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Journal of physics. A, Proceedings of the Physical Society. General

Motions in a Bose condensate. II. The oscillations of the rectilinear and large circular vortex lines

John R. Grant

Published Sep 1, 1971• 18 citations
Publisher

Abstract

For Pt.I, see abstr. A71357 of 1971. The linearized perturbation equations for the rectilinear and large circular vortex lines in a Bose condensate are studied. An asymptotic matching procedure is given whereby the frequency omega for the (bound state) oscillations of the rectilinear vortex, for wavelengths large compared with the healing length a, is found. A similar asymptotic procedure is employed to investigate the behaviour of the large ring vortex when its circular axis is slightly perturbed. It is found that the vortex, whose radius c is large compared with a, is stable for all small displacements (and executes simple harmonic vibrations).

DOI
10.1088/0305-4470/4/5/012

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Motions in a Bose condensate. I. The structure of the large circular vortex

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For Pt.I, see abstr. A71357 of 1971. The linearized perturbation equations for the rectilinear and large circular vortex lines in a Bose condensate are studied. An asymptotic matching procedure is given whereby the frequency omega for the (bound state) oscillations of the rectilinear vortex, for wavelengths large compared with the healing length a, is found. A similar asymptotic procedure is employed to investigate the behaviour of the large ring vortex when its circular axis is slightly perturbed. It is found that the vortex, whose radius c is large compared with a, is stable for all small displacements (and executes simple harmonic vibrations).
A procedure is given which in principle permits the exact evaluation of any term in the asymptotic expansion in a/c of the velocity U o and energy E of a circular vortex line of radius c in a Bose condensate (where a is the healing length). The procedure is used to obtain the two leading terms (previously only inaccurately determined by a variational calculation), namely U 0 = kappa /4 pi c(In(8c/a)-0.615); E= 1 / 2 rho kappa 2 c(1n(8c/a)-1.615); where kappa denotes the quantum of circulation, and rho denotes the fluid density at infinity.
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.
We derive a set of equations that describes the shape and behavior of a single perturbed vortex line in a Bose-Einstein condensate. Through the use of a matched asymptotic expansion and a unique coordinate transform, a relation for a vortex's velocity, anywhere along the line, is found in terms of the trapping, rotation, and distortion of the line at that location. This relation is then used to find a set of differential equations that give the line's specific shape and motion. This work extends a previous similar derivation by Svidzinsky and Fetter [Phys. Rev. A 62, 063617 (2000)], and enables a comparison with recent numerical results.
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.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.
We use the time-dependent Gross-Pitaevskii equation to study the motion of a vortex ring produced by phase imprinting on an elongated cloud of cold atoms. Our approach models the experiments of Yefsah et al. [Nature (London) 499, 426 (2013)] on 6{}^{6}6Li in the Bose-Einstein-condensate regime where the fermions are tightly bound into bosonic dimers. We find ring oscillation periods which are much larger than the period of the axial harmonic trap. Our results lend further strength to Bulgac et al.'s arguments (arXiv:1306.4266) that the ``heavy solitons'' seen in those experiments are actually vortex rings. We numerically calculate the periods of oscillation for the vortex rings as a function of interaction strength, trap aspect ratio, and minimum vortex ring radius. In the presence of axial anisotropies the rings undergo complicated internal dynamics where they break into sets of vortex lines, then later combine into rings. These structures oscillate with a similar frequency to simple axially symmetric rings.