Non-Abelian Braiding of Light

Many topological phenomena first proposed and observed in the context of electrons in solids have recently found counterparts in photonic and acoustic systems. In this work, we demonstrate that non-Abelian Berry phases can arise when coherent states of light are injected into “topological guided modes” in specially fabricated photonic waveguide arrays. These modes are photonic analogues of topological zero modes in electronic systems. Light traveling inside spatially well-separated topological guided modes can be braided, leading to the accumulation of non-Abelian phases, which depend on the order in which the guided beams are wound around one another. Notably, these effects survive the limit of large photon occupation, and can thus also be understood as wave phenomena arising directly from Maxwell’s equations, without resorting to the quantization of light. We propose an optical interference experiment as a direct probe of this non-Abelian braiding of light.

Figure 1

(a) Uniform Kekulé distortion in a hexagonal waveguide lattice at a slice of constant $z$. The faint circles represent the original hexagonal lattice, while darker circles represent the distorted lattice. Sublattice $A$ is colored red, while sublattice $B$ is colored blue. The overlaid vector field represents the magnitude and direction of the order parameter $\mathrm{\Delta }(\mathit{r})$. The background is an image of a low-lying eigenmode of Eq. (1). The intensity pattern represents the amplitude of the electric field in each waveguide. (b) Waveguide lattice in the presence of a vortex in the Kekulé pattern, with an overlaid order-parameter vector field showing circulation around the vortex core. The background is an intensity plot of the localized zero-mode wave function. We have chosen a vortex profile in which only sublattice $A$ is displaced.

Figure 2

Schematic of the proposed photonic non-Abelian interferometer. A coherent laser beam (one of three, one for each vortex in a lattice) passes through a $50/50$ beam splitter (blue) and simultaneously enters two separate photonic lattices fabricated with the two braids to be compared. The two output beams are reflected by mirrors (gray) into another beam splitter that interferes the two signals and outputs the sum and difference to separate screens (black). For the choice of braids used here, one screen shows a bright and two dark spots, while the other screen shows the “logical complement” of the first, with one dark and two bright spots.