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arXiv:cond-mat/0007139 · Physical Review A

Dynamics of a vortex in a trapped Bose-Einstein condensate

Anatoly A. Svidzinsky, Alexander L. Fetter

Published Nov 15, 2000• 129 citations
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Abstract

We consider a large condensate in a rotating anisotropic harmonic trap. Using the method of matched asymptotic expansions, we derive the velocity of an element of a vortex line as a function of the local gradient of the trap potential, the line curvature, and the angular velocity of the trap rotation. This velocity yields small-amplitude normal modes of the vortex for two-dimensional (2D) and 3D condensates. For an axisymmetric trap, the motion of the vortex line is a superposition of plane-polarized standing-wave modes. In a 2D condensate, the planar normal modes are degenerate, and their superposition can result in helical traveling waves, which differs from a 3D condensate. Including the effects of trap rotation allows us to find the angular velocity that makes the vortex locally stable. For a cigar-shaped condensate, the vortex curvature makes a significant contribution to the frequency of the lowest unstable normal mode; furthermore, additional modes with negative frequencies appear. As a result, it is considerably more difficult to stabilize a central vortex in a cigar-shaped condensate than in a disk-shaped one. Normal modes with imaginary frequencies can occur for a nonaxisymmetric condensate (in both 2D and 3D). In connection with recent JILA experiments, we consider the motion of a straight vortex line in a slightly nonspherical condensate. The vortex line changes its orientation in space at the rate proportional to the degree of trap anisotropy and can exhibit periodic recurrences.

DOI
10.1103/physreva.62.063617

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We consider a large condensate in a rotating anisotropic harmonic trap. Using the method of matched asymptotic expansions, we derive the velocity of an element of a vortex line as a function of the local gradient of the trap potential, the line curvature, and the angular velocity of the trap rotation. This velocity yields small-amplitude normal modes of the vortex for two-dimensional (2D) and 3D condensates. For an axisymmetric trap, the motion of the vortex line is a superposition of plane-polarized standing-wave modes. In a 2D condensate, the planar normal modes are degenerate, and their superposition can result in helical traveling waves, which differs from a 3D condensate. Including the effects of trap rotation allows us to find the angular velocity that makes the vortex locally stable. For a cigar-shaped condensate, the vortex curvature makes a significant contribution to the frequency of the lowest unstable normal mode; furthermore, additional modes with negative frequencies appear. As a result, it is considerably more difficult to stabilize a central vortex in a cigar-shaped condensate than in a disk-shaped one. Normal modes with imaginary frequencies can occur for a nonaxisymmetric condensate (in both 2D and 3D). In connection with recent JILA experiments, we consider the motion of a straight vortex line in a slightly nonspherical condensate. The vortex line changes its orientation in space at the rate proportional to the degree of trap anisotropy and can exhibit periodic recurrences.
Based on the method of matched asymptotic expansions and on a time-dependent variational analysis, we study the dynamics of a vortex in a three-dimensional disk-shaped nonaxisymmetric condensate in the Thomas-Fermi limit. Both methods show that a vortex in a trapped Bose-Einstein condensate has formally unstable normal mode(s) with positive normalization and negative frequency, corresponding to a precession of the vortex line around the center of the trap. In a rotating trap, the solution becomes stable above an angular velocity Omega(m), characterizing the onset of metastability with respect to small transverse displacements of the vortex from the central axis.
A hydrodynamic description is used to study the normal modes of a vortex in a zero-temperature Bose-Einstein condensate. In the Thomas-Fermi limit, the circulating superfluid velocity far from the vortex core provides a small perturbation that splits the originally degenerate normal modes of a vortex-free condensate. The relative frequency shifts are small in all cases considered (they vanish for the lowest dipole mode with ∣m∣=1|m|=1∣m∣=1), suggesting that the vortex is stable. The Bogoliubov equations serve to verify the existence of helical waves, similar to those of a vortex line in an unbounded weakly interacting Bose gas. In the large-condensate (small-core) limit, the condensate wave function reduces to that of a straight vortex in an unbounded condensate; the corresponding Bogoliubov equations have no bound-state solutions that are uniform along the symmetry axis and that decay exponentially far from the vortex core.
For a Bose-Einstein condensate placed in a rotating trap, we study the simplified energy of a vortex line derived by Aftalion and Riviere [Phys. Rev. A 64, 043611 (2001)] in order to determine the shape of the vortex line according to the rotational velocity and the elongation of the condensate. The energy reflects the competition between the length of the vortex, which needs to be minimized taking into account the anisotropy of the trap, and the rotation term, which pushes the vortex along the z axis. We prove that if the condensate has the shape of a pancake, the vortex stays straight along the z axis, while in the case of a cigar, the vortex is bent. We study the local stability of the straight vortex and find an estimate for the critical angular speed at which bent vortices are nucleated. When vortices are nucleated, we prove that they must have some finite length.
We consider a rotating axisymmetric trap containing a small Bose-Einstein condensate with a singly quantized vortex near the axis of symmetry. A time-dependent variational Lagrangian analysis yields the small-amplitude dynamics of the vortex and the condensate, directly determining the equations of motion of the coupled normal modes. As found previously from the Bogoliubov equations, there are two rigid dipole modes and one anomalous mode with a negative frequency when seen in the nonrotating laboratory frame.
The tensor-virial method is applied for a study of oscillation modes of uniformly rotating Bose-Einstein condensed gases, whose rigid-body rotation is supported by an vortex array. The second-order virial equations are derived in the hydrodynamic regime for an arbitrary external harmonic trapping potential assuming that the condensate is a superfluid at zero temperature. The axisymmetric equilibrium shape of the condensate is determined as a function of the deformation of the trap; its domain of stability is bounded by the constraint \ensuremath{\Omega}1 on the rotation rate (measured in units of the trap frequency ${\ensuremath{\omega}}_{0}).$ The oscillations of the axisymmetric condensate are stable with respect to the transverse-shear and toroidal modes of oscillations, corresponding to the l=2,∣m∣=1,2l=2,|m|=1,2l=2,∣m∣=1,2 surface deformations. The eigenfrequencies of the modes are real and represent undamped oscillations. The condensate is also stable against quasiradial pulsation modes (l=2,m=0),(l=2,m=0),(l=2,m=0), and its oscillations are undamped, if the superflow is assumed incompressible. In the compressible case we find that for a polytropic equation of state, the quasiradial oscillations are unstable when $\ensuremath{\gamma}(3\ensuremath{-}{\ensuremath{\Omega}}^{2})1\ensuremath{-}3{\ensuremath{\Omega}}^{2},$ and are stable otherwise. Thus, a dilute Bose gas, whose equation of state is polytropic with \ensuremath{\gamma}=2 to leading order in the diluteness parameter, is stable irrespective of the rotation rate. In nonaxisymmetric traps, the equilibrium constrains the (dimensionless) deformation in the plane orthogonal to the rotation to the domain ${A}_{2}{\ensuremath{\Omega}}^{2}$ with \ensuremath{\Omega}1. The second-harmonic-oscillation modes in nonaxisymmetric traps separate into two classes that have even or odd parity with respect to the direction of the rotation axis. Numerical solutions show that these modes are stable in the parameter domain where equilibrium figures exist.
In a uniform fluid, a quantized vortex line with circulation h∕Mh∕Mh∕M can support long-wavelength helical traveling waves $\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}$ with the well-known Kelvin dispersion relation ${\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity V\mathbf{V}V of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wave number k(z)k(z)k(z) upward near the trap center, and k(z)k(z)k(z) then increases still more toward the ends.