Quantum Dynamics of Polariton Condensates

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

Abstract

We illustrate the rich and fundamental physics that is accessible with the semiconductor implementation of the quantum superposition of light and matter: exciton–polaritons. The short lifetime of polaritons makes them an out-of-equilibrium system. Their dynamic is an important ingredient in their behaviour and properties. Their peculiar dispersion also allows a rich engineering of various processes, tuning the system from light- to matter-like. Finally, the exciton–exciton interaction turns them into a non-linear system. The interplay of all these factors makes polaritons one of today’s most versatile and fruitful research arena, both theoretically and experimentally. In this chapter we give a rather general picture of these specificities that we isolate in various dimensionalities (0, 1, and 2D). One of the most intensively researched area in the semiconductor implementation of the polariton physics is related to Bose–Einstein condensation. We solve exactly a configuration of relaxation from the Rayleigh circle into the ground state in the framework of quantum Boltzmann master equations and show how coherence builds up spontaneously in the system, by copying in a single quantum state statistical features characteristic of the macroscopic system. In this way, we extend to higher order correlations the historical reasoning of Einstein, who predicted the phenomenon by arguments on the mean populations. We show how lifetime and pumping allow a simpler treatment by reducing the required number of states, for which we present a full quantum treatment. We contrast this condensate build-up with the 0D case where the reduced complexity allows an exact numerical treatment. The coherence build-up in this cavity QED limit manifests as lasing with a sharp line in the cavity mode that produces a variation of the Mollow triplet in the exciton emission, as the cavity effectively replaces the laser in the conventional resonance fluorescence scenario. We show how lasing also arises in this case as a condensation of polaritons, and can be substituted in the case of vanishing intensities by a coherent field formed when strong coupling is optimum. This zero-dimensional limit also provides an exact picture of the transition from the quantum to the classical regime, a universal process of unsuspected complexity. Finally, we illustrate the recent development of polariton quantum hydrodynamics with propagation of coherent wave packets. The short lifetime allows a continuous observation of this dynamics in real space, a picture completed with the observation of their emission spectra in energy–momentum space. The peculiar polariton dispersion is the source of interesting behaviours even when described by the most fundamental and simplest equation of quantum physics: the Schrödinger equation.

Keywords

Wave Packet Coherent State Quantum Correlator Bose Condensation Cavity Photon 
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.
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Notes

Acknowledgements

I am indebted to Elena del Valle for constant exchanges and collaboration on most of the topics addressed here. Many of the results presented, and some of the most beautiful, are her own. I thank Daniele Sanvitto for his support with my contribution to this volume and for keeping this topic constantly exciting with his fresh experimental approach. I am grateful to Alexey Kavokin for having introduced me to the physics of microcavity polaritons which he also kept exciting, with his own peculiar approach. I also thank C. Tejedor, G. Malpuech, M. Glazov, Yu. Rubo, I. A. Shelykh, A. Gonzalez-Tudela, E. Cancellieri, A. Laucht, J. J. Finley and many other colleagues for discussions on many parts of this work, to which they often contributed to large extents. Support from the Marie Curie IEF “SQOD” is acknowledged.

References

  1. 1.
    A. Kavokin, J.J. Baumberg, G. Malpuech, F.P. Laussy, Microcavities, 2 edn. (Oxford University Press, Oxford, 2011)Google Scholar
  2. 2.
    J.J. Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals. Phys. Rev. 112, 1555 (1958)ADSMATHCrossRefGoogle Scholar
  3. 3.
    L.V. Keldysh, A.N. Kozlov, Collective properties of excitons in semiconductors. Sov. Phys. JETP 27, 521 (1968)ADSGoogle Scholar
  4. 4.
