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 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 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 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.
We derive a general and exact equation of motion for a quantized vortex in an inhomogeneous two-dimensional Bose-Einstein condensate. This equation expresses the velocity of a vortex as a sum of local ambient density and phase gradients in the vicinity of the vortex. We perform Gross-Pitaevskii simulations of single-vortex dynamics in both harmonic and hard-walled disk-shaped traps, and find excellent agreement in both cases with our analytical prediction. The simulations reveal that, in a harmonic trap, the main contribution to the vortex velocity is an induced ambient phase gradient, a finding that contradicts the commonly quoted result that the local density gradient is the only relevant effect in this scenario. We use our analytical vortex velocity formula to derive a point-vortex model that accounts for both density and phase contributions to the vortex velocity, suitable for use in inhomogeneous condensates. Although good agreement is obtained between Gross-Pitaevskii and point-vortex simulations for specific few-vortex configurations, the effects of nonuniform condensate density are in general highly nontrivial, and are thus difficult to efficiently and accurately model using a simplified point-vortex description.
The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.
We present a numerical calculation of the velocity field of an off-axis vortex within a harmonic trapping potential. We consider a condensate with a large number of particles which, in spite of being in the Thomas-Fermi regime, possesses a non-totally-negligible vortex core size. Via a simple, yet realistic, modeling of the vortex-state density, we derive, by treating the equation of continuity outside and inside the vortex core, respective equations for the velocity field and the vortex precession velocity. We find that the vortex precession velocity is given by two contributions: the background velocity field evaluated around the border of the core and another term which depends on the core shape. Our findings concerning the velocity field outside the vortex core are in good agreement with previous theoretical predictions in a narrow region around it, while far away from the vortex we observe a field with a large asymmetry with respect to the vortex position. We show that a better approximation may be obtained by adding the velocity field produced by an antivortex located outside the condensate.
One of the key issues related to superfluidity is the existence of quantized vortices. Following recent experiments on Bose–Einstein condensates exhibiting vortices, we investigate the behavior of the wave function which solves the Gross–Pitaevskii equation. For a rotating Bose–Einstein condensate in a harmonic trap, we give a simplified expression of the Gross–Pitaevskii energy in the Thomas–Fermi regime, which only depends on the number and shape of the vortex lines. This allows us to study in detail the structure of the line of a single quantized vortex, which has a U or S shape. S type vortices exist for all values of the angular velocity Ω but are not minimizers of the energy while U vortices are minimizers and exist only for Ω sufficiently large. Finally, we study the drag created by the movement of a detuned laser beam in a condensate and the nucleation of vortices in the low density region close to the boundary layer of the cloud.