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Optical Generation and Control of Quantum Coherence in Semiconductor Nanostructures pp 293–329Cite as

Keldysh Green’s function approach to coherence in a non-equilibrium steady state: connecting Bose-Einstein condensation and lasing

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Part of the book series: NanoScience and Technology ((NANO,volume 0))

Abstract

Solid state quantum condensates often differ from previous examples of condensates (such as Helium, ultra-cold atomic gases, and superconductors) in that the quasiparticles condensing have relatively short lifetimes, and so as for lasers, external pumping is required to maintain a steady state. On the other hand, compared to lasers, the quasiparticles are generally more strongly interacting, and therefore better able to thermalise. This leads to questions of how to describe such non-equilibrium condensates, and their relation to equilibrium condensates and lasers. This chapter discusses in detail how the non-equilibrium Green’s function approach can be applied to the description of such a non-equilibrium condensate, in particular, a system of microcavity polaritons, driven out of equilibrium by coupling to multiple baths. By considering the steady states, and fluctuations about them, it is possible to provide a description that relates both to equilibrium condensation and to lasing, while at the same time, making clear the differences from simple lasers.

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References

  1. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M.J. Keeling, F.M. Marchetti, M.H. Szymańska, R. André, J.L. Staehli, V. Savona, P.B. Littlewood, B. Deveaud, L.S. Dang, Nature 443(7110), 409 (2006)

    Article  ADS  Google Scholar 

  2. S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, B. Hillebrands, A.N. Slavin, Nature 443, 430 (2006)

    Article  ADS  Google Scholar 

  3. J.P. Eisenstein, A.H. MacDonald, Nature 432, 691 (2004)

    Article  ADS  Google Scholar 

  4. C. Rüegg, N. Cavadinin, A. Furrer, H.U. Güdel, K. Krämer, H. Mukta, A. Wildes, K. Habicht, P. Worderwisch, Nature 423, 62 (2003)

    Article  ADS  Google Scholar 

  5. L.V. Butov, C.W. Lai, A.L. Ivanov, A.C. Gossard, D.S. Chemla, Nature 417, 47 (2002)

    Article  ADS  Google Scholar 

  6. L.V. Butov, A.C. Gossard, D.S. Chemla, Nature 418, 751 (2002)

    Article  ADS  Google Scholar 

  7. D. Snoke, S. Denev, Y. Liu, L. Pfeiffer, K. West, Nature 418, 754 (2002)

    Article  ADS  Google Scholar 

  8. E.W. Streed, A.P. Chikkatur, T.L. Gustavson, M. Boyd, Y. Torii, D. Schneble, G.K. Campbell, D.E. Pritchard, W. Ketterle, Rev. Sci. Instrum. 77, 023106 (2006)

    Article  ADS  Google Scholar 

  9. A.T. Hammack, L.V. Butov, L. Mouchliadis, A.L. Ivanov, A.C. Gossard, Phys. Rev. B 76, 193308 (2007)

    Article  ADS  Google Scholar 

  10. H. Deng, D. Press, S. Götzinger, G.S. Solomon, R. Hey, K.H. Ploog, Y. Yamamoto, Phys. Rev. Lett. 97(14), 146402 (2006)

    Article  ADS  Google Scholar 

  11. M.S. Skolnick, T.A. Fisher, D.M. Whittaker, Semicond. Sci. Technol. 13, 645 (1998)

    Article  ADS  Google Scholar 

  12. V. Savona, C. Piermarocchi, A. Quattropani, P. Schwendimann, F. Tassone, Phase Transitions 68(1), 169 (1999)

    Article  Google Scholar 

  13. Y. Yamamoto, F. Tassone, H. Cao, Semiconductor Cavity Quantum Electrodynamics, Springer Tracts in Modern Physics, vol. 167 (Springer-Verlag, Berlin, 2000)

