Instability of a Bose-Einstein Condensate with Attractive Interaction.pdf
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Instability of a Bose-Einstein Condensate
with Attractive Interaction
Antonios Eleftheriou and Kerson Huang
Department of Physics and Center for Theoretical Physics
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
(February 1, 2008)
03.75.Fi, 42.65.Jx, 32.80.Pj MIT-CTP #2886
We study the stability of a Bose-Einstein condensate of harmonically
trapped atoms with negative scattering length, specifically 7Li. Our method
is to solve the time-dependent nonlinear Schro?dinger 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 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 by Ueda and Huang [16]. 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.
I. INTRODUCTION AND SUMMARY
Bose-Einstein condensation has been observed in magnetically trapped dilute vapors of
the alkali elements 87Rb [1], 23Na [2], 7Li [4], and 1H [3]. At the nanodegree temperatures
of these experiments, the systems would have frozen solid long ago were they in free space.
In the confining trap, however, zero-point motion keeps the atoms apart, and the systems
remain gaseous. The case of 7Li is special, however, in that the interatomic interaction is
predominantly attractive, as indicated by a negative scattering length. Thus, the condensate
in 7Li s
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