Modal input-output theory for quantum nanophotonics from the first-order Maxwell operator

We develop a quantum input-output theory for open photonic systems based on macroscopic quantum electrodynamics using the first-order electromagnetic Green's function. Quantum fields enter and leave the photonic system through waveguide ports, while the electromagnetic Green's function propagates the fields through the arbitrary interior of the photonic system, which may be dispersive and absorbing. The resulting relation contains a port-to-port scattering matrix and a Langevin-noise contribution from material absorption that together preserve the bosonic output commutation relations. For embedded emitters, the low-saturation and single-excitation scattering response is determined by Green's functions connecting the ports and emitters while retaining the non-Markovianity of the photonic environment. Using finite-difference time-domain simulation, we calculate the emitter-modified transmission for a nanophotonic cavity and inverse-designed coupler, providing a direct route from computed electromagnetic response to a quantum input-output model.