    Z.I. Alferov, Nobel lecture: The double heterostructure concept and its applications in physics, electronics, and technology. Rev. Mod. Phys. 73, 767 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    L.V. Butov, A.L. Ivanov, A. Ĭmamoḡlu, P.B. Littlewood, A.A. Shashkin, V.T. Dolgopolov, K.L. Campman, A.C. Gossard, Stimulated scattering of indirect excitons in coupled quantum wells: Signature of a degenerate Bose-gas of excitons. Phys. Rev. Lett. 86, 5608 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    M.G. Raizen, R.J. Thompson, R.J. Brecha, H.J. Kimble, H.J. Carmichael. Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity. Phys. Rev. Lett. 63, 240 (1989)ADSCrossRefGoogle Scholar
  8. 8.
    R.J. Thompson, G. Rempe, H.J. Kimble, Observation of normal-mode splitting for an atom in an optical cavity. Phys. Rev. Lett. 68, 1132 (1992)ADSCrossRefGoogle Scholar
  9. 9.
    K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Fălt, E. L. Hu, A. Ĭmamoḡlu. Quantum nature of a strongly coupled single quantum dot–cavity system. Nature 445, 896 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, Y. Yamamoto. Photon antibunching from a single quantum dot-microcavity system in the strong coupling regime. Phys. Rev. Lett. 98, 117402 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    R. Houdré, C. Weisbuch, R.P. Stanley, U. Oesterle, P. Pellandin, M. Ilegems, Measurement of cavity-polariton dispersion curve from angle-resolved photoluminescence experiments. Phys. Rev. Lett. 73, 2043 (1994)ADSCrossRefGoogle Scholar
  12. 12.
    P.G. Savvidis, J.J. Baumberg, R.M. Stevenson, M.S. Skolnick, D.M. Whittaker, J.S. Roberts, Angle-resonant stimulated polariton amplifier. Phys. Rev. Lett. 84, 1547 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    J.J. Baumberg, P.G. Savvidis, R.M. Stevenson, A.I. Tartakovskii, M.S. Skolnick, D.M. Whittaker, J.S. Roberts, Parametric oscillation in a vertical microcavity: A polariton condensate or micro-optical parametric oscillation. Phys. Rev. B 62, R16247 (2000)ADSCrossRefGoogle Scholar
  14. 14.
    A. Ĭmamoḡlu, R.J. Ram, S. Pau, Y. Yamamoto, Nonequilibrium condensates and lasers without inversion: Exciton-polariton lasers. Phys. Rev. A 53, 4250 (1996)ADSCrossRefGoogle Scholar
  15. 15.
    M. Kira, F. Jahnke, S.W. Koch, J.D. Berger, D.V. Wick, T.R. Nelson Jr., G. Khitrova, H.M. Gibbs, Quantum theory of nonlinear semiconductor microcavity luminescence explaining “Boser” experiments. Phys. Rev. Lett. 79, 5170 (1997)ADSCrossRefGoogle Scholar
  16. 16.
    L.V. Butov. A polariton laser. Nature 447, 540 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    H. Deng, G. Weihs, C. Santori, J. Bloch, Y. Yamamoto, Condensation of semiconductor microcavity exciton polaritons. Science 298, 199 (2002)ADSCrossRefGoogle Scholar
  18. 18.
    J. Bloch, B. Sermage, M. Perrin, P. Senellart, R. André, Le Si Dang. Monitoring the dynamics of a coherent cavity polariton population. Phys. Rev. B 71, 155311 (2005)Google Scholar
  19. 19.
    J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M.J. Keeling, F.M. Marchetti, M.H. Szymanska, R. André, J.L. Staehli, V. Savona, P.B. Littlewood, B. Deveaud, Le Si Dang, Bose–Einstein condensation of exciton polaritons. Nature 443, 409 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    A. Kavokin, G. Malpuech, F.P. Laussy, Polariton laser and polariton superfluidity in microcavities. Phys. Lett. A 306, 187 (2003)ADSCrossRefGoogle Scholar
  21. 21.
    I. Carusotto, C. Ciuti, Probing microcavity polariton superfluidity through resonant Rayleigh scattering. Phys. Rev. Lett. 93, 166401 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    K.G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. André, Le Si Dang, B. Deveaud-Plédran, Quantized vortices in an exciton-polariton condensate. Nat. Phys. 4, 706 (2008)Google Scholar
  23. 23.
