Dark-state-mediated topological response to coherence effects from two independent phases in single-photon transport

We unveil the inherent topological nature from a two-dimensional synthetic parameter space in a single-photon transport problem, with the simple configuration of one waveguide coupled with two four-level atoms driven by external fields. We find that, by tuning the system in the dark-state condition, the system holds two independent phase degrees of freedom from both the photon propagation and the atomic transition. As the direct output, the photon scattering characteristics show the total reflection with the frequency preservation or the complete frequency conversion at the specific choices of two phases. We uncover the underlying topological effect by analyzing the photon scattering amplitudes in the parameter space of two phases, which links to the singularity with nonzero winding numbers. Our results are verified by numerical simulations, and hence point to an alternative perspective for the research at the intersection of single-photon transport and topology, without using the topological geometric arrangement of atoms.