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arXiv:physics/0202074 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Globally linked vortex clusters in trapped wave fields

L.-C. Crasovan, Gabriel Molina‐Terriza, Juan P. Torres, Lluís Torner, Víctor M. Pérez‐García, DUMITRU BOGDAN MIHALACHE

Published Sep 23, 2002• 77 citations
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Abstract

We put forward the existence of a rich variety of fully stationary vortex structures, termed H clusters, made of an increasing number of vortices nested in paraxial wave fields confined by trapping potentials. However, we show that the constituent vortices are globally linked, rather than products of independent vortices. Also, they always feature a monopolar global wave front and exist in nonlinear systems, such as the Bose-Einstein condensates. Clusters with multipolar global wave fronts are nonstationary or, at best, flipping.

DOI
10.1103/physreve.66.036612

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We put forward the existence of a rich variety of fully stationary vortex structures, termed H clusters, made of an increasing number of vortices nested in paraxial wave fields confined by trapping potentials. However, we show that the constituent vortices are globally linked, rather than products of independent vortices. Also, they always feature a monopolar global wave front and exist in nonlinear systems, such as the Bose-Einstein condensates. Clusters with multipolar global wave fronts are nonstationary or, at best, flipping.
We show the existence of a rich variety of fully stationary vortex structures made of an increasing number of vortices nested in paraxial wave fields confined by symmetric and asymmetric harmonic trapping potentials in both static and rotating reference frames. In the static reference frame the clusters are built by means of Hermite-Gauss functions, being termed H-clusters, whereas in the rotating frame they are generated by using the Laguerre-Gauss functions, being thus termed L-clusters. These complex vortex-structures consist of globally linked vortices, rather than independent vortices and, in symmetric traps, they feature monopolar global wave front. Nonstationary or flipping circular vortex-clusters can be built in symmetric traps and they feature multipolar phase front. In asymmetric traps, the existing stationary vortex clusters can feature multipolar wave fronts, depending on the ratio of the trap frequencies. In the nonlinear case, corresponding to nonrotating Bose-Einstein condensates, the stationary vortex clusters also exist and some of these highly excited collective states display dynamical stability.
We show the existence of a rich variety of fully stationary vortex structures made of an increasing number of vortices nested in paraxial wave fields confined by symmetric and asymmetric harmonic trapping potentials in both static and rotating reference frames. In the static reference frame the clusters are built by means of Hermite-Gauss functions, being termed H-clusters, whereas in the rotating frame they are generated by using the Laguerre-Gauss functions, being thus termed L-clusters. These complex vortex-structures consist of globally linked vortices, rather than independent vortices and, in symmetric traps, they feature monopolar global wave front. Nonstationary or flipping circular vortex-clusters can be built in symmetric traps and they feature multipolar phase front. In asymmetric traps, the existing stationary vortex clusters can feature multipolar wave fronts, depending on the ratio of the trap frequencies. In the nonlinear case, corresponding to nonrotating Bose-Einstein condensates, the stationary vortex clusters also exist and some of these highly excited collective states display dynamical stability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest ppp-wave state of a particle in a harmonic trap with frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.
We numerically study the formation of vortex clusters in trapped Bose-Einstein condensates where vortices are initially imprinted in a line. We show that such a system exhibits a rich phenomenology depending on the distance at which the vortices are imprinted and their number. In particular we observe that it is possible to obtain systems of twin vortex clusters, twin vortex clusters with orbiting satellite vortices, and triplets of clusters. By using a clustering algorithm we are able to quantitatively describe the formation and dynamics of the clusters. We finally utilise an analytical model to determine the range of parameters for which the clustering occurs. Our work sets the stage for possible experimental implementations where the formation of vortex clusters and more exotic bound states of vortices could be observed.
We describe, analytically and numerically, topologically nontrivial stationary states with azimuthal modulation of density in a repulsive Bose-Einstein condensate (BEC) confined in a harmonic trap. We show that, depending on the number of density peaks and total topological charge, these states can have a sophisticated phase texture associated with a vortex cluster. For a given topological charge, the form of density distribution and the structure of phase singularities are uniquely linked, thus simultaneous phase and density modulation enforces the formation of a vortex cluster both in rotating and in nonrotating BEC clouds.
We numerically obtain stationary rings of vortices in pancake-shaped Bose-Einstein condensates confined in a three-dimensional nonrotating trap. For this purpose we use a static axisymmetric trapping potential that can sustain locally stable off-axis vortices. We analyze the maximum number of vortices the system can host as a function of the number of particles. We also show that this system provides a very suitable scenario for predicting vortex dynamics in inhomogeneous media. Specifically, on the one hand, we derive an exact and simple analytical expression for the velocity field in a particular vortex position due to the presence of the other vortices of the array. On the other hand, using the fact that in stationary conditions this field should balance all other contributions to the velocity, we investigate the applicability of approximated expressions for the precession velocity obtained in previous works for condensates with a single vortex.