Skip to main content
SpringerLink
Log in

The Condensed Fraction of a Homogeneous Dilute Bose Gas Within the Improved Hartree–Fock Approximation

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Motivated by the recent experiment (Lopes et al. in Phys Rev Lett 119:190404, 2017) with a homogeneous Bose gas, we investigate a homogeneous dilute Bose gas to calculate the quantum depletion density. By means of the Cornwall–Jackiw–Tomboulis effective action approach within an improved Hartree–Fock approximation, the condensed fraction is recovered in an alternative theoretical approach and compared with corresponding findings in experimental data. A good agreement in the extremely small region of gas parameter is demonstrated. Furthermore, the ground state energy is also reattained and discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. The experimental data is rescaled from Ref. [27] and was provided to us by Raphael Lopes.

References

  1. Bose, S.N.: Plancks Gesetz und Lichtquantenhypothese. Z. Phys. 26, 178–181 (1924). https://doi.org/10.1007/BF01327326

  2. Einstein, A.: Quantentheorie des Einatomigen Idealen Gases. Sitz. Ber. Preuss. Akad. Wiss. 22, 261 (1924)

  3. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198 (1995). https://doi.org/10.1126/science.269.5221.198

    Article  ADS  Google Scholar 

  4. Davis, K.B., Mewes, M.O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W.: Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969 (1995). https://doi.org/10.1103/PhysRevLett.75.3969

    Article  ADS  Google Scholar 

  5. Pethick, C.J., Smith, H.: Bose-Einstein Condensation in Dilute Gases, 2nd edn. Cambridge University Press, Cambridge (2008). https://doi.org/10.1017/CBO9780511802850

    Book  Google Scholar 

  6. Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  7. Gross, E.P.: Structure of a quantized vortex in boson systems. Nuovo Cim. 20, 454–477 (1961). https://doi.org/10.1007/BF02731494

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13, 451 (1961)

    MathSciNet  Google Scholar 

  9. Bogolyubov, N.N.: On the theory of superfluidity. J. Phys. 11, 23 (1947)

    MathSciNet  Google Scholar 

  10. Dalfovo, F., Giorgini, S., Guilleumas, M., Pitaevskii, L., Stringari, S.: Collective and single-particle excitations of a trapped Bose gas. Phys. Rev. A 56, 3840 (1997). https://doi.org/10.1103/PhysRevA.56.3840

  11. Stringari, S.: Quantum fluctuations and Gross-Pitaevskii theory. J. Expr. Theoret. Phys. 127, 844 (2018). https://doi.org/10.1134/S1063776118110195

    Article  ADS  Google Scholar 

  12. Carlen, E.A., Jauslin, I., Lieb, E.H.: Analysis of a simple equation for the ground state of the Bose gas II: monotonicity, convexity, and condensate fraction. SIAM J. Math. Anal. 53, 5322 (2021). https://doi.org/10.1137/20M1376820

    Article  MathSciNet  MATH  Google Scholar 

  13. Cornwall, J.M., Jackiw, R., Tomboulis, E.: Effective action for composite operators. Phys. Rev. D 10, 2428 (1974). https://doi.org/10.1103/PhysRevD.10.2428

    Article  MATH  ADS  Google Scholar 

  14. Andersen, J.O.: Theory of the weakly interacting Bose gas. Rev. Mod. Phys. 76, 599 (2004). https://doi.org/10.1103/RevModPhys.76.599

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Orland, J.W.N.H.: Quantum many-particle systems. In: Pines, D. (ed.) Westview Press (1998)

  16. Shi, H., Griffin, A.: Finite-temperature excitations in a dilute Bose-condensed gas. Phys. Rep. 304, 1 (1998). https://doi.org/10.1016/S0370-1573(98)00015-5

    Article  ADS  Google Scholar 

  17. Thu, N.V., Song, P.T.: Casimir effect in a weakly interacting Bose gas confined by a parallel plate geometry in improved Hartree-Fock approximation. Physica A 540, 123018 (2020). https://doi.org/10.1016/j.physa.2019.123018

    Article  MathSciNet  MATH  Google Scholar 

  18. Phat, T.H., Hoa, L.V., Anh, N.T., Van Long, N.: Bose-Einstein condensation in binary mixture of Bose gases. Ann. Phys. 324, 2074 (2009). https://doi.org/10.1016/j.aop.2009.07.003

    Article  MATH  ADS  Google Scholar 

  19. Goldstone, J.: Field theories with superconductor solutions. Nuovo Cim. 19, 154 (1961). https://doi.org/10.1007/BF02812722

    Article  MathSciNet  MATH  ADS  Google Scholar 

  20. Hugenholtz, N.M., Pines, D.: Ground-state energy and excitation spectrum of a system of interacting bosons. Phys. Rev. 116, 489 (1959). https://doi.org/10.1103/PhysRev.116.489

