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arXiv:1806.10783 · Journal of Experimental and Theoretical Physics

Quasi-Stable Configurations of Torus Vortex Knots and Links

V. P. Ruban

Published Sep 1, 2018• 5 citations
arXivHTMLPDF

Abstract

The dynamics of torus vortex configurations V n, p, q in a superfluid liquid at zero temperature ( n is the number of quantum vortices, p is the number of turns of each filament around the symmetry axis of the torus, and q is the number of turns of the filament around its central circle; radii R 0 and r 0 of the torus at the initial instant are much larger than vortex core width ξ) has been simulated numerically based on the regularized Biot–Savart law. The lifetime of vortex systems till the instant of their substantial deformation has been calculated with a small step in parameter B 0 = r 0 /R 0 for various values of parameter Λ = ln( R 0 /ξ). It turns out that for certain values of n, p , and q , there exist quasi-stability regions in the plane of parameters ( B 0 , Λ), in which the vortices remain almost invariable during dozens and even hundreds of characteristic lengths.

DOI
10.1134/s106377611809008x

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Cambridge University Press eBooks•Jan 7, 1988•0 citations
The dynamics of torus vortex configurations V n, p, q in a superfluid liquid at zero temperature ( n is the number of quantum vortices, p is the number of turns of each filament around the symmetry axis of the torus, and q is the number of turns of the filament around its central circle; radii R 0 and r 0 of the torus at the initial instant are much larger than vortex core width ξ) has been simulated numerically based on the regularized Biot–Savart law. The lifetime of vortex systems till the instant of their substantial deformation has been calculated with a small step in parameter B 0 = r 0 /R 0 for various values of parameter Λ = ln( R 0 /ξ). It turns out that for certain values of n, p , and q , there exist quasi-stability regions in the plane of parameters ( B 0 , Λ), in which the vortices remain almost invariable during dozens and even hundreds of characteristic lengths.
The dynamics of the simplest torus quantum vortex knots in a superfluid at zero temperature has been simulated with a regularized Biot–Savart law (the torus radii R 0 and r 0 for the initial vortex configuration are much larger than the width of the vortex core ). The evolution times of knots until their significant deformation have been calculated with a small step in the parameter B 0 = r 0 / R 0 for different values of the parameter Λ = log( R 0 /ξ). It has been found that regions of quasi-stability appear at Λ ≳ 3 in the range B 0 ≲ 0.2, which correspond to long knot lifetimes and very large traveling distances up to several hundred R 0 . This result is new and quite surprising because previously it was believed that the maximum lifetime of torus knots until reconnection does not exceed several typical periods. The opening of quasi-stable “windows” at increasing Λ is due to narrowing of main parametric resonances of the dynamic system in the parameter B 0 .
The long-time existence of topologically nontrivial configurations of quantum vortices in the form of torus knots and links in trapped Bose-Einstein condensates is demonstrated numerically within the three-dimensional Gross-Pitaevskii equation with an external anisotropic parabolic potential. We identify parametric domains of trap anisotropy, characterized by the axial over planar frequency ratio $\ensuremath{\lambda}\ensuremath{\approx}1.5--1.6$, where the lifetime of such quasistationary rotating vortex structures is many hundreds of typical rotation times. This suggests the potential experimental observability of the structures. We quantify the relevant lifetimes as a function of the model parameters (e.g., $\ensuremath{\lambda}$) and initial condition parameters of the knot profile.
The dynamics of the simplest vortex knots, “unknots,” and torus links in an atomic Bose condensate at zero temperature in an anisotropic harmonic trap has been simulated numerically within the three-dimensional Gross–Pitaevskii equation. It has been found that such quasistationary rotating vortex structures exist for a very long time in wide ranges of the parameters of the system. This new result is qualitatively consistent with a previous prediction based on a simplified one-dimensional model approximately describing the motion of knotted vortex filaments.
The dynamics of a quantum vortex toric knot TP,Q and other analogous knots in an atomic Bose condensate at zero temperature in the Thomas–Fermi regime is considered in the hydrodynamic approximation. The condensate has a spatially inhomogeneous equilibrium density profile ρ(z, r) due to the action of an external axisymmetric potential. It is assumed that z*= 0, r*= 1 is the point of maximum of function rρ(z, r), so that δ(rρ) ≈ –(α–)z2/2–(α + )(δr)2/2 for small z and δr. The geometrical configuration of a knot in the cylindrical coordinates is determined by a complex 2πP-periodic function A(ϕ, t) = Z(ϕ, t) + i[R(ϕ, t))–1]. When |A| ≪ 1, the system can be described by relatively simple approximate equations for P rescaled functions Wn(φ)∝A(2πn+φ):iWn,t=−(Wn,φφ+αWn−∈Wn∗)/2−∑j≠n1/(Wn∗−Wj∗){W_n}(\varphi ) \propto A(2\pi n + \varphi ):i{W_{n,t}} = - ({W_{n,\varphi \varphi }} + \alpha {W_n} - \in W_n^*)/2 - \sum\nolimits_{j \ne n} {1/(W_n^* - W_j^*)} Wn​(φ)∝A(2πn+φ):iWn,t​=−(Wn,φφ​+αWn​−∈Wn∗​)/2−∑j=n​1/(Wn∗​−Wj∗​) . For = 0, examples of stable solutions of type W n = θ n (ϕ–γt)exp(–iωt) with a nontrivial topology are found numerically for P = 3. In addition, the dynamics of various unsteady knots with P = 3 is modeled, and the tendency to the formation of a singularity over a finite time interval is observed in some cases. For P = 2 and small ≠ 0, configurations of type W0–W1 ≈ B0exp(iζ) + C(B0, α)exp(–iζ) + D(B0, α)exp(3iζ), where B0 > 0 is an arbitrary constant, ζ = k0ϕ–Ω0t + ζ0, k0 = Q/2, and Ω0 = (–α)/2–2/B02, which rotate about the z axis, are investigated. Wide stability regions for such solutions are detected in the space of parameters (α, B0). In unstable zones, a vortex knot may return to a weakly excited state.
The superfluid flow velocity is proportional to the gradient of the phase of the superfluid order parameter, leading to the quantization of circulation around a vortex core. In this work, we study the dynamics of a superfluid film on the surface of a torus. Such a compact surface allows only configurations of vortices with zero net vorticity. We derive analytic expressions for the flow field, the total energy, and the time-dependent dynamics of the vortex cores. The local curvature of the torus and the presence of noncontractable loops on this multiply connected surface alter both the superfluid flow and the vortex dynamics. Finally, we consider more general surfaces of revolution, called toroids.
In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems. Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955). In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state. Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.