Skip to main content
SpringerLink
Log in

Cold Quantum Gases and Bose–Einstein Condensation

  • Chapter
  • First Online:
Quantum Many Body Systems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2051))

Abstract

Bose–Einstein condensation (BEC) in cold atomic gases was first achieved experimentally in 1995 [1, 6]. After initial failed attempts with spin-polarized atomic hydrogen, the first successful demonstrations of this phenomenon used gases of rubidium and sodium atoms, respectively. Since then there has been a surge of activity in this field, with ingenious experiments putting forth more and more astonishing results about the behavior of matter at very cold temperatures.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It is possible to prove an upper bound on the critical temperature, however. That is, one can establish the absence of BEC for large enough temperature, see [31].

  2. 2.

    Strictly speaking, it is not the expectation value of the perturbation that is the relevant measure of its smallness, but rather the variance. Hence the condition for validity of perturbation theory is slightly more stringent than what is displayed here.

References

  1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)

    Google Scholar 

  2. N.N. Bogoliubov, On the theory of superfluidity. J. Phys. (USSR) 11, 23 (1947)

    Google Scholar 

  3. S.N. Bose, Plancks Gesetz und Lichtquantenhypothese. Z. Phys. 26, 178–181 (1924)

    Google Scholar 

  4. E. Buffet, J.V. Pulè, Fluctuation properties of the imperfect Bose gas. J. Math. Phys. 24, 1608–1616 (1983)

    Google Scholar 

  5. F. Calogero, Ground state of a one-dimensional N-body system. J. Math. Phys. 10, 2197–2200 (1969); Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971)

    Google Scholar 

  6. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995)

    Google Scholar 

  7. F.J. Dyson, Ground-state energy of a hard-sphere gas. Phys. Rev. 106, 20–26 (1957)

    Google Scholar 

  8. F.J. Dyson, E.H. Lieb, B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–383 (1978)

    Google Scholar 

  9. A. Einstein, Quantentheorie des einatomigen idealen Gases. Sitzungsber. Preuss. Akad. Wiss., Phys.-math. Klasse, 261–267 (1924). Zweite Abhandlung. Sitzungsber. Preuss. Akad. Wiss., Phys.-math. Klasse, 3–14 (1925)

    Google Scholar 

  10. J. Fröhlich, B. Simon, T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50, 79–85 (1976)

    Google Scholar 

  11. M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1, 516–523 (1960)

    Google Scholar 

  12. A. Giuliani, R. Seiringer, The ground state energy of the weakly interacting Bose gas at high density. J. Stat. Phys. 135, 915–934 (2009)

    Google Scholar 

  13. T.D. Lee, C.N. Yang, Many body problem in quantum mechanics and quantum statistical mechanics. Phys. Rev. 105, 1119–1120 (1957)

    Google Scholar 

  14. T.D. Lee, K. Huang, C.N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, 1135–1145 (1957)

    Google Scholar 

  15. M. Lewin, R. Seiringer, Strongly correlated phases in rapidly rotating Bose gases. J. Stat. Phys. 137, 1040–1062 (2009)

    Google Scholar 

  16. E.H. Lieb, Exact analysis of an interacting Bose gas, II. The excitation spectrum. Phys. Rev. 130, 1616–1624 (1963)

    Google Scholar 

  17. E.H. Lieb, W. Liniger, Exact Analysis of an interacting Bose gas, I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963)

    Google Scholar 

  18. E.H. Lieb, R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002)

    Google Scholar 

  19. E.H. Lieb, R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases. Comm. Math. Phys. 264, 505–537 (2006)

    Google Scholar 

  20. E.H. Lieb, J. Yngvason, Ground state energy of the low density Bose gas. Phys. Rev. Lett. 80, 2504–2507 (1998)

    Google Scholar 

  21. E.H. Lieb, J. Yngvason, The ground state energy of a dilute two-dimensional Bose gas. J. Stat. Phys. 103, 509 (2001)

    Google Scholar 

  22. E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2000)

    Google Scholar 

  23. E.H. Lieb, R. Seiringer, J.P. Solovej, Ground-state energy of the low-density Fermi gas. Phys. Rev. A 71, 053605 (2005)

    Google Scholar 

  24. E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, The Mathematics of the Bose Gas and its Condensation. Oberwolfach Seminars, vol. 34 (Birkhäuser, Boston, 2005). Also available at arXiv:cond-mat/0610117

    Google Scholar 

  25. E.H. Lieb, R. Seiringer, J. Yngvason, Yrast line of a rapidly rotating Bose gas: Gross-Pitaevskii regime. Phys. Rev. A 79, 063626 (2009)

    Google Scholar 

  26. O. Penrose, L. Onsager, Bose-Einstein condensation and liquid helium. Phys. Rev. 104, 576–584 (1956)

    Google Scholar 

  27. R. Seiringer, Ground state asymptotics of a dilute, rotating gas. J. Phys. A Math. Gen. 36, 9755–9778 (2003)

    Google Scholar 

  28. R. Seiringer, The thermodynamic pressure of a dilute Fermi gas. Comm. Math. Phys. 261, 729–758 (2006)

    Google Scholar 

  29. R. Seiringer, Free energy of a dilute Bose gas: Lower bound. Comm. Math. Phys. 279, 595–636 (2008)

    Google Scholar 

  30. R. Seiringer, The Excitation Spectrum for Weakly Interacting Bosons. Commun. Math. Phys. 306, 565–578 (2011)

    Google Scholar 

  31. R. Seiringer, D. Ueltschi, Rigorous upper bound on the critical temperature of dilute Bose gases. Phys. Rev. B 80, 014502 (2009)

    Google Scholar 

  32. B. Sutherland, Quantum many-body problem in one dimension: Ground state. J. Math. Phys. 12, 246–250 (1971); Quantum many-body problem in one dimension: Thermodynamics. J. Math. Phys. 12, 251–256 (1971)

    Google Scholar 

  33. D. Ueltschi, Feynman cycles in the Bose gas. J. Math. Phys. 47, 123303 (2006)

    Google Scholar 

  34. H.-T. Yau, J. Yin, The second order upper bound for the ground energy of a Bose gas. J. Stat. Phys. 136, 453–503 (2009)

    Google Scholar 

  35. J. Yin, Free Energies of Dilute Bose Gases. Upper Bound. J. Stat. Phys. 141, 683–726 (2010)

    Google Scholar 

Download references

Acknowledgements

Many thanks to Dana Mendelson, Alex Tomberg and Daniel Ueltschi for allowing me to use their figures in these notes. Financial support through the ERC starting grant CoMboS-239694 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    Robert Seiringer

Authors
  1. Robert Seiringer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Seiringer .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Seiringer, R. (2012). Cold Quantum Gases and Bose–Einstein Condensation. In: Quantum Many Body Systems. Lecture Notes in Mathematics(), vol 2051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29511-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation