[Paper Review] Upper bound for the quantum coupling between free electrons and photons

Zhexin Zhao|arXiv (Cornell University)|Apr 1, 2024

Electron and X-Ray Spectroscopy TechniquesMaterials Science3 citations

TL;DR

This paper derives the analytical upper bound for the quantum coupling coefficient between free electrons and photons using macroscopic quantum electrodynamics (MQED), showing that strong coupling is achievable under optimal conditions—particularly with sub-wavelength separation, low electron velocities (β ≈ 0.1–0.4), and high-permittivity materials like silicon. The upper bound peaks at sub-relativistic electron speeds and scales with interaction length and geometric factors.

ABSTRACT

The quantum interaction between free electrons and photons is fundamental to free-electron based light sources and free-electron quantum optics applications. A large coupling between free electrons and photons is generally desired. In this manuscript, I obtain the upper bound for the quantum coupling between free electrons and photons. The upper bound has a straightforward expression and can be applied to a broad range of optical materials, especially widely used low-loss photonic materials. The upper bound depends on the optical medium, the free-electron velocity, and the separation between the free electron and the optical medium. With simple structures, the numerically calculated coupling coefficient can reach ~99% of the upper bound. This study provides simple and practical guidance to reach the strong coupling between free electrons and photons.

Motivation & Objective

• To establish the fundamental theoretical limit of quantum coupling between free electrons and photons in photonic systems.
• To analyze the upper bound of the coupling coefficient in two regimes: continuous photonic spectra and discrete modes in lossless cavities.
• To guide experimental design by identifying optimal system parameters—electron velocity, separation distance, material permittivity, and interaction length—for achieving strong coupling.
• To connect the continuous and discrete spectral cases through low-loss approximations, enabling broader applicability to realistic photonic structures.

Proposed method

• Formulates the interaction Hamiltonian within macroscopic quantum electrodynamics (MQED), treating both electrons and photons quantum mechanically.
• Derives the quantum coupling coefficient using the interaction Hamiltonian and field operators, incorporating electron wave functions and photonic Green’s functions.
• Introduces a geometric factor $ g^2_{\text{geo}} $ that depends on electron velocity and transverse separation from the photonic structure, capturing spatial overlap effects.
• Applies the analytical upper bound to realistic systems, including silicon and SiN photonic structures, with varying electron-beam separation and interaction length.
• Uses numerical simulations to demonstrate the dependence of the upper bound on electron velocity $ \beta = v/c $, separation $ d $, and material properties.
• Validates the upper bound using a guided mode in a silicon sub-wavelength grating, achieving $ |g_{\text{Qu}}| = 0.1 $, indicating strong coupling potential at $ L > 95\lambda $

Experimental results

Research questions

• RQ1 What is the fundamental upper limit of the quantum coupling coefficient between free electrons and photons in arbitrary photonic environments?
• RQ2 How does the coupling upper bound depend on electron velocity, transverse separation from the photonic structure, and material permittivity?
• RQ3 What is the connection between the upper bound in continuous-spectrum systems and discrete-mode cavities under low-loss conditions?
• RQ4 Can the derived upper bound be experimentally approached, and under what conditions is strong coupling most feasible?

Key findings

• The upper bound for the quantum coupling coefficient $ g_{\text{ub}} $ is maximized at sub-relativistic electron velocities ($ \beta \approx 0.1–0.4 $) and decreases with increasing electron speed.
• For $ d = 0.02\lambda $, $ L = \lambda $, and silicon ($ \varepsilon_r \approx 11.9 $), $ g_{\text{ub}} > 1 $, indicating a strong coupling regime is theoretically accessible.
• The coupling upper bound is twice as large for silicon compared to SiN due to silicon’s higher permittivity.
• The geometric factor $ g^2_{\text{geo}} $, which captures spatial overlap, peaks at low $ \beta $ and decays with increasing separation $ d $, especially when $ d/\lambda < 0.06 $.
• A numerical example with a silicon sub-wavelength grating achieves $ |g_{\text{Qu}}| = 0.1 $ at $ \beta = 0.25 $, implying strong coupling is possible with interaction lengths $ L > 95\lambda $.
• The analytical upper bound remains valid when the electron beam size is smaller than the separation $ d $, suggesting practical feasibility in transversely confined beams.