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Physical Review A

Collective oscillations in a two-dimensional Bose-Einstein condensate with a quantized vortex state

Arup Banerjee, B. Tanatar

Published Nov 18, 2005• 5 citations
Publisher

Abstract

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 $N$. 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.

DOI
10.1103/physreva.72.053620

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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.
Using a sum rule approach we calculate the frequency shifts of the quadrupole oscillations of a harmonically trapped Bose gas due to the presence of a quantized vortex. Analytic results are obtained for positive scattering lengths and large NNN where the shift relative to excitations of opposite angular momentum is found to be proportional to the quantum circulation of the vortex and to decrease as ${N}^{\ensuremath{-}2/5}$. Results are also given for smaller values of NNN covering the transition between the ideal gas and the Thomas-Fermi limit. The splitting of the collective frequencies in toroidal configurations is also discussed.
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 study the properties of vortex states of a trapped Bose gas in the large-gas-parameter regime. To test the validity of the Gross-Pitaevskii theory in this regime for vortex states, we compare the results of the Gross-Pitaevskii and the modified Gross-Pitaevskii calculations for the total energy, chemical potential, density profile, and frequency shift of the quadrupole modes of the collective oscillations of a harmonically trapped condensate carrying a single quantized vortex. We find that in the large-gas-parameter regime, two calculations give substantially different results for all the properties mentioned above.
We study the response of a trapped Bose-Einstein condensate to a sudden turn on of a rotating drive by numerically solving the two-dimensional Gross-Pitaevskii equation. A weakly anisotropic rotating potential excites a quadrupole shape oscillation and its time evolution is analyzed by a quasiparticle projection method. In a quadrupolar resonant regime, which depends on the trap anisotropy, simple periodic oscillations in surface-mode populations disappear and the system exhibits stochastic dynamics. In the presence of the phenomenological dissipation, an initially irrotational condensate is found to undergo damped elliptic deformation followed by unstable surface ripple excitations, some of which develop into quantized vortices that eventually form a lattice. Recent experimental results on the vortex nucleation should be explained not only by the dynamical instability but also by the Landau instability; the latter is necessary for the vortices to penetrate into the condensate.
The dynamical instabilities and ensuing dynamics of singly and doubly quantized vortex states of Bose-Einstein condensates with attractive interactions are investigated using full three-dimensional numerical simulations of the Gross-Pitaevskii equation. With increasing the strength of attractive interactions, a series of dynamical instabilities such as quadrupole, dipole, octupole, and monopole instabilities emerge. The most prominent instability depends on the strength of interactions, the geometry of the trapping potential, and deviations from the axisymmetry due to external perturbations. Singly quantized vortices split into two clusters and subsequently undergo split-merge cycles in a pancake-shaped trap, whereas the split fragments immediately collapse in a spherical trap. Doubly quantized vortices are always unstable to disintegration of the vortex core. If we suddenly change the strength of interaction to within a certain range, the vortex splits into three clusters, and one of the clusters collapses after a few split-merge cycles. The vortex split can be observed using a current experimental setup of Leanhardt et al. [Phys. Rev. Lett. 89, 190403 (2002)].
We study the normal modes of a two-dimensional rotating Bose-Einstein condensate confined in a quadratic plus quartic trap. Hydrodynamic theory and sum rules are used to derive analytical predictions for the collective frequencies in the limit of high angular velocities Omega where the vortex lattice produced by the rotation exhibits an annular structure. We predict a class of excitations with frequency sqrt[6]Omega in the rotating frame, irrespective of the mode multipolarity m, as well as a class of low energy modes with frequency proportional to |m|/Omega. The predictions are in good agreement with results of numerical simulations based on the 2D Gross-Pitaevskii equation. The same analysis is also carried out at even higher angular velocities, where the system enters the giant vortex regime.