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arXiv:cond-mat/0002393 · Physical Review Letters

Intermittent Implosion and Pattern Formation of Trapped Bose-Einstein Condensates with an Attractive Interaction

Hiroki Saito, Masahito Ueda

Published Feb 19, 2001• 112 citations
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Abstract

The collapsing dynamics of a trapped Bose-Einstein condensate with attractive interaction are revealed to exhibit two previously unknown phenomena. During the collapse, the condensate undergoes a series of rapid implosions that occur intermittently within a very small region. When the sign of the interaction is suddenly switched from repulsive to attractive, e.g., by the Feshbach resonance, density fluctuations grow to form various patterns such as a shell structure.

DOI
10.1103/physrevlett.86.1406

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The collapsing dynamics of a trapped Bose-Einstein condensate with attractive interaction are revealed to exhibit two previously unknown phenomena. During the collapse, the condensate undergoes a series of rapid implosions that occur intermittently within a very small region. When the sign of the interaction is suddenly switched from repulsive to attractive, e.g., by the Feshbach resonance, density fluctuations grow to form various patterns such as a shell structure.
The dynamics of collapsing and exploding Bose-Einstein condensates caused by a sudden switch of interactions from repulsive to attractive are studied by numerically integrating the Gross-Pitaevskii equation with atomic loss for an axially symmetric trap. We investigate the decay rate of condensates and the phenomena of bursts and jets of atoms, and compare our results with those of the experiments performed by E. A. Donley et al. [Nature (London) 412, 295 (2001)]. Our study suggests that the condensate decay and the burst production is due to local intermittent implosions in the condensate, and that atomic clouds of bursts and jets are coherent. We also predict nonlinear pattern formation caused by the density instability of attractive condensates.
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.
Summary form only given. Quantum theory predicts that Bose-Einstein condensation (BEC) of a spatially homogeneous gas with attractive interactions is precluded by a conventional phase transition into either a liquid or solid. When confined to a trap, however, such a condensate can form provided that its occupation number No does not exceed a limiting value. The stability limit is determined by a balance between self-attraction and a repulsion arising from position-momentum uncertainty under conditions of spatial confinement. Near the stability limit, self-attraction can overwhelm the repulsion, causing the condensate to collapse. Growth of the condensate, therefore, is punctuated by intermittent collapses, which are triggered either by macroscopic quantum tunneling or thermal fluctuation. The data presented here are the first direct observations of the growth and collapse of Bose Einstein condensates with attractive interactions.
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.
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