Scattering of few photons by a ladder-type quantum system

We consider the inelastic scattering of few photons by a ladder-type quantum system residing in a semi-infinite waveguide. Two scenarios, namely 2-photon scattering and 3-photon scattering, are investigated. The single photons are described by continuous temporal pulse functions, according to which the exact forms of the stationary output field states can be derived. Based on the exact analysis of the nonlinear dynamics, the spectral entanglement among the output photons mediated by the ladder-type quantum system is simulated. In the 2-photon scattering case, strong correlation between the output photons can be observed if each input photon is in resonance with a transition frequency of the system. In particular, there exists a weak two-photon process for the non-resonance case, where two input photons can couple to the transition between the first and third levels of the system if a two-photon resonance condition is satisfied. In the 3-photon scattering case, the presence of an ancillary photon could significantly influence the correlation pattern. There exist two nonlinear terms, one is for 2-photon inelastic scattering and the other for 3-photon inelastic scattering, in the 3-photon output state. As a result, both two-photon and three-photon processes can be observed in the pattern of spectral entanglement.