    A. Amo, D. Sanvitto, F.P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaître, J. Bloch, D.N. Krizhanovskii, M.S. Skolnick, C. Tejedor, L. Viña, Collective fluid dynamics of a polariton condensate in a semiconductor microcavity. Nature 457, 291 (2009)ADSCrossRefGoogle Scholar
  24. 24.
    M.D. Fraser, G. Roumpos, Y. Yamamoto, Vortex–antivortex pair dynamics in an exciton–polariton condensate. New J. Phys. 11, 113048 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    A. Amo, J. Lefrère, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, A. Bramati, Superfluidity of polaritons in semiconductor microcavities. Nat. Phys. 5, 805 (2009)CrossRefGoogle Scholar
  26. 26.
    K.G. Lagoudakis, T. Ostatnický, A.V. Kavokin, Y.G. Rubo, R. André, B. Deveaud-Plédran, Observation of half-quantum vortices in an exciton-polariton condensate. Science 326, 974 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    D. Sanvitto, F.M. Marchetti, M.H. Szymańska, G. Tosi, M. Baudisch, F.P. Laussy, D.N. Krizhanovskii, M.S. Skolnick, L. Marrucci, A. Lemaître, J. Bloch, C. Tejedor, L. Viña, Persistent currents and quantized vortices in a polariton superfluid. Nat. Phys. 6, 527 (2010)CrossRefGoogle Scholar
  28. 28.
    G. Roumpos, M.D. Fraser, A. Löffler, S. Höfling, A. Forchel, Y. Yamamoto, Single vortex-antivortex pair in an exciton-polariton condensate. Nat. Phys. 7, 129 (2011)CrossRefGoogle Scholar
  29. 29.
    H.T.C. Stoof, Nucleation of Bose–Einstein condensation. Phys. Rev. A 45, 8398 (1992)ADSCrossRefGoogle Scholar
  30. 30.
    D.V. Semikoz, I.I. Tkachev, Kinetics of Bose condensation. Phys. Rev. Lett. 74, 3093 (1995)ADSCrossRefGoogle Scholar
  31. 31.
    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, 3331 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    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, 2902 (1997)ADSCrossRefGoogle Scholar
  33. 33.
    L. Banyai, P. Gartner, Real-time Bose–Einstein condensation in a finite volume with a discrete spectrum. Phys. Rev. Lett. 88, 210404 (2002)ADSCrossRefGoogle Scholar
  34. 34.
    F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Bottleneck effects in the relaxation and photoluminescence of microcavity polaritons. Phys. Rev. B 56, 7554 (1997)ADSCrossRefGoogle Scholar
  35. 35.
    D. Porras, C. Ciuti, J.J. Baumberg, C. Tejedor, Polariton dynamics and Bose–Einstein condensation in semiconductor microcavities. Phys. Rev. B 66, 085304 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    G. Malpuech, A. Di Carlo, A. Kavokin, J.J. Baumberg, M. Zamfirescu, P. Lugli, Room-temperature polariton lasers based on GaN microcavities. Appl. Phys. Lett. 81, 412 (2002)ADSCrossRefGoogle Scholar
  37. 37.
    Yu. G. Rubo, F.P. Laussy, G. Malpuech, A. Kavokin, P. Bigenwald, Dynamical theory of polariton amplifiers. Phys. Rev. Lett. 91, 156403 (2003)ADSCrossRefGoogle Scholar
  38. 38.
    F.P. Laussy, G. Malpuech, A. Kavokin, Spontaneous coherence buildup in a polariton laser. Phys. Stat. Sol. C 1, 1339 (2004)CrossRefGoogle Scholar
  39. 39.
    H.T. 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, 245325 (2004)ADSCrossRefGoogle Scholar
  40. 40.
    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, 205306 (2008)ADSCrossRefGoogle Scholar
  41. 41.
    M. Wouters, V. Savona, Stochastic classical field model for polariton condensates. Phys. Rev. B 79, 165302 (2009)ADSCrossRefGoogle Scholar
  42. 42.
