First order Maxwell operator formalism for macroscopic quantum electrodynamics

• URL: http://arxiv.org/abs/2603.27475v1
• Date: Sun, 29 Mar 2026 02:18:57 GMT
• Title: First order Maxwell operator formalism for macroscopic quantum electrodynamics
• Abstract summary: Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms.<n>We develop a first-order electromagnetic operator approach that retains both $mathbfE$ and $mathbfH$ and leads to a quantum input-output formalism.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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