Multiphoton pulses interacting with multiple emitters in a one-dimensional waveguide

We derive a generalized master equation for multiphoton pulses interacting with multiple emitters in a waveguide-quantum electrodynamics system where the emitter frequency can be modulated and the effects of nonguided modes can also be considered. Based on this theory, we can calculate the real-time dynamics of an array of interacting emitters driven by an incident photon pulse which can be vacuum, a coherent state, a Fock state, or their superpositions. Moreover, we also derive generalized input-output relations to calculate the reflectivity and transmissivity of this system. We can also calculate the output photon pulse shapes. Our theory can find important applications in the study of waveguide-based quantum systems.