    I.G. Savenko, E.B. Magnusson, I.A. Shelykh, Density-matrix approach for an interacting polariton system. Phys. Rev. B 83, 165316 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    J. Keeling, P.R. Eastham, M.H. Szymanska, P.B. Littlewood, Polariton condensation with localized excitons and propagating photons. Phys. Rev. Lett. 93, 226403 (2004)ADSCrossRefGoogle Scholar
  44. 44.
    J. Keeling, P.R. Eastham, M.H. Szymanska, P.B. Littlewood, BCS–BEC crossover in a system of microcavity polaritons. Phys. Rev. B 72, 115320 (2005)ADSCrossRefGoogle Scholar
  45. 45.
    M.H. Szymanska, J. Keeling, P.B. Littlewood, Nonequilibrium quantum condensation in an incoherently pumped dissipative system. Phys. Rev. Lett. 96, 230602 (2006)ADSCrossRefGoogle Scholar
  46. 46.
    F.M. Marchetti, J. Keeling, M.H. Szymanska, P.B. Littlewood, Thermodynamics and excitations of condensed polaritons in disordered microcavities. Phys. Rev. Lett. 96, 066405 (2006)ADSCrossRefGoogle Scholar
  47. 47.
    E.T. Jaynes, F.W. Cummings, Comparison of quantum and semiclassical radiation theory with application to the beam maser. Proc. IEEE 51, 89 (1963)CrossRefGoogle Scholar
  48. 48.
    Y. Zhu, D.J. Gauthier, S.E. Morin, Q. Wu, H.J. Carmichael, T.W. Mossberg, Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations. Phys. Rev. Lett. 64, 2499 (1990)ADSCrossRefGoogle Scholar
  49. 49.
    G. Khitrova, H.M. Gibbs, F. Jahnke, M. Kira, S.W. Koch, Nonlinear optics of normal-mode-coupling semiconductor microcavities. Rev. Mod. Phys 71, 1591 (1999)ADSCrossRefGoogle Scholar
  50. 50.
    C.N. Cohen-Tannoudji, Manipulating atoms with photons. Rev. Mod. Phys. 70, 707 (1998)ADSCrossRefGoogle Scholar
  51. 51.
    R.H. Dicke, Coherence in spontaneous radiation processes. Phys. Rev. 93, 99 (1954)ADSMATHCrossRefGoogle Scholar
  52. 52.
    J.P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L.V. Keldysh, V.D. Kulakovskii, T.L. Reinecker, A. Forchel, Strong coupling in a single quantum dot–semiconductor microcavity system. Nature, 432, 197 (2004)ADSCrossRefGoogle Scholar
  53. 53.
    T. Yoshie, A. Scherer, J. Heindrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, D.G. Deppe, Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity. Nature 432, 200 (2004)ADSCrossRefGoogle Scholar
  54. 54.
    E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J.M. Gérard, J. Bloch, Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity. Phys. Rev. Lett. 95, 067401 (2005)ADSCrossRefGoogle Scholar
  55. 55.
    E. del Valle, F.P. Laussy, C. Tejedor, Luminescence spectra of quantum dots in microcavities. II. Fermions. Phys. Rev. B 79, 235326 (2009)ADSCrossRefGoogle Scholar
  56. 56.
    E. del Valle, F.P. Laussy, Mollow triplet under incoherent pumping. Phys. Rev. Lett. 105, 233601 (2010)ADSCrossRefGoogle Scholar
  57. 57.
    E. del Valle, F.P. Laussy, Regimes of strong light-matter coupling under incoherent excitation. Phys. Rev. A 84, 043816 (2011)ADSCrossRefGoogle Scholar
  58. 58.
    D. Sanvitto, A. Amo, F.P. Laussy, A. Lemaître, J. Bloch, C. Tejedor, L. Viña, Polariton condensates put in motion. Nanotechnology 21, 134025 (2010)Google Scholar
  59. 59.
