Field-Based Macroscopic QED

• Field-based macroscopic QED is a rigorous framework that quantizes both electromagnetic fields and material responses in dispersive, absorbing media.
• It employs a first-order Maxwell operator formalism to analyze dual electric and magnetic fields and facilitate input–output mappings via Green tensor propagation.
• The approach underpins advanced applications such as complex photonic structures, quantum nanophotonics, and nonreciprocal device modeling with enforced fluctuation–dissipation relations.

Field-based macroscopic quantum electrodynamics (QED) is a rigorous operator-theoretic and Hamiltonian framework for describing quantized electromagnetic fields in the presence of arbitrary, spatially structured, dispersive, and absorbing media. Unlike microscopic QED, which begins from canonical quantization in vacuum and adds matter perturbatively, the macroscopic theory quantizes the full, constrained electromagnetic field$+$medium system, respecting material response functions at the operator level. The formalism is built around operator-valued fields, bosonic reservoir variables, and Green tensor propagation, and is tailored to encompass complex photonic, plasmonic, and dissipative environments as well as input/output boundaries. Central to this approach are fluctuation–dissipation relations enforced at the operator algebra level, explicit symplectic and duality symmetries, and input–output mappings by first-order Maxwell operator structures. Field-based macroscopic QED provides the foundation for the quantization of complex photonic structures, multi-emitter quantum nanophotonics, vacuum-induced forces, quantum friction, and nonreciprocal/effective-magnetic device modeling (Agarwal et al., 29 Mar 2026, Westerberg et al., 2021, Horsley, 2022, Oue, 25 Feb 2026, Philbin, 2010, Buhmann et al., 2011).

1. Dual-Field First-Order Maxwell Operator Formalism

Modern field-based macroscopic QED employs a first-order Maxwell operator formalism, which treats the electric and magnetic fields as a combined six-component “dual” field,

$$|\mathcal{E}\rangle \equiv \begin{bmatrix} \mathbf{E} \ Z_0\,\mathbf{H} \end{bmatrix},$$

alongside a dual source vector

$$|\mathcal{J}\rangle \equiv \begin{bmatrix} Z_0\,\mathbf{J}_E \ \mathbf{J}_M \end{bmatrix}.$$

Here, $Z_0$ is the vacuum impedance. The frequency-domain Maxwell equations are then recast as an operator equation,

$$\left(i\,J\otimes(\nabla\times)-k_0\,\bar\varepsilon(\mathbf r,\omega)\right)|\mathcal{E}\rangle = i\,|\mathcal{J}\rangle,$$

with a symplectic generator $$J = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}$$ and a material tensor $$\bar\varepsilon = \text{diag}\{\varepsilon(\mathbf r,\omega),\,\mu(\mathbf r,\omega)\}$$. The resulting Maxwell operator $$\mathcal{M} \equiv iJ\otimes(\nabla\times) - k_0\bar\varepsilon$$ yields a self-adjoint structure under appropriate inner product, encoding both bulk laws and boundary conditions (Agarwal et al., 29 Mar 2026). This construction is essential for formulating quantum input–output theory in open geometries, as the propagator (Green operator) describes transmission between surfaces and not just field points.

2. Green Operators, Propagation, and Reciprocity

The field-based approach is grounded in the existence of a retarded Green operator (g(\omega)\