Dasbach, M. Schwab, M. Bayer, A. Forchel, Parametric polariton scattering in microresonators with three-dimensional optical confinement. Phys. Rev. B 64(20), 201309 (2001)Google Scholar
  14. 14.
    M.J. Davis, S.A. Morgan, K. Burnett, Simulations of bose fields at finite temperature. Phys. Rev. Lett. 87(16), 160402 (2001)Google Scholar
  15. 15.
    H. Deng, H. Haug, Y. Yamamoto, Exciton-polariton bose–einstein condensation. Rev. Mod. Phys. 82(2), 1489–1537 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    H. Deng, D. Press, S. Gotzinger, G.S. Solomon, R. Hey, K.H. Ploog, Y. Yamamoto, Quantum degenerate exciton-polaritons in thermal equilibrium. Phys. Rev. Lett. 97(14), 146402 (2006)Google Scholar
  17. 17.
    H. Deng, G.S. Solomon, R. Hey, K.H. Ploog, Y. Yamamoto, Spatial coherence of a polariton condensate. Phys. Rev. Lett. 99(12), 126403 (2007)Google Scholar
  18. 18.
    T.D. Doan, H. Thien Cao, D.B. Tran Thoai, H. Haug, Coherence of condensed microcavity polaritons calculated within boltzmann-master equations. Phys. Rev. B 78(20), 205306 (2008)Google Scholar
  19. 19.
    T.D. Doan, H. Thien Cao, D.B. Tran Thoai, H. Haug, Condensation kinetics of microcavity polaritons with scattering by phonons and polaritons. Phys. Rev. B 72(8), 085301 (2005)Google Scholar
  20. 20.
    L. Ferrier, S. Pigeon, E. Wertz, M. Bamba, P. Senellart, Isabelle Sagnes, Aristide Lemaitre, Cristiano Ciuti, and Jacqueline Bloch. Polariton parametric oscillation in a single micropillar cavity. Appl. Phys. Lett. 97(3), 031105 (2010)Google Scholar
  21. 21.
    L. Ferrier, E. Wertz, R. Johne, D.D. Solnyshkov, P. Senellart, I. Sagnes, A. Lemaitre, G. Malpuech, J. Bloch, Interactions in confined polariton condensates. Phys. Rev. Lett. 106(12), 126401 (2011)Google Scholar
  22. 22.
    A.L. Fetter, Rotating trapped bose–einstein condensates. Rev. Mod. Phys. 81(2), 647–691 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    C.W. Gardiner, M.J. Davis, J. Phys. B 36, 4731 (2003)ADSCrossRefGoogle Scholar
  24. 24.
    C.W. Gardiner, P. Zoller, Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting bose gas without a trapping potential. Phys. Rev. A 55(4), 2902–2921 (1997)ADSCrossRefGoogle Scholar
  25. 25.
    N.A. Gippius, S.G. Tikhodeev, V.D. Kulakovskii, D.N. Krizhanovskii, A.I. Tartakovskii, Nonlinear dynamics of polariton scattering in semiconductor microcavity: Bistability vs. stimulated scattering. Europhys. Lett. 67(6), 997–1003 (2004)Google Scholar
  26. 26.
    M. Gurioli, F. Bogani, D.S. Wiersma, P. Roussignol, G. Cassabois, G. Khitrova, H. Gibbs, Experimental study of disorder in a semiconductor microcavity. Phys. Revi. B 64(16), 165309 (2001)Google Scholar
  27. 27.
    Z. Hadzibabic, P. Krueger, M. Cheneau, B. Battelier, J. Dalibard, Berezinskii - kosterlitz - thouless crossover in a trapped atomic gas. Nature 441, 1118 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    H. Haug, H. Thien Cao, D.B. Tran Thoai, Coherence and decoherence of a polariton condensate. Phys. Rev. B 81(24), 245309 (2010)Google Scholar
  29. 29.
    H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors. (Springer, Berlin, 1997)Google Scholar
  30. 30.
    R. Johne, I.A. Shelykh, D.D. Solnyshkov, G. Malpuech, Polaritonic analogue of datta and das spin transistor. Phys. Rev. B 81(12), 125327 (2010)Google Scholar
  31. 31.