    E. Wertz, L. Ferrier, D.D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A.V. Kavokin, P. Senellart, G. Malpuech an J. Bloch, Spontaneous formation and optical manipulation of extended polariton condensates. Nat. Phys. 6, 860 (2010)Google Scholar
  60. 60.
    F.P. Laussy, M.M. Glazov, A. Kavokin, D.M. Whittaker, G. Malpuech, Statistics of excitons in quantum dots and their effect on the optical emission spectra of microcavities. Phys. Rev. B 73, 115343 (2006)ADSCrossRefGoogle Scholar
  61. 61.
    E. del Valle, Microcavity Quantum Electrodynamics. (VDM Verlag, 2010)Google Scholar
  62. 62.
    V. Savona, Z. Hradil, A. Quattropani, P. Schwendimann, Quantum theory of quantum-well polaritons in semiconductor microcavities. Phys. Rev. B 49, 8774 (1994)ADSCrossRefGoogle Scholar
  63. 63.
    D.W. Snoke, The quantum boltzmann equation in semiconductor physics. Annalen der Physik 523, 87 (2010)MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    E.A. Uehling, G.E. Uhlenbeck, Transport phenomena in Einstein–Bose and Fermi–Dirac gases. I. Phys. Rev. 43, 552 (1933)ADSMATHGoogle Scholar
  65. 65.
    C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, and P. Schwendimann. Role of the exchange of carriers in elastic exciton-exciton scattering in quantum wells. Phys. Rev. B 58, 7926 (1998)ADSCrossRefGoogle Scholar
  66. 66.
    G. Malpuech, A. Kavokin, A. Di Carlo, J.J. Baumberg, Polariton lasing by exciton-electron scattering in semiconductor microcavities. Phys. Rev. B 65, 153310 (2002)ADSCrossRefGoogle Scholar
  67. 67.
    J. Kasprzak, D.D. Solnyshkov, R. André, Le Si Dang, G. Malpuech, Formation of an exciton polariton condensate: Thermodynamic versus kinetic regimes. Phys. Rev. Lett. 101, 146404 (2008)Google Scholar
  68. 68.
    V.E. Hartwell, D.W. Snoke, Numerical simulations of the polariton kinetic energy distribution in GaAs quantum-well microcavity structures. Phys. Rev. B 82, 075307 (2010)ADSCrossRefGoogle Scholar
  69. 69.
    R.J. Glauber, Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766 (1963)MathSciNetADSCrossRefGoogle Scholar
  70. 70.
    R. Hanbury Brown, R.Q. Twiss, A test of a new type of stellar interferometer on Sirius. Nature 178, 1046 (1956)ADSCrossRefGoogle Scholar
  71. 71.
    L.D. Landau, Zur Theorie der Phasenumwandlungen I, II. Zh. Ekspr. Teoret. Fiz 19, 627 (1937)Google Scholar
  72. 72.
    F.P. Laussy, Y.G. Rubo, G. Malpuech, A. Kavokin, P. Bigenwald, Dissipative quantum theory of polariton lasers. Phys. Stat. Sol. C 0, 1476 (2003)Google Scholar
  73. 73.
    Yu. G. Rubo, Kinetics of the polariton condensate formation in a microcavity. Phys. Stat. Sol. A 201, 641 (2004)ADSCrossRefGoogle Scholar
  74. 74.
    D. Sarchi, V. Savona, Long-range order in the Bose–Einstein condensation of polaritons. Phys. Rev. B 75, 115326 (2007)ADSCrossRefGoogle Scholar
  75. 75.
    F.P. Laussy, G. Malpuech, A. Kavokin, P. Bigenwald, Spontaneous coherence buildup in a polariton laser. Phys. Rev. Lett. 93, 016402 (2004)ADSCrossRefGoogle Scholar
  76. 76.
    D. Jaksch, C.W. Gardiner, P. Zoller, Quantum kinetic theory. II. Simulation of the quantum Boltzmann master equation. Phys. Rev. A 56, 575 (1997)Google Scholar
  77. 77.
