A Green's Function Approach to Quantum Input–Output Formalism in an Arbitrary Magneto-Dielectric Medium

We present a unified Green's function formalism for quantum electrodynamics in dispersive and lossy dielectric media containing arrays of quantum emitters. Our approach demonstrates that relevant quantum observables like transmission spectra, photon correlations, and collective emitter dynamics can be computed directly from the classical electromagnetic Green's functions obtained through standard numerical solvers, including FDTD/finite-element packages.

Building on the Modified Langevin Noise formalism, we employ a symplectic six-vector representation of electromagnetic fields that yields compact identities connecting material absorption through fluctuation-dissipation relations and input fields through boundary contributions. The output field operators emerge as linear combinations of input modes and Langevin noise sources, and scattering matrices are expressed in terms of the Green's tensor. By establishing rigorous connection between computational electromagnetics and quantum optics, our approach extends beyond conventional cavity and waveguide paradigms and is applicable to arbitrary geometries including inverse-designed structures, thereby opening new possibilities for engineering atom-photon interfaces in complex photonic environments where analytical methods are intractable.