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Physical Review A

Stationary vortex clusters in nonrotating Bose-Einstein condensates

Mikko Möttönen, Suvi Mirjami Virtanen, Tomoya Isoshima, M.M. Salomaa

Published Mar 24, 2005• 80 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.71.033626

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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 study stationary clusters of vortices and antivortices in dilute pancake-shaped Bose-Einstein condensates confined in nonrotating harmonic traps. Previous theoretical results on the stability properties of these topologically nontrivial excited states are seemingly contradicting. We clarify this situation by a systematic stability analysis. The energetic and dynamic stability of the clusters is determined from the corresponding elementary excitation spectra obtained by solving the Bogoliubov equations. Furthermore, we study the temporal evolution of the dynamically unstable clusters. The stability of the clusters and the characteristics of their destabilizing modes only depend on the effective strength of the interactions between particles and the trap anisotropy. For certain values of these parameters, there exist several dynamical instabilities, but we show that there are also regions in which some of the clusters are dynamically stable. Moreover, we observe that the dynamical instability of the clusters does not always imply their structural instability, and that for some dynamically unstable states annihilation of the vortices is followed by their regeneration, and revival of the cluster.
We review the theory of vortices in trapped dilute Bose-Einstein condensates and compare theoretical predictions with existing experiments. Mean-field theory based on the time-dependent Gross-Pitaevskii equation describes the main features of the vortex states, and its predictions agree well with available experimental results. We discuss various properties of a single vortex, including its structure, energy, dynamics, normal modes, and stability, as well as vortex arrays. When the nonuniform condensate contains a vortex, the excitation spectrum includes unstable (`anomalous') mode(s) with negative frequency. Trap rotation shifts the normal-mode frequencies and can stabilize the vortex. We consider the effect of thermal quasiparticles on vortex normal modes as well as possible mechanisms for vortex dissipation. Vortex states in mixtures and spinor condensates are also discussed.
We study numerically the stability of axially symmetric vortex lines in trapped dilute gases subject to rotation. For this purpose, we solve numerically both the Gross-Pitaevskii and the Bogoliubov equations for a three-dimensional condensate in spherically and cylindrically symmetric traps, from small to very large nonlinearities. In the stationary case we find that the vortex states with m=1m=1m=1 and m=2m=2m=2 are energetically unstable. In the rotating trap it is found that this energetic instability may only be suppressed for the m=1m=1m=1 vortex line, and that the multicharged vortices are never a local minimum of the energy functional. This result implies that the absolute minimum of the energy is not an eigenstate of the Lz{L}_{z}Lz​ operator, when the angular speed is above a certain value.
Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.
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Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.