Topological states of quantum light have recently attracted much attention due to their prospects for topologically protected quantum information processing and quantum computation. While first realizations of single-photon topological states are now starting to appear, even richer and initially unexpected physics emerges in the interacting regime, when the effective photon-photon interaction is driven by the nonlinearity of the medium. As an illuminating example, repulsive Kerr-type nonlinearity of cavities arranged into a periodic array can give rise to repulsively bound photon pairs (doublons). As we prove, tailoring the parameters of the lattice one may realize the topological states of doublons, which can coexist with the continuum of two-photon scattering states, thus providing a realization of many-body bound state in the continuum (BIC).
Furthermore, using a one-to-one correspondence between quantum two-particle one-dimensional system and classical two-dimensional setup, we elaborate a protocol to observe topological states of bound photon pairs and to reconstruct the topological eigenmodes experimentally. Finally, we perform a proof-of-concept microwave experiment demonstrating interaction-induced topological states of bound photon pairs for the first time.