  We derive in the Heisenberg picture a widely used phenomenological coupling
element to treat feedback effects in quantum optical platforms. Our derivation
is based on a microscopic Hamiltonian, which describes the mirror-emitter
dynamics based on a dielectric, a mediating fully quantized electromagnetic
field, and a single two-level system in front of the dielectric. The dielectric
is modeled as a a system of identical two-state atoms. The Heisenberg equation
yields a system of describing differential operator equations, which we solve
in the Weisskopf-Wigner limit. Due to a finite round-trip time between emitter
and dielectric, we yield delay differential operator equations. Our derivation
motivates and justifies the typical phenomenological assumed coupling element
and allows, furthermore, a generalization to a variety of mirrors, such as
dissipative mirrors or mirrors with gain dynamics.
