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Motion of Vortices in a Trapped Bose-Einstein Condensate

Peter Mason

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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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 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.
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 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 have performed numerical simulations of various vortex configurations in a trapped BoseEinstein condensate by solving the two-dimensional Gross-Pitaevskii equation in the presence of a simple model of interaction between the condensate and the finite temperature thermal cloud that surrounds it. In that interaction the non-condensed thermal cloud acts as a source of dissipation with a damping effect of excitations. In the case ora single vortex and a vortex - anti vortex pair, we have found that the path of the vortices depends on the initial position, the initial separation distance if the case of two vortices and dissipation. This motion is periodic and it was found that sound waves are created by vortex motion; the intensity was stronger when the initial vortex separation distance was smaller. We have calculated the sound energy as the difference between the kinetic energy and the vortex energy. With no dissipation the vortices followed the same path with a slight oscillation due to the sound waves. We found that the smaller the initial vortex separation distance do is, the larger the sound production. The period, frequency, translation speed, sound energy and vortex energy were measured for different initial separation distances do and for different dissipation parameters 'Y. In the case of motion of one vortex, the connection between the dissipation 'Y and the friction coefficients, a and a' was studied as well. To create a simple turbulent state, we put eight pairs of vortex - anti vortex at random positions in the condensate with initial separation distance do =1.8 between them. 'Ve have studied the decay rate of the total energy, kinetic energy, quantum energy, trap energy and the z - component of the angular momentum together with the increase rate of the internal energy. Finally, we finished our investigation by putting randomly vortex - anti vortex pairs and studied the decrease of the number of vortices with time t. We found that the decrease is exponential.
There are many efforts to understand better the dipolar long-range interaction, the shell potential confinement, and the vortices in Bose-Einstein condensates (BECs). Furthermore, combining these three areas suggests that it can give rise to new phenomena. In this thesis, we studied the dynamics of vortices in a shell-trapped dipolar BEC. The long-range dipolar interaction induces an inhomogeneous density distribution on the shell, affecting the vortex velocity field. We analyzed the dynamics of one and two pairs of vortices symmetrically located in the shell to separate the vortex-vortex interaction contribution from that of the density gradient due to the inhomogeneous background. In addition to the numerical studies, a Lagrangian formalism allowed us to derive semi-analytical expressions which are in agreement with the results of the Gross-Pitaevskii equation (GPE) simulations, giving us better insights into the influence of the core in the vortex dynamics outside the validity of the point-vortex model. In the second part of this thesis, we describe an actual experimental implementation of the shell BEC. Using the Stochastic Projected GPE simulations, we determine the configuration of a finite temperature vortex lattice produced in this potential over the action of gravity. More specifically, we studied the transition from a perfectly hexagonal (triangular) lattice to the hexatic and liquid phases by detecting the vortex positions to calculate the angular and pair correlation functions of this system for different values of temperatures and trap rotation frequencies. Our results for the vortex lattice melting are consistent with the hexatic to liquid transition in the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario and can be extended to predict the melting temperature of other finite-size systems.
We analyze the effect of a hard wall trapping potential on the dynamics of a vortex molecule in a two-component Bose-Einstein condensate with linear coherent coupling. A vortex molecule consists of a vortex of the same charge in each component condensate connected by a domain wall of the relative phase. In a previous paper [S. Choudhury and J. Brand, Phys. Rev. A 106, 043319 (2022)] we described the interaction of a vortex molecule with the boundary using the method of images by separately treating each component vortex as a point vortex, in addition to a Magnus force effect from the surface tension of the domain wall. Here we extend the model by considering a continuous distribution of image vorticity reflecting the effect of the domain wall on the vortex molecule phase structure. In the case of a precessing centered vortex molecule in an isotropic trap, distributing the image vorticity weakens its contribution to the precession frequency. We test the model predictions against numerical simulations of the coupled Gross-Pitaevskii equations in a two-dimensional circular disk and find support for the improved model.