  The purpose of this paper is to study the dynamics of a quantum coherent
feedback network, where an $N$-level atom is coupled with a cavity and the
cavity is also coupled with single or multiple parallel waveguides. When the
atom is initialized at the highest energy level, it can emit multiple photons
into the cavity, and the photons can be further transmitted to the waveguides
and re-interact with the cavity quantum electrodynamics (cavity-QED) system.
The transmission of photons in the waveguide can construct a feedback channel
with a delay determined by the length of the waveguide. We model the dynamics
of the atomic and photonic states of the cavity-QED system as a linear control
system with delay. By tuning the control parameters such as the coupling
strengths among the atom, cavity and waveguide, the eigenstates of the quantum
system can be exponentially stable or unstable, and the exponential stability
of the linear quantum control system with delay is related with the generation
of single- and multi-photon states. Besides, when the cavity-QED system is
coupled with multiple parallel waveguides, the emitted photons oscillate among
different waveguides and the stability of quantum states is influenced by the
feedback loop length and coupling strengths among waveguides.
