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Communications in Theoretical Physics

Spatial Chaos of Bose–Einstein Condensates in a Cigar-Shaped Trap

Li Fei, Zhongzhou Ren, Luo Hai-Lu, Shu Wei-Xing, Qin Wu

Published Jul 1, 2007• 10 citations
Publisher

Abstract

The spatial chaos of Bose–Einstein condensates in a cigar-shaped trap is studied. For a system with a steady current, we construct the general solution of the 1st-order equation. From the boundedness condition of the general solution, we obtain the Melnikov function predicting the onset of chaos. The unpredictability of the system's distribution of atom density is also theoretically analyzed. For a 23 Na system meeting the perturbation condition, numerical simulations show the existence of chaos, which is in accordance with our analytical results. Numerical simulations of a 87 Rb system dissatisfying the perturbation condition also demonstrate that there exists chaos in the system. The case without a current is also investigated.

DOI
10.1088/0253-6102/48/1/023

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The spatial chaos of Bose–Einstein condensates in a cigar-shaped trap is studied. For a system with a steady current, we construct the general solution of the 1st-order equation. From the boundedness condition of the general solution, we obtain the Melnikov function predicting the onset of chaos. The unpredictability of the system's distribution of atom density is also theoretically analyzed. For a 23 Na system meeting the perturbation condition, numerical simulations show the existence of chaos, which is in accordance with our analytical results. Numerical simulations of a 87 Rb system dissatisfying the perturbation condition also demonstrate that there exists chaos in the system. The case without a current is also investigated.
The spatially chaotic attractor in an elongated cloud of Bose-Einstein condensed atoms perturbed by a weak optical lattice potential is studied. The analytical insolvability and numerical incomputability of the atomic number density are revealed by a perturbed solution that illustrates the unpredictability of the deterministic chaos. Although this could lead the nonphysical explosion and unboundedness to the numerical solution, the theoretical analysis offers a criterion to avoid them. Moreover, the velocity field is investigated that exhibits the superfluid property of the chaotic system.
This paper studies the regular and chaotic spatial distribution of Bose-Einstein condensed atoms with a space-dependent nonlinear interaction.There exists a steady atomic current in the system due to the space-dependent phase of condensate.For the system with a positive chemical potential and repulsive interatomic interaction,we construct the general solution of the 1st-order equation,whose boundedness conditions contain the famous Mel'nikov chaotic criterion.When the system doesn't satisfy the perturbation conditions,numerical simulations reveal that increasing the atomic current can destroy the spatial symmetry of the distributional structure of condensed atoms,whether the condensed atoms in a chaotic or regular distribution.
Abstract We investigate the chaotic and regular spatial structures of Bose–Einstein condensates (BECs) with a spatially modulated atom-atom interaction and without an external trapping potential. A BEC with a spatially modulated atom-atom interaction is equivalent to being constrained by a nonlinear optical lattice. Theoretical analyses show the existence of a steady atomic current in the BEC with a spatially varying phase. Under perturbative conditions, the Melnikov chaos criteria of BECs with a spatially varying phase and a constant one are theoretically obtained, respectively. When the perturbative conditions cannot be satisfied, for a repulsive BEC with a spatially varying phase, numerical simulations demonstrate that changing the initial condition can eliminate the chaotic spatial structure and then the system transitions into a biperiodic spatial structure. Increasing the chemical potential can result in a transition from the biperiodic spatial structure to a single-periodic spatial structure. For an attractive BEC with a spatially varying phase, numerical simulations show that decreasing the chemical potential can lead to a high atomic density, but when the wave number of the laser inducing the optical Feshbach resonance exceeds a critical value, the atomic density falls back to a finite range. Regardless of whether the BEC has a spatially varying phase or a constant one, modulating the laser wave number can effectively suppress the chaotic spatial structure in the BEC and then force it into a regular spatial structure.
The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg--de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.
For a Bose-Einstein condensate (BEC) confined in a double lattice consisting of two weak laser standing waves we find the Melnikov chaotic solution and chaotic region of parameter space by using the direct perturbation method. In the chaotic region, spatial evolutions of the chaotic solution and the corresponding distribution of particle number density are bounded but unpredictable between their superior and inferior limits. It is illustrated that when the relation k 1 ≈ k 2 between the two laser wave vectors is kept, the adjustment from k 2 < k 1 to k 2 ⩾ k 1 can transform the chaotic region into regular one or the other way round. This suggests a feasible scheme for generating and controlling chaos, which could lead to an experimental observation in the near future.
In this paper, we study the spatially chaotic distribution of atoms in a Bose-Einstein condensate system, trapped in an asymmetric periodic potential. For a constant phase of condensate, without atom currents in the system, the space distributed structure of condensated atoms can be described by an undamped Duffing equation with double drivers. Through theoretical analyses, the Mel'nikov chaotic criterion for the system with a repulsive interatomic interaction is presented. Numerical simulations show that an increasing chemical potential can exert considerable suppression on the chaotic distribution of condensated atoms and even completely eliminate chaos. For a system with an attractive interatomic interaction, under some specific parametric conditions, adjusting the ratio between optical lattice potential amplitudes will force the condensated atoms from a periodic state into a spatially chaotic distribution; with the increase of chemical potential, the spatially chaotic distribution is completely suppressed.