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arXiv:cond-mat/9908229 · Physical Review A

Instability of a Bose-Einstein condensate with an attractive interaction

Antonios Eleftheriou, Kerson Huang

Published Mar 2, 2000• 55 citations
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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 Schr\"odinger 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 $N>{N}_{c},$ 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 ${N}_{c}\ensuremath{\approx}1251.$ 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 ${N}_{c}\ensuremath{\approx}1260$ as the maximum value in the oscillations.

DOI
10.1103/physreva.61.043601

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We study the stability of a Bose-Einstein condensate of harmonically trapped atoms with negative scattering length, specifically 7Li.{}^{7}\mathrm{Li}.7Li. Our method is to solve the time-dependent nonlinear Schr"odinger 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.{N}_{c}.Nc​. For N>Nc,N>{N}_{c},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 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 ${N}_{c}\ensuremath{\approx}1251.$ 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 Nc\ensuremath≈1260{N}_{c}\ensuremath{\approx}1260Nc​\ensuremath≈1260 as the maximum value in the oscillations.
Evolution and collapse of a trapped Bose condensate with negative scattering length are predetermined by 3-body recombination of Bose-condensed atoms and by feeding of the condensate from the nonequilibrium thermal cloud. The collapse, starting once the number of condensate atoms reaches the critical value, ceases and turns to expansion when the density becomes so high that the recombination losses dominate over attractive interparticle interaction. As a result, we obtain a sequence of collapses, each of them followed by dynamic oscillations of the condensate. In every collapse the 3-body recombination burns only a part of the Bose-condensed atoms.
The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r1/r1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.
We study the Bose-Einstein condensation for a cloud of 7Li{}^{7}\mathrm{Li}7Li atoms with attractive nonlocal (finite-range) interaction in a harmonic trap. In addition to the low-density metastable branch, which is present also in the case of local interaction, a new stable branch appears at higher densities. For a large number of atoms, the size of the cloud in the stable high-density branch is independent of the trap size and the atoms are in a macroscopic quantum self-trapped configuration. We analyze the macroscopic quantum tunneling between the low-density metastable branch and the high-density one by using the instanton technique. Moreover, we consider the decay rate of the Bose condensate due to inelastic two- and three-body collisions.
We investigate the structure and stability of Bose-Einstein condensates of 7Li^{7}\mathrm{Li}7Li atoms with realistic van der Waals interactions by using the potential harmonic expansion method. Besides the known low-density metastable solution with a contact $\ensuremath{\delta}$-function interaction, we find a stable branch at a higher density which corresponds to the formation of an atomic cluster. Comparison with the results of a nonlocal effective interaction is also presented. We analyze the effect of trap size on the transition between the two branches of solutions. We also compute the loss rate of a Bose condensate due to two- and three-body collisions.
Numerical solution of nonlinear Schrodinger equation (NLSE) can be used to describe an inhomogeneous,weakly interact Bose-Einstein condensate in a small harmonic trap potential at zero temperature. In one dimension situation, we find solutions for the NLSE for ground-state condensate wave function, and obtain the ground-state condensate energy per particle. Furthermore, with the time-dependent NLSE, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering length in one-dimensional (1D) system for a fixed number of atoms, and the negative scattering length solutions have solitonlike properties.