    Google Scholar 

  14. C. Ciuti, P. Schwendimann, A. Quattropani, Semicond. Sci. Technol. 18, S279 (2003)

    Article  ADS  Google Scholar 

  15. B. Deveaud (ed.). Special Issue: Physics of Semiconductor Microcavities, Phys. Stat. Sol. (b), vol. 242 (2005)

    Google Scholar 

  16. J. Keeling, F.M. Marchetti, M.H. Szymańska, P.B. Littlewood, Semicond. Sci. Technol. 22(5), R1 (2007)

    Article  ADS  Google Scholar 

  17. A.V. Kavokin, J.J. Baumberg, G. Malpuech, F.P. Laussy, Microcavities (Oxford Univeristy Press, Oxford, 2007)

    Book  Google Scholar 

  18. H. Cao, J. Phys. A: Math. Gen. 38(49), 10497 (2005)

    Article  ADS  Google Scholar 

  19. H.E. Türeci, L. Ge, S. Rotter, A.D. Stone, Science 320, 643 (2008)

    Article  ADS  Google Scholar 

  20. M.O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, W. Ketterle, Phys. Rev. Lett. 78, 582(1997)

    Article  ADS  Google Scholar 

  21. I. Bloch, T.W. Hänsch, T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999)

    Article  ADS  Google Scholar 

  22. E.W. Hagley, L. Deng, M. Kozuma, J. Wen, S.L. Rolston, W.D. Phillips, Science 283, 1706 (1999)

    Article  ADS  Google Scholar 

  23. M.H. Szymanska, P.B. Littlewood, Solid State Commun. 124, 103 (2002)

    Article  ADS  Google Scholar 

  24. M.H. Szymanska, P.B. Littlewood, B.D. Simons, Phys. Rev. A 68, 013818 (2003)

    Article  ADS  Google Scholar 

  25. A.A. Abrikosov, L.P. Gor'kov, Sov. Phys. JETP 12, 1243 (1960)

    Google Scholar 

  26. P.W. Anderson, J. Phys. Chem. Solids 11, 26 (1959)

    Article  ADS  Google Scholar 

  27. F.M. Marchetti, B.D. Simons, P.B. Littlewood, Phys. Rev. B 70, 155327 (2004)

    Article  ADS  Google Scholar 

  28. M.H. Szymańska, J. Keeling, P.B. Littlewood, Phys. Rev. Lett. 96, 230602 (2006)

    Article  ADS  Google Scholar 

  29. M.H. Szymańska, J. Keeling, P.B. Littlewood, Phys. Rev. B 75(19), 195331 (2007)

    Article  ADS  Google Scholar 

  30. A. Kamenev, in Nanophysics: Coherence and transport, Les Houches, vol. LXXXI, ed. by H. Bouchiat, Y. Gefen, S. Guéron, G. Montambaux, J. Dalibard (Elsevier, Amsterdam, 2005), p. 177

    Chapter  Google Scholar 

  31. L.V. Keldysh, JETP 20, 1018 (1965)

    MathSciNet  Google Scholar 

  32. P. Danielewicz, Ann. Phys. 152, 239 (1984)

    Article  ADS  Google Scholar 

  33. E.M. Lifshitz, L.P. Pitaevskii, Physical Kinetics, Course of theoretical Physics, vol. 10 (Butterworth-Heinemann, Oxford, 1981)

