The theoretical study of low temperature quantum gases found in book is often inspired by the mean field approximation and second quantization methods, which are most appropriate for condensed systems where every atoms interacts simultaneously with several others, so that some averaging takes place.  However, they do not remain so appropriate for gases, where interactions occur primarily through binary collisions.  In this case, one must take into account the detail of the correlations which are created during the collision, in particular the short range correlations introduced by the repulsive part of the interaction potential - in other words, one must consider distortions of the wave functions which are ignored within a mean field theory. The Ursell operator method [16] was introduced and developed precisely for this purpose.

This method has now been applied to various situations and has provided, in particular,  corrections to the ideal gas Bose-Einstein condensation temperature. When interactions are present, the condensation temperature varies linearly (and with a positive proportionality coefficient) with the s-wave scattering length [17]. Those results were first obtained numerically with a Path Integral Monte-Carlo simulation [18].

More recently, non-analytical higher-order corrections have been determined [19,20]. The relative variation of critical temperature (as compared to the ideal gas value) is (in the dilute limit) :

where a is the scattering length and  n is the density.

Recently, the Ursell operator theory has been extended to dense fluids,  which allows one to make the link with the theory of classical liquids in the clusters expansion approach [21].

Currently, the focus is on spin waves in dilute polarized gases, especially on understanding the experiment of E. Cornell’s group at JILA on « spin state segregation » [22].

Another subject of investigations deals with exchange cycle statistics in a Bose gas and their link with the condensate fraction.The goal is to obtain a useful formula for PIMC simulations, much in the spirit of what Pollock and Ceperley [23] did for the superfluid fraction.

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(updated on November 27th, 2001 by P.J.Nacher )  

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References

[16] P. Grüter et F. Laloë, Journal de Physique 5 (1995) 181 ; 5 (1995) 1255 ; 7 (1997) 485.

[17] M. Holzmann, P. Grüter et F. Laloë, Eur. Phys. J.  B10, 739 (1999).

[18] P. Grüter, D. Ceperley et F. Laloë, Phys. Rev. Lett.  79, 3549 (1997).

[19] M. Holzmann, G. Baym, J.P. Blaizot et F. Laloë, Phys. Rev. Lett.  87, 120403 (2001).

[20] G. Baym, J.P. Blaizot, M. Holzmann, F. Laloë et D. Vautherin, cond-mat/0107129 , accepté pour publication dans Eur. Phys. J.  B.

[21] J.N. Fuchs, M. Holzmann et  F. Laloë, cond-mat/0109265, soumis à Eur. Phys. J.  B.

[22] H.J. Lewandowski, D.M. Harber, D.L. Whitaker et E. Cornell,  cond-mat/0109476 (2001).

[23] D.M. Ceperley, Rev. Mod. Phys.  67, 1601 (1995).

 

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