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Field Theory for Trapped Atomic Gases

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Coherent atomic matter waves

Part of the book series: Les Houches - Ecole d’Ete de Physique Theorique ((LHSUMMER,volume 72))

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Abstract

In this course we give a selfcontained introduction to the quantum field theory for trapped atomic gases, using functional methods throughout. We consider both equilibrium and nonequilibrium phenomena. In the equilibrium case, we first derive the appropriate Hartree—Fock theory for the properties of the gas in the normal phase. We then turn our attention to the properties of the gas in the superfluid phase, and present a microscopic derivation of the Bogoliubov and Popov theories of Bose-Einstein condensation and the Bardeen-Cooper-Schrieffer theory of superconductivity. The former are applicable to trapped bosonic gases such as rubidium, lithium, sodium and hydrogen, and the latter in particular to the fermionic isotope of atomic lithium. In the nonequilibrium case, we discuss various topics for which a field-theoretical approach is especially suited, because they involve physics that is not contained in the Gross-Pitaevskii equation. Examples are quantum kinetic theory, the growth and collapse of a Bose condensate, the phase dynamics of bosonic and fermionic superfluids, and the collisionless collective modes of a Bose gas below the critical temperature.

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References

  1. E. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1.

    Article  ADS  Google Scholar 

  2. K.G. Wilson and J. Kogut, Phys. Rep. 12 (1974) 75.

    Article  ADS  Google Scholar 

  3. P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435.

    Article  ADS  Google Scholar 

  4. S. Sachdev, in Proceedings of the 19th IUPAP International Conference on Statistical Physics, edited by B.-L. Hao (World Scientific, Singapore, 1996).

    Google Scholar 

  5. P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287.

    Article  ADS  Google Scholar 

  6. P.W. Anderson, Phys. Rev. 109 (1958) 1492.

    Article  ADS  Google Scholar 

  7. R.B. Laughlin, in The Quantum Hall Effect, edited by R.E. Prange and S.M. Girvin (Springer-Verlag, New York, 1990).

    Google Scholar 

  8. X.G. Wen, Phys. Rev. B 43 (1991) 11025.

    Article  ADS  Google Scholar 

  9. M.L. Mehta, Random Matrices (Academic Press, New York, 1991).

    MATH  Google Scholar 

  10. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Sci 269 (1995) 198.

    Article  ADS  Google Scholar 

  11. C.C. Bradley, C.A. Sackett, J.J. Tollett and R.G. Hulet, Phys. Rev. Lett. 75 (1995) 1687; C.C. Bradley, C.A. Sackett and R.G. Hulet, ibid. 78 (1997) 985.

    Article  ADS  Google Scholar 

  12. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn and W. Ketterle, Phys. Rev. Lett. 75 (1995) 3969.

    Article  ADS  Google Scholar 

  13. A. Einstein, Sitz. Kgl. Preuss. Akad. Wiss. (Berlin) (1925) 3.

    Google Scholar 

  14. I.F. Silvera and J.T.M. Walraven, Phys. Rev. Lett. 44 (1980) 164.

    Article  ADS  Google Scholar 

  15. A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971). Note that these authors use the more common notation ân,α and â †nα for the bosonic creation and annihilation operators.

    Google Scholar 

  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995).

    Google Scholar 

  17. J.W. Negele and H. Orland, Quantum M any-Particle Systems (Addison-Wesley, New York, 1988).

    Google Scholar 

  18. D.J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 1984).

    Google Scholar 

  19. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford, New York, 1989).

    Google Scholar 

  20. F.J. Dyson, Phys. Rev. 75 (1949) 1736.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. R.L. Stratonovich, Sov. Phys. Dok. 2 (1958) 416.

    Google Scholar 

  22. J. Hubbard, Phys. Rev. Lett. 3 (1959) 77.

    Article  ADS  Google Scholar 

  23. D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Kleppner and T.J. Greytak, Phys. Rev. Lett. 81 (1998) 3811.

    Article  ADS  Google Scholar 

  24. N.N. Bogoliubov, J. Phys. (U.S.S.R.) 11 (1947) 23.

    Google Scholar 

  25. L.P. Pitaevskii, Sov. Phys. JETP 13 (1961) 451 and E.P. Gross, J. Math. Phys. 4 (1963) 195.

    MathSciNet  Google Scholar 

  26. D.S. Jin, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Phys. Rev. Lett. 77 (1996) 420; D.S. Jin, M.R. Matthews, J.R. Ensher, C.E. Wieman and E.A. Cornell, ibid. 78 (1997) 764.

