Two-photon topological states in the array of qubits caused by the effective photon-photon interaction

We investigate two-photon topological states in the array of nearest-neighbor coupled qubits for two model systems based on the extended version of the Bose-Hubbard Hamiltonian, one of which includes direct two-photon tunneling and another one incorporates density-dependent hopping. For both models, we demonstrate that the effective photon-photon interaction gives rise to the topological bandgap and topologically protected edge states of bound photon pairs. We design effective density-dependent hopping by inserting auxiliary frequency-detuned qubits in the appropriate positions in the array, which is feasible to realize, for instance, in existing arrays of transmon qubits. Furthermore, we map our system onto the paradigmatic Su-Schrieffer-Heeger model in the limiting cases illustrating our conclusions by the results of numerical simulations.