We study the instability of collective excitations of a three-dimensional Bose-Einstein condensate with repulsive and attractive interactions in a shallow trap designed as a quadratic plus a quartic potential. By using a correlated many-body theory, we determine the excitation modes and probe the critical behavior of collective modes, having a crucial dependence on the anharmonic parameter. We examine the power-law behavior of monopole frequency near criticality. In Gross-Pitaevskii variational treatment [Phys. Rev. Lett. 80, 1576 (1998)] the power-law exponent is determined as one-fourth power of $(1-\frac{A}{A_{\mathit{cr}}})$, $A$ is the number of condensate atoms and $A_{\mathit{cr}}$ is the critical number near collapse. We observe that the power-law exponent becomes $\frac{1}{6}$ in our calculation for the pure harmonic trap and it becomes $\frac{1}{7}$, for traps with a small anharmonic distortion. However for large anharmonicity the power law breaks down.

• Received 15 April 2010

DOI:https://doi.org/10.1103/PhysRevA.82.043614