🔬 mesoscale physics

Unity-order coupling between free electrons and multiphoton waveguided Fock states

This paper theoretically demonstrates that electrostatic steering of grazing electrons around a biased silicon waveguide enables unity-order coupling, achieving strong, tunable interaction with multiphoton waveguided Fock states while suppressing energy-loss channels.

Original authors: L. Prelat, S. Abdullah, C. I. Velasco, F. J. García de Abajo

Published 2026-05-28

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CC BY 4.0

Original authors: L. Prelat, S. Abdullah, C. I. Velasco, F. J. García de Abajo

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

Imagine you have a tiny, super-fast electron beam acting like a microscopic flashlight. Normally, if you shine this "flashlight" past a glass wire (a silicon waveguide) that carries light, the electron zips by so quickly that it barely has time to interact with the light inside the wire. It's like a race car driver speeding past a pit crew; they are too close for too short a time to really connect.

This paper proposes a clever trick to solve that problem: Electrostatic Steering.

Here is the breakdown of their idea using simple analogies:

1. The Problem: The "Fly-By" Effect

In standard setups, the electron travels in a straight line. To get it to interact with the light inside the wire, it has to get very close. But if it gets too close, it might crash into the wire or cause unwanted "noise" (like creating extra energy that isn't useful light). If it stays too far away, it doesn't transfer enough energy to create light. It's a difficult balancing act.

2. The Solution: The "Magnetic Slide" (But with Electricity)

The researchers suggest using an electric field to gently push the electron beam, making it curve.

The Analogy: Imagine a skier coming down a mountain. Instead of skiing in a straight line, they approach a gentle, curved slope that forces them to slow down, turn, and glide along the side of the mountain for a longer time before heading back up.
In the Paper: They use a "biased" silicon waveguide (essentially giving it an electric charge) and place electrodes nearby. This creates an invisible electric "wall" that repels the electron. As the electron approaches the wire, the electric push gets stronger, forcing the electron to stop moving closer, turn around, and glide away.

3. The "Turning Point" Advantage

This turning point is the magic ingredient.

Closer is Better (but not too close): Because the electron is forced to turn around at a specific, controlled distance, it can get much closer to the wire than it would in a straight-line crash.
More Time: Because it has to curve and turn, it spends more time "hanging out" near the wire. This gives it plenty of time to transfer its energy to the light waves inside the wire.
Selective Tuning: By adjusting the angle at which the electron arrives or the strength of the electric push (the voltage), the researchers can control exactly how close the electron gets. This allows them to "tune" which specific light colors (modes) get excited, like tuning a radio to a specific station while ignoring the static.

4. The Result: A Photon Factory

The paper claims that by using this steering method with 100 keV electrons (very fast ones), they can generate a huge amount of light.

The Numbers: They predict that for every single electron that goes through this process, it will create an average of more than ten photons (particles of light) inside the waveguide.
Clean Energy: Because the electron never actually touches the wire (it stays at a safe distance), it avoids creating messy, high-energy waste. It only creates the specific, useful light waves the researchers want.

5. The "Ghost" Force (Image Potential)

There is one tricky part the paper had to account for. When an electron gets close to a surface, it creates an invisible "ghost" attraction (called an image force) that tries to pull it into the surface, like a magnet sticking to a fridge.

The Fix: The researchers calculated that if the electric repulsion (the steering force) is strong enough, it can overcome this ghost pull. This ensures the electron turns around safely without crashing into the wire.

Summary

In short, the paper demonstrates a way to use electric fields to steer a fast electron beam so it "grazes" a silicon wire, turns around, and glides away. This controlled dance allows the electron to dump a lot of its energy into the wire, creating a burst of light (multiphoton states) without crashing or making a mess. It turns a quick, ineffective fly-by into a productive, tunable light-generating event.

Technical Summary: Unity-order coupling between free electrons and multiphoton waveguided Fock states

Problem Statement
Free electron beams offer a powerful platform for the localized excitation and probing of optical modes in dielectric waveguides, enabling the generation of nonclassical light, including multiphoton Fock states. However, the coupling efficiency between free electrons and waveguided modes is typically constrained by the short interaction times inherent in straight-line trajectories with fixed impact parameters. In conventional configurations, the coupling strength is determined by the electron's kinetic energy and a fixed distance from the waveguide. This limitation often results in the simultaneous excitation of multiple degenerate modes and undesired lossy, high-energy channels (such as interband transitions above the material's band gap), making it difficult to achieve spectral and modal selectivity. While trajectory engineering has been proposed theoretically, its practical implementation for mode-selective excitation in integrated platforms remains largely unexplored.

