  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.