    C.W. Gardiner, P. Zoller, Quantum kinetic theory. III. quantum kinetic master equation for strongly condensed trapped systems. Phys. Rev. A 58, 536 (1998)Google Scholar
  78. 78.
    D. Jaksch, C.W. Gardiner, K.M. Gheri, P. Zoller, Quantum kinetic theory. IV. intensity and amplitude fluctuations of a Bose–Einstein condensate at finite temperature including trap loss. Phys. Rev. A 58, 1450 (1998)Google Scholar
  79. 79.
    C.W. Gardiner, P. Zoller, Quantum kinetic theory. V. quantum kinetic master equation for mutual interaction of condensate and noncondensate. Phys. Rev. A 61, 033601 (2000)Google Scholar
  80. 80.
    M.D. Lee, C.W. Gardiner, Quantum kinetic theory. VI. the growth of a Bose–Einstein condensate. Phys. Rev. A 62, 033606 (2000)Google Scholar
  81. 81.
    M.J. Davis, C.W. Gardiner, R.J. Ballagh, Quantum kinetic theory. VII. the influence of vapor dynamics on condensate growth. Phys. Rev. A 62, 063608 (2000)Google Scholar
  82. 82.
    A. Einstein, Quantentheorie des einatomigen idealen Gases. Sitzungsberichte der Preussischen Akademie der Wissenschaften. 1, 3–14 (1925)Google Scholar
  83. 83.
    R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics. 2nd edn. (Addison-Wesley Longman, Amsterdam, 1994)Google Scholar
  84. 84.
    F.P. Laussy, I.A. Shelykh, G. Malpuech, A. Kavokin, Effects of Bose–Einstein condensation of exciton polaritons in microcavities on the polarization of emitted light. Phys. Rev. B 73, 035315 (2006)ADSCrossRefGoogle Scholar
  85. 85.
    E. del Valle, D. Sanvitto, A. Amo, F.P. Laussy, R. André, C. Tejedor, L. Viña. Dynamics of the formation and decay of coherence in a polariton condensate. Phys. Rev. Lett. 103, 096404 (2009)ADSCrossRefGoogle Scholar
  86. 86.
    E. del Valle, S. Zippilli, F.P. Laussy, A. Gonzalez-Tudela, G. Morigi, C. Tejedor. Two-photon lasing by a single quantum dot in a high-Q microcavity. Phys. Rev. B 81, 035302 (2010)ADSCrossRefGoogle Scholar
  87. 87.
    F.P. Laussy, E. del Valle, C. Tejedor, Luminescence spectra of quantum dots in microcavities. I. Bosons. Phys. Rev. B 79, 235325 (2009)ADSCrossRefGoogle Scholar
  88. 88.
    F.P. Laussy, E. del Valle, J.J. Finley, Lasing in strong coupling (2011). http://arxiv.org/pdf/1106.0509.pdf
  89. 89.
    Y. Mu, C.M. Savage, One-atom lasers. Phys. Rev. A 46, 5944 (1992)ADSGoogle Scholar
  90. 90.
    P. Gartner, Two-level laser: Analytical results and the laser transition, Phys. Rev. A 84, 053804 (2011)Google Scholar
  91. 91.
    S. Strauf, K. Hennessy, M.T. Rakher, Y.S. Choi, A. Badolato, L.C. Andreani, E.L. Hu, P.M. Petroff, and D. Bouwmeester, Self-tuned quantum dot gain in photonic crystal lasers. Phys. Rev. Lett. 96, 127404 (2006)ADSCrossRefGoogle Scholar
  92. 92.
    Z. G. Xie, S. Götzinger, W. Fang, H. Cao, G.S. Solomon, Influence of a single quantum dot state on the characteristics of a microdisk laser. Phys. Rev. Lett. 98, 117401 (2007)ADSCrossRefGoogle Scholar
  93. 93.
    S.M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, P. Michler, Photon statistics of semiconductor microcavity lasers. Phys. Rev. Lett. 98, 043906 (2007)ADSCrossRefGoogle Scholar
  94. 94.
