Skip to main content
SpringerLink
Log in

One-Dimensional Behavior of Dilute, Trapped Bose Gases

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Recent experimental and theoretical work has shown that there are conditions in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is the size of an atom – as might have been supposed – but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the one-dimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, N, transverse and longitudinal dimensions of the trap, r and L, and the scattering length a of the interaction potential. Our main result is that if r/L→0 and N→∞ the ground state energy and density can be obtained by minimizing a one-dimensional density functional involving the Lieb-Liniger energy density with coupling constant ∼a/r 2. This density functional simplifies in various limiting cases and we identify five asymptotic parameter regions altogether. Three of these, corresponding to the weak coupling regime, can also be obtained as limits of a three-dimensional Gross-Pitaevskii theory. We also show that Bose-Einstein condensation in the ground state persists in a part of this regime. In the strong coupling regime the longitudinal motion of the particles is strongly correlated. The Gross-Pitaevskii description is not valid in this regime and new mathematical methods come into play.

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

Similar content being viewed by others

References

  1. Astrakharchik, G.E., Giorgini, S.: Quantum Monte Carlo study of the three- to one-dimensional crossover for a trapped Bose gas. Phys. Rev. A 66, 053614-1–6 (2002)

    Article  Google Scholar 

  2. Baumgartner, B., Solovej, J.P., Yngvason, J.: Atoms in Strong Magnetic Fields: The High Field Limit at Fixed Nuclear Charge. Commun. Math. Phys. 212, 703–724 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blume, D.: Fermionization of a bosonic gas under highly elongated confinement: A diffusion quantum Monte Carlo study. Phys. Rev. A 66, 053613-1–8 (2002)

    Article  Google Scholar 

  4. Bongs, K., Burger, S., Dettmer, S., Hellweg, D., Artl, J., Ertmer, W., Sengstok, K.: Waveguides for Bose-Einstein condensates. Phys. Rev. A 63, 031602 (2001)

    Article  Google Scholar 

  5. Cornell, E.A., Wieman, C.E.: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. In: Les Prix Nobel 2001, Stockholm: The Nobel Foundation, 2002, pp. 87–108. Reprinted in: Rev. Mod. Phys. 74, 875–893 (2002); Chem. Phys. Chem. 3, 476–493 (2002)

    Article  Google Scholar 

  6. Das, K.K.: Highly anisotropic Bose-Einstein condensates: Crossover to lower dimensionality. Phys. Rev. A 66, 053612-1–7 (2002)

    Article  Google Scholar 

  7. Das, K.K., Girardeau, M.D., Wright, E.M.: Crossover from One to Three Dimensions for a Gas of Hard-Core Bosons. Phys. Rev. Lett. 89, 110402-1–4 (2002)

    Article  Google Scholar 

  8. Dunjko, V., Lorent, V., Olshanii, M.: Bosons in Cigar-Shaped Traps: Thomas-Fermi Regime, Tonks-Girardeau Regime, and In Between. Phys. Rev. Lett. 86, 5413–5316 (2001)

    Article  Google Scholar 

  9. Dyson, F.J.: Ground-State Energy of a Hard-Sphere Gas. Phys. Rev. 106, 20–26 (1957)

    Article  MATH  Google Scholar 

  10. Forrester, P.J., Frankel, N.E., Garoni, T.M., Witte, N.S.: Finite one dimensional impenetrable Bose systems: Occupation numbers. Phys. Rev. A 67, 043607 (2003); Forrester, P.J., Frankel, N.E., Garoni, T.M.: Random Matrix Averages and the impenetrable Bose Gas in Dirichlet and Neumann Boundary Conditions, J. Math. Phys. 44, 4157 (2003)

    Article  Google Scholar 

  11. Gangardt, D.M., Shlyapnikov, G.V.: Local correlations in a strongly interacting 1D Bose gas. New J. Phys. 5, 79 (2003)

    Article  Google Scholar 

  12. Girardeau, M.: Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension. J. Math. Phys. 1, 516–523 (1960)

    MATH  Google Scholar 

  13. Girardeau, M.D., Wright, E.M.: Bose-Fermi variational Theory for the BEC-Tonks Crossover. Phys. Rev. Lett. 87, 210401-1–4 (2001)

    Article  Google Scholar 

  14. Girardeau, M.D., Wright, E.M., Triscari, J.M.: Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic trap. Phys. Rev. A 63, 033601-1–6 (2001)

    Article  Google Scholar 

  15. Görlitz, A., Vogels, J.M., Leanhardt, A.E., Raman, C., Gustavson, T.L., Abo-Shaeer, J.R., Chikkatur, A.P., Gupta, S., Inouye, S., Rosenband, T., Ketterle, W.: Realization of Bose-Einstein Condensates in Lower Dimension. Phys. Rev. Lett. 87, 130402-1–4 (2001)

    Article  Google Scholar 

  16. Greiner, M., Bloch, I., Mendel, O., Hänsch, T., Esslinger, T.: Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates. Phys. Rev. Lett. 87, 160405 (2001)

    Article  Google Scholar 

  17. Jackson, A.D., Kavoulakis, G.M.: Lieb Mode in a Quasi-One-Dimensional Bose-Einstein Condensate of Atoms. Phys. Rev. Lett. 89, 070403 (2002)

    Article  Google Scholar 

  18. Ketterle, W.: When atoms behave as waves: Bose-Einstein condensation and the atom laser. In: Les Prix Nobel 2001, Stockholm: The Nobel Foundation, 2002, pp. 118–154. Reprinted in: Rev. Mod. Phys. 74, 1131–1151 (2002); Chem. Phys. Chem. 3, 736–753 (2002)

