Research ArticlePHYSICS

Synthetic electromagnetic knot in a three-dimensional skyrmion

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Science Advances  02 Mar 2018:
Vol. 4, no. 3, eaao3820
DOI: 10.1126/sciadv.aao3820
  • 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 Embedded Image cm−2. For (C) and (E), these quantities are 738 μm × 246 μm and Embedded Image cm−2, and for (D) and (F), these quantities are 657 μm × 219 μm and Embedded Image 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 Embedded Image, 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 Embedded Image. (C) As in (B), but with quantization along +x.

  • Fig. 4 Synthetic electromagnetic quantities.

    (A and B) Dimensionless synthetic electric field Embedded Image 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 Embedded Image. (C and D) As in (A) and (B) but for the dimensionless synthetic magnetic field Embedded Image. (E and F) As in (A) and (B) but for the dimensionless Maxwellian synthetic electric current density associated with the synthetic magnetic field, Embedded Image and (G and H) with the synthetic electric field, Embedded Image. (I) As in (A) but for the dimensionless Maxwellian synthetic electric charge density Embedded Image. (J and K) Simulation of the dimensionless synthetic vector potential Embedded Image, 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.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/3/eaao3820/DC1

    Condensate order parameter

    Spin texture

    Synthetic gauge potentials

    Synthetic electric field

    Synthetic magnetic field

    Maxwellian synthetic electric charge density

    Maxwellian synthetic electric current density

    Superfluid velocity

    fig. S1. Detailed temporal evolution of the condensate spin texture during skyrmion creation, side view, quantized along +z.

    fig. S2. Detailed temporal evolution of the condensate spin texture during skyrmion creation, top view, quantized along +z.

    fig. S3. Detailed temporal evolution of the condensate spin texture during skyrmion creation, side view, quantized along +x.

    fig. S4. Detailed temporal evolution of the condensate spin texture during skyrmion creation, side view, quantized along +y.

    fig. S5. Detailed temporal evolution of the condensate spin texture during skyrmion creation, top view, quantized along +y.

  • Supplementary Materials

    This PDF file includes:

    • Condensate order parameter
    • Spin texture
    • Synthetic gauge potentials
    • Synthetic electric field
    • Synthetic magnetic field
    • Maxwellian synthetic electric charge density
    • Maxwellian synthetic electric current density
    • Superfluid velocity
    • fig. S1. Detailed temporal evolution of the condensate spin texture during skyrmion creation, side view, quantized along +z.
    • fig. S2. Detailed temporal evolution of the condensate spin texture during skyrmion creation, top view, quantized along +z.
    • fig. S3. Detailed temporal evolution of the condensate spin texture during skyrmion creation, side view, quantized along +x.
    • fig. S4. Detailed temporal evolution of the condensate spin texture during skyrmion creation, side view, quantized along +y.
    • fig. S5. Detailed temporal evolution of the condensate spin texture during skyrmion creation, top view, quantized along +y.

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