Topological cascade of quantum Borromean rings

The evolution and the topological cascade of quantum vortices forming Borromean rings are studied for the first time. The initial configuration of the system is given by three elliptical planar loops linked together, and the evolution is governed by the numerical implementation of the Gross–Pitaevskii equation. It is found that the topological cascade is not unique, but it depends crucially on the initial geometric configuration. Quantum vortices undergo a series of spontaneous reconnections, resulting in various degenerative pathways characterized by different topology and structural complexity triggered by the different inclination of one of the initial ellipses. Typical decaying routes are given by the successive creation of a Whitehead link, a connected sum of two Hopf links, a trefoil knot, a Hopf link, and the final formation of unknotted, unlinked loops. By structural complexity analysis, we show that the generic trend of the vortex decay goes through a series of topological simplifications, resulting in the formation of small-scale planar loops (rings). During the later stage of evolution, the inverse cascade and topological cycles involving the interaction of unknotted loops become more common, remaining sub-dominant to the overall topological simplification process. These results pave the way to investigate the fundamental relations between structural complexity and energy contents.