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Kluwer Academic Publishers eBooks

Globally-Linked Vortex Clusters

L.-C. Crasovan, Vadym E. Vekslerchik, DUMITRU BOGDAN MIHALACHE, Juan P. Torres, Víctor M. Pérez‐García, Lluís Torner

Published Mar 1, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1007/1-4020-2190-9_6

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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 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 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 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 investigate the recently found stationary vortex cluster states in dilute atomic Bose-Einstein condensates confined by a nonrotating trap, and also present a stationary three-vortex cluster. We find the stationary states by minimizing directly an error norm for the stationary Gross-Pitaevskii equation, and study the dynamic and energetic stability of the resulting states by solving the corresponding Bogoliubov equations for the elementary excitations. The results are verified by integrating the time-dependent Gross-Pitaevskii equation. Contrary to previously reported results, the stationary states were observed to be both energetically and dynamically unstable. The dynamical decay rate of the clusters is typically very slow, but it should be experimentally observable. The most promising circumstances to experimentally generate and observe these structures and their dynamics is in weakly dissipative condensate systems, using phase-imprinting techniques.
We construct three-dimensional structures of topological defects hosted in trapped wave fields, in the form of vortex stars, vortex cages, parallel vortex lines, perpendicular vortex rings, and parallel vortex rings, and we show that the latter exist as robust stationary, collective states of nonrotating Bose-Einstein condensates. We discuss the stability properties of excited states containing several parallel vortex rings hosted by the condensate, including their dynamical and structural stability.
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.