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Physical Review Letters

Solitary Waves on Vortex Lines in Ginzburg-Landau Models for the Example of Bose-Einstein Condensates

Natalia G. Berloff

Published Jan 7, 2005• 9 citations
Publisher

Abstract

Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum ($p$)-substitution energy ($\stackrel{^}{\mathcal{E}}$) plane with $p\ensuremath{\rightarrow}0.09\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/{c}^{2}$, $\stackrel{^}{\mathcal{E}}\ensuremath{\rightarrow}0.091\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/c$ as $U\ensuremath{\rightarrow}c$, where $\ensuremath{\rho}$ is the density, $c$ is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and $U$ is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particular, it is shown that during collapse the vortex core becomes significantly compressed, and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.

DOI
10.1103/physrevlett.94.010403

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Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum ($p$) -- substitution energy ($\hat{\cal E}$) plane with $p\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/c$ as $U \to c$, where $ρ$ is the density, $c$ is the speed of sound, $κ$ is the quantum of circulation and $U$ is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particularly, it is shown that during collapse the vortex core becomes significantly compressed and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.

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Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (ppp)-substitution energy (\stackrel{^}{\mathcal{E}}) plane with p\ensuremath→0.09\ensuremathρ\ensuremathκ3/c2p\ensuremath{\rightarrow}0.09\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/{c}^{2}p\ensuremath→0.09\ensuremathρ\ensuremathκ3/c2, \stackrel{^}{\mathcal{E}}\ensuremath{\rightarrow}0.091\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/c as U\ensuremath→cU\ensuremath{\rightarrow}cU\ensuremath→c, where \ensuremathρ\ensuremath{\rho}\ensuremathρ is the density, ccc is the speed of sound, \ensuremathκ\ensuremath{\kappa}\ensuremathκ is the quantum of circulation, and UUU is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particular, it is shown that during collapse the vortex core becomes significantly compressed, and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.
Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (ppp) -- substitution energy (E^\hat{\cal E}E^) plane with p→0.09ρκ3/c2,E^→0.091ρκ3/cp\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/cp→0.09ρκ3/c2,E^→0.091ρκ3/c as U→cU \to cU→c, where ρρρ is the density, ccc is the speed of sound, κκκ is the quantum of circulation and UUU is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particularly, it is shown that during collapse the vortex core becomes significantly compressed and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.
For pt.III see ibid., vol.7, p.260 (1974). Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 rho kappa 3 /c 2 and minimum energy about 0.145 rho kappa 3/c , where rho is density, c is the speed of sound and kappa is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation kappa ; as p to infinity its radius omega approximately (p/ pi kappa ) 1/2 becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p to infinity ; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be delta E/ delta p, the derivative being taken along the family.
Axisymmetric three-dimensional solitary waves in uniform two-component mixture Bose-Einstein condensates are obtained as solutions of the coupled Gross-Pitaevskii equations with equal intracomponent but varying intercomponent interaction strengths. Several families of solitary wave complexes are found: (1) vortex rings of various radii in each of the components; (2) a vortex ring in one component coupled to a rarefaction solitary wave of the other component; (3) two coupled rarefaction waves; (4) either a vortex ring or a rarefaction pulse coupled to a localized disturbance of a very low momentum. The continuous families of such waves are shown in the momentum-energy plane for various values of the interaction strengths and the relative differences between the chemical potentials of two components. Solitary wave formation, their stability, and solitary wave complexes in two dimensions are discussed.
There are several ways to create the vorticity-free solitary waves --- rarefaction pulses --- in condensates: by the process of strongly nonequilibrium condensate formation in a weakly interacting Bose gas, by creating local depletion of the condensate density by a laser beam, and by moving a small object with supercritical velocities. Perturbations created by such waves colliding with vortices are studied in the context of the Gross-Pitaevskii model. We find that the effect of the interactions consists of two competing mechanisms: the creation of vortex line as rarefaction waves acquire circulation in a vicinity of a vortex core and the loss of the vortex line to sound due to Kelvin waves that are generated on vortex lines by rarefaction pulses. When a vortex ring collides with a rarefaction wave, the ring either stabilizes to a smaller ring after emitting sound through Kelvin-wave radiation or the entire energy of the vortex ring is lost to sound if the radius of the ring is of the order of the healing length. We show that during the time evolution of a tangle of vortices, the interactions with rarefaction pulses provide an important dissipation mechanism enhancing the decay of superfluid turbulence.
Pade approximants are used to find approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials. Rational function and generalised rational function approximations of axisymmetric solitary waves of the Gross-Pitaevskii equation are obtained in two and three dimensions. These approximations are used to establish a new mechanism of vortex nucleation as a result of solitary wave interactions.
Abstract We investigate solitary vortex evolution in two-dimensional Bose-Einstein condensates based on the Gross-Pitaevskii equation model. Through the variational method, together with the novel Gaussian ansatz incorporating asymmetric perturbation effects, we arrive at the analytical solitary vortex solution with two typical forms: a symmetric quasi-stable solution under certain parametric settings and a diverging propagation case arising from an initial asymmetric perturbation. The derived pictorial evolutionary patterns of the solitary vortices are compared with those from a pure numerical analysis, and by identifying the key qualitative features, we show the applicability of the theoretical treatment presented here.