Low-temperature Bose-Einstein condensates in time-dependent traps: Beyond the [Formula Presented] symmetry-breaking approach

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 57, Issue 4, 1998, Pages 3008-3021

  Castin, Y ^a   Dum, R ^a  

^a ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001053337     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.57.3008     Document Type: Article

References (30)

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20. For a formally correct definition of this expansion see Eq. (20).
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22. In fact the (Formula presented) distribution in the pseudopotential does not lead to a well-defined scattering problem and has to be regularized as discussed in 21, that is, (Formula presented) has to be replaced by (Formula presented) for a wave function (Formula presented). The corresponding scattering amplitude for a relative momentum (Formula presented) between the particles is then (Formula presented) for the (Formula presented) wave and vanishes for the other waves. The pseudopotential therefore has the same binary scattering properties as the true interaction potential in the limit (Formula presented). Note that to the order of the calculations performed in this paper this regularization is not essential. The use of the pseudopotential is correct only in the dilute gas limit, where the three-body interactions are negligible (which imposes (Formula presented) on the spatial density (Formula presented).
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26. We have to verify that numerically and find indeed for (Formula presented) real eigenvalues. Physically this means that the condensate is stable with respect to quantum fluctuations.
27. For a positive scattering length (Formula presented) one can prove indeed that (Formula presented) is positive.
28. As (Formula presented) is calculated up to order (Formula presented) a careful reader may argue that the contribution of (Formula presented) defined in Eq. (21) is relevant. In fact, it vanishes: as (Formula presented) solves Eq. (41), the quantity (Formula presented) does not vary to first order in any norm-preserving change of (Formula presented).
29. The term with the trace in Eq. (70) and the sum in Eq. (71) are in fact infinite. They can be made finite using the regularization of 19.
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