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First order Maxwell operator formalism for macroscopic quantum electrodynamics

Ishita Agarwal, Ankit Kundu, Christian M. Lange, Jonathan D. Hood

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.

First order Maxwell operator formalism for macroscopic quantum electrodynamics

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 and 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 =, whose first-order Green operator 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 , 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.

Paper Structure

This paper contains 36 sections, 148 equations, 4 figures.

Table of Contents

  1. Introduction
  2. First-order symplectic formalism
  3. First-Order Maxwell Operators
  4. Second-order Maxwell operator.
  5. Green's Operator and Its Kernel
  6. Operator representation.
  7. Kernel representation.
  8. Second-Order Green's Function and Relation to the First Order
  9. Introduction to adjoint identities
  10. Energy inner product and adjoint operators
  11. Adjoint of the Green's Operator
  12. Green kernel singularity.
  13. Interior Field Representation
  14. Generalized Optical Theorem
  15. Connection to the algebraic resolvent identity.
  16. ...and 21 more sections

Figures (4)

  • Figure 1: Field decomposition geometry and system consideration. A lossy, dispersive medium with material tensor $\bar{\varepsilon}(\mathbf{r},\omega)$ occupies a region $V$ enclosed by a surface $S$. The retarded Green operator $g^{(+)}(\mathbf{r},\mathbf{r}',\omega)$ propagates between interior points, while the incoming boundary field $(\bar{n}\times)\mathcal{E}_\mathrm{in}(s)$ enters through $S$. The outgoing radiation fields satisfy the Sommerfeld condition on $S_\infty$, rendering unique solutions.
  • Figure 2: Surface-volume decomposition of the Green tensor. The direct response between $\mathbf r_1$ and $\mathbf r_2$ is related to two mediated pathways: a volume contribution through an interior point $\mathbf r$ in the lossy region $\bar{\varepsilon}_I(\mathbf r)$, and a surface contribution through a boundary point $\mathbf r_S$. The inset indicates the LDOS at $\mathbf r$.
  • Figure 3: Plane-to-plane transfer of the tangential electromagnetic state. The trace $|\Psi_1\rangle$ on $S_1$ propagates to $|\Psi_2\rangle$ on $S_2$ through the transfer operator $T_{21}$, and then to $|\Psi_3\rangle$ on $S_3$ through $T_{32}$. The net propagation from $S_1$ to $S_3$ is therefore given by the exact composition law $T_{31}=T_{32}T_{21}$.
  • Figure 4: Quantization scheme. The total Hamiltonian couples three subsystems: the free electromagnetic field $\hat{H}_{\mathrm{EM}}$, the material polarization $\hat{H}_P$, and the bosonic bath $\hat{H}_{\mathrm{bath}}$. The field and polarization interact via the dipole coupling $-\hat{\mathbf{P}}(\mathbf{r})\cdot\hat{\mathbf{E}}(\mathbf{r})$, while the polarization and bath are coupled with spectrum $\kappa(\mathbf{r},\omega)$. The field and bath are never coupled directly.