Vortex knots in a Bose-Einstein condensate

We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how vortex knots break up into vortex rings.

Figure 1

The density field $\rho$ around an axisymmetric two-dimensional vortex. Radial distances $R$ are in units of the healing length $\xi$.

Figure 2

Construction of the trefoil knot ${\mathcal{T}}_{2,3}$.

Figure 3

Positions of the four vortices on $\phi =0$ used to construct the wave function of the trefoil knot ${\mathcal{T}}_{2,3}$. Clockwise and anticlockwise arrows describe the motion of vortex positions with respect to the function $f\left(\phi \right)=3\phi /2$.

Figure 4

Construction of the trefoil knot ${\mathcal{T}}_{3,2}$.

Figure 5

Positions of the six vortices on $\phi =0$ used to construct the wave function of the knot ${\mathcal{T}}_{3,2}$. Clockwise and anticlockwise arrows describe the motion of vortex positions with respect to the function $g\left(\phi \right)=2\phi /3$.

Figure 6

Isosurfaces of the density field at the threshold level ${\rho }_{\mathrm{th}}=0.2$ for ${\mathcal{T}}_{2,3}$ knots of various knot ratios $R_1/R_0$ (see Table ). Snapshots at times $t=0,400,800,1200$. Unstable knots are not shown. For complete evolution movies refer to [37].

Figure 7

Isosurfaces of the density field at the threshold level ${\rho }_{\mathrm{th}}=0.2$ for ${\mathcal{T}}_{3,2}$ knots of various knot ratios $R_1/R_0$ (see Table ). Snapshots at times to $t=0,400,800$. Unstable knots are not shown. For complete evolution movies refer to [37].

Figure 8

Position along the $z$ axis (in units of the healing length) of the center of mass of ${\mathcal{T}}_{2,3}$ knots of various knot ratios $R_1/R_0$ as a function of time. The filled points denote the position where vortex knots break up. The horizontal lines denote, respectively, the distance where boundary effects become non-negligible and the finite system size along $z$.

Figure 9

As in Fig. 8 but for ${\mathcal{T}}_{3,2}$.

Figure 10

Velocity component $v_z$ of ${\mathcal{T}}_{2,3}$ versus time of knots with various knot ratios $R_1/R_0$ before destroying (filled points). Velocities are expressed in units of vortex ring velocity (7) with quantum number $n=1$ and radius $R=R_0$.

Figure 11

As in Fig. 10 but for ${\mathcal{T}}_{3,2}$.

Figure 12

Averaged velocity components $v_z$ of ${\mathcal{T}}_{2,3}$ and ${\mathcal{T}}_{3,2}$ vortex knots with various knot ratios $R_1/R_0$. Velocities are expressed in units of vortex ring velocity (7) with quantum number $n=1$ and radius $R=R_0$. Error bars correspond to one standard deviation.

Figure 13

Three successive snapshots showing how the ${\mathcal{T}}_{2,3}$ vortex knot with knot ratio $R_1/R_0=2/5$ breaks up into two vortex rings. Here we plot two perspectives (up and to the side of the vortex propagation) of the isosurfaces of the density field corresponding to the threshold level ${\rho }_{\mathrm{th}}=0.2$. For a complete evolution movie refer to [37].

Figure 14

Two sequences of three successive snapshots showing how the ${\mathcal{T}}_{3,2}$ vortex knot with knot ratio $R_1/R_0=3/5$ breaks up into three vortex rings. Here we plot two perspectives (up and to the side of the vortex propagation) of the isosurfaces of the density field corresponding to the threshold level ${\rho }_{\mathrm{th}}=0.2$. For a complete evolution movie refer to [37].

Figure 15

Three successive snapshots showing how the ${\mathcal{T}}_{3,2}$ knot with knot ratio $R_1/R_0=2/5$ breaks into three vortex rings. Here we plot two perspectives (up and to the side of the vortex propagation) of the isosurfaces of the density field corresponding to the threshold ${\rho }_{\mathrm{th}}=0.2$. For a complete evolution movie refer to [37].