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arXiv:cond-mat/9903353 · Physical Review A

Stability of vortices in inhomogeneous Bose condensates subject to rotation: A three-dimensional analysis

Juan José García‐Ripoll, Víctor M. Pérez‐García

Published Dec 1, 1999• 66 citations
arXivHTMLPDF

Abstract

We study numerically the stability of axially symmetric vortex lines in trapped dilute gases subject to rotation. For this purpose, we solve numerically both the Gross-Pitaevskii and the Bogoliubov equations for a three-dimensional condensate in spherically and cylindrically symmetric traps, from small to very large nonlinearities. In the stationary case we find that the vortex states with $m=1$ and $m=2$ are energetically unstable. In the rotating trap it is found that this energetic instability may only be suppressed for the $m=1$ vortex line, and that the multicharged vortices are never a local minimum of the energy functional. This result implies that the absolute minimum of the energy is not an eigenstate of the ${L}_{z}$ operator, when the angular speed is above a certain value.

DOI
10.1103/physreva.60.4864

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We consider a rotating Bose-Einstein condensate in a harmonic trap and investigate numerically the behavior of the wave function which solves the Gross-Pitaevskii equation. Following recent experiments [P. Rosenbuch, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 89, 200403 (2002)], we study in detail the line of a single quantized vortex, which has a U or S shape. We find that a single vortex can lie only in the $x\ensuremath{-}z$ or $y\ensuremath{-}z$ plane. S-type vortices exist for all values of the angular velocity $\ensuremath{\Omega}$ while U vortices exist for $\ensuremath{\Omega}$ sufficiently large. We compute the energy of the various configurations with several vortices and study the three-dimensional structure of vortices.
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A second-order expansion of the Gross-Pitaevskii equation in the interaction parameter determines the thermodynamic critical angular velocity ${\ensuremath{\Omega}}_{c}$ for the creation of a vortex in a small axisymmetric condensate. Similarly, a second-order expansion of the Bogoliubov equations determines the (negative) frequency ${\ensuremath{\omega}}_{a}$ of the anomalous mode. Although ${\ensuremath{\Omega}}_{c}=\ensuremath{-}{\ensuremath{\omega}}_{a}$ through first order, the second-order contributions ensure that the absolute value $|{\ensuremath{\omega}}_{a}|$ is always smaller than the critical angular velocity ${\ensuremath{\Omega}}_{c}.$ With increasing external rotation $\ensuremath{\Omega},$ the dynamical instability of the condensate with a vortex disappears at ${\ensuremath{\Omega}}^{*}=|{\ensuremath{\omega}}_{a}|,$ whereas the vortex state becomes energetically stable at the larger value ${\ensuremath{\Omega}}_{c}.$ Both second-order contributions depend explicitly on the axial anisotropy of the trap. The appearance of a local minimum of the free energy for a vortex at the center determines the metastable angular velocity ${\ensuremath{\Omega}}_{m}.$ A variational calculation yields ${\ensuremath{\Omega}}_{m}=|{\ensuremath{\omega}}_{a}|$ to first order (hence ${\ensuremath{\Omega}}_{m}$ also coincides with the critical angular velocity ${\ensuremath{\Omega}}_{c}$ to this order). Qualitatively, the scenario for the onset of stability in the weak-coupling limit is the same as that found in the strong-coupling (Thomas-Fermi) limit.
Abstract We study three-dimensional vortex lattice structures in purely dipolar Bose–Einstein condensate (BEC). By using the mean-field approximation, we obtain a stability diagram for the vortex states in purely dipolar BECs as a function of harmonic trap aspect ratio ( λ ) and dipole–dipole interaction strength ( D ) under rotation. Rotating the condensate within the unstable region leads to collapse while in the stable region furnishes stable vortex lattices of dipolar BECs. We analyse stable vortex lattice structures by solving the three-dimensional time-dependent Gross–Pitaevskii equation in imaginary time. Further, the stability of vortex states is examined by evolution in real-time. We also investigate the distribution of vortices in a fully anisotropic trap by increasing eccentricity of the external trapping potential. We observe the breaking up of the condensate in two parts with an equal number of vortices on each when the trap is sufficiently weak, and the rotation frequency is high.
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