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arXiv:2610.04043

Modal input-output theory for quantum nanophotonics from the first-order Maxwell operator

Ankit Kundu, Ishita Agarwal, Adhyyan S. Mansukhani, Jonathan D. Hood

quant-phphysics.opticsSubmitted Oct 2, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

We develop a quantum input-output theory for open photonic systems based on macroscopic quantum electrodynamics using the first-order electromagnetic Green's function. Quantum fields enter and leave the photonic system through waveguide ports, while the electromagnetic Green's function propagates the fields through the arbitrary interior of the photonic system, which may be dispersive and absorbing. The resulting relation contains a port-to-port scattering matrix and a Langevin-noise contribution from material absorption that together preserve the bosonic output commutation relations. For embedded emitters, the low-saturation and single-excitation scattering response is determined by Green's functions connecting the ports and emitters while retaining the non-Markovianity of the photonic environment. Using finite-difference time-domain simulation, we calculate the emitter-modified transmission for a nanophotonic cavity and inverse-designed coupler, providing a direct route from computed electromagnetic response to a quantum input-output model.

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Using the Green-function approach to the problem of quantization of the phenomenological Maxwell theory, the propagation of quantized radiation through dispersive and absorptive multilayer dielectric plates is studied. Input-output relations are derived, with special emphasis on the determination of the quantum noise generators associated with the absorption of radiation inside the dielectric matter. The input-output relations are used to express arbitrary correlation functions of the outgoing field in terms of correlation functions of the incoming field and those of the noise generators. To illustrate the theory, photons at dielectric tunneling barriers are considered. It is shown that inclusion in the calculations of losses in the photonic band gaps may substantially change the barrier traversal times. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Aug 1, 1996•123 citations
We develop a quantum input-output theory for open photonic systems based on macroscopic quantum electrodynamics using the first-order electromagnetic Green's function. Quantum fields enter and leave the photonic system through waveguide ports, while the electromagnetic Green's function propagates the fields through the arbitrary interior of the photonic system, which may be dispersive and absorbing. The resulting relation contains a port-to-port scattering matrix and a Langevin-noise contribution from material absorption that together preserve the bosonic output commutation relations. For embedded emitters, the low-saturation and single-excitation scattering response is determined by Green's functions connecting the ports and emitters while retaining the non-Markovianity of the photonic environment. Using finite-difference time-domain simulation, we calculate the emitter-modified transmission for a nanophotonic cavity and inverse-designed coupler, providing a direct route from computed electromagnetic response to a quantum input-output model.
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\mathbf{E}E and H\mathbf{H}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\mathit{E}E=[E,H]T[\mathbf{E},\mathbf{H}]^T[E,H]T, whose first-order Green operator ggg 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†]∝Im g[{\mathit{E}},{\mathit{E}}^\dagger]\propto \mathrm{Im}\,g[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.
This work presents a full-wave Green's function framework for modeling collective and coherent single-photon emission from multiple quantum emitters embedded in complex electromagnetic structures. Starting from a transverse modal completeness relation of modified Langevin noise formalism, we derive a closed set of coupled equations for population dynamics and frequency-resolved field amplitudes in the single-excitation regime. Since the electromagnetic reservoir is not traced out at the level of the dynamical amplitudes, the emitted single-photon dynamics can be modeled within the same closed set of equations without Markovian approximation in open and dissipative environments. We demonstrate that finite-bandwidth truncation of the spectral density leads to systematic deviations in coherent dispersive interactions, even when dissipative rates appear converged. To restore causal consistency, we introduce a counter-term compensation scheme that restores the missing dispersive contributions without modifying the retained non-Markovian memory kernel. To validate the scheme and demonstrate the practicality of the proposed framework, we present numerical examples ranging from benchmark configurations to a three-dimensional dispersive ring-resonator structure via finite element method. These capabilities provide a practical route for rigorously incorporating full-wave electromagnetic simulations into non-Markovian multi-emitter quantum electrodynamics, enabling predictive modeling of collective emission, coherent energy exchange, and single-photon radiation in realistic open structures.
We formulate a quantization scheme for open electromagnetic systems with arbitrary passive boundary conditions. Rather than specifying reservoirs phenomenologically, the method identifies them from the dissipation geometry of the Maxwell operator. Factoring the imaginary part of the Maxwell operator gives a bosonic realization of the field operator and separates the fluctuation channels into medium-assisted reservoirs from material absorption and boundary-assisted reservoirs from exchange through the open boundary. Depending on the boundary condition, the latter become free-space radiation modes, impedance-load channels, guided port modes, or more general boundary channels. Green-function input-output relations then follow as an application, yielding frequency-dependent scattering and noise kernels without Markov or single-mode assumptions. To illustrate the practical application, we consider a lossy structure with mixed impedance and outgoing boundaries, and photonic integrated circuit configurations with waveguide port boundaries.
Recent advances are enabling quantum emitters to interact with photonic crystals, whose electromagnetic modes exhibit complex dispersion relations, spatial mode structure, and polarization textures. However, modeling light-matter behavior in these systems faces a persistent trade-off: electromagnetic approaches based on Maxwell-equation solvers provide realistic vectorial descriptions but are difficult to integrate with quantum many-body and non-perturbative methods, whereas simplified quantum-optical lattice models are tractable but typically rely on scalar and spatially independent light-matter couplings that miss essential features of these structured photonic environments. Here, we introduce a constructive framework to derive quantum-optical lattice descriptions that overcome this trade-off. Combining symmetry-constrained tight-binding constructions with numerically computed photonic band structures and field profiles, our method yields minimal, symmetry-enforced lattice Hamiltonians that reproduce the target photonic dispersion while retaining the mode-resolved (position- and polarization-dependent) structure of the light-matter coupling. We show that these models recover Green's-function-based emitter dynamics in the perturbative regime, while providing access to non-perturbative quantum dynamical simulations beyond emitter-only descriptions. As a proof of principle, we apply the framework to a two-dimensional photonic crystal and show that it captures polarization-dependent directional emission inaccessible to scalar models, while enabling the analysis of non-Markovian light-matter dynamics and entanglement. Our results provide a practical bridge between classical electromagnetic simulation tools and quantum-optical many-body and non-Markovian modeling in photonic crystal settings.
We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.
Using the Green-function approach to the problem of quantization of the phenomenological Maxwell theory, the propagation of quantized radiation through dispersive and absorptive multilayer dielectric plates is studied. Input-output relations are derived, with special emphasis on the determination of the quantum noise generators associated with the absorption of radiation inside the dielectric matter. The input-output relations are used to express arbitrary correlation functions of the outgoing field in terms of correlation functions of the incoming field and those of the noise generators. To illustrate the theory, photons at dielectric tunneling barriers are considered. It is shown that inclusion in the calculations of losses in the photonic band gaps may substantially change the barrier traversal times. \textcopyright{} 1996 The American Physical Society.