    M. Witzany, R. Roßbach, W.-M. Schulz, M. Jetter, P. Michler, T.-L. Liu, E. Hu, J. Wiersig, and F. Jahnke, Lasing properties of InP/(Ga0. 51In0. 49P) quantum dots in microdisk cavities. Phys. Rev. B 83, 205305 (2011)Google Scholar
  95. 95.
    F.P. Laussy, A. Laucht, E. del Valle, J.J. Finley, J.M. Villas-Bôas, Luminescence spectra of quantum dots in microcavities. III. Multiple quantum dots, Phys. Rev. B 84, 195313 (2011)Google Scholar
  96. 96.
    F.P. Laussy, E. del Valle, Optical spectra of the Jaynes-Cummings ladder. AIP Conf. Proc. 1147, 46 (2009)ADSCrossRefGoogle Scholar
  97. 97.
    B.R. Mollow, Power spectrum of light scattered by two-level systems. Phys. Rev. 188, 1969 (1969)ADSCrossRefGoogle Scholar
  98. 98.
    M. Löffler, G.M. Meyer, H. Walther, Spectral properties of the one-atom laser. Phys. Rev. A 55, 3923 (1997)ADSCrossRefGoogle Scholar
  99. 99.
    A.N. Poddubny, M.M. Glazov, N.S. Averkiev, Nonlinear emission spectra of quantum dots strongly coupled to a photonic mode. Phys. Rev. B 82, 205330 (2010)ADSCrossRefGoogle Scholar
  100. 100.
    E. del Valle, Strong and weak coupling of two coupled qubits. Phys. Rev. A 81, 053811, (2010)ADSCrossRefGoogle Scholar
  101. 101.
    L.D. Landau, Theory of the superfluidity of helium II. Phys. Rev. 60, 356 (1941)ADSMATHCrossRefGoogle Scholar
  102. 102.
    N.N. Bogoliubov, Theory of the weakly interacting Bose gas. J. Phys. (Moscow) 11, 23 (1947)Google Scholar
  103. 103.
    R.P. Feynman, Application of quantum mechanics to liquid helium. Progr. Low Temp. Phys. 1, 17 (1955)CrossRefGoogle Scholar
  104. 104.
    E. Cancellieri, F.M. Marchetti, M.H. Szymanska, C. Tejedor, Superflow of resonantly driven polaritons against a defect. Phys. Rev. B 82, 224512 (2010)ADSCrossRefGoogle Scholar
  105. 105.
    S. Ianeselli, C. Menotti, A. Smerzi, Beyond the Landau criterion for superfluidity. J. phys. B.: At. Mol. Phys. 39, S135 (2006)Google Scholar
  106. 106.
    W.F. Wreszinski, On translational superfluidity and the Landau criterion for Bose gases in the Gross–Pitaevski limit. J. phys. A.: Math. Gen. 41, 392006 (2008)Google Scholar
  107. 107.
    R.H. Stuewer, Resource letter Sol-1: Solitons. Am. J. Phys. 66, 486 200 (1998)Google Scholar
  108. 108.
    S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, M. Lewenstein, Dark solitons in Bose–Einstein condensates. Phys. Rev. Lett. 83, 5198 (1999)ADSCrossRefGoogle Scholar
  109. 109.
    K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Formation and propagation of matter-wave soliton trains. Nature 417, 150 (2002)ADSCrossRefGoogle Scholar
  110. 110.
    O.A. Egorov, D.V. Skryabin, A.V. Yulin, F. Lederer, Bright cavity polariton solitons. Phys. Rev. Lett. 102, 153904 (2009)ADSCrossRefGoogle Scholar
  111. 111.
    A. Amo, S. Pigeon, D. Sanvitto, V.G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Leménager, R. Houdré, E. Giacobino, C. Ciuti, A. Bramati, Polariton superfluids reveal quantum hydrodynamic solitons. Science 332, 1167 (2011)ADSCrossRefGoogle Scholar
  112. 112.
    J. Keeling, N.G. Berloff, Going with the flow. Nature 457, 273 (2009)Google Scholar
  113. 113.
