Exact Formulation of the Resonant Coupling of a Deep-Subwavelength Particle in a Rectangular Conducting Cavity: Beyond the Jaynes-Cummings Model

We investigate the resonant interaction between a deep-subwavelength particle and a perfectly conducting rectangular cavity, with potential applications in cavity classical and quantum electrodynamics and wave physics. The particle may behave as a harmonic oscillator, such as a semiconductor plasmonic nanoparticle, a ferrite particle, or as a two-level system, such as a Rydberg atom or trapped ion operating in the microwave regime, where metals can be modeled as perfect conductors at microwave frequencies. Our primary objective is to determine the joint resonances (eigenfrequencies) of the coupled cavity-particle system without imposing restrictions on the coupling strength. Unlike conventional quantum optics models, such as the Jaynes-Cummings model, which typically consider interactions with a limited number of cavity modes and may suffer from causality violations, our formulation fully accounts for the complete mode spectrum of the cavity, ensuring strict causality. From a mathematical standpoint, we introduce a novel singularity extraction technique (renormalization) for the local field Green's function, $G^{\text{loc}}(r)$, which represents the field at the particle's location due to backscattering of its own radiation. This function is defined as the difference between the cavity Green's function and its free-space counterpart, resulting in a numerically challenging $\infty - \infty$ form. To precisely calculate this limit, we develop a stable and efficient recursive approach tailored for a rectangular cavity, which can be easily applied to other structured cavities and waveguides [...]