Photonic-Fock-state scattering in a waveguide-QED system and their correlation functions

We investigate the problem of arbitrary photonic-Fock-state scattering in a waveguide-QED system which consists of a one-dimensional waveguide coupled to a two-level system. By imposing the open boundary conditions that directly describe the physical settings, we construct a complete set of eigenstates of the system, and elucidate the mathematical structures of the eigenstates. In particular, we show that the eigenstates include a set of multiphoton extended states and multiphoton bound states, formed by all possible partitions of the photon number N. The total number of the eigenstates is exactly described by the integer number partition function Z(N). Using the set of eigenstates, we form the scattering matrix, which facilitates the calculations of the scattered photon states, for the scattering processes. With the scattered photon states, we compute the photon correlation functions that manifestly exhibit the bunching and antibunching behaviors in the scattered photon states. As a concrete example, we discuss in detail with a focus on the three-photon Fock state. Such a capability to generate photonic entanglement from unentangled Fock states will have broad applications in quantum information processing.