    E. Schrödinger. Der stetige übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664 (1926)ADSCrossRefGoogle Scholar
  114. 114.
    C.G. Darwin, Free motion in the wave mechanics. Proc. Roy. Soc A117, 258 (1928)ADSGoogle Scholar
  115. 115.
    J.R. Klein, Do free quantum-mechanical wave packets always spread? Am. J. Phys. 48, 1035 (1980)ADSCrossRefGoogle Scholar
  116. 116.
    M. Berry. Quantum physics on the edge of chaos. New Scientist 116(1587), 44 (1987)Google Scholar
  117. 117.
    J. Scott Russell, Report on waves. Fourteenth meeting of the British Association for the Advancement of Science, (1844)Google Scholar
  118. 118.
    D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philosophical Mag. 39, 422 (1895)Google Scholar
  119. 119.
    C.C. Yan, Soliton like solutions of the schrödinger equation for simple harmonic oscillator. Am. J. Phys. 62, 147 (1994)ADSCrossRefGoogle Scholar
  120. 120.
    D.F. Walls, Squeezed states of light. Nature 306, 141 (1983)ADSGoogle Scholar
  121. 121.
    M.V. Berry, N.L. Balazs, Nonspreading wave packets. Am. J. Phys. 47, 264 (1979)Google Scholar
  122. 122.
    F.P. Laussy, Propagation of polariton wavepackets. ICSCE4 conference, Cambridge, http://www.tcm.phy.cam.ac.uk/BIG/icsce4/talks/laussy.pdf, 2008
  123. 123.
    I.A. Shelykh, A.V. Kavokin, Yu.G. Rubo, T.C.H. Liew, G. Malpuech, Polariton polarization-sensitive phenomena in planar semiconductor microcavities. Semicond. Sci. Technol. 25, 013001 (2010)ADSCrossRefGoogle Scholar
  124. 124.
    T.S. Raju, C. Nagaraja Kumar, P.K. Panigrahi, On exact solitary wave solutions of the nonlinear Schrödinger equation with a source. J. phys. A.: Math. Gen. 38, L271 (2005)Google Scholar
  125. 125.
    V.M. Vyas, T.S. Raju, C.N. Kumar, P.K. Panigrahi, Soliton solutions of driven nonlinear schrödinger equation. J. phys. A.: Math. Gen. 39, 9151 (2006)Google Scholar
  126. 126.
    C. Ciuti, I. Carusotto, Quantum fluid effects and parametric instabilities in microcavities. Phys. Stat. Sol. B 242, 2224 (2005)ADSCrossRefGoogle Scholar
  127. 127.
    M.H. Szymanska, F.M. Marchetti, D. Sanvitto, Propagating wave packets and quantized currents in coherently driven polariton superfluids. Phys. Rev. Lett. 105, 236402 (2010)ADSCrossRefGoogle Scholar
  128. 128.
    C. Ciuti, P. Schwendimann, B. Deveaud, A. Quattropani, Theory of the angle-resonant polariton amplifier. Phys. Rev. B 62, R4825 (2000)ADSCrossRefGoogle Scholar
  129. 129.
    C. Ciuti, P. Schwendimann, A. Quattropani, Parametric luminescence of microcavity polaritons. Phys. Rev. B 63, 041303 (2001)ADSCrossRefGoogle Scholar
  130. 130.
    D.M. Whittaker, Effects of polariton-energy renormalization in the microcavity optical parametric oscillator. Phys. Rev. B 71, 115301 (2005)ADSCrossRefGoogle Scholar
  131. 131.
    A.E. Garriz, A. Sztrajman, D. Mitnik, Running into trouble with the time-dependent propagation of a wavepacket. Eur. J. Phys. 31, 785 (2010)CrossRefGoogle Scholar
  132. 132.
    H. Shao, Z. Wang, Numerical solutions of the time-dependent Schrödinger equation: Reduction of the error due to space discretization. Phys. Rev. E 79, 056705 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Walter Schottky InstitutTechnische Universität MünchenGarchingGermany

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