Bose-Einsein condensate with attractive interaction
Many experimental groups have
achieved Bose-Einstein condensation in cold trapped atoms, at temperatures so low
that the atoms would have solidified in free space. They remain in a gaseous
state in the trap because of the quantum zero-point motion, which reduces the
effectiveness of the attractive part of the interatomic potential.
For atoms with a postive scattering
length, such as Rb and Na, the condensate is stable. For the
case of negative scattering length, as in Li, the attractive potential
has a stronger effect, and the condensate is metastable, with a lifetime of the
order of minutes. What becomes of the condensate?
We study the
stability by solving the time-dependent nonlinear Schrodinger equation
numerically, for Li atoms trapped in a harmonic potential. 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 Nc. For N
> Nc, 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 1
trap period. Within numerical uncertainty, the results are consistent with the
formation of a ``black hole'' of infinite density fluctuations, as predicted
earlier by Ueda and Huang. We obtain numerically Nc = 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 Nc = 1260 as the maximum value in the
oscillations.
References:
A. Elefthrious and K. Huang,
cond-mat/9908229 16 Aug 1999.
M. Ueda and K. Huang, Phys. Rev.
A (in prress); cond-mat/9807359 27 Jul 1998.