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arXiv:cond-mat/0303244 · Journal of Physics B Atomic Molecular and Optical Physics

Free expansion of two-dimensional condensates with a vortex

Onur Hosten, Patrizia Vignolo, Anna Minguzzi, B Tanatar, M P Tosi

Published Jun 3, 2003• 10 citations
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Abstract

We study the free expansion of a pancake-shaped Bose-condensed gas, which is initially trapped under harmonic confinement and containing a vortex at its centre. In the case of a radial expansion holding the axial confinement fixed we consider various models for the interactions, depending on the thickness of the condensate relative to the value of the scattering length. We are thus able to evaluate different scattering regimes ranging from quasi-three-dimensional (Q3D) to strictly two-dimensional (2D). We find that as the system goes from Q3D to 2D the expansion rate of the condensate increases whereas that of the vortex core decreases. In the Q3D scattering regime we also examine a fully free expansion in 3D and find oscillatory behaviour for the vortex core radius: an initial fast expansion of the vortex core is followed by a slowing down. Such a nonuniform expansion rate of the vortex core implies that the timing of its observation should be chosen appropriately.

DOI
10.1088/0953-4075/36/12/306

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We study the free expansion of a pancake-shaped Bose-condensed gas, which is initially trapped under harmonic confinement and containing a vortex at its centre. In the case of a radial expansion holding the axial confinement fixed we consider various models for the interactions, depending on the thickness of the condensate relative to the value of the scattering length. We are thus able to evaluate different scattering regimes ranging from quasi-three-dimensional (Q3D) to strictly two-dimensional (2D). We find that as the system goes from Q3D to 2D the expansion rate of the condensate increases whereas that of the vortex core decreases. In the Q3D scattering regime we also examine a fully free expansion in 3D and find oscillatory behaviour for the vortex core radius: an initial fast expansion of the vortex core is followed by a slowing down. Such a nonuniform expansion rate of the vortex core implies that the timing of its observation should be chosen appropriately.
We observe that a density-depleted vortex core evolves into concentric density ripples in a freely-expanding quasi-two-dimensional Bose-Einstein condensate. The atomic density of the expanding condensate rapidly reduces due to the fast expansion along the tightly confined axial direction; thus, the transverse expansion of the condensate is not significantly affected by atom-atom interactions. We find that the observed density profiles of the vortex cores are in good quantitative agreement with numerical simulations assuming no interactions. We analyze the defocused images of the vortex cores, where defocussing is caused by the free fall of the condensate during the expansion time. In the defocused images, the vortex core becomes magnified and appears to be filled after a certain expansion time that depends on the vortex charge number.
We observe that a density-depleted vortex core evolves into concentric density ripples in a freely expanding quasi-two-dimensional Bose-Einstein condensate. The atomic density of the expanding condensate rapidly reduces due to the fast expansion along the tightly confining axial direction, and thus the transverse expansion of the condensate is not significantly affected by the atom-atom interactions. We find that the observed density profiles of the vortex cores are in good quantitative agreement with numerical simulations assuming no interactions. We analyze the defocused images of the vortex cores, where defocussing is caused by the free fall of the condensate during the expansion time. In the defocused images, the vortex core becomes magnified and appears to be filled after a certain expansion time which depends on the vortex charge number.
Free expansion of attractive and repulsive Bose-Einstein condensed vortex states formed in an axially symmetric trap is investigated using the numerical solution of the time-dependent Gross-Pitaevskii equation. In a repulsive condensate the vortex-core radius is much smaller than the radial root-mean-square (rms) radius, which makes the experimental observation of the vortex core difficult. The opposite is found to be true in an attractive condensate that makes it a better candidate for experimental observation. Also, in all cases the ratio of vortex-core radius to radial rms radius increases as the angular momentum of the vortex increases. This makes the vortex states with higher angular momenta more suitable for experimental confirmation.
The expansion of Bose-Einstein condensates with quantized vortices is studied by solving numerically the time-dependent Gross-Pitaevskii equation at zero temperature. For a condensate initially trapped in a spherical harmonic potential, we confirm previous results obtained by means of variational methods showing that, after releasing the trap, the vortex core expands faster than the radius of the atomic cloud. This could make the detection of vortices feasible, by observing the depletion of the density along the axis of rotation. We find that this effect is significantly enhanced in the case of anisotropic disk-shaped traps. The results obtained as a function of the anisotropy of the initial configuration are compared with the analytic solution for a noninteracting gas in three dimensions as well as with the scaling law predicted for an interacting gas in two dimensions.
We study the effect of lower dimensional geometry on the frequency splitting of the quadrupole oscillations of a harmonically trapped Bose-Einstein condensate due to the presence of a quantized vortex. To study the effect of two-dimensional geometry we consider a pancake-shaped condensate and employ various models for the coupling parameter depending on the thickness of the condensate relative to the value of the scattering length. Using these models and the sum-rule approach we obtain analytical expressions for the frequency splitting. These expressions are valid for positive scattering length and large NNN. We show that the frequency splitting of the quadrupole oscillations are significantly altered by the reduced dimensionality and also study the evolution of the splitting as the system makes transition from one scattering regime to the other.
In this work, we analyze the free expansion of Bose-Einstein condensates containing multicharged vortices. The atomic cloud is initially confined in a three-dimensional asymmetric harmonic trap. We apply both approximate variational solutions and numerical simulations of the Gross-Pitaevskii equation. The data obtained provide a way to establish the presence as well as the multiplicity of vortices based only on the properties of the expanded cloud which can be obtained via time-of-flight measurements. In addition, several features such as the evolution of the vortex core size and the asymptotic velocity during free expansion were studied considering the atomic cloud as being released from different harmonic trap configurations.