⚛️ quantum physics

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

This paper develops a quantum input-output theory for open, dispersive, and absorbing nanophotonic systems by utilizing the first-order Maxwell operator and electromagnetic Green's functions to derive scattering matrices and noise terms that preserve bosonic commutation relations while enabling the modeling of non-Markovian emitter interactions and direct translation of classical electromagnetic simulations into quantum models.

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

Published 2026-10-06

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light behaves in two distinct ways depending on how closely we look at it. To the naked eye, or when we design a standard flashlight, light acts like a wave, rippling through space and bouncing off mirrors. But when we shrink our focus to the scale of atoms and the tiny particles of light called photons, the rules change. At this microscopic level, light is also a stream of individual particles, and the way it interacts with matter becomes a delicate quantum dance of probabilities. Scientists have long sought a reliable map to predict how these single photons move through complex structures, especially when those structures are made of materials that absorb or scatter light. This is crucial for building the next generation of quantum computers and ultra-sensitive sensors, which rely on controlling light at the single-particle level. However, existing maps often break down when the terrain gets complicated, such as when light travels through materials that are not perfectly clear or when the structures are too intricate to be described by simple, repeating patterns.

A team of researchers at Purdue University has now drawn a new, more versatile map for this microscopic world. They developed a fresh way to describe how light enters and leaves a photonic system, one that works even when the interior is messy, absorbing, or highly complex. Instead of trying to force every structure into a simple, idealized shape, their method treats the light's journey as a continuous flow governed by the fundamental laws of electromagnetism. They found that by using a specific mathematical tool known as a Green's function—which acts like a universal guide for how a disturbance spreads through a medium—they could predict exactly how photons would behave as they passed through a device. This approach allows them to account for the fact that real materials often soak up some energy, turning it into heat or random noise, without losing the ability to track the photons that make it through.

The core of their discovery is a new set of rules that connects the light coming in to the light going out. Imagine a photon entering a device through a specific channel, like a waveguide. As it travels through the interior, it encounters the material's unique properties. If the material absorbs some energy, it doesn't just disappear; it creates a kind of background static, or noise, that is an unavoidable part of the quantum world. The researchers showed that their new equations perfectly balance this loss. The light that enters, the light that scatters, and the noise generated by the material all fit together in a way that preserves the fundamental quantum nature of the light. This means that even in a lossy, imperfect system, the output light remains a valid quantum state, ready to be used for information processing.

To prove their theory works, the team applied it to two very different types of nanophotonic structures. First, they looked at a "fishbone" cavity, a tiny trap for light that is designed to hold photons in a specific spot. Second, they examined an inverse-designed coupler, a structure created by a computer algorithm to guide light in a very specific, non-intuitive way. In both cases, they used powerful computer simulations to calculate how light would move through these devices. They then inserted a theoretical atom, or emitter, into the mix to see how it would change the light's path. The results were striking: their method could accurately predict how the presence of the emitter would alter the transmission of light, including subtle shifts in frequency and changes in how quickly the light decayed. They were able to separate the effects of the device's shape from the effects of the material's absorption, providing a clear picture of what was happening inside.

One of the most significant aspects of this work is that it does not require the researchers to first break the complex structure down into simple, idealized modes. Traditional methods often struggle with irregular shapes or materials that change their properties depending on the frequency of the light. This new approach bypasses that hurdle entirely. It takes the raw electromagnetic response of the device, as calculated by standard simulation tools, and translates it directly into a quantum model. This means that engineers can now take a design for a complex, lossy device and immediately know how it will behave with single photons, without having to build a separate, simplified model first.

The researchers also demonstrated that their theory holds up even when the light interacts with multiple emitters. They showed that the collective behavior of these atoms, and how they talk to each other through the surrounding material, could be predicted with high precision. This is particularly important for future technologies that might rely on groups of atoms working together to process information. By keeping the full frequency dependence of the system, their method captures effects that simpler models miss, such as the memory the system retains about past interactions, a phenomenon known as non-Markovian behavior.

In the end, this work provides a direct bridge between the classical world of electromagnetic simulations and the quantum world of single photons. It offers a practical toolkit for scientists and engineers who are designing the intricate optical circuits of tomorrow. By showing that they can predict the quantum behavior of light in any arbitrary, lossy structure using only the classical response of the material, the team has removed a major barrier to the development of robust quantum devices. Their findings suggest that the path to building complex quantum networks is clearer than previously thought, as the tools to model these systems are now more powerful and universally applicable.

Technical Summary: Modal Input-Output Theory for Quantum Nanophotonics from the First-Order Maxwell Operator

Problem Statement
Quantum input-output (QIO) theory is essential for relating incident light in specified channels to the fields exiting a photonic system, particularly when embedded emitters are present. Conventional approaches, such as cavity and waveguide quantum electrodynamics (QED), typically rely on expanding fields into selected resonances or guided modes. However, extending these descriptions to complex, open, dispersive, and absorbing structures is challenging. Furthermore, macroscopic quantum electrodynamics (QED) has traditionally described field propagation via classical electromagnetic Green's functions to capture dispersion and absorption, often requiring specific boundary treatments or noise formulations that do not seamlessly integrate with modal input-output frameworks for arbitrary open systems. There is a need for a unified theory that connects waveguide ports to arbitrary interior media without requiring a pre-defined modal expansion of the internal field.

