Stable and Unstable Vortex Knots in a Trapped Bose Condensate

Abstract

The dynamics of a quantum vortex toric knot TP,Q and other analogous knots in an atomic Bose condensate at zero temperature in the Thomas–Fermi regime is considered in the hydrodynamic approximation. The condensate has a spatially inhomogeneous equilibrium density profile ρ(z, r) due to the action of an external axisymmetric potential. It is assumed that z*= 0, r*= 1 is the point of maximum of function rρ(z, r), so that δ(rρ) ≈ –(α–)z2/2–(α + )(δr)2/2 for small z and δr. The geometrical configuration of a knot in the cylindrical coordinates is determined by a complex 2πP-periodic function A(ϕ, t) = Z(ϕ, t) + i[R(ϕ, t))–1]. When |A| ≪ 1, the system can be described by relatively simple approximate equations for P rescaled functions $${W_n}(\varphi ) \propto A(2\pi n + \varphi ):i{W_{n,t}} = - ({W_{n,\varphi \varphi }} + \alpha {W_n} - \in W_n^*)/2 - \sum\nolimits_{j \ne n} {1/(W_n^* - W_j^*)} $$ . For = 0, examples of stable solutions of type W n = θ n (ϕ–γt)exp(–iωt) with a nontrivial topology are found numerically for P = 3. In addition, the dynamics of various unsteady knots with P = 3 is modeled, and the tendency to the formation of a singularity over a finite time interval is observed in some cases. For P = 2 and small ≠ 0, configurations of type W0–W1 ≈ B0exp(iζ) + C(B0, α)exp(–iζ) + D(B0, α)exp(3iζ), where B0 > 0 is an arbitrary constant, ζ = k0ϕ–Ω0t + ζ0, k0 = Q/2, and Ω0 = (–α)/2–2/B02, which rotate about the z axis, are investigated. Wide stability regions for such solutions are detected in the space of parameters (α, B0). In unstable zones, a vortex knot may return to a weakly excited state.