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arXiv:cond-mat/9611207 · Physical Review A

Quantum dephasing of normal modes of a Bose-Einstein condensate in a magnetic trap

Anatoly B. Kuklov, N. Chencinski, Alfred M. Levine, W. M. Schreiber, Joseph L. Birman

Published May 1, 1997• 8 citations
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Abstract

Quantum evolution of a collective mode of a Bose-Einstein condensate containing a finite number N of particles can demonstrate the phenomena of collapses and revivals. The characteristic collapse time depends on the scattering length, the initial amplitude of the mode, and N. The corresponding time values have been derived analytically under certain approximations and numerically for the parabolic atomic trap. The revival of the mode amplitude at a time of several seconds, as direct evidence of the effect, can occur if the normal component is significantly suppressed.

DOI
10.1103/physreva.55.r3307

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arXiv:cond-mat/0208569•Aug 28, 2002•7 citations
Quantum evolution of a collective mode of a Bose-Einstein condensate containing a finite number N of particles can demonstrate the phenomena of collapses and revivals. The characteristic collapse time depends on the scattering length, the initial amplitude of the mode, and N. The corresponding time values have been derived analytically under certain approximations and numerically for the parabolic atomic trap. The revival of the mode amplitude at a time of several seconds, as direct evidence of the effect, can occur if the normal component is significantly suppressed.
We present a quantum theory of low-lying excitations in a trapped Bose condensate with finite particle numbers. We find that even at zero temperature condensate number fluctuations and/or fluctuations of the excitation frequency due to quantum uncertainties of the mode occupation lead to a collapse of the collective modes due to dephasing. Coherent revivals of the collective excitations are predicted on a much longer time scale. Depletion of collective modes due to second-harmonic generation is discussed.
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 discuss some recent work on the dephasing of Bose-Einstein condensates of interacting atoms due to fluctuations in the chemical potential μ. At vanishing temperature such fluctuations are due to the uncertainty of the number of particles in the condensate caused by the depletion effect. The dephasing is in this case a non-diffusive (and in principle reversible) effect called ‘collapse’ (and ‘revival’). Above a certain cross-over temperature the dephasing proceeds mainly via a diffusive process which follows in time a short and ineffective phase collapse. The phase-diffusion constant is dominated by the temperature dependent condensate-number fluctuations caused by the scattering of low-energy excitations off the condensate. We also discuss the (in general subdominant) contribution of the occupation-number fluctuations in quasi-particle states in the thermal cloud. For simplicity we discuss here results for a box-like trapping potential.
Thermal fluctuations of the normal component induce dephasing---reversible damping of the low-energy collective modes of a confined Bose-Einstein condensate. The dephasing rate is calculated for the isotropic oscillator trap, where the Landau damping is expected to be suppressed. This rate is characterized by a steep temperature dependence. It is weakly amplitude dependent, and is sensitive to the total number of atoms in the trap. The value of the rate belongs to the range of the damping rates observed by Jin et al. [Phys. Rev. Lett. 77, 420 (1996)]. We suggest that a reversible nature of the damping caused by the thermal dephasing in the isotropic trap can be tested by the echo effect. A reversible nature of the Landau damping is also discussed, and a possibility of observing the echo effect in an anisotropic trap is considered as well. The parameters of the echo for the isotropic trap are calculated in the weak echo limit. Results of the numerical simulations of the echo are also presented.
In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.
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-