Methodology
The authors develop a QIO theory based on macroscopic QED using the first-order electromagnetic Green's function. The approach proceeds as follows:

First-Order Formalism: The electric ( $\mathbf{E}$ ) and magnetic ( $\mathbf{H}$ ) fields are combined into an impedance-normalized dual field $\mathbf{E} = [\mathbf{E}, Z_0\mathbf{H}]^T$ . Maxwell's equations are cast in a compact first-order form $\mathbf{M}(\omega)\mathbf{E} = \mathbf{S}$ , where $\mathbf{M}$ is the first-order Maxwell operator. The retarded Green's function $\mathbf{g}$ is defined as the inverse of this operator.
Boundary Decomposition: The scattering region is enclosed by planar input and output reference surfaces. The interior medium is arbitrary (dispersive, absorbing, structured), while the regions near the boundaries are assumed lossless and translationally invariant to define waveguide modes.
Field Quantization: The quantum field operator inside the region is expressed as the sum of a volume contribution from Langevin polarization noise (associated with material absorption) and a boundary contribution from the incoming field, both propagated by $\mathbf{g}$ .
Modal Projection: The boundary fields are expanded into a complete, flux-orthogonal basis of waveguide modes. By projecting the propagated field onto these output modes, the authors derive a modal input-output relation.
Emitter Coupling: The theory is extended to include $N$ embedded two-level emitters. In the low-saturation regime (or single-excitation subspace), the emitter dynamics are linearized. The total output field is derived by solving for the emitter operators driven by the background field and material noise, then projecting the radiated field onto the output modes.

Key Contributions

General Input-Output Relation: The paper derives a fundamental relation $\hat{b}_m(\omega) = \sum_n S_{mn}(\omega)\hat{a}_n(\omega) + \hat{\xi}_m(\omega)$ , where $\hat{a}_n$ and $\hat{b}_m$ are input and output channel operators. The scattering matrix $S_{mn}$ is calculated directly from the Green's function propagating fields between ports, and $\hat{\xi}_m$ represents noise from material absorption. This formulation preserves bosonic commutation relations even in lossy systems.
Recovery of Standard QED: The authors demonstrate that familiar cavity-QED and waveguide-QED input-output relations are recovered by inserting appropriate modal expansions of the Green's function into their general formalism.
Emitter-Mediated Scattering: A closed-form expression is derived for the scattering matrix of a system with embedded emitters: $S^{(N)}_{mn} = S_{mn} + (\eta^{out}_m)^T \mathbf{M}^{-1} \eta^{in}_n$ . Here, $\mathbf{M}$ is a collective response matrix containing self-energies and inter-emitter interactions mediated by the Green's function. This formulation retains the non-Markovian nature of the photonic environment.
Numerical Recipe: The theory provides a direct route from classical electromagnetic simulations (e.g., Finite-Difference Time-Domain, FDTD) to a quantum input-output model. It requires only a set of Green's functions connecting ports to ports, ports to emitters, and emitters to emitters, avoiding the need to construct a separate modal model of the device.

Results
Using FDTD simulations (via Tidy3D), the authors validated the theory on two nanophotonic structures:

Fishbone Cavity: A lossless dielectric cavity with air cladding.
Inverse-Designed Dipole Coupler: A bidirectional coupler designed via inverse methods.

For both structures, the authors reconstructed the emitter-modified transmission spectra. They calculated the bare transmission, the emitter-modified transmission at various detunings, the total electromagnetic decay rate enhancement ( $\Gamma_A/\Gamma_0$ ), the resonant frequency shift ( $\Delta_{res}$ ), and the phase-referenced coupling parameter. The results illustrate that enhanced decay rates do not necessarily imply deeper transmission dips, highlighting the importance of interference between the emitter radiation and the background transmitted field. The simulations successfully reconstructed the complex transmission amplitude using classical Green's functions derived from dipole source simulations.

Significance and Claims
The paper claims to provide a robust framework for quantum nanophotonics that is applicable to complex structures where traditional modal expansions are difficult or impossible to define. Specifically:

It handles arbitrary open, dispersive, and absorptive media characterized by $\bar{\varepsilon}(\mathbf{r}, \omega)$ without requiring an interior modal expansion.
It naturally treats magnetodielectric media and extends to chiral media with magnetoelectric coupling because the first-order formulation retains both electric and magnetic fields.
It is suitable for structures with photonic band edges, coupled resonances, and systems where loss and dispersion are central (e.g., plasmonics, 2D materials, inverse-designed geometries).
It offers a "direct numerical recipe" for quantum photonics: a small set of classical Green-function calculations determines the quantum optical response in the linear regime, bypassing the need to quantize a separate modal model of the device.

The authors note that while the current work focuses on the linear, low-saturation regime, the formalism lays the groundwork for future extensions to multiphoton scattering, nonlinear emitter dynamics, and strongly absorbing systems using causal material models.

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