    Article  Google Scholar 

  19. Kolomeisky, E.B., Newman, T.J., Straley, J.P., Qi, X.: Low-Dimensional Bose Liquids: Beyond the Gross-Pitaevskii Approximation. Phys. Rev. Lett. 85, 1146–1149 (2000). Bhaduri, R.K., Sen, D.: Comment on ‘‘Low-Dimensional Bose Liquids: Beyond the Gross-Pitaevskii Approximation’’. Phys. Rev. Lett. 86, 4708 (2001). Reply, Phys. Rev. Lett. 86, 4709 (2001)

    Article  Google Scholar 

  20. Komineas, S., Papanicolaou, N.: Vortex Rings and Lieb Modes in a Cylindrical Bose-Einstein Condensate. Phys. Rev. Lett. 89, 070402 (2002)

    Article  Google Scholar 

  21. Lenard, A.: Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys. 5, 930–943 (1964)

    Google Scholar 

  22. Lieb, E.H., Liniger, W.: Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605–1616 (1963)

    Article  MATH  Google Scholar 

  23. Lieb, E.H.: Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum. Phys. Rev. 130, 1616–1624 (1963)

    Article  MATH  Google Scholar 

  24. Lieb, E.H., Loss, M.: Analysis. Second edition, Providence, RI: American Mathematical Society, 2001

  25. Lieb, E.H., Seiringer, R.: Proof of Bose-Einstein Condensation for Dilute Trapped Gases. Phys. Rev. Lett. 88, 170409-1–4 (2002)

    Article  Google Scholar 

  26. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Ground State of the Bose Gas. In: Current Developments in Mathematics, 2001, Cambridge: International Press, 2002, pp. 131–178

  27. Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602-1–13 (2000)

    Article  Google Scholar 

  28. Lieb, E.H., Seiringer, R., Yngvason, J.: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-dimensional Bose Gas. Commun. Math. Phys. 224, 17–31 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lieb, E.H., Seiringer, R., Yngvason, J.: One-dimensional Bosons in Three-dimensional Traps. arXiv:cond-mat/0304071, Phys. Rev. Lett. 91, 150401 (2003)

    Article  Google Scholar 

  30. Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of Heavy Atoms in High Magnetic Fields. I : Lowest Landau Band Regions. Commun. Pure and Appl. Math. 47, 513–593 (1994)

    MATH  Google Scholar 

  31. Lieb, E.H., Yngvason, J.: Ground State Energy of the Low Density Bose Gas. Phys. Rev. Lett. 80, 2504–2507 (1998)

    Article  Google Scholar 

  32. Menotti, C., Stringari, S.: Collective Oscillations of a 1D Trapped Bose gas. Phys. Rev. A 66, 043610 (2002)

    Article  Google Scholar 

  33. Olshanii, M.: Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons. Phys. Rev. Lett. 81, 938–941 (1998)

    Article  Google Scholar 

  34. Papenbrock, T.: Ground-state properties of hard-core bosons in one-dimensional harmonic traps. Phys. Rev. A 67, 041601(R) (2003)

    Article  Google Scholar 

  35. Petrov, D.S., Shlyapnikov, G.V., Walraven, J.T.M.: Regimes of Quantum Degeneracy in Trapped 1D Gases. Phys. Rev. Lett. 85, 3745–3749 (2000)

    Article  Google Scholar 

  36. Pitaevskii, L., Stringari, S.: Uncertainty Principle, Quantum Fluctuations, and Broken Symmetries. J. Low Temp. Phys. 85, 377–388 (1991)

    Google Scholar 

  37. Robinson, D.W.: The Thermodynamic Pressure in Quantum Statistical Mechanics. Lecture Notes in Physics, Vol. 9, Berlin-Heidelberg-New York: Springer, 1971

  38. Schreck, F., Khaykovich, L., Corwin, K.L., Ferrari, G., Boudriel, T., Cubizolles, J., Salomon, C.: Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea. Phys. Rev. Lett. 87, 080403 (2001)

    Article  Google Scholar 

  39. Tanatar, B., Erkan, K.: Strongly interacting one-dimensional Bose-Einstein condensates in harmonic traps. Phys. Rev. A 62, 053601-1–6 (2000). Girardeau, M.D., Wright, E.M.: Comment on ‘‘Strongly interacting one-dimensional Bose-Einstein condensates in harmonic traps’‘. arXiv:cond-mat/0010457

    Article  Google Scholar 

  40. Temple, G.: The theory of Rayleigh’s Principle as Applied to Continuous Systems. Proc. Roy. Soc. London A 119, 276–293 (1928)

    Google Scholar 

  41. Tonks, L.: The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres. Phys. Rev. 50, 955–963 (1936)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Jadwin Hall, Princeton University, 708, Princeton, NJ 08544, USA

    Elliott H. Lieb & Robert Seiringer

  2. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090, Vienna, Austria

    Robert Seiringer & Jakob Yngvason

Authors
  1. Elliott H. Lieb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Seiringer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jakob Yngvason
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Aizenman

© 2003 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Work partially supported by U.S. National Science Foundation grant PHY 01-39984.

Erwin Schrödinger Fellow, supported by the Austrian Science Fund.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lieb, E., Seiringer, R. & Yngvason, J. One-Dimensional Behavior of Dilute, Trapped Bose Gases. Commun. Math. Phys. 244, 347–393 (2004). https://doi.org/10.1007/s00220-003-0993-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0993-3

Keywords

Search

Navigation