Canonical Quantization of Cylindrical Waveguides: A Gauge-Based Approach

TL;DR

This paper develops a gauge-based canonical quantization framework for electromagnetic modes in cylindrical waveguides, unifying the treatment of different guided modes and enabling precise quantum descriptions of measurable quantities.

Contribution

It introduces a novel gauge-based formalism for quantizing cylindrical waveguide modes, extending previous Cartesian approaches and deriving explicit Hamiltonians and mode-specific parameters.

Findings

Unified treatment of cylindrical and Cartesian guided modes.

Explicit Hamiltonian derived from Maxwell's equations.

Quantization of mode-specific voltage and current operators.

Abstract

We present a canonical quantization of electromagnetic modes in cylindrical waveguides, extending a gauge-based formalism previously developed for Cartesian geometries [1]. By introducing the two field quadratures $X,Y$ of TEM (transverse electric-magnetic), but also of TM (transverse magnetic) and TE (transverse electric) traveling modes, we identify for each a characteristic one-dimensional scalar field (a generalized flux $\varphi$) governed by a Klein-Gordon type equation. The associated Hamiltonian is derived explicitly from Maxwell's equations, allowing the construction of bosonic ladder operators. The generalized flux is directly deduced from the electromagnetic potentials $A,V$ by a proper gauge choice, generalizing Devoret's approach [2]. Our analysis unifies the treatment of cylindrical and Cartesian guided modes under a consistent and generic framework, ensuring both theoretical insight and experimental relevance. We derive mode-specific capacitance and inductance from the field profiles and express voltage and current in terms of the canonical field variables. Measurable quantities are therefore properly defined from the mode quantum operators, especially for the non-trivial TM and TE ones. The formalism shall extend in future works to any other type of waveguides, especially on-chip coplanar geometries particularly relevant to quantum technologies.