Methodology
The authors theoretically investigate the coupling between a free electron beam and a silicon (Si) waveguide, employing a framework based on Electron Energy-Loss Spectroscopy (EELS). The study proceeds in three stages:

Reference Configuration: The authors first analyze a planar, infinitely extended Si waveguide with an electron moving parallel to the surface at a fixed distance $b$ . They calculate the momentum-resolved EELS probability to identify the dispersion of guided modes (TE and TM) and quantify the suppression of interband transitions as the impact parameter increases.
Trajectory Engineering via Electrostatic Steering: To overcome the limitations of fixed trajectories, the authors propose shaping the electron path using external DC electric fields. They model the electron motion under the influence of both the applied electrostatic potential and the attractive image force generated by the dielectric interface. The electron trajectory is calculated by solving the equation of motion, identifying a "turning point" where the repulsive electrostatic force balances the image attraction, creating a grazing trajectory with a tunable minimum separation ( $b$ ) from the waveguide.
Configuration Analysis:
- Vertical Gating: A tutorial model using a uniform DC field (parallel plate capacitor geometry) to produce a parabolic trajectory.
- Lateral Gating: A more practical design involving a 1D Si waveguide on a sapphire substrate flanked by two Si linear gates. This configuration generates a spatially varying repulsive field. The authors use the Boundary Element Method (BEM) to calculate the EELS probability for the 1D geometry and self-consistently solve for the DC potential distribution to determine the electron trajectory.

The total excitation probability is obtained by integrating the EELS probability per unit path length along the curved electron trajectory. The analysis accounts for relativistic effects (via the Lorentz factor $\gamma$ ) and image-charge interactions.

Key Contributions and Results

Electrostatic Steering Mechanism: The paper demonstrates that electrostatic steering introduces a tunable turning point in grazing electron trajectories. This mechanism allows for dynamic control over the minimum electron–waveguide separation ( $b$ ) and the interaction length, which are critical parameters for coupling strength and modal selectivity.
Suppression of Lossy Channels: By maintaining a relatively large impact parameter (tens of nanometers) via the turning point, the study shows that the excitation of lossy high-energy channels (interband transitions across the Si band gap) is strongly suppressed. This preserves the excitation of long-lived, low-loss waveguided modes.
High Photon Yield: Using a practical lateral gating configuration with 100 keV electrons, the authors predict an average photon yield exceeding ten photons per electron. Specifically, for a minimum separation of $b = 50$ nm and an incidence angle of $\sim 0.6$ mrad (corresponding to a bias voltage difference of $\sim 45$ mV), the first guided mode yields $\sim 15$ photons per electron, and the second yields $\sim 5$ photons.
Role of Image Forces: The study quantifies the impact of the image force, which is attractive and can pull the electron toward the surface. The authors define a threshold distance ( $b_{th}$ ) below which the electron would collide with the waveguide. They demonstrate that appropriately designed electrostatic fields can overcome this image attraction, enabling stable grazing trajectories. The inclusion of image forces in the trajectory calculation is shown to enhance the photon yield by smoothing the potential energy profile and increasing the dwell time near the waveguide.
Modal Selectivity: The coupling strength and spectral distribution are shown to be tunable through the electron incidence angle, kinetic energy, and applied electrostatic bias.

Significance and Claims
The paper establishes electrostatic steering as a practical route for engineering and enhancing free-electron coupling to waveguided photonic modes. The authors claim that their results provide a mechanism to achieve "unity-order coupling," where the interaction probability is of order unity, leading to the generation of multiphoton Fock states in integrated nanophotonic devices.

The study clarifies that the analysis is performed within a classical EELS framework, where the calculated probabilities represent mean photon numbers. In a quantum description, this corresponds to the electron and the waveguided field forming an entangled state in the photon-number basis. The work suggests that electrically controlled electron paths can be used to enhance and tune free-electron coupling, opening a route toward deterministic or near-deterministic generation of waveguided photons and multiphoton Fock states without the need for complex optical coupling schemes like gratings or edge coupling. The proposed lateral gating geometry is highlighted as being amenable to conventional nanofabrication methods such as electron-beam lithography.