    D. Kadio, M. Gajda, K. Rzazewski, Phase fluctuations of a bose–einstein condensate in low-dimensional geometry. Phys. Rev. A 72(1), 013607 (2005)Google Scholar
  32. 32.
    Yu. Kagan, V.A. Kashurnikov, A.V. Krasavin, N.V. Prokof’ev, B.V. Svistunov, Quasicondensation in a two-dimensional interacting bose gas. Phys. Rev. A 61(4), 043608 (2000)Google Scholar
  33. 33.
    Yu. Kagan, B.V. Svistunov, Evolution of correlation properties and appearance of broken symmetry in the process of bose–einstein condensation. Phys. Rev. Lett. 79(18), 3331–3334 (1997)ADSCrossRefGoogle Scholar
  34. 34.
    R. Idrissi Kaitouni, O. El Daif, A. Baas, M. Richard, T. Paraiso, P. Lugan, T. Guillet, F. Morier-Genoud, J.D. Ganiere, J.L. Staehli, V. Savona, B. Deveaud, Engineering the spatial confinement of exciton polaritons in semiconductors. Phys. Rev. B 74(15) 155311 (2006)Google Scholar
  35. 35.
    J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M.J. Keeling, F.M. Marchetti, M.H. Szymanska, R. Andre, J.L. Staehli, V. Savona, P.B. Littlewood, B. Deveaud, Le Si Dang, Bose–einstein condensation of exciton polaritons. Nature 443(7110), 409–414 (2006)ADSCrossRefGoogle Scholar
  36. 36.
    K.V. Kavokin, M.A. Kaliteevski, R.A. Abram, A.V. Kavokin, S. Sharkova, I.A. Shelykh, Stimulated emission of terahertz radiation by exciton-polariton lasers. Appl. Phys. Lett. 97(20), 1111 (2010)Google Scholar
  37. 37.
    J. Keeling, F.M. Marchetti, M.H. Szymanska, P.B. Littlewood, Collective coherence in planar semiconductor microcavities. Semicond. Sci. Tech. 22(5), R1–R26 (2007)ADSCrossRefGoogle Scholar
  38. 38.
    J. Keeling, N.G. Berloff, Spontaneous rotating vortex lattices in a pumped decaying condensate. Phys. Rev. Lett. 100(25), 250401 (2008)Google Scholar
  39. 39.
    J. Keeling, N.G. Berloff, Spontaneous rotating vortex lattices in a pumped decaying condensate. Phys. Rev. Lett. 100(25), 250401 (2008)Google Scholar
  40. 40.
    J.M. Kosterlitz, D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973)ADSCrossRefGoogle Scholar
  41. 41.
    D.N. Krizhanovskii, D.M. Whittaker, R.A. Bradley, K. Guda, D. Sarkar, D. Sanvitto, L. Vina, E. Cerda, P. Santos, K. Biermann, R. Hey, M.S. Skolnick, Effect of interactions on vortices in a nonequilibrium polariton condensate. Phys. Rev. Lett. 104(12), 126402 (2010)Google Scholar
  42. 42.
    K.G. Lagoudakis, F. Manni, B. Pietka, M. Wouters, T.C.H. Liew, V. Savona, A.V. Kavokin, R. Andre, B. Deveaud-Pledran, Probing the dynamics of spontaneous quantum vortices in polariton superfluids. Phys. Rev. Lett. 106(11), 115301 (2011)Google Scholar
  43. 43.
    K.G. Lagoudakis, T. Ostatnicky, A.V. Kavokin, Y.G. Rubo, R. Andre, B. Deveaud-Pledran, Observation of half-quantum vortices in an exciton-polariton condensate. Science 326(5955), 974–976 (2009)ADSCrossRefGoogle Scholar
  44. 44.
    K.G. Lagoudakis, B. Pietka, M. Wouters, R. Andre, B. Deveaud-Pledran, Coherent oscillations in an exciton-polariton josephson junction. Phys. Rev. Lett. 105(12), 120403 (2010)Google Scholar
  45. 45.