    Google Scholar 

  34. F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani, P. Schwendimann, Phys. Rev. B 56, 7554 (1997)

    Article  ADS  Google Scholar 

  35. F. Tassone, Y. Yamamoto, Phys. Rev. B 59, 10830 (1999)

    Article  ADS  Google Scholar 

  36. G. Malpuech, A. Di Carlo, A. Kavokin, J.J. Baumberg, M. Zamfirescu, P. Lugli, Appl. Phys. Lett. 81, 412 (2002)

    Article  ADS  Google Scholar 

  37. G. Malpuech, A. Kavokin, A. Di Carlo, J.J. Baumberg, Phys. Rev. B 65(15), 153310 (2002)

    Article  ADS  Google Scholar 

  38. D. Porras, C. Ciuti, J.J. Baumberg, C. Tejedor, Phys. Rev. B 66, 085304 (2002)

    Article  ADS  Google Scholar 

  39. T.D. Doan, H.T. Cao, D.B. Tran Thoai, H. Haug, Phys. Rev. B 72, 085301 (2005)

    Article  ADS  Google Scholar 

  40. T.D. Doan, H.T. Cao, D.B. Tran Thoai, H. Haug, Phys. Rev. B 74, 115316 (2006)

    Article  ADS  Google Scholar 

  41. T.D. Doan, H.T. Cao, D.B. Tran Thoai, H. Haug, Phys. Rev. B 78, 205306 (2008)

    Article  ADS  Google Scholar 

  42. B. Mieck, H. Haug, Phys. Rev. B 66, 075111 (2002)

    Article  ADS  Google Scholar 

  43. I. Carusotto, C. Ciuti, Phys. Rev. B 72, 125335 (2005)

    Article  ADS  Google Scholar 

  44. M. Wouters, V. Savona, Phys. Rev. B 79, 165302 (2009)

    Article  ADS  Google Scholar 

  45. M. Wouters, I. Carusotto, Phys. Rev. Lett. 99(14), 140402 (2007)

    Article  ADS  Google Scholar 

  46. M. Wouters, I. Carusotto, C. Ciuti, Phys. Rev. B 77, 115340 (2008)

    Article  ADS  Google Scholar 

  47. J. Keeling, N.G. Berloff, Phys. Rev. Lett. 100, 250401 (2008)

    Article  ADS  Google Scholar 

  48. L. Kadanoff, G. Baym, Quantum Statistical Mechanics (W. A. Benjamin, New York, 1962)

    MATH  Google Scholar 

  49. M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, 1997)

    Google Scholar 

  50. G.W. Ford, R.F. O'Connel, Phys. Rev. Lett. 77, 798 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  51. P.R. Eastham, P.B. Littlewood, Solid State Commun. 116, 357 (2000)

    Article  ADS  Google Scholar 

  52. J. Keeling, P.R. Eastham, M.H. Szymanska, P.B. Littlewood, Phys. Rev. Lett. 93, 226403 (2004)

    Article  ADS  Google Scholar 

  53. F.M. Marchetti, J. Keeling, M.H. Szymańska, P.B. Littlewood, Phys. Rev. Lett. (2006)

    Google Scholar 

  54. E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part II, Course of theoretical Physics, vol. 5 (Butterworth-Heinemann, Oxford, 1980)

    Google Scholar 

  55. A. Abrikosov, L. Gorkov, I. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1975)

    Google Scholar 

  56. M. Randeria, in Bose-Einstein Condensation, ed. by A. Griffin, D. Snoke, S. Stringari (Cambridge University Press, Cambridge, 1995), p. 355

    Google Scholar 

  57. M.H. Szymańska, Bose condensation and lasing in optical microstructures. Ph.D. thesis, University of Cambridge (2002). ArXiv:cond-mat/0204294

    Google Scholar 

  58. N.M. Hugenholtz, D. Pines, Phys. Rev. 116, 489 (1959)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  59. V.N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (D. Reidel, Dordrecht, 1983)

    MATH  Google Scholar 

  60. M. Wouters, I. Carusotto, Phys. Rev. A 76, 043807 (2007)

    Article  ADS  Google Scholar 

  61. I. Carusotto, C. Ciuti, Phys. Rev. Lett. 93, 166401 (2004)

    Article  ADS  Google Scholar 

  62. C. Ciuti, I. Carusotto, Phys. Stat. Sol. (b) 242, 2224 (2005)

    Article  ADS  Google Scholar 

  63. A. Amo, J. Lefrére, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, A. Bramati, Nature Phys. 5, 805 (2009)

    Article  ADS  Google Scholar 

  64. D.M. Whittaker, P.R. Eastham, EPL (Europhysics Letters) 87(2), 27002 (2009)

    Article  ADS  Google Scholar 

  65. R. Kubo, J. Phys. Soc. Jap 9, 935 (1954)

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom

    Jonathan Keeling & Peter B. Littlewood

  2. Department of Physics, University of Warwick and London Centre for Nanotechnology, London, United Kingdom

    Marzena H. Szymańska

Authors
  1. Jonathan Keeling
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  2. Marzena H. Szymańska
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  3. Peter B. Littlewood
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Correspondence to Jonathan Keeling .

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Editors and Affiliations

  1. , The Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2BZ, United Kingdom

    Gabriela Slavcheva

  2. Labo. Pierre Aigrain, Ecole Normale Supérieure, rue Lhomond 24, Paris CX 05, 75231, France

    Philippe Roussignol

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Keeling, J., Szymańska, M.H., Littlewood, P.B. (2010). Keldysh Green’s function approach to coherence in a non-equilibrium steady state: connecting Bose-Einstein condensation and lasing. In: Slavcheva, G., Roussignol, P. (eds) Optical Generation and Control of Quantum Coherence in Semiconductor Nanostructures. NanoScience and Technology, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12491-4_12

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