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arXiv:cond-mat/0111523 · Physical Review A

Exact results on the dynamics of a multicomponent Bose-Einstein condensate

Pijush K. Ghosh

Published Apr 15, 2002• 16 citations
arXivHTMLPDF

Abstract

We study the time evolution of a two-dimensional multicomponent Bose-Einstein condensate in an external harmonic trap with arbitrary time-dependent frequency. We show analytically that the time evolution of the total mean-square radius of the wave packet is determined in terms of the same solvable equation as in the case of a single-component condensate. The dynamics of the total mean-square radius is also the same for a rotating as well as a nonrotating multicomponent condensate. We determine the criteria for the collapse of the condensate at a finite time. Generalizing our previous work on a single-component condensate, we show explosion-implosion duality in the multicomponent condensate.

DOI
10.1103/physreva.65.053601

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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 investigate the expansion dynamics of a Bose–Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the nonuniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.
Abstract The modulational instability of two-component Bose–Einstein condensates (BECs) under an external parabolic potential is discussed. Based on the trapped two-component Gross–Pitaevskill equations, a time-dependent dispersion relation is obtained analytically by means of the modified lens-type transformation and linear stability analysis. It is shown that a modulational unstable time scale exists for trapped two-component BECs. The modulational properties—which are determined by the wave number, external trapping parameter, intra- and inter-species atomic interactions—are modified significantly. The analytical results are confirmed by direct numerical simulation. Our results provide a criterion for judging the occurrence of instability of the trapped two-component BECs in experiment.
We experimentally investigate how the collapse dynamics of a Cr-52 Bose-Einstein condensate depends on the external harmonic trap geometry. When the collapse is initiated by reducing the s-wave scattering length below its critical value, a complex dynamics is observed, involving a d-wave symmetric explosion. We find good agreement between our experiments and simulations of the Gross-Pitaevskii equation including 3-body losses. In order to probe the phase-coherence of collapsed condensates we induce the collapse in several condensates simultaneously and let them interfere.
In this paper, we derive equations for the dynamics of ring dark solitons in an expanding cloud of a two-dimensional Bose-Einstein condensate. Assuming that the soliton's width is much smaller than its radius, we obtain the Hamilton equations for its evolution. Then they are transformed into the Newton equation, which is more convenient for applications. The general theory is illustrated by the solution of the Newton equation for the case of the axially symmetric condensate cloud, which expands after switching off a harmonic trap. The validity of our approximate analytical approach is confirmed by comparison with the results of numerical simulations of the Gross-Pitaevskii equation.