We show how to calculate the first- and second-order statistics of the scattered fields for an arbitrary intensity coherent-state light field interacting with a two-level system in a waveguide geometry. Specifically, we calculate the resonance fluorescence from the qubit, using input-output formalism. We derive the transmission and reflection coefficients, and illustrate the bunching and antibunching of light that is scattered in the forward and backward directions, respectively. Our results agree with previous calculations on one- and two-photon scattering as well as those that are based on the master equation approach.
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
E and
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
E=
[E,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
[E,E†]∝Img, 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.