Canonical Quantization of Cylindrical Waveguides: A Gauge-Based Approach

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 $φ$) 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.

💡 Research Summary

The manuscript presents a comprehensive canonical quantization scheme for electromagnetic modes propagating in cylindrical waveguides, extending the gauge‑based formalism previously introduced for Cartesian (parallel‑plate) geometries. Starting from Maxwell’s equations in cylindrical coordinates, the authors systematically derive the field profiles for three families of guided waves—TEM, TM, and TE—both for coaxial structures (inner radius a, outer radius b) and for hollow cylindrical pipes (radius a). The radial dependence of the fields is expressed in terms of Bessel functions of the first and second kind, Jₙ and Yₙ, with the appropriate boundary conditions leading to discrete dispersion relations for the transverse wavenumber k_c. For TM modes the condition Jₙ(k_c a)Yₙ(k_c b)−Jₙ(k_c b)Yₙ(k_c a)=0 determines the allowed k_c values, while TE modes satisfy J′ₙ(k_c a)Y′ₙ(k_c b)−J′ₙ(k_c b)Y′ₙ(k_c a)=0. Each mode is indexed by an azimuthal integer n and a radial mode number m, and the longitudinal propagation constant β follows from k² = k_c² + β², giving a cutoff frequency ω_c = c k_c.

A central contribution is the introduction of a generalized flux variable φ(z,t), defined through a specific gauge choice for the vector potential A and scalar potential V. This flux is shown to obey a one‑dimensional Klein‑Gordon equation (∂_t² φ − v_φ² ∂_z² φ + ω_c² φ = 0) and serves as the canonical coordinate for each guided mode. The conjugate momentum Π(z,t) = C ∂_t φ, where C is the mode‑specific effective capacitance per unit length, together with φ satisfy the canonical commutation relation