Realizing anomalous Floquet non-Abelian band topology in photonic scattering networks

The concept of multi-gap topology has recently been shown to give rise to uncharted phases beyond conventional single-gap classifications. These phases relate to band nodes with non-Abelian quaternion charges and momentum-space braiding processes characterized by new invariants such as paradigmatic Euler class, phenomena that intrinsically require at least two spatial dimensions. Extending such phases into the non-equilibrium regime is predicted to unlock even richer multi-gap topologies beyond static settings, yet their experimental realization has remained elusive due to the stringent requirements on dimensionality, symmetry, and dynamical control. Here, we theoretically demonstrate and, for the first time, experimentally realize two-dimensional (2D) Floquet non-Abelian band topology in photonic scattering networks. Within this platform, we uncover a sequence of topological phenomena unique to 2D multi-gap systems far from equilibrium, including anomalous multi-gap phases interconnected by band nodes, Floquet Euler transfer, gapped phases with anomalous Dirac string configurations, and Floquet-induced non-Abelian braiding of band nodes. In addition, we observe Floquet-periodic anomalous edge states across multiple gaps, providing experimental signatures of these sought-after 2D multi-gap Floquet topological phases. Our results establish photonic scattering networks as a practical and versatile route to non-Abelian Floquet systems, opening avenues for dynamical topological physics with braiding capability and robust photonic functionalities.