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Quantum Electronics and Laser Science Conference

Collapsing dynamics of trapped Bose-Einstein condensates with attractive interactions

Hiroki Saito, Masahito Ueda

Published Nov 13, 2002• 0 citations
Publisher

Abstract

Summary form only given. Bose-Einstein condensation (BEC) of trapped atomic vapor has been realized in several atomic species. The static and dynamical properties of BEC crucially depend on the sign of the interatomic interaction. When the interaction is attractive, BEC in a spatially uniform 3D system is unstable to collapse into a denser phase. In a spatially confined system, however, the zero-point energy serves as a kinetic obstacle against collapse, allowing metastable BEC to be formed if the number of BEC atoms is below a certain critical number. Above the critical number of atoms, BEC collapses. The BEC with attractive interactions, therefore, has been restricted to a small number of atoms (/spl sim/1000) and the number has not been controllable.

DOI
10.1109/qels.2001.962164

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Summary form only given. Bose-Einstein condensation (BEC) of trapped atomic vapor has been realized in several atomic species. The static and dynamical properties of BEC crucially depend on the sign of the interatomic interaction. When the interaction is attractive, BEC in a spatially uniform 3D system is unstable to collapse into a denser phase. In a spatially confined system, however, the zero-point energy serves as a kinetic obstacle against collapse, allowing metastable BEC to be formed if the number of BEC atoms is below a certain critical number. Above the critical number of atoms, BEC collapses. The BEC with attractive interactions, therefore, has been restricted to a small number of atoms (/spl sim/1000) and the number has not been controllable.
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Among the species that have Bose-condensed, lithium 7 and rubidium 85 form unique condensates in that atoms interact attractively and that the energy barrier that stabilizes the gaseous Bose-Einstein condensates originates from the zero-point motion of atoms. This paper reviews some unique features of these new states of matter that are distinguished from those of other species with repulsive interactions. These include the density instability, macroscopic quantum tunneling, intermittent implosion, shellstructure formation, and partial quantization of circulation. 1. METASTABILITY AND DENSITY INSTABILITY OF THE CONDENSATE Atoms with attractive interactions are believed not to undergo Bose-Einstein condensation in the uniform 3D system. It is therefore appropriate to begin by discussing why attractive Bose-Einstein condensates (BEes) have been realized in trapped systems [1, 2]. A trapped BEe has three characteristic energies, namely, the kinetic, potential and interaction energies. The kinetic energy arises from the uncertainty principle, and is proportional to R-2 where R is the linear dimension of BEe. The trapping potential is usually harmonic, and therefore the potential energy is proportional to R2. The mean-field interaction energy is proportional to NoaR-3, where No and a denote the number of condensate atoms and the s-wave scattering length, respectively. The interaction energy and hence a are negative for attractive systems. The sum of these three energies F = No(AR-2 + BR2 CNolaIR-3), where A, B, Care positive constants, gives the total energy of the system which has a metastable minimum if No is smaller than a certain critical value Nc.lt is at this local minimum where a metastable BEe is believed to be formed. If No is larger than Nc' there is no metastable minimum and the condensate is bound to collapse to the R = 0 state, which is a metallic phase. It is clear from the above argument that the zero-point energy due to confinement serves as a kinetic obstacle against collapse, allowing the metastable BEe to be formed. For the case of attractive interactions, the lowest-energy collective mode is the
The critical behavior of collective modes and the collapsing dynamics of trapped Bose-Einstein condensates with attractive interactions are studied analytically and numerically. The time scales of these dynamics both below and above the critical point of the collapse are found to obey power laws with a single parameter of ${N/N}_{c}\ensuremath{-}1,$ where N is the number of condensate atoms and Nc{N}_{c}Nc​ is the critical number. The collapsing condensate eventually undergoes rapid implosion, which occurs several times intermittently, and then the implosion turns to an explosion. The release energy of the explosion is found to be proportional to the square of the interaction strength, inversely proportional to the three-body recombination rate, and independent of the number of condensate atoms and the trap frequency.
We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to 1/rb1/{r}^{b}1/rb and arbitrary angular dependence. It is shown exactly that collapse of a Bose-Einstein condensate without contact interactions is possible only for $b\ensuremath{\geqslant}2$. The case b=2b=2b=2 is critical and requires the number of particles to exceed a critical value to allow collapse. The critical collapse in that case is a strong one, trapping into the collapsing region a finite number of particles. The case b>2b>2b>2 is supercritical with an expected weak collapse which traps a rapidly decreasing number of particles during approach to collapse. For b<2b<2b<2 a singularity at r=0r=0r=0 is not strong enough to allow collapse but an attractive 1/rb1/{r}^{b}1/rb interaction admits stable self-trapping (a stable three-dimensional soliton solution) even in the absence of an external trapping potential.
The point of instability of a Bose-Einstein condensate (BEC) due to attractive interactions was studied. Stable 85Rb BECs were created and then caused to collapse by slowly changing the atom-atom interaction from repulsive to attractive using a Feshbach resonance. At a critical value, an abrupt transition was observed in which atoms were ejected from the condensate. By measuring the onset of this transition as a function of number and attractive interaction strength, we determined the stability condition to be N(absolute value of a) / a(ho) = 0.459+/-0.012+/-0.054, slightly lower than the predicted value of 0.574.
We consider the dynamics of a quantum degenerate trapped gas of 7Li{}^{7}\mathrm{Li}7Li atoms. Because the atoms have a negative sss-wave scattering length, a Bose condensate of 7Li{}^{7}\mathrm{Li}7Li becomes mechanically unstable when the number of condensate atoms approaches a maximum value. We calculate the dynamics of the collapse that occurs when the unstable point is reached. In addition, we use the quantum Boltzmann equation to investigate the nonequilibrium kinetics of the atomic distribution during and after evaporative cooling. The condensate is found to undergo many cycles of growth and collapse before a stationary state is reached.