Knots in a Spinor Bose-Einstein Condensate

We show that knots of spin textures can be created in the polar phase of a spin-1 Bose-Einstein condensate, and discuss experimental schemes for their generation and probe, together with their lifetime.

Figure 1

(a) Three-dimensional configuration of the $\mathit{n}$ field of a knot with Hopf charge $Q=1$, expressed as $\mathit{n}=e^{-i\alpha (r)\widehat{\mathit{v}}·\tilde{\mathit{F}}}(0,0,1{)}^T$, where $\alpha (r)=2\pi [1-\tanh (r/{\xi }_{\mathrm{knot}})]$ and $\widehat{\mathit{v}}=(-\sin \theta \cos \varphi ,-\sin \theta \sin \varphi ,\cos \theta )$. Note that $\mathit{n}=(n_x,n_y,n_z{)}^T$ is related to the order parameter $\mathbf{\Psi }$ through Eq. (2). In the figure, only $\mathit{n}$ on the $x$, $y$, and $z$ axes is shown as arrows. The dashed line traces the point $\mathit{n}=(0,0,-1{)}^T$, which is a circle on the $xy$ plane with the radius $[{tanh}^{-1}(1/2)]{\xi }_{\mathrm{knot}}\simeq 0.55{\xi }_{\mathrm{knot}}$. The color on the arrows represents the value of $n_z$ (see the scale). (b) The torus shape on the left side of the figure shows the isopycnic surface of $|{\psi }_{-1}{|}^2=0.47$, which is equivalent to the region where $|n_z|\le 0.24$. The color on the surface shows the phase of ${\psi }_{-1}$, that is equivalent to $\arctan (n_y/n_x)$. The white and black tubes are the preimages of $\mathit{n}\simeq (0,0,-1{)}^T(n_z<-0.95)$ and $\mathit{n}\simeq (1,0,0{)}^T(n_x>0.95)$, respectively, which are reproduced on the right side of the figure. The two tubes cross once (linking number 1), which is consistent with $Q=1$. (c) Cross sections of the density of the $m=\pm 1$ (left) and $m=0$ (right) components in the $xy$ plane. The size of each panel is $4{\xi }_{\mathrm{knot}}\times 4{\xi }_{\mathrm{knot}}$. The torus shape of the $m=\pm 1$ components appears as double rings in the cross section, which can serve as the signature of a knot.

Figure 2

Dynamics of the creation and destruction of knots in a spherical trap under a quadrupolar magnetic field. (a)–(d) The left sides of the figure show the isopycnic surface of $|{\psi }_{-1}{|}^2/\rho =0.4$, and the color on the surface shows the phase of ${\psi }_{-1}$ at (a) $t=0.5T_L$, (b) $1.1T_L$, (b) $2.2T_L$, and (d) $3.3T_L$, where $T_L=2h/({\mu }_B{b}^{\prime }{R}_{\mathrm{TF}})$. The white and black tubes on the right are the preimages of $\mathit{n}\simeq (0,0,-1{)}^T(n_z<-0.95)$ and $\mathit{n}\simeq (1,0,0{)}^T(n_x>0.95)$, respectively, and the solid circles represent the periphery of the BEC [$\rho /\rho (\mathit{x}=0)=0.01$] in the $xy$ plane. (e)–(h) Cross sections of the density normalized by $\rho (\mathit{x}=0)$ for $m=-1$ (top) and $m=0$ (bottom) components in the $xy$ plane at (e) $t=0.5T_L$, (f) $1.1T_L$, (g) $2.2T_L$, and (h) $3.3T_L$.

Figure 3

Time evolution of the maximum of the local polarization in a spherical trap with $R_{\mathrm{TF}}=4.3{\xi }_{\mathrm{knot}}$, starting from the knot shown in Fig. 1. The time is measured in units of $M{\xi }_{\mathrm{knot}}^2/\hslash$. The dashed line shows $|\mathit{f}(\mathit{x}){|}_{\max }=t/{\tau }_{\mathrm{knot}}$ (see text), which agrees well with the numerical result. The insets show $|\mathit{f}(\mathit{x})|$ in the $xz$ plane, and the number below each panel shows the time elapsed in units of $M{\xi }_{\mathrm{knot}}^2/\hslash$. The size of each panel is $1.9{\xi }_{\mathrm{knot}}\times 3.8{\xi }_{\mathrm{knot}}$.