Quantum Fluctuations and Gross-Pitaevskii Theory


Using the linearized version of the time dependent Gross–Pitaevskii equation, we calculate the dynamic response of a Bose–Einstein condensed gas to periodic density and particle perturbations. The zero temperature limit of the fluctuation—dissipation theorem is used to evaluate the corresponding quantum fluctuations induced by the elementary excitations in the ground state. In uniform conditions the predictions of Bogoliubov theory, including the infrared divergency of the particle distribution function and the quantum depletion of the condensate, are exactly reproduced by Gross–Pitaevskii theory. Results are also given for the crossed particle-density response function and the extension of the formalism to nonuniform systems is discussed. The generalization of the Gross–Pitaevskii equation to include beyond mean field effects is finally considered and an explicit result for the chemical potential is found, in agreement with the prediction of Lee–Huang–Yang theory.

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  1. 1.

    At finite temperature the function sgn(ω) should be replaced by cot(β\(\hbar \)ω/2).

  2. 2.

    Corrections to the Gross–Pitaevskii equation accounting for beyond mean field effects were also discussed in [19].


  1. 1

    N. N. Bogoliubov, J. Phys. USSR 11, 23 (1947).

    Google Scholar 

  2. 2

    E. P. Gross, Nuovo Cimento 20, 454 (1961).

    Article  Google Scholar 

  3. 3

    L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).

    MathSciNet  Google Scholar 

  4. 4

    R. Kubo, Rep. Prog. Phys. 29, 255 (1966).

    ADS  Article  Google Scholar 

  5. 5

    L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity (Oxford Univ. Press, Oxford, 2016).

    Google Scholar 

  6. 6

    J. Gavoret and Ph. Nozieres, Ann. Phys. (N.Y.) 28, 349 (1964).

    ADS  Article  Google Scholar 

  7. 7

    T. D. Lee and K. Huang, Phys. Rev. 105, 1119 (1957).

    ADS  Article  Google Scholar 

  8. 8

    T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 136, 1135 (1957).

    ADS  Article  Google Scholar 

  9. 9

    L. P. Pitaevskii and S. Stringari, J. Low Temp. Phys. 85, 377 (1991).

    ADS  Article  Google Scholar 

  10. 10

    R. Lopes, Ch. Eigen, Nir Navon, D. Clement, R. P. Smith, and Z. Hadzibabic, Phys. Rev. Lett. 119, 190404 (2017).

    ADS  Article  Google Scholar 

  11. 11

    S. Stringari, Phys. Rev. B 46, 2974 (1992).

    ADS  Article  Google Scholar 

  12. 12

    L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 90, 250403 (2003).

    ADS  Article  Google Scholar 

  13. 13

    L. Chomaz et al., Nat. Phys. 14, 442 (2018).

    Article  Google Scholar 

  14. 14

    M. Jona-Lasinio, K. Lakomy, and L. Santo, Phys. Rev. A 88, 025603 (2013).

    ADS  Article  Google Scholar 

  15. 15

    M. Greiner et al., Nature (London, U.K.) 415, 39 (2002).

    ADS  Article  Google Scholar 

  16. 16

    R. Chang, Q. Bouton, H. Cayla, C. Qu, A. Aspect, C. I. Westbrook, and D. Clément, Phys. Rev. Lett. 117, 235303 (2016).

    ADS  Article  Google Scholar 

  17. 17

    Chunlei Qu, L. Pitaevskii, and S. Stringari, Phys. Rev. A 94, 063635 (2016).

    ADS  Article  Google Scholar 

  18. 18

    E. Timmermans, P. Tommasini, and K. Huang, Phys. Rev. A 55, 3645 (1997).

    ADS  MathSciNet  Article  Google Scholar 

  19. 19

    Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).

    ADS  Article  Google Scholar 

Download references


It is a great pleasure to thank long-standing scientific collaborations and stimulating discussions with Lev Pitaevskii, which started 30 years ago, after my first visit to the Kapitza Institute for Physical Problems in Moscow and are still now continuing successfully in Trento.

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Correspondence to S. Stringari.

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Contribution for the JETP special issue in honor of L.P. Pitaevskii’s 85th birthday

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Stringari, S. Quantum Fluctuations and Gross-Pitaevskii Theory. J. Exp. Theor. Phys. 127, 844–850 (2018). https://doi.org/10.1134/S1063776118110195

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