    K.G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. Andre, L.E.S.I. Dang, B. Deveaud-Pledran, Quantized vortices in an exciton-polariton condensate. Nat. Phys. 4(9), 706–710 (2008)CrossRefGoogle Scholar
  46. 46.
    W. Langbein, J.M. Hvam, Elastic scattering dynamics of cavity polaritons: Evidence for time-energy uncertainty and polariton localization. Phys. Rev. Lett. 88(4), 047401 (2002)Google Scholar
  47. 47.
    F.P. Laussy, A.V. Kavokin, I.A. Shelykh, Exciton-polariton mediated superconductivity. Phys. Rev. Lett. 104(10), 106402 (2010)Google Scholar
  48. 48.
    F.P. Laussy, G. Malpuech, A. Kavokin, P. Bigenwald, Spontaneous coherence buildup in a polariton laser. Phys. Rev. Lett. 93(1), 016402 (2004)Google Scholar
  49. 49.
    C. Leyder, T.C.H. Liew, A.V. Kavokin, I.A. Shelykh, M. Romanelli, J. Ph. Karr, E. Giacobino, A. Bramati, Interference of coherent polariton beams in microcavities: Polarization-controlled optical gates. Phys. Rev. Lett. 99(19), 196402 (2007)Google Scholar
  50. 50.
    T.C.H. Liew, A.V. Kavokin, T. Ostatnicky, M. Kaliteevski, I.A. Shelykh, R.A. Abram, Exciton-polariton integrated circuits. Phys. Rev. B 82(3), 033302 (2010)Google Scholar
  51. 51.
    T.C.H. Liew, A.V. Kavokin, I.A. Shelykh, Optical circuits based on polariton neurons in semiconductor microcavities. Phys. Rev. Lett. 101(1), 016402 (2008)Google Scholar
  52. 52.
    T.C.H. Liew, Y.G. Rubo, A.V. Kavokin, Generation and dynamics of vortex lattices in coherent exciton-polariton fields. Phys. Rev. Lett. 101(18), 187401 (2008)Google Scholar
  53. 53.
    A.P.D. Love, D.N. Krizhanovskii, D.M. Whittaker, R. Bouchekioua, D. Sanvitto, S. Al Rizeiqi, R. Bradley, M.S. Skolnick, P.R. Eastham, R. André, Le Si Dang. Intrinsic decoherence mechanisms in the microcavity polariton condensate. Phys. Rev. Lett. 101(6), 067404 (2008)Google Scholar
  54. 54.
    G. Malpuech, A. Kavokin, A. Di Carlo, J.J. Baumberg, Polariton lasing by exciton-electron scattering in semiconductor microcavities. Phys. Rev. B 65(15), 153310 (2002)Google Scholar
  55. 55.
    C. Mora, Y. Castin, Extension of bogoliubov theory to quasicondensates. Phys. Rev. A 67(5), 053615 (2003)Google Scholar
  56. 56.
    N. Na, Y. Yamamoto, Massive parallel generation of indistinguishable single photons via the polaritonic superfluid to mott-insulator quantum phase transition. New J. Phys. 12(12), 123001 (2010)Google Scholar
  57. 57.
    G. Nardin, K.G. Lagoudakis, B. Pietka, F. Morier-Genoud, Y. Leger, B. Deveaud-Pledran, Selective photoexcitation of confined exciton-polariton vortices. Phys. Rev. B 82(7), 073303 (2010)Google Scholar
  58. 58.
    T. Nikuni, E. Zaremba, A. Griffin, Two-fluid dynamics for a bose–einstein condensate out of local equilibrium with the noncondensate. Phys. Rev. Lett. 83(1), 10–13 (1999)ADSCrossRefGoogle Scholar
  59. 59.
    T. Ostatnicky, I.A. Shelykh, A.V. Kavokin, Theory of polarization-controlled polariton logic gates. Phys. Rev. B 81(12), 125319 (2010)Google Scholar
  60. 60.
