Critical number of atoms for attractive Bose-Einstein condensates with cylindrically symmetrical traps

We calculated, within the Gross-Pitaevskii formalism, the critical number of atoms for Bose-Einstein condensates with two-body attractive interactions in cylindrical traps with different frequency ratios. In particular, by using the trap geometries considered by Roberts et al. [Phys. Rev. Lett. 86, 4211 (2001)], we show that the theoretical maximum critical numbers are given approximately by ${N}_{c}{=0.55(l}_{0}/|a|).$ Our results also show that, by exchanging the frequencies ${\ensuremath{\omega}}_{z}$ and ${\ensuremath{\omega}}_{\ensuremath{\rho}},$ the geometry with ${\ensuremath{\omega}}_{\ensuremath{\rho}}l{\ensuremath{\omega}}_{z}$ favors the condensation of larger number of particles. We also simulate the time evolution of the condensate when changing the ground state from $a=0$ to $al0$ using a 200 ms ramp. A conjecture on higher-order nonlinear effects is also added in our analysis with an experimental proposal to determine its signal and strength.