Quantum Theory of Light in a Dispersive Structured Linear Dielectric: a Macroscopic Hamiltonian Tutorial Treatment

TL;DR

This tutorial presents a macroscopic Hamiltonian approach to quantizing light in dispersive, nonuniform dielectrics, emphasizing mode normalization, nonorthogonality, and quantum optical applications without detailed medium physics.

Contribution

It introduces a macroscopic Hamiltonian framework for quantizing light in dispersive, nonuniform dielectrics, avoiding Lagrangian methods and providing new mode normalization insights.

Findings

Derived mode normalization condition

Demonstrated nonorthogonality of modes

Applied to waveguide quantum optics

Abstract

These notes, intended to be self contained and tutorial, present a direct, macroscopic approach to quantizing light inside a linear-response dielectric material when both spectral dispersion and spatial nonuniformity are present, but the spectral region of interest is optically transparent so that explicit treatment of the underlying physics of the medium is not needed. The approach taken is based on the macroscopic Maxwell equations and a corresponding Hamiltonian, without the use of Lagrangians or any dynamical model for the medium, and uses a standard mode-based quantization method. The treatment covers: energy density and flux in a dispersive dielectric; a summary of the inverse permittivity formalism; a new derivation of the mode normalization condition; a direct proof of the nonorthogonality of the modes; examples of quantized field expressions for the general case and various special cases; the relationship between group velocity and energy flux; the band approximation and the continuum limit; and quantum optical treatment of waveguide modes.