Canonical Quantization of Cylindrical Waveguides: A Gauge-Based Approach

TL;DR

This work extends canonical quantization of electromagnetic fields from Cartesian to cylindrical waveguides, introducing a generalized flux $φ$ and two quadratures $X$ and $Y$ to describe TEM, TM, and TE traveling modes. By fixing an appropriate gauge that links $φ$ to the potentials $\mathbf{A}$ and $V$, the authors derive mode-specific Klein-Gordon or wave equations and construct a Hamiltonian with geometry-dependent capacitance and inductance densities, enabling bosonic quantization with $b$ and $b^ abla$ operators. The framework provides explicit modal profiles via Bessel functions, handles virtual electrodes for certain TE modes, and yields measurable quantities such as voltage and current from the canonical variables, thereby unifying cylindrical and Cartesian treatments. The results pave the way for integrating cylindrical-waveguide quantization into on-chip quantum technologies and quantum-limited devices, including a natural connection to transmission-line theory through a generalized telegrapher’s equation.

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.

Figures (3)

• Figure 1: Geometries with cylindrical symmetry. a) coaxial line. b) hollow cylinder. The axes corresponding to the coordinate system $r,\theta,z$ are displayed. Real electrodes are shown in blue ($in$ and $out$ labels for coaxial, and $front$, $back$ for hollow pipe). The virtual electrodes (labeled vir top and vir bottom) are shown in green. See text for details.
• Figure 2: Transverse currents of TE modes with $n=0$. a) coaxial case. b) hollow cylinder. The direction of the outer current circulation depends on the mode number $m$; for the coaxial line, it is also a function of the ratio $b/a$ (but the behavior becomes equivalent to the hollow guide at $b/a \gg 1$). Considering this limit, we represent in red the $m$ even case; the outer current is reversed for $m$ odd (yellow).
• Figure 3: Difference between the gauge profile Eq. (\ref{['eqgaugerTEcoax']}) and the $g_{vir}(r)$ function (no units, to be compared to 1). a) For mode $m=1$, different $b/a$ ratios. b) For $b/a=2.$, different mode numbers $m$. See text for details.