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In a preceding paper we introduced a formalism to study the scattering of lowintensity fields from a system of multi-level emitters embedded in a $3$Ddielectric medium. Here we show how this photon-scattering relation can be usedto analyze the scattering of single photons and weak coherent states from anygeneric multi-level quantum emitter coupled to a $1$D waveguide. The reductionof the photon-scattering relation to $1$D waveguides provides for the firsttime a direct solution of the scattering problem involving low intensity fieldsin the waveguide QED regime. To show how our formalism works, we considerexamples of multi-level emitters and evaluate the transmitted and reflectedfield amplitude. Furthermore, we extend our study to include the dynamicalresponse of the emitters for scattering of a weak coherent photon pulse. As ourphoton-scattering relation is based on the Heisenberg picture, it is quiteuseful for problems involving photo-detection in the waveguide architecture. Weshow this by considering a specific problem of state generation byphoto-detection in a multi-level emitter, where our formalism exhibits its fullpotential. Since the considered emitters are generic, the $1$D results apply toa plethora of physical systems like atoms, ions, quantum dots, superconductingqubits, and nitrogen-vacancy centers coupled to a $1$D waveguide ortransmission line.

Photon Scattering from a System of Multi-Level Quantum Emitters. II. Application to Emitters Coupled to a 1D Waveguide