Synthetic electromagnetic knot in a three-dimensional skyrmion

Abstract

Classical electromagnetism and quantum mechanics are both central to the modern understanding of the physical world and its ongoing technological development. Quantum simulations of electromagnetic forces have the potential to provide information about materials and systems that do not have conveniently solvable theoretical descriptions, such as those related to quantum Hall physics, or that have not been physically observed, such as magnetic monopoles. However, quantum simulations that simultaneously implement all of the principal features of classical electromagnetism have thus far proved elusive. We experimentally realize a simulation in which a charged quantum particle interacts with the knotted electromagnetic fields peculiar to a topological model of ball lightning. These phenomena are induced by precise spatiotemporal control of the spin field of an atomic Bose-Einstein condensate, simultaneously creating a Shankar skyrmion-a topological excitation that was theoretically predicted four decades ago but never before observed experimentally. Our results reveal the versatile capabilities of synthetic electromagnetism and provide the first experimental images of topological three-dimensional skyrmions in a quantum system.

Fig. 1. Creation of the Shankar skyrmion and its synthetic electromagnetic fields.

(A and B) External magnetic field (white field lines) applied to the condensate (blue ellipsoid) just before (A) and after (B) the commencement of the creation process. The field has a continuous rotational symmetry about the z axis. The spin vectors are initially aligned with the magnetic field depicted in (A) but are reoriented as they precess about the new field lines (B) at their local Larmor frequency. (C) Cutaway octant of the created Shankar skyrmion texture exhibiting its continuous and topologically nontrivial triad texture. The inset shows a triad, for which the local spin direction S is marked by a green-tipped arrow. The blue- and red-tipped arms define the rotation angle of the triad about its spin vector. The colored curves are example contours, for which the spin vector points in a common direction, and along which the rotation angle winds about the direction of the spin by 4π. (D and E) Examples of the synthetic magnetic (D) and electric (E) field lines arising from the spin texture in (C), for which the colors are to guide the eye.

Fig. 2. Comparison of experiment with theory.

Side (A) and top (B) column particle densities of the three spin states in the trap 508 μs after the nonadiabatic ramp, calculated using the experimental parameters (see Materials and Methods). (C to F) Calculated (C and D) and experimentally measured (E and F) post-expansion density profiles for an in-trap evolution time of 508 μs. The quantization axis is along +z. Throughout, the particle number is 2 × 10⁵. For (A) and (B), the field of view is 39 μm × 13 μm, and the maximum pixel intensity corresponds to column densities in excess of ${\tilde{\mathit{n}}}_{\mathrm{p}}=3.8\times {10}^{11}$ cm⁻². For (C) and (E), these quantities are 738 μm × 246 μm and ${\tilde{\mathit{n}}}_{\mathrm{p}}=8.5\times {10}^8$ cm⁻², and for (D) and (F), these quantities are 657 μm × 219 μm and ${\tilde{\mathit{n}}}_{\mathrm{p}}=1.0\times {10}^9$ cm⁻², respectively.

Fig. 3. Full quantum character of the skyrmion.

(A) Simulated (top) and experimental (bottom) particle column densities in different spin states for 508 μs after the nonadiabatic ramp and quantization axis along +y. The maximum pixel intensity corresponds to column densities in excess of ${\tilde{\mathit{n}}}_{\mathrm{p}}=1.0\times {10}^9{\text{cm}}^{-2}$, and the field of view is 657 μm × 438 μm. (B) As in (A), but viewed from the side, with field of view 738 μm × 492 μm and ${\tilde{\mathit{n}}}_{\mathrm{p}}=8.5\times {10}^8{\text{cm}}^{-2}$. (C) As in (B), but with quantization along +x.

Fig. 4. Synthetic electromagnetic quantities.

(A and B) Dimensionless synthetic electric field $\tilde{\mathbf{E}}*=\frac{{\mathit{a}}_{\mathit{r}}{\mathit{q}}_{\mathrm{e}}^{*}}{\mathrm{\hslash }{\mathrm{\omega }}_{\mathit{r}}}\mathbf{E}*$ depicted in sections through the xz and xy planes, respectively, with the analytic result in the left half of the panel and the simulation result in the right half. The values are expressed in terms of the radial harmonic oscillator length ${\mathit{a}}_{\mathit{r}}=\sqrt{\frac{\mathrm{\hslash }}{\mathit{m}{\mathrm{\omega }}_{\mathit{r}}}}$. (C and D) As in (A) and (B) but for the dimensionless synthetic magnetic field $\tilde{\mathbf{B}}*=\frac{{\mathit{a}}_{\mathit{r}}^2{\mathit{q}}_{\mathrm{e}}^{*}}{\mathrm{\hslash }}\mathbf{B}*$. (E and F) As in (A) and (B) but for the dimensionless Maxwellian synthetic electric current density associated with the synthetic magnetic field, ${\tilde{\mathbf{J}}*}_{\mathrm{B}}=\frac{\mathrm{\mu }*{\mathit{a}}_{\mathit{r}}^3{\mathit{q}}_{\mathrm{e}}^{*}}{\mathrm{\hslash }}{\mathbf{J}}_{\nabla \times \mathbf{B}*}^{*}$ and (G and H) with the synthetic electric field, ${\tilde{\mathbf{J}}}_{\mathrm{E}}^{*}=\frac{{\mathit{a}}_{\mathit{r}}{\mathit{q}}_{\mathrm{e}}^{*}}{\mathrm{ϵ}*\mathrm{\hslash }{\mathrm{\omega }}_{\mathit{r}}^2}{\mathbf{J}}_{\partial \mathbf{E}*/\partial \mathit{t}}^{*}$. (I) As in (A) but for the dimensionless Maxwellian synthetic electric charge density $\tilde{\mathrm{\rho }}*=\frac{{\mathit{a}}_{\mathit{r}}^2{\mathit{q}}_{\mathrm{e}}^{*}}{\mathrm{ϵ}*\mathrm{\hslash }{\mathrm{\omega }}_{\mathit{r}}}\mathrm{\rho }*$. (J and K) Simulation of the dimensionless synthetic vector potential $\tilde{\mathbf{A}}*=\frac{{\mathit{a}}_{\mathit{r}}{\mathit{q}}_{\mathrm{e}}^{*}}{\mathrm{\hslash }}\mathbf{A}*$, sections in the xz and xy planes, respectively. The field of view in all panels is 13 μm × 13 μm. Background colors in (A), (C), (E), (G), and (J) and (B), (D), (F), (H), and (K) represent the y components and z components of the corresponding vector field, respectively. The fields and sources are evaluated 508 μs after the nonadiabatic ramp using the experimental parameters.