    Article  ADS  Google Scholar 

  27. M.-O. Mewes, M.R. Anderson, N.J. van Druten, D.M. Kurn, D.S. Durfee, C.G. Townsend and W. Ketterle, Phys. Rev. Lett. 77 (1996) 988.

    Article  ADS  Google Scholar 

  28. L.V. Hau, B.D. Busch, C. Liu, Z. Dutton, M.M. Burns and J.A. Golovchenko, Phys. Rev. A 58 (1998) R54.

    Article  ADS  Google Scholar 

  29. K.G. Singh and D.S. Rokhsar, Phys. Rev. Lett. 77 (1996) 1667.

    Article  ADS  Google Scholar 

  30. M. Edwards, P.A. Ruprecht, K. Burnett, R.J. Dodd and C.W. Clark, Phys. Rev. Lett. 77 (1996) 1671.

    Article  ADS  Google Scholar 

  31. A.L. Fetter, Ann. Phys. 70 (1972) 67.

    Article  ADS  Google Scholar 

  32. M. Lewenstein and L. You, Phys. Rev. Lett. 77 (1996) 3489.

    Article  ADS  Google Scholar 

  33. V.N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983) and references therein.

    MATH  Google Scholar 

  34. F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Rev. Mod. Phys. 71 (1999) 463.

    Article  ADS  Google Scholar 

  35. D.A.W. Hutchinson, E. Zaremba and A. Griffin, Phys. Rev. Lett. 78 (1997) 1842.

    Article  ADS  Google Scholar 

  36. R.J. Dodd, M. Edwards, C.W. Clark and K. Burnett, Phys. Rev. A 57 (1998) R32.

    Article  ADS  Google Scholar 

  37. B.A. Lippmann and J. Schwinger, Phys. Rev. 79 (1950) 469.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. M. Bijlsma and H.T.C. Stoof, Phys. Rev. A 54 (1996) 5085.

    Article  ADS  Google Scholar 

  39. H. Shi and A. Griffin, Phys. Rep. 304 (1998) 1.

    Article  ADS  Google Scholar 

  40. J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. H. Kleinert, Forts. Phys. 26 (1978) 565.

    Article  Google Scholar 

  42. E.R.I. Abraham, W.I. McAlexander, J.M. Gerton, R.G. Hulet, R. Côté and A. Dalgarno, Phys. Rev. A 55 (1997) R3299.

    Article  ADS  Google Scholar 

  43. H.T.C. Stoof, M. Houbiers, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 76 (1996) 10; M. Houbiers, R. Ferwerda, H.T.C. Stoof, W.I. McAlexander, C.A. Sackett and R.G. Hulet, Phys. Rev. A 56 (1997) 4864.

    Article  ADS  Google Scholar 

  44. For an equilibrium argument at zero temperature see L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1958) and P. Nozières, in Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke and S. Stringari (Cambridge, New York, 1995) p. 15. Nonzero temperatures are discussed in H.T.C. Stoof, Phys. Rev. A 49 (1995) 4704.

    MATH  Google Scholar 

  45. M. Houbiers and H.T.C. Stoof, Phys. Rev. A 54 (1996) 5055.

    Article  ADS  Google Scholar 

  46. T. Bergeman, Phys. Rev. A 55 (1997) 3658.

    Article  ADS  Google Scholar 

  47. See, for instance, S. Coleman in Aspects of Symmetry (Cambridge, New York, 1985).

    Google Scholar 

  48. J.A. Freire and D.P. Arovas, Phys. Rev. A 59 (1999) 1461.

    Article  ADS  Google Scholar 

  49. S. Stringari, Phys. Rev. Lett. 77 (1996) 2360.

    Article  ADS  Google Scholar 

  50. Y. Castin and R. Dum, Phys. Rev. Lett. 77 (1996) 5315.

    Article  ADS  Google Scholar 

  51. V.M. Perez-Garcia, H. Michinel, J.I. Cirac, M. Lewenstein and P. Zoller, Phys. Rev. Lett. 77 (1996) 5320.

    Article  ADS  Google Scholar 

  52. R.J. Dodd, M. Edwards, C.J. Williams, C.W. Clark, M.J. Holland, P.A. Ruprecht and K. Burnett, Phys. Rev. A 54 (1996) 661.

