Wave mixing effects in cascade systems of superconducting qubits

In present work, we numerically simulated the nonlinear quantum-optical effect, -- wave mixing of propagating wave packets (two coherent and antibunched with a coherent signal) in a cascade system of superconducting atoms -- qubits. As a possible realization, we consider qubits in the transmon architecture interacting with the waveguide in the strong coupling regime, which allows us to study the scattering in the cascade system at the level of single quanta of electromagnetic radiation. The obtained wave mixing spectra demonstrate sensitivity to photon statistics in the waveguide -- the detuning of the peak from the center frequency corresponds to the number of photons involved in the process.