    T.K. Paraiso, M. Wouters, Y. Leger, F. Morier-Genoud, B. Deveaud-Pledran, Multistability of a coherent spin ensemble in a semiconductor microcavity. Nat. Mater. 9(8), 655–660 (2010)ADSCrossRefGoogle Scholar
  61. 61.
    S. Pigeon, I. Carusotto, C. Ciuti, Hydrodynamic nucleation of vortices and solitons in a resonantly excited polariton superfluid. Phys. Rev. B 83(14), 144513 (2011)Google Scholar
  62. 62.
    S. Pilati, S. Giorgini, N. Prokof’ev, Critical temperature of interacting bose gases in two and three dimensions. Phys. Rev. Lett. 100(14), 140405 (2008)Google Scholar
  63. 63.
    L. Pitaevskii, S. Stringari, Bose–Einstein Condensation. (Oxford University Press, Oxford, 2003)Google Scholar
  64. 64.
    D. Porras, C. Ciuti, J.J. Baumberg, C. Tejedor, Polariton dynamics and bose–einstein condensation in semiconductor microcavities. Phys. Rev. B 66(8), 085304 (2002)Google Scholar
  65. 65.
    A. Posazhennikova, Colloquium: Weakly interacting, dilute bose gases in 2d. Rev. Mod. Phys. 78(4), 1111–1134 (2006)ADSCrossRefGoogle Scholar
  66. 66.
    N. Prokof’ev, O. Ruebenacker, B. Svistunov, Critical point of a weakly interacting two-dimensional bose gas. Phys. Rev. Lett. 87(27), 270402 (2001)Google Scholar
  67. 67.
    A. Recati, N. Pavloff, I. Carusotto, Bogoliubov theory of acoustic hawking radiation in bose–einstein condensates. Phys. Rev. A 80(4), 043603 (2009)Google Scholar
  68. 68.
    G. Roumpos, M.D. Fraser, A. Loffler, S. Hofling, A. Forchel, Y. Yamamoto, Single vortex-antivortex pair in an exciton-polariton condensate. Nat. Phys. 7(2), 129–133 (2011)CrossRefGoogle Scholar
  69. 69.
    D. Sarchi, V. Savona, Long-range order in the Bose–Einstein condensation of polaritons. Phys. Rev. B 75(11), 115326 (2007)Google Scholar
  70. 70.
    D. Sarkar, S.S. Gavrilov, M. Sich, J.H. Quilter, R.A. Bradley, N.A. Gippius, K. Guda, V.D. Kulakovskii, M.S. Skolnick, D.N. Krizhanovskii, Polarization bistability and resultant spin rings in semiconductor microcavities. Phys. Rev. Lett. 105(21), 216402 (2010)Google Scholar
  71. 71.
    V. Savona, Effect of interface disorder on quantum well excitons and microcavity polaritons. J. Phys. Condens. Matter 19(29), 295208 (2007)Google Scholar
  72. 72.
    M.O. Scully, M.S. Zubairy, Quantum optics. (Cambridge University Press, Cambridge, 1997)Google Scholar
  73. 73.
    I.A. Shelykh, R. Johne, D.D. Solnyshkov, G. Malpuech, Optically and electrically controlled polariton spin transistor. Phys. Rev. B 82(15), 153303 (2010)Google Scholar
  74. 74.
    A. Sinatra, C. Lobo, Y. Castin, The truncated wigner method for bose-condensed gases: limits of validity and applications. J. Phys. B Atom. Mol. Opt. Phys. 35(17), 3599 (2002)Google Scholar
  75. 75.
    D. Snoke, Polariton condensates - a feature rather than a bug. Nat. Phys. 4(9), 674–675 (2008)CrossRefGoogle Scholar
  76. 76.
    D. Snoke, P. Littlewood, Polariton condensates. Phys. Today 63(8), 42–47 (2010)CrossRefGoogle Scholar
  77. 77.
    R.M. Stevenson, V.N. Astratov, M.S. Skolnick, D.M. Whittaker, M. Emam-Ismail, A.I. Tartakovskii, P.G. Savvidis, J.J. Baumberg, J.S. Roberts, Continuous wave observation of massive polariton redistribution by stimulated scattering in semiconductor microcavities. Phys. Rev. Lett. 85(17), 3680–3683 (2000)ADSCrossRefGoogle Scholar
  78. 78.
