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arXiv:cond-mat/0409636

Vorticity Knot in Two-component Bose-Einstein Condensates

Y. M. Cho

Published Sep 24, 2004• 0 citations
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Abstract

We demonstrate the existence of the helical vortex solution in two-component Bose-Einstein condensates which can be identified as a twisted vorticity flux. Based on this we argue that the recently proposed knot in two-component Bose-Einstein condensates can be interpreted as a vorticity knot, a vortex ring made of the helical vortex. This picture shows that the knot is made of two quantized vorticity fluxes linked together, whose topology $π_3(S^2)$ is fixed by the linking number of two vorticity fluxes. Due to the helical structure the knot has both topological and dynamical stability. We estimate the energy of the lightest knot to be about $3\times 10^{-3} eV$.

DOI
10.48550/arxiv.cond-mat/0409636

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We demonstrate the existence of the helical vortex solution in two-component Bose-Einstein condensates which can be identified as a twisted vorticity flux. Based on this we argue that the recently proposed knot in two-component Bose-Einstein condensates can be interpreted as a vorticity knot, a vortex ring made of the helical vortex. This picture shows that the knot is made of two quantized vorticity fluxes linked together, whose topology π3(S2)π_3(S^2)π3​(S2) is fixed by the linking number of two vorticity fluxes. Due to the helical structure the knot has both topological and dynamical stability. We estimate the energy of the lightest knot to be about 3×10−3eV3\times 10^{-3} eV3×10−3eV.
We study the topological objects in two-component Bose-Einstein condensates. We compare two competing theories of two-component Bose-Einstein condensates, the popular Gross-Pitaevskii theory, and the recently proposed gauge theory of two-component Bose-Einstein condensate which has an induced vorticity interaction. We show that two theories produce very similar topological objects, in spite of the obvious differences in dynamics. Furthermore we show that the gauge theory of two-component Bose-Einstein condensates, with the U(1)U(1)U(1) gauge symmetry, is remarkably similar to the Skyrme theory. Just like the Skyrme theory this theory admits the non-Abelian vortex, the helical vortex, and the vorticity knot. We construct the lightest knot solution in two-component Bose-Einstein condensates numerically, and discuss how the knot can be constructed in the spin-12\frac{1}{2}21​ condensate of 87Rb^{87}\mathrm{Rb}87Rb atoms.
We propose two types of topologically stable knot solitons in condensed matters, one in two-component Bose-Einstein condensates and one in two-gap superconductors. We identify the knot in Bose-Einstein condensates as a twisted vorticity flux ring and the knot in two-gap superconductors as a twisted magnetic flux ring. In both cases we show that there is a remarkable interplay between topology and dynamics which transforms the topologcal stability to the dynamical stability, and vise versa. We discuss how these knots can be constructed in the spin-1/2 condensate of 87Rb^{87}{\rm Rb}87Rb atoms and in two-gap superconductor of MgB2MgB_2MgB2​.
The recent experimental creation of knotted solitons in spin-1 Bose-Einstein condensates opens an interesting avenue for exploring the stability and dynamics of knot solitons. Knotted and helical vortices can be accommodated in spin-orbit-coupled antiferromagnetic spin-1 Bose-Einstein condensates. We demonstrate the stability of these topological structures via numerical simulations of the corresponding Gross-Pitaevskii equation. By changing the strength of the spin-orbit coupling, we find that the ground state undergoes a change from a knotted spin texture to a helical spin texture. Our results reveal that these types of topological object are experimentally realizable in spin-1 Bose-Einstein condensates.
In this paper, we consider the dynamics of two interacting point vortex rings in a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose – Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented.