LAUSR

We identify a class of dressed atom-photon states forming at the same energy of the atom at any coupling strength. As a hallmark, their photonic component is an eigenstate of the bare photonic bath with a vacancy in place of the atom. The picture accommodates waveguide-QED phenomena where atoms behave as perfect mirrors, connecting in particular dressed bound states (BSs) in the continuum with geometrically confined photonic modes. When applied to photonic lattices, the framework establishes a one-to-one correspondence between topologically robust dressed states and topologically robust photonic BSs seeded by a vacancy. This is used to predict new classes of dressed BSs in the photonic Creutz-ladder and Haldane models. In the latter case, states with nonzero local photon flux occur in which an atom is dressed by a photon orbiting around it.