    Article  ADS  Google Scholar 

  53. J. Javanainen, Phys. Rev. A 54 (1996) 3722.

    Article  ADS  Google Scholar 

  54. L. You, W. Hoston and M. Lewenstein, Phys. Rev. A 55 (1997) 1581.

    Article  ADS  Google Scholar 

  55. P. Ohberg, E.L. Surkov, I. Tittonen, S. Stenholm, M. Wilkens and G.V. Shlyapnikov, Phys. Rev. A 56 (1997) R3346.

    Article  ADS  Google Scholar 

  56. For the anisotropic generalization see, for instance, G. Baym and C.J. Pethick, Phys. Rev. Lett. 76 (1996) 6.

    Article  ADS  Google Scholar 

  57. A.L. Fetter, Phys. Rev. A 53 (1996) 4245. The exact result is obtained in P.A. Ruprecht, M.J. Holland, K. Burnett and M. Edwards, Phys. Rev. A 51 (1995) 4704.

    Article  ADS  Google Scholar 

  58. H.T.C. Stoof, J. Stat. Phys. 87 (1997) 1353. For a different calculation of the tunneling rate that however neglects the phase fluctuations of the condensate, see E.V. Shuryak, Phys. Rev. A 54 (1996) 3151.

    Article  ADS  Google Scholar 

  59. C.A. Sackett, C.C. Bradley, M. Welling and R.G. Hulet, Appl. Phys. B 65 (1997) 433.

    Article  ADS  Google Scholar 

  60. C.A. Sackett, H.T.C. Stoof and R.G. Hulet, Phys. Rev. Lett. 80 (1998) 2031.

    Article  ADS  Google Scholar 

  61. L.P. Pitaevskii, Phys. Lett. A 221 (1996) 14.

    Article  ADS  Google Scholar 

  62. C.A. Sackett, J.M. Gerton, M. Welling and R.G. Hulet, Phys. Rev. Lett. 82 (1999) 876.

    Article  ADS  Google Scholar 

  63. P.W. Anderson, Phys. Rev. 112 (1958) 1900.

    Article  ADS  MathSciNet  Google Scholar 

  64. E. Abrahams and T. Tsuneto, Phys. Rev. 152 (1966) 416.

    Article  ADS  Google Scholar 

  65. In principle the effective action for a neutral superconductor contains also a topological term, but this does not affect the final result of the following argument because it only leads to a constant shift in the total number of particles. For more details, see M. Stone, Int. J. Mod. Phys. B 9 (1995) 1359 and references therein.

    Article  ADS  Google Scholar 

  66. This may be compared directly with the resultsof E.M. Wright, D.F. Walls and J.C. Garrison, Phys. Rev. Lett. 77 (1996) 2158 and K. Mølmer, Phys. Rev. A 58 (1998) 566.

    Article  ADS  Google Scholar 

  67. V.V. Goldman, I.F. Silvera and A.J. Leggett, Phys. Rev. B 24 (1981) 2870.

    Article  ADS  Google Scholar 

  68. L.H. Thomas, Proc. Camb. Phil. Soc. 23 (1927) 542 and E. Fermi, Mat. Nat. 6 (1927) 602.

    Article  MATH  Google Scholar 

  69. See, however, T.-L. Ho and V.B. Shenoy, Phys. Rev. Lett. 77 (1996) 2595.

    Article  ADS  Google Scholar 

  70. T.-L. Ho, Phys. Rev. Lett. 81 (1998) 742.

    Article  ADS  Google Scholar 

  71. M. Houbiers and H.T.C. Stoof, Phys. Rev. A 59 (1999) 1556.

    Article  ADS  Google Scholar 

  72. See, for example, K. Huang, Statistical Mechanics (Wiley, New York, 1987).

    MATH  Google Scholar 

  73. J. Schwinger, J. Math. Phys. 2 (1961) 407.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  74. L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys.-JETP 20 (1965) 1018].

    Google Scholar 

  75. P. Danielewicz, Ann. Phys. (N. Y.) 152 (1984) 239.

    Article  ADS  Google Scholar 

  76. B. de Wit (private communication).

    Google Scholar 

  77. P. Carruthers and K.S. Dy, Phys. Rev. 147 (1966) 214.

    Article  ADS  Google Scholar 

  78. H. Kleinert, Path Integrals in Quantum Mechanics Statistics and Polymer Physics (World Scientific, Singapore, 1994).

    Google Scholar 

  79. A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211 and A.O. Caldeira and A.J. Leggett, Ann. Phys. (N. Y.) 149 (1983) 374; ibid. 153, (1984) 445.

    Article  ADS  Google Scholar 

  80. L.P. Kadanoff and G. Baym, Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Nonequilibrium Problems (Addison-Wesley, New York, 1962).

