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arXiv:2007.02239 · Physical Review A

Topological transition from superfluid vortex rings to isolated knots and links

Wen-Kai Bai, Tao Yang, Wu‐Ming Liu

Published Dec 16, 2020• 19 citations
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Abstract

Knots and links are fundamental topological objects that play a key role in both classical and quantum fluids. In this paper, we propose a scheme to generate torus vortex knots and links through the reconnections of vortex rings perturbed by Kelvin waves in trapped Bose-Einstein condensates, which makes it possible to generate topologically complex structures from topologically trivial objects controllably. The transfer of helicity between knots, links, and coils occur with different pathways which can be controlled through designing specific initial states. The generation of a knot or link can be achieved by setting the parity of the Kelvin wave number. The stability of knots and links can be greatly improved with tunable parameters, including the ideal relative angle and the minimal distance between the initial vortex rings.

DOI
10.1103/physreva.102.063318

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Knots and links are fundamental topological objects that play a key role in both classical and quantum fluids. In this paper, we propose a scheme to generate torus vortex knots and links through the reconnections of vortex rings perturbed by Kelvin waves in trapped Bose-Einstein condensates, which makes it possible to generate topologically complex structures from topologically trivial objects controllably. The transfer of helicity between knots, links, and coils occur with different pathways which can be controlled through designing specific initial states. The generation of a knot or link can be achieved by setting the parity of the Kelvin wave number. The stability of knots and links can be greatly improved with tunable parameters, including the ideal relative angle and the minimal distance between the initial vortex rings.
Lord Kelvin proposed that atoms form hydrodynamic vortex knots. However, they typically untie through reconnections, i. e., local cut-and-slice events, unlike stable vortex unknots such as smoke rings. The same holds in superfluids--quantum fluids with zero viscosity--where vortices have quantized circulation, making them topologically stable. For over 150 years, hydrodynamically stable vortex knots have been sought both experimentally and theoretically. Here, we present the first demonstration of hydrodynamically stable vortex knots and links in experimentally realizable Bose-Einstein condensates of ultracold atomic gases and confirm it through dynamic simulations. Our method creates stable knotted vortex structures in systems where reconnections are prohibited, with potential relevance to neutron star interiors. Additionally, we anticipate our mathematical framework could have applications in quantum computation, quantum turbulence, and DNA dynamics, particularly where reconnections are restricted.
Helicity is a topological invariant that measures the linkage and knottedness of lines, tubes, and ribbons. As such, it has found myriads of applications in astrophysics, fluid dynamics, atmospheric sciences, and biology. In quantum flows, where topology-changing reconnection events are a staple, helicity appears as a key quantity to study. However, the usual definition of helicity is not well posed in quantum vortices, and its computation based on counting links and crossings of centerline vorticity can be downright impossible to apply in complex and turbulent scenarios. We present a definition of helicity which overcomes these problems and which gives the expected result in the large-scale limit. With it, we show that certain reconnection events can excite Kelvin waves and other complex motions of the centerline vorticity, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum fluids. This process also results in the depletion of helicity in a fully turbulent quantum flow, in a way reminiscent of the decay of helicity in classical fluids.
We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how vortex knots break up into vortex rings.
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.
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 consider the reconnection of two untwisted, linked quantised vortex rings in a Bose-Einstein condensate. We show that the reconnection is capable of transferring helicity, from the links present in the initial configuration, to a twisting of the resulting vortex ring, and hence a rotation of the phase plane along the vortex ring. As velocities in a quantum fluid are the gradient of the phase, a twisting of the phase along the vortices leads to an axial flow along the vortices. Hence one would expect that the dynamics of quantum turbulence are strongly influence by the helicity of the initial conditions or the system forcing. Our results also provide an important link between quantised vortex reconnections and reconnections in classical and magnetised fluids.