Instability of a Bose-Einstein condensate with an attractive interaction

Abstract

We study the stability of a Bose-Einstein condensate of harmonically trapped atoms with negative scattering length, specifically ${}^7\mathrm{Li}.$ Our method is to solve the time-dependent nonlinear Schrödinger equation numerically. For an isolated condensate, with no gain or loss, we find that the system is stable (apart from quantum tunneling) if the particle number N is less than a critical number $N_c.$ For $N>\mathrm{}{N}_c,$ the system collapses to high-density clumps in a region near the center of the trap. The time for the onset of collapse is on the order of one trap period. Within numerical uncertainty, the results are consistent with the formation of a “black hole” of infinite density fluctuations, as predicted by Ueda and Huang [Phys. Rev. A (to be published)]. We numerically obtain $N_c\approx 1251.$ We then include gain-loss mechanisms, i.e., the gain of atoms from a surrounding “thermal cloud,” and the loss due to two- and three-body collisions. The number N now oscillates in a steady state, with a period of about 145 trap periods. We obtain $N_c\approx 1260$ as the maximum value in the oscillations.