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Collapse of a Bose-Einstein Condensate as Squeezing of the Vacuum

Esteban Calzetta, B. L. Hu

Published Aug 28, 2002• 0 citations
OpenAlex

Abstract

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-

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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-
We propose an explanation of the phenomena of Bose Novae, the controlled collapse of a Bose-Einstein condensate described in the experiment of Donley et al [1], as a consequence of the squeezing and amplification of the 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. Bose Novae are observed when 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, and a fraction of the condensate atoms are seen to oscillate within the magnetic trap which contains the gas (see below and [1]). These atoms are said to belong to a “burst”. In the experiments described by Donley et al., the interaction is again suddenly turned off after a time τevolve. For a certain range of values of τevolve, new emissions of atoms from the condensate are observed, the so-called “jets”. Jets are distinct from bursts: they are colder, weaker, and have a characteristic disk-like shape. To date, the most comprehensive theoretical approach to Bose Novae is based on the Gross-Pitaevsky equation with explicitly time-dependent nonlinear terms. Loss mechanisms are incorporated by adding imaginary terms to the Hamiltonian [3]. This approach is analyzed in ref. [4]. There is some experimental evidence that atom recombination into molecules cannot fully explain the observed behavior [5]. We claim, in contrast to the emphasis placed on the dynamics of the condensate alone or the kinet-
We analyze the phenomena of condensate collapse, as described by Donley et al. [Nature 412, 295 (2001)] and N. Claussen [Ph. D thesis, University of Colorado, 2003 (unpublished)] by focusing on the behavior of excitations or fluctuations above the condensate, as driven by the dynamics of the condensate, rather than the dynamics of the condensate alone or the kinetics of the atoms. The dynamics of the condensate squeezes and amplifies the quantum excitations, mixing the positive and negative frequency components of their wave functions thereby creating particles that appear as bursts and jets. By analyzing the changing amplitude and particle content of these excitations, our simple physical picture explains well the overall features of the collapse phenomena and provides excellent quantitative fits with experimental data on several aspects, such as the scaling behavior of the collapse time and the number of particles in the jet. The prediction of the bursts at this level of approximation is less than satisfactory but may be improved by including the backreaction of the excitations on the condensate. The mechanism behind the dominant effect---parametric amplification of vacuum fluctuations and freezing of modes outside of horizon---is similar to that of cosmological particle creation and structure formation in a rapid quench (which is fundamentally different from Hawking radiation in black holes). This shows that Bose-Einstein condensate dynamics is a promising venue for doing ``laboratory cosmology.''
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.
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 perform numerical simulations based on the time-dependent mean-field Gross-Pitaevskii equation to understand some aspects of a recent experiment by Donley et al. [Nature (London) 412, 295 (2001)] on the dynamics of collapsing and exploding Bose-Einstein condensates of 85Rb{}^{85}\mathrm{Rb}85Rb atoms. These authors manipulated the atomic interaction by an external magnetic field via a Feshbach resonance, thus changing the repulsive condensate into an attractive one, and vice versa. In the actual experiment they suddenly changed the scattering length of atomic interaction from a positive to a large negative value on a preformed condensate in an axially symmetric trap. Consequently, the condensate collapsed and ejected atoms via explosion. We find that the present mean-field analysis can explain some aspects of the dynamics of the collapsing and exploding Bose-Einstein condensates.
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.