    M.H. Szymanska, J. Keeling, P.B. Littlewood, Nonequilibrium quantum condensation in an incoherently pumped dissipative system. Phys. Rev. Lett. 96(23), 230602 (2006)Google Scholar
  79. 79.
    A.I. Tartakovskii, M. Emam-Ismail, R.M. Stevenson, M.S. Skolnick, V.N. Astratov, D.M. Whittaker, J.J. Baumberg, J.S. Roberts, Relaxation bottleneck and its suppression in semiconductor microcavities. Phys. Rev. B 62(4), R2283–R2286 (2000)ADSCrossRefGoogle Scholar
  80. 80.
    F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Bottleneck effects in the relaxation and photoluminescence of microcavity polaritons. Phys. Rev. B 56(12), 7554–7563 (1997)ADSCrossRefGoogle Scholar
  81. 81.
    F. Tassone, Y. Yamamoto, Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons. Phys. Rev. B 59(16), 10830–10842 (1999)ADSCrossRefGoogle Scholar
  82. 82.
    M. Trujillo-Martinez, A. Posazhennikova, J. Kroha, Nonequilibrium josephson oscillations in bose–einstein condensates without dissipation. Phys. Rev. Lett. 103(10), 105302 (2009)Google Scholar
  83. 83.
    S. Utsunomiya, L. Tian, G. Roumpos, C.W. Lai, N. Kumada, T. Fujisawa, M. Kuwata-Gonokami, A. Loeffler, S. Hoefling, A. Forchel, Y. Yamamoto, Observation of bogoliubov excitations in exciton-polariton condensates. Nat. Phys. 4(9), 700–705 (2008)CrossRefGoogle Scholar
  84. 84.
    D.F. Walls, G.J. Milburn, Quantum Optics. Springer-Verlag, Berlin (2008).MATHCrossRefGoogle Scholar
  85. 85.
    E. Wertz, L. Ferrier, D.D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaitre, I. Sagnes, R. Grousson, A.V. Kavokin, P. Senellart, G. Malpuech, J. Bloch, Spontaneous formation and optical manipulation of extended polariton condensates. Nat. Phys. 6(11), 860–864 (2010)CrossRefGoogle Scholar
  86. 86.
    E. Wertz, L. Ferrier, D.D. Solnyshkov, P. Senellart, D. Bajoni, A. Miard, A. Lemaitre, G. Malpuech, J. Bloch, Spontaneous formation of a polariton condensate in a planar gaas microcavity. Appl. Phys. Lett. 95(5), 051108 (2009)Google Scholar
  87. 87.
    M. Wouters, T.C.H. Liew, V. Savona, Energy relaxation in one-dimensional polariton condensates. Phys. Rev. B 82(24), 245315 (2010)Google Scholar
  88. 88.
    M. Wouters, I. Carusotto, Excitations in a nonequilibrium bose–einstein condensate of exciton polaritons. Phys. Rev. Lett. 99(14), 140402 (2007)Google Scholar
  89. 89.
    M. Wouters, I. Carusotto, Superfluidity and critical velocities in nonequilibrium bose–einstein condensates. Phys. Rev. Lett. 105(2), 020602 (2010)Google Scholar
  90. 90.
    M. Wouters, I. Carusotto, C. Ciuti, Spatial and spectral shape of inhomogeneous nonequilibrium exciton-polariton condensates. Phys. Rev. B 77(11), 115340 (2008)Google Scholar
  91. 91.
    M. Wouters, V. Savona, Stochastic classical field model for polariton condensates. Phys. Rev. B 79(16), (2009)Google Scholar
  92. 92.
    M. Wouters, V. Savona, Superfluidity of a nonequilibrium Bose–Einstein condensate of polaritons. Phys. Rev. B 81(5), 054508 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.TQC, Universiteit AntwerpenAntwerpenBelgium
  2. 2.Institute of Theoretical Physics, Ecole Polytechnique Fédérale de Lausanne EPFLLausanneSwitzerland

Personalised recommendations