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Physical Review

Vortices in an Imperfect Bose Gas. IV. Translational Velocity

Alexander L. Fetter

Published Nov 4, 1966• 89 citations
Publisher

Abstract

The translational velocity of a vortex with circulation $\frac{h}{m}$ in an imperfect Bose gas is calculated by considering the time dependence of the condensate wave function corresponding to a given initial configuration of vortices. Each vortex in a system of rectilinear vortices is shown to move with the local fluid velocity at its core; a vortex ring of radius $R$ is shown to move with velocity $(\frac{h}{4\ensuremath{\pi}\mathrm{mR}})\mathrm{ln}(\frac{8R}{\ensuremath{\alpha}})$, where $\ensuremath{\alpha}$ is a length the order of the core size. Both results agree with the predictions of classical hydrodynamics.

DOI
10.1103/physrev.151.100

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Journal of Physics A Mathematical and General•Jul 20, 1999•56 citations
The translational velocity of a vortex with circulation hm\frac{h}{m}mh​ in an imperfect Bose gas is calculated by considering the time dependence of the condensate wave function corresponding to a given initial configuration of vortices. Each vortex in a system of rectilinear vortices is shown to move with the local fluid velocity at its core; a vortex ring of radius RRR is shown to move with velocity (h4\ensuremathπmR)ln(8R\ensuremathα)(\frac{h}{4\ensuremath{\pi}\mathrm{mR}})\mathrm{ln}(\frac{8R}{\ensuremath{\alpha}})(4\ensuremathπmRh​)ln(\ensuremathα8R​), where \ensuremathα\ensuremath{\alpha}\ensuremathα is a length the order of the core size. Both results agree with the predictions of classical hydrodynamics.
A theoretical study of rectilinear vortices in an imperfect Bose gas shows a close correspondence with classical hydrodynamics. The energy and momentum of a vortex pair in an unbounded fluid are calculated. The similarity between a vortex pair and a vortex ring leads to an estimate of the critical velocity vc{v}_{c}vc​ of liquid He II in a tube of radius RRR that includes the effect of the walls ${v}_{c}=\frac{C\ensuremath{\hbar}}{2mR}$, where CCC is a constant of order unity. A variational treatment of a system of many identical vortices in a container shows that the energy is lowest for a uniform distribution, and that the number of vortices per unit area $\ensuremath{\nu}$ agrees with Feyman's result $\ensuremath{\nu}=\frac{2m\ensuremath{\omega}}{h}$. In the classical limit ($h\ensuremath{\rightarrow}0$), the angular momentum and energy approach the values for solid-body rotation.
We derive, from the Gross-Pitaevskii equation, an exact expression for the velocity of any vortex in a Bose-Einstein condensate, in equilibrium or not, in terms of the condensate wave function at the center of the vortex. In general, the vortex velocity is a sum of the local superfluid velocity, plus a correction related to the density gradient near the vortex. A consequence is that in rapidly rotating, harmonically trapped Bose-Einstein condensates, unlike in the usual situation in slowly rotating condensates and in hydrodynamics, vortices do not move with the local fluid velocity. We indicate how Kelvin's conservation of circulation theorem is compatible with the velocity of the vortex center being different from the local fluid velocity. Finally, we derive an exact wave function for a single vortex near the rotation axis in a weakly interacting system, from which we derive the vortex precession rate.
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.
Vortex rings are considered as flow patterns of a phenomenological Hartree-type fluid with parameters fitted to the observed phonon-roton spectrum. The basis of the calculations is variational, and density patterns which are constant except in vortex cores are employed. The energy of vortex lines and rings is expressed in terms of a core radius ccc, defined as the distance from the singular line of the vortex to the point of half the asymptotic density. The core size is determined by minimizing the energy and is found to be independent of the ring radius when the latter is larger than ten times the core radius. Comparing the results with the data of Rayfield and Reif, one finds c=1.7c=1.7c=1.7 \AA{}, while the phenomenological model yields half this value. The critical velocity associated with ring formation in a channel is considered following the lines of Fetter's earlier calculation. In the present calculation it is possible to incorporate the Fineman-Chase analysis of channel wall effects. Agreement with experiment is considerably improved.
The scattering of single-particle excitatins by a vortex in the condensate of an imperfect Bose gas is studied using the Bogoliubov approximation. The SSS matrix is calculated as in conventional scattering theory, and the phase shifts are determined with the Schwinger variational principle. The differential cross section in the long-wavelength limit is found to be $\frac{d\ensuremath{\sigma}}{d\ensuremath{\chi}}=\frac{1}{2}\ensuremath{\pi}{(\frac{\ensuremath{\hbar}}{\mathrm{mc}})}^{2}k{cot}^{2}(\frac{1}{2}\ensuremath{\chi})$, where mmm is the mass of one atom and ccc is the speed of sound. This cross section agrees precisely with that found previously for scattering of sound by a classical vortex with circulation hm\frac{h}{m}mh​.
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.