Diagonalization of Hamiltonian for finite-sized dispersive media: Canonical quantization with numerical mode-decomposition (CQ-NMD)

TL;DR

This paper introduces a novel modeling approach combining canonical quantization with numerical mode-decomposition to analyze quantum interactions in finite-sized dispersive media, surpassing previous methods like Fano-diagonalization.

Contribution

The paper develops a new mathematical framework that integrates Hamiltonian mechanics and computational electromagnetics for quantum modeling of dispersive media.

Findings

Successfully models quantum effects like non-local dispersion cancellation.

Simulates Hong-Ou-Mandel effect in dispersive beam splitters.

Provides a computational method for finite-sized dispersive media.

Abstract

We present a new math-physics modeling approach, called canonical quantization with numerical mode-decomposition, for capturing the physics of how incoming photons interact with finite-sized dispersive media, which is not describable by the previous Fano-diagonalization methods. The main procedure is to (1) study a system where electromagnetic (EM) fields are coupled to non-uniformly distributed Lorentz oscillators in Hamiltonian mechanics, (2) derive a generalized Hermitian eigenvalue problem for conjugate pairs in coordinate space, (3) apply computational electromagnetics methods to find a countably/finite set of time-harmonic eigenmodes that diagonalizes the Hamiltonian, and (4) perform the subsequent canonical quantization with mode-decomposition. Moreover, we provide several numerical simulations that capture the physics of full quantum effects, impossible by classical Maxwell's equations, such as non-local dispersion cancellation of an entangled photon pair and Hong-Ou-Mandel (HOM) effect in a dispersive beam splitter.