    MATH  Google Scholar 

  81. D.C. Langreth and J.W. Wilkins, Phys. Rev. B 6 (1972) 3189.

    Article  ADS  Google Scholar 

  82. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

    MATH  Google Scholar 

  83. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, Reading, 1975).

    Google Scholar 

  84. To make a connection with a large body of knowledge in quantum optics, we note that equation (3.104) explicitly shows that the probability distribution P[φ*,φ;t ] does not correspond to a P or Q representation of the density matrix, but to a Wigner representation instead. The “diffusion matrix” in the Fokker-Planck equation for P[φ*,φ;t] is therefore diagonal and positive, as is explained by C.W. Gardiner in Quantum Noise (Springer, Berlin, 1991), Chapter 6. The same nice feature will also appear in the case of an interacting Bose gas.

    MATH  Google Scholar 

  85. M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849 and references therein.

    Article  ADS  Google Scholar 

  86. See also H.T.C. Stoof in Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke and S. Stringari (Cambridge, New York, 1995) p. 226.

    Google Scholar 

  87. O.J. Luiten, M.W. Reynolds and J.T.M. Walraven, Phys. Rev. A 53 (1996) 381.

    Article  ADS  Google Scholar 

  88. N.P. Proukakis and K. Burnett, J. Res. Natl. Inst. Stand. Technol. 101 (1996) 457; N.P. Proukakis, K. Burnett and H.T.C. Stoof, Phys. Rev. A 57 (1998) 1230.

    Google Scholar 

  89. C.W. Gardiner, P. Zoller, R.J. Ballagh and M.J. Davis, Phys. Rev. Lett. 79 (1997) 1793.

    Article  ADS  Google Scholar 

  90. H. Risken, Z. Phys. 186 (1965) 85; ibid. 191 (1966) 302.

    Article  ADS  Google Scholar 

  91. H.-J. Miesner, D.M. Stamper-Kurn, M.R. Andrews, D.S. Durfee, S. Inouye and W. Ketterle, Sci. 279 (1998) 1005.

    Article  ADS  Google Scholar 

  92. For a recent extension of their theory see C.W. Gardiner, M.D. Lee, R.J. Ballagh, M.J. Davis and P. Zoller, Phys. Rev. Lett. 81 (1998) 5266.

    Article  ADS  Google Scholar 

  93. M. Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner, Phys. Rep. 106 (1984) 121.

    Article  ADS  MathSciNet  Google Scholar 

  94. D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74 (1995) 3093.

    Article  ADS  Google Scholar 

  95. H.T.C. Stoof, J. Low Temp. Phys. 114 (1999) 11.

    Article  Google Scholar 

  96. E. Zaremba, A. Griffin and T. Nikuni, Phys. Rev. A 57 (1998) 4695. See also A. Griffin, W.-C. Wu and S. Stringari, Phys. Rev. Lett. 78 (1997) 1838; G.M. Kavoulakis, C.J. Pethick and H. Smith, Phys. Rev. A 57 (1998) 2938; T. Nikuni and A. Griffin, J. Low. Temp. Phys. 111 (1998) 793, and V. Shenoy and T.-L. Ho, Phys. Rev. Lett. 80 (1998) 3895.

    Article  ADS  Google Scholar 

  97. W. Kohn, Phys. Rev. 123 (1961) 1242.

    Article  MATH  ADS  Google Scholar 

  98. T.R. Kirkpatrick and J.R. Dorfman, J. Low Temp. Phys. 58 (1985) 301; ibid. 58 (1985) 399.

    Article  ADS  Google Scholar 

  99. E. Zaremba, T. Nikuni and A. Griffin, J. Low Temp. Phys. 116 (1999) 277.

    Article  Google Scholar 

  100. K. Damle, S.N. Majumdar and S. Sachdev, Phys. Rev. A 54 (1996) 5037.

    Article  ADS  Google Scholar 

  101. M.J. Bijlsmaand H.T.C. Stoof, Phys. Rev. A 60 (1999).

    Google Scholar 

  102. E.A. Cornell (private communication).

    Google Scholar 

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  1. Institute for Theoretical Physics, University of Utrecht, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    H. T. C. Stoof

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R. Kaiser C. Westbrook F. David

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Stoof, H.T.C. (2001). Field Theory for Trapped Atomic Gases. In: Kaiser, R., Westbrook, C., David, F. (eds) Coherent atomic matter waves. Les Houches - Ecole d’Ete de Physique Theorique, vol 72. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45338-5_3

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