Exciton Polaritons in Microcavities pp 267-288 | Cite as
Truncated Wigner Approximation for Nonequilibrium Polariton Quantum Fluids
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 PolaritonReferences
- 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.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.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.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.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.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.A. Baas, J.P. Karr, H. Eleuch, E. Giacobino, Optical bistability in semiconductor microcavities. Phys. Rev. A 69(2) 023809 (2004)Google Scholar
- 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.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.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.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.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.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.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.H. Deng, H. Haug, Y. Yamamoto, Exciton-polariton bose–einstein condensation. Rev. Mod. Phys. 82(2), 1489–1537 (2010)ADSCrossRefGoogle Scholar
- 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.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.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.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.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.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.A.L. Fetter, Rotating trapped bose–einstein condensates. Rev. Mod. Phys. 81(2), 647–691 (2009)ADSCrossRefGoogle Scholar
- 23.C.W. Gardiner, M.J. Davis, J. Phys. B 36, 4731 (2003)ADSCrossRefGoogle Scholar
- 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.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.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.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.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.H. Haug, A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors. (Springer, Berlin, 1997)Google Scholar
- 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.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.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.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.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.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.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.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.J. Keeling, N.G. Berloff, Spontaneous rotating vortex lattices in a pumped decaying condensate. Phys. Rev. Lett. 100(25), 250401 (2008)Google Scholar
- 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.J.M. Kosterlitz, D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973)ADSCrossRefGoogle Scholar
- 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.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.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.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.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.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.F.P. Laussy, A.V. Kavokin, I.A. Shelykh, Exciton-polariton mediated superconductivity. Phys. Rev. Lett. 104(10), 106402 (2010)Google Scholar
- 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.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.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.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.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.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.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.C. Mora, Y. Castin, Extension of bogoliubov theory to quasicondensates. Phys. Rev. A 67(5), 053615 (2003)Google Scholar
- 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.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.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.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.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.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.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.L. Pitaevskii, S. Stringari, Bose–Einstein Condensation. (Oxford University Press, Oxford, 2003)Google Scholar
- 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.A. Posazhennikova, Colloquium: Weakly interacting, dilute bose gases in 2d. Rev. Mod. Phys. 78(4), 1111–1134 (2006)ADSCrossRefGoogle Scholar
- 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.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.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.D. Sarchi, V. Savona, Long-range order in the Bose–Einstein condensation of polaritons. Phys. Rev. B 75(11), 115326 (2007)Google Scholar
- 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.V. Savona, Effect of interface disorder on quantum well excitons and microcavity polaritons. J. Phys. Condens. Matter 19(29), 295208 (2007)Google Scholar
- 72.M.O. Scully, M.S. Zubairy, Quantum optics. (Cambridge University Press, Cambridge, 1997)Google Scholar
- 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.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.D. Snoke, Polariton condensates - a feature rather than a bug. Nat. Phys. 4(9), 674–675 (2008)CrossRefGoogle Scholar
- 76.D. Snoke, P. Littlewood, Polariton condensates. Phys. Today 63(8), 42–47 (2010)CrossRefGoogle Scholar
- 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.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.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.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.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.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.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.D.F. Walls, G.J. Milburn, Quantum Optics. Springer-Verlag, Berlin (2008).MATHCrossRefGoogle Scholar
- 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.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.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.M. Wouters, I. Carusotto, Excitations in a nonequilibrium bose–einstein condensate of exciton polaritons. Phys. Rev. Lett. 99(14), 140402 (2007)Google Scholar
- 89.M. Wouters, I. Carusotto, Superfluidity and critical velocities in nonequilibrium bose–einstein condensates. Phys. Rev. Lett. 105(2), 020602 (2010)Google Scholar
- 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.M. Wouters, V. Savona, Stochastic classical field model for polariton condensates. Phys. Rev. B 79(16), (2009)Google Scholar
- 92.M. Wouters, V. Savona, Superfluidity of a nonequilibrium Bose–Einstein condensate of polaritons. Phys. Rev. B 81(5), 054508 (2010)Google Scholar