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Quantum Knots that Never Come Untied

Michikazu Kobayashi, Yuta Nozaki, Yuya Koda, Muneto Nitta

Abstract

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.

Quantum Knots that Never Come Untied

Abstract

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.

Paper Structure

This paper contains 6 sections, 5 theorems, 13 equations, 6 figures, 1 table.

Table of Contents

  1. Vortices in Spinor BECs
  2. Cyclic phase in spin-2 BEC
  3. $D_4$-BN phases in spin-2 BEC
  4. Classification of vortex knots and links
  5. Mathematical backgrounds
  6. Simulations

Key Result

Proposition 2.1

In cyclic spin-2 BECs, there are exactly eight MNCOPBs whose colors correspond to the elements of conjugacy classes (IV) and (VII), for up to two-component links. These braids are trefoil knots, with specific colors as shown in the left panel of Fig. fig:mnob_cyclic. The colorings $$ correspond to o The four colorings 1--4 (and similarly 5--8) are equivalent to each other up to simultaneous conjug

Figures (6)

  • Figure 1: Upper panel: Image of Lord Kelvin's vortex atom model, where atoms are represented as vortex knots in the aether fluid. Lower panel: Dynamics of stable and metastable vortex knots and links in spin-2 spinor BECs, corresponding to the vortex atom model shown in the upper panel. (a): A trefoil knot ($(2,3)$-torus knot), (b): A $(2,6)$-torus link, (c): A $(2,9)$-torus knot in the cyclic phase, and (d): A $(2,4)$-torus link in the nematic phase (see Supplementary Movies 4, 9, 11, and 19, respectively). All vortex knots and links move from bottom to top over time at constant velocities. The blue curves represent hydrodynamic vortices, while the green curves depict non-hydrodynamic (rung) vortices. (a) is the unique stable knot. (b) and (c) are constructed by repeating the braid for the trefoil in panel (a) twice and three times, respectively, and taking their closures. (d) is metastable.
  • Figure 2: Dynamics of two point vortices in the 2-dimensional $xy$-plane and a vortex braid in the $(2+1)$-dimensional $xyt$-spacetime. When two point vortices have hydrodynamic circulations with positive orientation, they rotate around each other counterclockwise (clockwise) toward the future (past) in panel (a) (panel (b)), forming a positive (negative) braid in $xyt$-spacetime. The braid's orientation reverses for vortices with negative hydrodynamic circulation.
  • Figure 3: Dynamics of positive (negative) braiding of two vortex lines in panel (a) (panel (b)). Dashed arrows show the direction of the braid evolution.
  • Figure 4: Examples of (a) a trefoil vortex knot, (c) a coherently oriented vortex link, (e) an incoherently oriented vortex link, and (g) a figure-eight vortex knot. The knots and links in panels (a), (c), (e), and (g) are constructed by connecting the upper and lower parts of the vortex braids shown in panels (b), (d), (f), and (h), respectively. The orientations of the hydrodynamic circulations for each vortex are indicated by arrows. Different arcs are represented in different colors.
  • Figure 5: The MNCOPBs (left) and incoherently oriented MNCPB (right)
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem A.1: Alexander, cf. Kassel_2008
  • Lemma A.2
  • proof
  • Proposition A.3
  • proof