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Physics of Fluids

Topological cascade of quantum Borromean rings

Hao Guan, Simone Zuccher, Xin Liu

Published Feb 1, 2025• 3 citations
Publisher

Abstract

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.

DOI
10.1063/5.0252708

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•Jan 1, 2019•0 citations
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.
By numerically solving the three-dimensional Gross-Pitaevskii equation we analyze the cascade process associated with the evolution and decay of a pair of linked vortex rings. The system decays through a series of reconnections to produce finally three unlinked, unfolded, almost planar vortex loops. Total helicity, initially zero, remains unchanged throughout the process. The gradual transfer from writhe (due to initial linking) to twist helicity, followed by a continuous relaxation of twist across scales during the evolution is shown to be a generic mechanism that consistently takes place on each individual component.
We demonstrate that a single reconnection of two quantum vortices can lead to the creation of a cascade of vortex rings. Our analysis, motivated by the analytical results in localized induction approximation, involved high resolution Biot-Savart and Gross-Pitaevskii simulations. The latter showed that the rings cascade starts on the atomic scale, with rings diameters orders of magnitude smaller than the characteristic line spacing in the tangle, and thus capable of penetrating the tangle and annihilating on the boundaries. That way such process can be an efficient decay mechanism for sparse or moderately dense vortex tangle at very low temperatures.
Borromean states refer to a class of bound states with special topological structures, generally thought to exist only in microscopic systems. In this paper, we investigate vortex bound states in three-component Bose-Einstein condensates using coupled Gross-Pitaevskii equations. Specifically, through the competition between two-body and three-body interactions, we identify the emergence of vortex Borromean states, where only three vortices form a stable bound state while any two-vortex subsystems remain unbound. Our theoretical analysis and numerical simulations provide a comprehensive phase diagram of the vortex Borromean states. Additionally, we uncover an unconventional topological phase---a vortex anti-Borromean state---in which two vortices bind together but separate upon the addition of a third vortex. This study highlights the complex role of three-body interactions in macroscopic systems and offers different insights into the topology and phase transitions of multicomponent vortex systems.
A quantum vortex is a topological singularity with quantized circulation, unlike a classical vortex with its continuous circulation strength.Quantum turbulence, envisaged as strong tangle of quantum vortices, of a Bose-Einstein condensate is examined by developing a unitary qubit lattice algorithm for the solution of the Gross-Pitaevskii equation.Earlier, it was shown that a certain class of initial conditions had a very short Poincare recurrence time for this Hamiltonian system.Here it is shown quantitatively that increasing the internal energy of the initial state leads to a systematic degradation of this class of solutions.Coupled Bose-Einstein condensate systems are investigated for a Hopf link class of initial conditions in which a vortex ring core is threaded by a linear vortex core that then closes toroidal around the vortex ring.These states are known as skyrmions and play a role in particle physics, astrophysics and condensed matter physics.
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.
Topology provides deep conceptual links between the various branches of physics. Bose–Einstein condensates with spin degrees of freedom are among the most accessible quantum systems available for studying topological structures. A wide range of topological defects and textures available in condensates, such as vortices, monopoles, knots, and skyrmions, are analogous to those predicted in electromagnetism, high-energy physics, and cosmology. In this dissertation, we numerically and experimentally investigate novel creation methods for topological structures and study their dynamical properties. Specifically, we experimentally observe the evolution of an isolated monopole into a Dirac monopole in the presence of a quadrupole magnetic field. The Dirac monopole appears in the synthetic magnetic field of the condensate and is accompanied by spontaneously emerging nodal lines. We observe the decay of a quantum knot into a polar-core spin vortex in the presence of a uniform magnetic field. Furthermore, we observe that a decaying coreless-vortex state gives rise to a pair of singular SO(3) vortices. Many of the studied creation methods for topological structures in the condensate rely on adiabatic control of the external magnetic field. The counterdiabatic protocol offers a way to speed up the otherwise slow magnetic field driving required for adiabatic dynamics with a correcting magnetic field. Using this method, we numerically implement a scheme for fast vortex pumping which leads to the vortex with highest angular momentum reported to date in Bose–Einstein condensates with experimentally feasible methods. We further use the counterdiabatic protocol in a novel way to create quantum knots in the condensate. Another focal point of this dissertation is the study on different types of skyrmions in spinor condensates. Our simulations of two-dimensional skyrmions are in a quantitative agreement with an experiment carried out elsewhere, explaining the experimentally observed instabilities. We numerically analyze the exotic spin-2 skyrmions available in the cyclic and biaxial nematic phases. Importantly, we present the first experimental observations of Shankar skyrmions in spin-1 condensates and analyze their synthetic electromagnetic properties. This dissertation addresses an extensive amount of topological structures available in the condensate, but many different structures await future studies. In addition, the precise computational characterization of the elementary excitations of monopoles, knots, and skyrmions is of great interest. The results of this dissertation form a sturdy basis for future experimental studies on the dynamics of topological structures in spinor condensates.; Topologia tarjoaa syvallisia kasitteellisia yhteyksia fysiikan eri haarojen valilla. Bosen–Einsteinin kondensaatit spin-vapausasteilla ovat yksia monipuolisimmista systeemeista topologisten rakenteiden tutkimukseen. Laaja kirjo kondensaatissa esiintyvia topologisia rakenteita, kuten kvanttipyorteet, monopolit, solmut ja skyrmionit, ovat…