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arXiv:cond-mat/0605648 · Physical Review A

Motion of a vortex line near the boundary of a semi-infinite uniform condensate

Peter Mason, Natalia G. Berloff, Alexander L. Fetter

Published Oct 18, 2006• 31 citations
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Abstract

We consider the motion of a vortex in an asymptotically homogeneous condensate bounded by a solid wall where the wave function of the condensate vanishes. For a vortex parallel to the wall, the motion is essentially equivalent to that generated by an image vortex, but the depleted surface layer induces an effective shift in the position of the image compared to the case of a vortex pair in an otherwise uniform flow. Specifically, the velocity of the vortex can be approximated by $U\ensuremath{\approx}(\ensuremath{\hbar}∕2m){({y}_{0}\ensuremath{-}\sqrt{2}\ensuremath{\xi})}^{\ensuremath{-}1}$, where ${y}_{0}$ is the distance from the center of the vortex to the wall, $\ensuremath{\xi}$ is the healing length of the condensate, and $m$ is the mass of the boson.

DOI
10.1103/physreva.74.043611

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We consider the motion of a vortex in an asymptotically homogeneous condensate bounded by a solid wall where the wave function of the condensate vanishes. For a vortex parallel to the wall, the motion is essentially equivalent to that generated by an image vortex, but the depleted surface layer induces an effective shift in the position of the image compared to the case of a vortex pair in an otherwise uniform flow. Specifically, the velocity of the vortex can be approximated by U\ensuremath≈(\ensuremathℏ∕2m)(y0\ensuremath−2\ensuremathξ)\ensuremath−1U\ensuremath{\approx}(\ensuremath{\hbar}∕2m){({y}_{0}\ensuremath{-}\sqrt{2}\ensuremath{\xi})}^{\ensuremath{-}1}U\ensuremath≈(\ensuremathℏ∕2m)(y0​\ensuremath−2​\ensuremathξ)\ensuremath−1, where y0{y}_{0}y0​ is the distance from the center of the vortex to the wall, \ensuremathξ\ensuremath{\xi}\ensuremathξ is the healing length of the condensate, and mmm is the mass of the boson.
The theory of vortex motion in a dilute superfluid of inhomogeneous density demands a boundary layer approach, in which different approximation schemes are employed close to and far from the vortex, and their results matched smoothly together. The most difficult part of this procedure is the hydrodynamic problem of the velocity field, many healing lengths away from the vortex core. This paper derives and exploits an exact solution of this problem in the two-dimensional case of a linear trapping potential, which is an idealization of the surface region of a large condensate. It thereby shows that vortices in inhomogeneous clouds are effectively ``dressed'' by a nontrivial distortion of their flow fields, that image vortices are not relevant to Thomas-Fermi surfaces, and that for condensates large compared to their surface depths, the energetic barrier to vortex penetration disappears at the Landau critical velocity for surface modes.
The motion of quantized vortices on two-dimensional inhomogeneous density backgrounds with boundaries is considered numerically and asymptotically. We show that a Hamiltonian group relation together with the method of images and an approximation for the density background is useful to understand vortex motion. We analyze the vortex motion on a variety of background configurations motivated by experiments on trapped Bose-Einstein condensates including linear and quadratic traps. We show that close to the center of the condensate cloud in the Thomas-Fermi regime the vortex is moving predominantly due to the image vortex that is effectively shifted because of the large density depletion of the condensate at the boundary. Close to the boundary the vortex moves mostly due to a large background density gradient across the vortex core. The vortex velocity is a nonlinear combination of these effects and of the global shape of the background density. We find the complete families of the traveling coherent structures in an infinitely long two-dimensional channel and derive an approximation of the velocity of a single vortex moving close to the center of the channel in the Thomas-Fermi regime.
The center-of-mass dynamics of a vortex in the surface region of a Bose-Einstein condensate is investigated both analytically using a variational calculation and numerically by solving the time-dependent Gross-Pitaevskii equation. We find, in agreement with previous works, that away from the Thomas-Fermi surface, the vortex moves parallel to the surface of the condensate with a constant velocity. We obtain an expression for this velocity in terms of the distance of the vortex core from the Thomas-Fermi surface that fits accurately with the numerical results. We find also that, coupled to its motion parallel to the surface, the vortex oscillates along the direction normal to the surface around a minimum point of an effective potential.
The mechanisms behind the motion of vortices has recently been re-examined. The findings have given a more detailed and complete view as to how vortices move in Bose-Einstein condensation. Specifically, vortex propagation is not seen as purely the result of a density gradient which would arise because of the gradient of the trapping potential, but is also made up of a contribution from the boundary eects. I will consider the motion of vortices in an asymptotically inhomogeneous condensate under various trapping potentials. By first considering a linear potential trap and then a semi-infinite quadratic trap, the eects that govern the motion can be separated. It will be shown that for large density gradient backgrounds the motion can be analysed from the energy-momentum dispersion relationship, whereas for low density gradient backgrounds the motion is dominated by an eective shift in the image term caused by the eects of the boundary. Finally, a separate case of the motion of vortices in an infinitely long two-dimensional channel will be presented where the interaction strength plays a key role in the dynamics.
The Bose condensate model is used to analyse the superfluid flow around an ion (modelled as a solid sphere) and to elucidate the mechanism of vortex ring emission from the sphere that occurs if its velocity exceeds a critical value. An asymptotic expansion is developed for the steady subcritical flow, using the ratio of the healing length to the radius of the sphere as a small parameter. This expansion allows for the compressibility of the condensate, and converges well enough for the critical ion velocity to be calculated accurately. The flow for supercritical ion velocities is computed numerically. Particular attention is paid to the question of why the vortex rings are emitted at a preferred location on the sphere's surface.
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.