  While designing the energy-momentum relation of photons is key to many
linear, non-linear, and quantum optical phenomena, a new set of light-matter
properties may be realized by employing the topology of the photonic bath
itself. In this work we investigate the properties of superconducting qubits
coupled to a metamaterial waveguide based on a photonic analog of the
Su-Schrieffer-Heeger model. We explore topologically-induced properties of
qubits coupled to such a waveguide, ranging from the formation of directional
qubit-photon bound states to topology-dependent cooperative radiation effects.
Addition of qubits to this waveguide system also enables direct quantum control
over topological edge states that form in finite waveguide systems, useful for
instance in constructing a topologically protected quantum communication
channel. More broadly, our work demonstrates the opportunity that topological
waveguide-QED systems offer in the synthesis and study of many-body states with
exotic long-range quantum correlations.
