Skyrmionic polarization textures in structured dielectric planar media

TL;DR

The paper demonstrates that skyrmionic polarization textures can emerge in the eigenpolarizations of light propagating through spatially structured planar dielectrics, realized with three liquid-crystal metasurfaces that define a 2D synthetic Brillouin zone. By mapping the system to a two-band Bloch Hamiltonian with H_eff(q) = E(q) n(q) · σ, it shows that the Skyrme number ν_s coincides with the Chern number ν_c, signaling a photonic Chern-insulator topology. Using neural-network-assisted quantum process tomography, the authors reconstruct eigenpolarizations across the Brillouin zone, observe skyrmions for certain birefringence settings (ν_s ≈ 1), and extract Berry curvature and quantum metric as local geometric observables. They further simulate an all-optical quantum Hall effect, observing a transverse drift proportional to the topological invariant, thereby linking abstract topology to measurable dynamics and highlighting potential for robust, all-optical topological photonics in structured dielectrics.

Abstract

Skyrmionic patterns of optical fields have recently emerged across diverse photonic platforms. Here, we show that such textures also arise in the polarization eigenstates of light propagation through flat dielectric devices with an engineered, space-dependent optic-axis orientation. We focus on two-dimensional periodic structures, where propagation through multiple devices maps onto quantum dynamics on a synthetic optical lattice. Adopting the condensed-matter framework, a spatial period defines an effective Brillouin zone, and polarization eigenstates can be grouped in two bands, with the role of energy played by the opposite phase delay. When such eigenstates exhibit skyrmionic textures, the corresponding lattice model shows the topology of a Chern insulator. These structures result from the interaction between the optical field and the medium and do not reflect a topological structure of the medium itself. We validate these concepts in a system of three tunable liquid-crystal metasurfaces. Using quantum process tomography based on supervised machine learning, we reconstruct the polarization eigenmodes over one spatial period. We identify configurations of the devices' parameters that lead to topologically non-trivial bands, where we directly observe skyrmionic eigenpolarization textures. Along the analogy with condensed matter, we also extract local observables of lattice models, such as the Berry curvature and the quantum metric. We finally report a numerical simulation of an all-optical quantum Hall effect emerging when light propagates through a sequence of such devices, arranged so as to mimic the effect of an external force on the lattice.

Figures (5)

• Figure 1: Eigenpolarization of structured planar media. (a) Three adjacent liquid-crystal metasurfaces define a 2D optical operator $U$, implementing a complex polarization transformation. Its action is periodic across a characteristic distance $\Lambda$, which defines a Brillouin zone (BZ) in real space (see Sec. \ref{['sec:bloch']} for more details). Accordingly, each transverse position on the metasurfaces' plane identifies a quasi-momentum value according to the mapping $\textbf{q}=-2\pi\textbf{r}/\Lambda$. Each local polarization transformation $U(x_0,y_0)$ thus corresponds to the Bloch-diagonal form of a discrete lattice evolution operator at a given quasi-momentum. The associated polarization eigenstate, $\ket{\textbf{n}(x_0,y_0)}$, can be visualized on the Bloch-Poincaré sphere, with the components of $\textbf{n}$ giving the Stokes parameters $\textbf{S}=(S_1,S_2,S_3)$. (b) Eigenpolarization textures for three representative cases along the curve $\delta_x=\delta_y\equiv\delta$. Arrows give the orientation of the local eigenpolarization on the sphere. The patterns preserve the orientation in the cases ${\delta=\pi/8}$ and ${\delta=7\pi/30}$, while a skyrmion appears at ${\delta=\pi/2}$, where the eigenpolarization flips at the center. (c) Topological phase diagram for the operator ${U=T_y(\delta_y)T_x(\delta_x)W}$ as a function of $\delta_x$ and $\delta_y$. For ${\delta=\pi/8}$ (orange) and ${\delta=7\pi/30}$ (purple), the eigenpolarization field $\textbf{n}(\textbf{q})$ does not fully cover the Bloch sphere, which corresponds to a vanishing Skyrme number (${\nu_\text{s}=0}$). In contrast, for ${\delta=\pi/2}$ (red), the Bloch sphere is completely covered as $\textbf{q}$ varies across the BZ, indicating a topologically non-trivial phase (${\nu_\text{s}=1}$).
• Figure 2: Experimental process tomography. (a) A laser beam is expanded with two lenses (L) and spatially filtered with a pinhole (Ph) placed in the focal plane. Three liquid-crystal metasurfaces implement a space-dependent polarization transformation. The system topology is tuned by applying an AC voltage to each device. Polarimetric measurements are realized by preparing and projecting onto the desired polarization states with a linear polarizer (P), a half-wave plate (H), and a quarter-wave plate (Q). (b) As an example, set of polarimetric images taken for the case ${\delta=\pi/2}$. (c) Such images constitute the input layer of a fully-connected neural network, pretrained to output the model eigenstructure from polarimetric data. The reconstructed patterns can be visualized as arrows overlapping with the local eigenpolarization on the Bloch sphere at each quasi-momentum value. Arrow colors indicate the corresponding energy eigenvalue.
• Figure 3: Skyrmionic eigenpolarization textures. Tomographic reconstructions of energy bands $E$ and eigenpolarizations $\textbf{n}$. Experimental patterns obtained via the machine-learning-based tomography (ML) are compared with theoretical (Th.) predictions. The three cases (a) $\delta=\pi/8$, (b) $\delta=5\pi/12$, and (c) $\delta=\pi/2$ are considered. The eigenpolarization pattern is also plotted as arrows, whose orientation gives the direction of the local polarization eigenstate on the Bloch sphere across the BZ. The skyrmionic feature is absent in the topologically trivial phase (a), and only appears in the topologically non-trivial phases (b)-(c). Average fidelities: (a) $(98\pm1)\%$ , (b) $(97\pm1)\%$, (c) $(96\pm3)\%$
• Figure 4: Extracting the quantum geometric tensor. From the reconstructed eigenpolarizations (see Fig. \ref{['fig:fig3']}), we extract the Berry curvature $B_z$ and the quantum metric components $g_{ij}$, representing the imaginary and real parts of the quantum geometric tensor, respectively. The case ${\delta=\pi/2}$ is shown here. Machine-learning-based (ML) experimental reconstructions are compared with theoretical (Th.) predictions.
• Figure 5: Detecting topology through dynamics. Center-of-mass trajectory under the effect of an external force along the $x$ axis in two simulated experiments at (a) ${\delta=\pi/8}$ and (b) ${\delta=\pi/2}$. In the case of topologically non-trivial bands, a non-vanishing anomalous transverse displacement $\braket{k_y}$ is revealed, as predicted by the adiabatic approximation.