Radiative topological biphoton states in modulated qubit arrays

We study topological properties of bound pairs of photons in spatially-modulated qubit arrays (arrays of two-level atoms) coupled to a waveguide. While bound pairs behave like Bloch waves, they are topologically nontrivial in the parameter space formed by the center-of-mass momentum and the modulation phase, where the latter plays the role of a synthetic dimension. In a superlattice where each unit cell contains three two-level atoms (qubits), we calculate the Chern numbers for the bound-state photon bands, which are found to be $(1,-2,1)$. For open boundary condition, we find exotic topological bound-pair edge states with radiative losses. Unlike the conventional case of the bulk-edge correspondence, these novel edge modes not only exist in gaps separating the bound-pair bands, but they also may merge with and penetrate into the bands. By joining two structures with different spatial modulations, we find long-lived interface states which may have applications in storage and quantum information processing.