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The purpose of this paper is to study the coherent feedback control dynamicsbased on the network that an $N$-level atom is coupled with a cavity and thecavity is coupled with a single or multiple parallel waveguides through twosemitransparent mirrors. When initially the atom is excited at the highestenergy level, it can emit multiple photons into the cavity via the spontaneousemission, and the photons in the cavity can be transmitted into the waveguideand then re-interact with the cavity quantum electrodynamics (cavity-QED)system through the feedback channel. When the cavity is coupled with a singlewaveguide, the generation of multi-photon states in the waveguide can becharacterized by the exponential stability of the linear control system withfeedback delays determined by the feedback loop length. By tuning the feedbackloop length, there can be zero or multiple photons in the waveguide. Besides,when the cavity-QED system is coupled with multiple parallel waveguides, theemitted photons oscillate among different waveguides and this process isinfluenced by the feedback loop length and coupling strengths among waveguides.

Quantum coherent feedback control of an N-level atom with multiple excitations