Stable and unstable vortex knots in a trapped Bose-Einstein condensate

Abstract

The dynamics of a quantum vortex torus knot ${\cal T}_{P,Q}$ and similar knots in an atomic Bose-Einstein condensate at zero temperature in the Thomas-Fermi regime has been considered in the hydrodynamic approximation. The condensate has a spatially nonuniform equilibrium density profile $ρ(z,r)$ due to an external axisymmetric potential. It is assumed that $z_*=0$, $r_*=1$ is a maximum point for function $rρ(z,r)$, with $δ(rρ)\approx-(α-ε) z^2/2 -(α+ε) (δr)^2/2$ at small $z$ and $δr$. Configuration of knot in the cylindrical coordinates is specified by a complex $2πP$-periodic function $A(φ,t)=Z(φ,t)+i [R(φ,t)-1]$. In the case $|A|\ll 1$ the system is described by relatively simple approximate equations for re-scaled functions $W_n(φ)\propto A(2πn+φ)$, where $n=0,\dots,P-1$, and $iW_{n,t}=-(W_{n,φφ}+αW_n -εW_n^*)/2-\sum_{j\neq n}1/(W_n^*-W_j^*)$. At $ε=0$, numerical examples of stable solutions as $W_n=θ_n(φ-γt)\exp(-iωt)$ with non-trivial topology have been found for $P=3$. Besides that, dynamics of various non-stationary knots with $P=3$ was simulated, and in some cases a tendency towards a finite-time singularity has been detected. For $P=2$ at small $ε\neq 0$, rotating around $z$ axis configurations of the form $(W_0-W_1)\approx B_0\exp(iζ)+εC(B_0,α)\exp(-iζ) + εD(B_0,α)\exp(3iζ)$ have been investigated, where $B_0>0$ is an arbitrary constant, $ζ=k_0φ-Ω_0 t+ζ_0$, $k_0=Q/2$, $Ω_0=(k_0^2-α)/2-2/B_0^2$. In the parameter space $(α, B_0)$, wide stability regions for such solutions have been found. In unstable bands, a recurrence of the vortex knot to a weakly excited state has been noted to be possible.