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

Abstract

The dynamics of a quantum vortex torus knot ${\mathcal{T}}_{P,Q}$ 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 $\rho (z,r)$ due to an external axisymmetric potential. It is assumed that $z_{\ast }=0$, $r_{\ast }=1$ is a maximum point for function $r\rho (z,r)$, with $\delta (r\rho )\approx -(\alpha -\epsilon )z^2/2-(\alpha +\epsilon )(\delta r{)}^2/2$ at small $z$ and $\delta r$. Configuration of knot in the cylindrical coordinates is specified by a complex $2\pi P$-periodic function $A(\phi ,t)=Z(\phi ,t)+i[R(\phi ,t)-1]$. In the case $|A|\ll 1$ the system is described by relatively simple approximate equations for re-scaled functions $W_n(\phi )\propto A(2\pi n+\phi )$, where $n=0,\dots ,P-1$, and $i{W}_{n,t}=-(W_{n,\phi \phi }+\alpha W_n-\epsilon W_n^{\ast })/2-{\sum }_{j\ne n}1/(W_n^{\ast }-W_j^{\ast })$. At $\epsilon =0$, numerical examples of stable solutions as $W_n={\theta }_n(\phi -\gamma t)\exp (-i\omega t)$ 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 $\epsilon \ne 0$, rotating around $z$ axis configurations of the form $(W_0-W_1)\approx B_0\exp (i\zeta )+\epsilon C(B_0,\alpha )\exp (-i\zeta )+\epsilon D(B_0,\alpha )\exp (3i\zeta )$ have been investigated, where $B_0>0$ is an arbitrary constant, $\zeta =k_0\phi -{\Omega }_{0}t+{\zeta }_0$, $k_0=Q/2$, ${\Omega }_0=(k_0^2-\alpha )/2-2/B_0^2$. In the parameter space $(\alpha ,B_0)$, 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.

7 pages, 10 figures, English translation of Russian text submitted to JETP

References in corpus (4)

• Bending-wave Instability of a Vortex Ring in a Trapped Bose-Einstein Condensate
• Parametric instability of oscillations of a vortex ring in a $z$-periodic Bose-Einstein condensate and the recurrence to starting state
• Hamilton's equations of motion of a vortex filament in the rotating Bose-Einstein condensate and their "soliton" solutions
• Some exact solutions of the local induction equation for motion of a vortex in a Bose-Einstein condensate with Gaussian density profile

Cited by in corpus (5)

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• Quasi-stable quantum vortex knots and links in anisotropic harmonically trapped Bose-Einstein condensates
• Three-dimensional numerical simulation of long-lived quantum vortex knots and links in a trapped Bose-Einstein condensate
• Discrete vortices on spatially nonuniform two-dimensional electric networks
• Vortex knots on three-dimensional lattices of nonlinear oscillators coupled by space-varying links