Decay of a Quantum Knot

We experimentally study the dynamics of quantum knots in a uniform magnetic field in spin-1 Bose-Einstein condensates. The knot is created in the polar magnetic phase, which rapidly undergoes a transition toward the ferromagnetic phase in the presence of the knot. The magnetic order becomes scrambled as the system evolves, and the knot disappears. Strikingly, over long evolution times, the knot decays into a polar-core spin vortex, which is a member of a class of singular SO(3) vortices. The polar-core spin vortex is stable with an observed lifetime comparable to that of the condensate itself. The structure is similar to that predicted to appear in the evolution of an isolated monopole defect, suggesting a possible universality in the observed topological transition.

Figure 1

(a) Schematic representation of isosurfaces and densities in the different spinor components of the quantum knot in the scaled coordinate system $(x^{\prime },y^{\prime },z^{\prime })=(x,y,2z)$. The densities are revealed by partially cutting the regions in the spinor components. The red-colored isosurfaces and the color gradient minimum (dark blue) correspond to the value $|{\zeta }_{\alpha }{|}_{\min }^2=0.29$ while the maximum gradient color (dark red) corresponds to $|{\zeta }_{\alpha }{|}_{\max }^2=1$. (b) Schematic representation of the cylindrical spinor structure of the polar-core spin vortex in the $z=0$ plane resulting from a long-time evolution of the quantum knot. The red, blue, and green arrows in the triads represent $\mathbf{m}$, $\mathbf{n}$, and $\mathbf{s}$ vectors, respectively. The large gray circle denotes the region with nonvanishing total particle density, whereas red, blue, and green circles enclose the regions in which the predominant spinor component is ${\zeta }_{+1}$, ${\zeta }_0$, and ${\zeta }_{-1}$, respectively. The ${\varphi }_0$, ${\varphi }_0^{\prime }$, and ${\varphi }_0^{\prime \prime }$ are the reference regional phases of ${\zeta }_{-1}$, ${\zeta }_{+1}$, and ${\zeta }_0$, respectively, showing the $\pi$ phase difference of the adjacent regions in the ${\zeta }_{\pm 1}$ components. For this schematic, we choose the phases ${\varphi }_0=\pi /2$, ${\varphi }_0^{\prime }=0$, and ${\varphi }_0^{\prime \prime }=\pi /2$.

Figure 2

(a),(b) Postexpansion column particle densities of the three spinor components integrated along $y$ from experiments (top row) and simulations (bottom row) showing the decay of the knot for evolution times (a) $T=0\mathrm{ms}$ and (b) 4 ms. (c)–(e) In-trap density isosurfaces showing the decay dynamics of the initially overlapping hollow vortex rings in ${\zeta }_{+1}$ (red) and ${\zeta }_{-1}$ (green) components associated with the quantum knot. The evolution times are (c) $T=0\mathrm{ms}$, (d) 1 ms, and (e) 4 ms. (f) The spin density isosurface at $T=4\mathrm{ms}$. In each of the panels in (a) and (b), the peak particle density is ${\tilde{n}}_p=8.5\times {10}^8{\mathrm{cm}}^{-2}$ and the field of view is $230\times 270{\mu \mathrm{m}}^2$. The particle density isosurfaces in (c)–(e) correspond to ${\tilde{n}}_{\min }=3.5\times {10}^{13}{\mathrm{cm}}^{-3}$ with the maximum densities (c) ${\tilde{n}}_{\max }=2.6\times {10}^{14}{\mathrm{cm}}^{-3}$, (d) $3.2\times {10}^{14}{\mathrm{cm}}^{-3}$, and (e) $4.4\times {10}^{14}{\mathrm{cm}}^{-3}$. The normalized spin density isosurface corresponds to ${\tilde{s}}_{\min }=0.5$ with the gradient-maximum ${\tilde{s}}_{\max }=1$.

Figure 3

(a)–(c) Experimental column particle densities along $z$ of different spinor components of the polar-core spin vortex emerging from the evolution of the quantum knot with the evolution time (a) $T=1.0\mathrm{s}$ and (b),(c) 1.5 s. (e),(f) Simulated particle densities integrated along $z$ and phases in the $z=0$ plane of different spinor components with $T=0.5\mathrm{s}$ evolution time. The dashed circular shapes are guides for the eye towards the regions with high particle densities. (g),(h) Expectation value of spin, $\mathbf{s}={\zeta }^{†}\mathbf{F}\zeta$, in the (g) $z=0$ and (h) $x=0$ planes, with the arrows depicting its planar projection and the color denoting the magnitude $|\mathbf{s}|$. (i) Triad representation of the order parameter in the $z=0$ plane, where $\mathbf{s}$, $\mathbf{m}$, and $\mathbf{n}$ vectors are represented with green, red, and blue arrows, respectively. The column particle density of ${\zeta }_0$ is shown for reference. (d) Experimental and (j) simulated column particle densities integrated along the $z$ axis in a $\pi /2$-rotated spinor basis with evolution times 1.0 s in experiments and 0.5 s in simulations. The particle densities in (j) are obtained by expressing the spinor in (e) in $x$-quantized basis. The peak particle density is ${\tilde{n}}_p=1.0\times {10}^9{\mathrm{cm}}^{-2}$ and the fields of view of each panel in (a)–(g),(i),(j) are $225\times 225{\mu \mathrm{m}}^2$ and in (h) $225\times 270{\mu \mathrm{m}}^2$.