First order Maxwell operator formalism for macroscopic quantum electrodynamics

TL;DR

This paper introduces a first-order operator formalism for macroscopic quantum electrodynamics that retains boundary terms and provides an exact quantum input-output framework applicable to complex photonic structures.

Contribution

It develops a novel first-order Maxwell operator approach that includes boundary effects and yields a comprehensive quantum input-output formalism for macroscopic QED.

Findings

Derived a first-order Green operator propagating electromagnetic states.

Established an exact commutation relation consistent with fluctuation-dissipation theorem.

Extended the quantum input-output framework to complex structures with numerically obtained Green's functions.

Abstract

Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both $\mathbf{E}$ and $\mathbf{H}$ and keeps those boundary terms, naturally leading to a quantum input-output formalism. We recast Maxwell's equations as an operator equation for the dual field $\mathit{E}$=$[\mathbf{E},\mathbf{H}]^T$, whose first-order Green operator $g$ propagates the electromagnetic state between surfaces. Symmetries of the Maxwell operator under energy and reciprocal inner products yield the propagation formula, Lorentz reciprocity, and a generalized optical theorem, with minimal vector calculus. Quantizing via a Heisenberg-Langevin approach for absorptive, dispersive media yields two independent quantum noise sources: bulk Langevin operators from material absorption and input-output field operators at the boundary. Expressing the interior field in terms of these operators and the Green propagator yields an exact closed commutation relation $[{\mathit{E}},{\mathit{E}}^\dagger]\propto \mathrm{Im}\,g$, consistent with the fluctuation-dissipation theorem. This identity holds even when dielectrics extend to the boundary, as in waveguide input-output problems, and enables quantum input-output descriptions of complex photonic structures where the Green's function is obtained numerically, extending the framework beyond cavities and waveguides.