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arXiv:cond-mat/0305242 · Journal of the Physical Society of Japan

A Consistent Picture of a Collapsing Bose–Einstein Condensate

Masahito Ueda, Hiroki Saito

Published Jan 1, 2003• 24 citations
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Abstract

We present a consistent picture of the recent JILA experiments on the collapse of a Bose-Einstein condensate (BEC) [E. A. Donley, et al., Nature 412, 295 (2001)] based on a generalized Gross-Pitaevskii equation that incorporates the effect of a three-body loss. The questions we address in this paper are: (1) Why does a BEC survive, albeit partially, after the collapse? (2) Why does the number of the remnant BEC atoms observed at JILA exceed the critical value? (3) What is the origin of the atom burst? (4) What is the origin of the atom jet and is it coherent? We review our predictions on pattern formation in the course of collapse and on the split instability of vortices that can test our collapsing theory.

DOI
10.1143/jpsjs.72sc.127

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We present a consistent picture of the recent JILA experiments on the collapse of a Bose-Einstein condensate (BEC) [E. A. Donley, et al., Nature 412, 295 (2001)] based on a generalized Gross-Pitaevskii equation that incorporates the effect of a three-body loss. The questions we address in this paper are: (1) Why does a BEC survive, albeit partially, after the collapse? (2) Why does the number of the remnant BEC atoms observed at JILA exceed the critical value? (3) What is the origin of the atom burst? (4) What is the origin of the atom jet and is it coherent? We review our predictions on pattern formation in the course of collapse and on the split instability of vortices that can test our collapsing theory.
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.
We numerically study the behaviour of collapsing and exploding condensates using the parameters of the experiments by Donley et al (2001 Nature 412 295). Our studies are based on a full three-dimensional numerical solution of the Gross–Pitaevskii equation (GPE) including three-body loss. We determine the three-body loss rate from the number of remnant condensate atoms and collapse times, and obtain only one possible value which fits with the experimental results. We then study the formation of jet atoms by interrupting the collapse, and find very good agreement with the experiment. Furthermore, we investigate the sensitivity of the jets to the initial conditions. According to our analysis, the dynamics of the burst atoms is not described by the GPE with three-body loss incorporated.
We perform a systematic numerical study, based on the time-dependent Gross-Pitaevskii equation, of jet formation in collapsing and exploding Bose-Einstein condensates as in the experiment by Donley et al [2001 Nature 412 295]. In the actual experiment, via a Feshbach resonance, the scattering length of atomic interaction was suddenly changed from positive to negative on a pre-formed condensate. Consequently, the condensate collapsed and ejected atoms via explosion. On a disruption of collapse by suddenly changing the scattering length to zero a radial jet of atoms was formed in the experiment. We present a satisfactory account of jet formation under the experimental conditions as well as make predictions beyond experimental conditions which can be verified in future experiments.
Phenomena associated with the controlled collapse of a Bose-Einstein condensate described in the experiment of Donley et al [1] are explained here as a consequence of the squeezing and amplification of quantum fluctuations above the condensate by the condensate dynamics. In analyzing the changing amplitude and particle contents of these excitations, our simple physical picture provides excellent quantitative fits with experimental data on the scaling behavior of the collapse time and the amount of particles emitted in the jets. In the experiments described by Donley et al. [1], a Bose-Einstein condensate (BEC) [2] in a cold (3nK) gas of Rubidium atoms is rendered unstable by a sudden inversion of the sign of the interaction between atoms. After a waiting time tcollapse, the condensate implodes (called Bose-Nova), and a fraction of the condensate atoms are seen to oscillate within the magnetic trap which contains the gas. These atoms are said to belong to a ‘burst’. After a time τevolve the interaction is suddenly turned off. For a certain range of values of τevolve, new emissions of atoms from the condensate are observed. They are called ‘jets’. Jets are distinct from bursts: they are colder, weaker, and have a characteristic disk-like shape. In the specific conditions of the experiment described in [1], the interaction is attractive in the relevant regime, corresponding to a negative scattering length, whose magnitude is typically tens of the critical value when collapse is first observed. This is different from a later JILA experiment [3] which explores the regime of positive scattering lengths thousands of times the critical one. There is a variety of theoretical explanations to these experiments, ranging from mean field dynamics of the condensate [4] to bosonic amplification [5]. The second experiment may already have a satis-
Based on the time-dependent Gross-Pitaevskii equation we study the evolution of a collapsing and exploding Bose-Einstein condensate in different trap symmetries to see the effect of confinement on collapse and subsequent explosion, which can be verified in future experiments. We make a prediction for the evolution of the shape of the condensate and the number of atoms in it for different trap symmetries (cigar to pancake) as well as in the presence of an optical lattice potential. We also make a prediction for the jet formation in different cases when the collapse is suddenly terminated by changing the scattering length to zero via a Feshbach resonance. In addition to the usual global collapse to the center of the condensate, in the presence of an optical-lattice potential one could also have in certain cases independent collapse of parts of the condensate to local centers, which could be verified in experiments.
Collapsing Bose-Einstein condensates are rich and complex quantum systems for which quantitative explanation by simple models has proved elusive. We present experimental data on the collapse of high-density 85{}^{85}85Rb condensates with attractive interactions and find quantitative agreement with the predictions of the Gross-Pitaevskii equation. The collapse data and measurements of the decay of atoms from our condensates allow us to put new limits on the value of the 85{}^{85}85Rb three-body loss coefficient K3{K}_{3}K3​ at small positive and negative scattering lengths.