Topological Quantum Electrodynamics in Synthetic Non-Abelian Gauge Fields

Quantum electrodynamics (QED), a cornerstone framework that describes light-matter interactions rooted in Abelian symmetries, renders the harnessing of synthetic non-Abelian gauge fields as a fundamental yet uncharted frontier. Here, we develop a general theory of light-matter interaction of quantum emitters embedded in non-Abelian photonic lattices. Based on analytical solutions to the non-Abelian Landau dressed states beyond the continuum limit, we reveal chiral photon emission and vortices with emergent nonreciprocity enabled by selective coupling between emitters and spin-momentum-locked bands. When coexisting with Abelian and non-Abelian magnetic fields, emitters hybridize with Landau dressed orbits to form spin-polarized, squeezed Landau polaritons that carry quantized angular momenta, with Rabi frequencies tunable via Landau levels and pseudospin interactions. Multiemitter dynamics further exhibit collective phenomena governed by real-space staggered phases induced by nonsymmorphic crystalline symmetry. These results bridge non-Abelian physics with quantum optics and establish non-Abelian gauge fields as a versatile tool for synthesizing topological quantum optical states, angular momentum transfer, and controlling photon-mediated correlations in QED systems, relevant for applications in quantum simulations and chiral quantum optical networks.