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arXiv:cond-mat/0402208 · Physical Review A

Kelvin mode of a vortex in a nonuniform Bose-Einstein condensate

Alexander L. Fetter

Published Apr 23, 2004• 50 citations
arXivHTMLPDF

Abstract

In a uniform fluid, a quantized vortex line with circulation $h∕M$ 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 $\mathbf{V}$ 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 $k(z)$ upward near the trap center, and $k(z)$ then increases still more toward the ends.

DOI
10.1103/physreva.69.043617

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In a uniform fluid, a quantized vortex line with circulation h∕Mh∕Mh∕M can support long-wavelength helical traveling waves \ensuremath∝ei(kz\ensuremath−\ensuremathωkt)\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}\ensuremath∝ei(kz\ensuremath−\ensuremathωk​t) with the well-known Kelvin dispersion relation \ensuremathωk\ensuremath≈(\ensuremathℏk2∕2M)ln(1∕\ensuremath∣k\ensuremath∣\ensuremathξ){\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})\ensuremathωk​\ensuremath≈(\ensuremathℏk2∕2M)ln(1∕\ensuremath∣k\ensuremath∣\ensuremathξ), where \ensuremathξ\ensuremath{\xi}\ensuremathξ 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.
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.
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.
Ultra-cold quantum turbulence is expected to decay through a cascade of Kelvin waves. These helical excitations couple vorticity to the quantum fluid causing long wavelength phonon fluctuations in a Bose-Einstein condensate. This interaction is hypothesized to be the route to relaxation for turbulent tangles in quantum hydrodynamics. The local induction approximation is the lowest order approximation to the Biot-Savart velocity field induced by a vortex line and, because of its integrability, is thought to prohibit energy transfer by Kelvin waves. Using the Biot-Savart description, we derive a generalization to the local induction approximation which predicts that regions of large curvature can reconfigure themselves as Kelvin wave packets. While this generalization preserves the arclength metric, a quantity conserved under the Eulerian flow of vortex lines, it also introduces a non-Hamiltonian structure on the geometric properties of the vortex line. It is this non-Hamiltonian evolution of curvature and torsion which provides a resolution to the missing Kelvin wave motion. In this work, we derive corrections to the local induction approximation in powers of curvature and state them for utilization in vortex filament methods. Using the Hasimoto transformation, we arrive at a nonlinear integro-differential equation which reduces to a modified nonlinear Schrödinger type evolution of the curvature and torsion on the vortex line. We show that this modification seeks to disperse localized curvature profiles. At the same time, the non-Hamiltonian break in integrability bolsters the deforming curvature profile and simulations show that this dynamic results in Kelvin wave propagation along the dispersive vortex medium.
The local induction equation, approximately describing dynamics of a quantized vortex filament in a trapped Bose-Einstein condensate in the Thomas-Fermi regime on a spatially nonuniform density background ρ(r)ρ({\bf r})ρ(r) and taking dimensionless form Rt=ϰb+[∇ln⁡ρ(R)×τ]{\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]Rt​=ϰb+[∇lnρ(R)×τ] (where ϰ\varkappaϰ is a local curvature of the filament, b{\mathbf b}b is the unit binormal vector, and τ{\boldsymbol τ}τ is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian, ρ∝exp⁡(−∣r∣2/2)ρ\propto\exp(-|{\bf r}|^2/2)ρ∝exp(−∣r∣2/2). The reduction corresponds to a geometrically perfect vortex ring centered at position A(t){\bf A}(t)A(t), with orientation and size both determined by a vector B(t){\bf B}(t)B(t). Parameters A{\bf A}A and B{\bf B}B exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D: A˙=B/∣B∣2−B\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}A˙=B/∣B∣2−B, and B˙=A\dot {\bf B}={\bf A}B˙=A.
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.