    Article  MathSciNet  MATH  ADS  Google Scholar 

  21. Hohenberg, P., Martin, P.: Microscopic theory of superfluid helium. Ann. Phys. 34, 291 (1965). https://doi.org/10.1016/0003-4916(65)90280-0

    Article  ADS  Google Scholar 

  22. Hutchinson, D.A.W., Dodd, R.J., Burnett, K.: Gapless finite- \({T}\) theory of collective modes of a trapped gas. Phys. Rev. Lett. 81, 2198 (1998). https://doi.org/10.1103/PhysRevLett.81.2198

    Article  Google Scholar 

  23. Griffin, A.: Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures. Phys. Rev. B 53, 9341 (1996). https://doi.org/10.1103/PhysRevB.53.9341

    Article  ADS  Google Scholar 

  24. Ivanov, Y.B., Riek, F., Knoll, J.: Gapless Hartree-Fock resummation scheme for the \(O(N)\) model. Phys. Rev. D 71, 105016 (2005). https://doi.org/10.1103/PhysRevD.71.105016

    Article  Google Scholar 

  25. Schmitt, A.: Dense Matter in Compact Stars. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-12866-0

    Book  Google Scholar 

  26. Wu, T.T.: Ground state of a Bose system of hard spheres. Phys. Rev. 115, 1390 (1959). https://doi.org/10.1103/PhysRev.115.1390

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. Lopes, R., Eigen, C., Navon, N., Clément, D., Smith, R.P., Hadzibabic, Z.: Quantum depletion of a homogeneous Bose-Einstein condensate. Phys. Rev. Lett. 119, 190404 (2017). https://doi.org/10.1103/PhysRevLett.119.190404

    Article  ADS  Google Scholar 

  28. Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.J., Stamper-Kurn, D.M., Ketterle, W.: Observation of Feshbach resonances in a Bose-Einstein condensate. Nature 392, 151 (1998). https://doi.org/10.1038/32354

    Article  ADS  Google Scholar 

  29. Lee, T.D., Huang, K., Yang, C.N.: Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, 1135 (1957). https://doi.org/10.1103/PhysRev.106.1135

    Article  MathSciNet  MATH  ADS  Google Scholar 

  30. Yau, H.-T., Yin, J.: The second order upper bound for the ground energy of a Bose gas. J. Stat. Phys. 136, 453 (2009). https://doi.org/10.1007/s10955-009-9792-3

    Article  MathSciNet  MATH  ADS  Google Scholar 

  31. Giorgini, S., Boronat, J., Casulleras, J.: Ground state of a homogeneous Bose gas: a diffusion Monte Carlo calculation. Phys. Rev. A 60, 5129 (1999). https://doi.org/10.1103/PhysRevA.60.5129

    Article  ADS  Google Scholar 

  32. Carlen, E.A., Holzmann, M., Jauslin, I., Lieb, E.H.: Simplified approach to the repulsive Bose gas from low to high densities and its numerical accuracy. Phys. Rev. A 103, 053309 (2021). https://doi.org/10.1103/PhysRevA.103.053309

    Article  MathSciNet  ADS  Google Scholar 

  33. Lieb, E.H.: Simplified approach to the ground-state energy of an imperfect Bose gas. Phys. Rev. 130, 2518 (1963). https://doi.org/10.1103/PhysRev.130.2518

    Article  ADS  Google Scholar 

  34. Navon, N., Piatecki, S., Günter, K., Rem, B., Nguyen, T.C., Chevy, F., Krauth, W., Salomon, C.: Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas. Phys. Rev. Lett. 107, 135301 (2011). https://doi.org/10.1103/PhysRevLett.107.135301

    Article  ADS  Google Scholar 

  35. Lee, T.D., Yang, C.N.: Many-body problem in quantum statistical mechanics. V. Degenerate phase in Bose-Einstein condensation. Phys. Rev. 117, 897 (1960). https://doi.org/10.1103/PhysRev.117.897

    Article  MathSciNet  MATH  ADS  Google Scholar 

Download references

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2018.02. We are very grateful to M. Guilleumas for the useful conversations and to L. Raphael for extensive discussions about the experimental data.

Author information

Authors and Affiliations

  1. Department of Physics, Hanoi Pedagogical University 2, Hanoi, 100000, Vietnam

    Nguyen Van Thu

  2. Institute for Theoretical Physics, KU Leuven, 3001, Leuven, Belgium

    Jonas Berx

Authors
  1. Nguyen Van Thu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonas Berx
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonas Berx.

Ethics declarations

Conflict of interest

All of the authors declare that we have no conflict of interest.

Additional information

Communicated by Alessandro Giuliani.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Thu, N., Berx, J. The Condensed Fraction of a Homogeneous Dilute Bose Gas Within the Improved Hartree–Fock Approximation. J Stat Phys 188, 16 (2022). https://doi.org/10.1007/s10955-022-02944-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-022-02944-0

Keywords

Search

Navigation