Stability criterion for attractive Bose-Einstein condensates

Experimental observation of Bose-Einstein condensation ~BEC! in ultracold atomic clouds @1# has stimulated a new direction in the study of macroscopic quantum phenomena. Basically, the interaction between two confined bosons in a condensate is determined by the s-wave scattering length a and it can be either repulsive (a.0) or attractive (a,0). Although first BEC experiments were commonly realized with gases promoting a positive scattering length, trapped Li atom gases, which are characterized by a negative scattering length, have raised an increasing interest @2# justified by the rich and complex dynamics mixing instability and generation of solitonlike structures, which substantially alters the formation of condensates. Furthermore, experimental results @3# suggested the possibility of using so-called Feshbach resonances to continuously detune a from positive to negative values by means of an external magnetic field, which brings insight into the experimental realization of BEC’s with attractive interactions. The dynamical behaviors of gases with negative scattering length are especially interesting, because in the absence of trapping, the condensate is described by the nonlinear Schrödinger ~NLS! equation, whose localized, multidimensional solutions are unstable and may collapse in finite time ~for a review see, e.g., Ref. @4#!. Similarly, collapse also occurs in confined condensates with attractive interactions, as emphasized by many theoretical works @5–7#, and sequences of collapse events have been experimentally detected in BEC’s of Li gas @8#, in which the condensate was observed to shrink on time scales of the trap oscillation period. General conditions for collapse follow from the virial theorem applied to the Gross-Pitaevskii ~GP! equation