Abstract We present a canonical quantization of electromagnetic modes in cylindrical waveguides, extending a gauge-based formalism previously developed for Cartesian geometries (Collin and Delattre 2025 New J. Phys. 27 093502). By introducing the two field quadratures X , Y of 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 (Devoret 1997 Les Houches, Session LXIII). Our analysis unifies the treatment of cylindrical and Cartesian guided modes under a consistent and generic framework, ensuring both theoretical insight and experimental relevance. Especially, we find out that TE n = 0 modes break a fundamental gauge symmetry that is observed by all other families. 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.
Abstract We present a canonical quantization of electromagnetic modes in cylindrical waveguides, extending a gauge-based formalism previously developed for Cartesian geometries (Collin and Delattre 2025 New J. Phys. 27 093502). By introducing the two field quadratures X , Y of 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 (Devoret 1997 Les Houches, Session LXIII). Our analysis unifies the treatment of cylindrical and Cartesian guided modes under a consistent and generic framework, ensuring both theoretical insight and experimental relevance. Especially, we find out that TE n = 0 modes break a fundamental gauge symmetry that is observed by all other families. 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.
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. Especially, we find out that TEn=0 modes break a fundamental gauge symmetry that is observed by all other families. 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.
Abstract Solid state quantum devices, operated at dilution cryostat temperatures, are relying on microwave signals to both drive and read-out their quantum states. These signals are transmitted into the cryogenic environment, out of it towards detection devices, or even between quantum systems by well-designed waveguides , almost lossless when made of superconducting materials. Here we report on the quantum theory that describes the simplest Cartesian-type geometries: parallel plates, and rectangular tubes. The aim of the article is twofold: first on a technical and pragmatic level, we provide a full and compact quantum description of the different traveling wave families supported by these guides. Second, on an ontological level, we interpret the results and discuss the nature of the light fields corresponding to each mode family. The concept of potential difference is extended from transverse electric-magnetic (TEM) waves to all configurations, by means of a specific gauge fixing . The generalized flux φ ^ introduced in the context of quantum electronics becomes here essential: it is the scalar field, conjugate of a charge Q ^ , that confines light within the electrodes, let them be real or virtual . The gap in the dispersion relations of non-TEM waves turns out to be linked either to a potential energy necessary for the photon confinement , or to a kinetic energy arising from a photon mass . We finally compute the field zero-point fluctuations in every configuration. The theory is predictive: the lowest transverse magnetic modes should have smaller quantum noise than the higher ones, which at large wavevectors recover a conventional value similar to TEM and transverse electric ones. Such low-noise modes might be particularly useful for the routing of quantum information.
We report on the mapping of classical microwave transmission line theory onto the quantum scattering matrix description. By means of a generalized flux formalism a la Devoret, we derive a charge-current vector living on the confining electrodes, to which a Hodge-like decomposition is applied. The presented theory holds equally well for all types of waves traveling within Cartesian and cylindrical guide geometries (the usual Transverse Electric Magnetic, but also Transverse Magnetic and Transverse Electric ones). We then demonstrate how a generic load decomposes into a series of complex coefficients Zload(α), for each mode α propagating along the line. This decomposition leads to a particular classification of traveling waves: gradient-type and curl-type charge-currents. This distinction shines a new light on a broken gauge symmetry exhibited by a class of Transverse Electric modes. Our results provide a universal theoretical framework, which finds a direct usage in microwave-based quantum information transfer.
The notion that the electromagnetic field is quantized is usually inferred from observations such as the photoelectric effect and the black-body spectrum. However accounts of the quantization of this field are usually mathematically motivated and begin by introducing a vector potential, followed by the imposition of a gauge that allows the manipulation of the solutions of Maxwell's equations into a form that is amenable for the machinery of canonical quantization. By contrast, here we quantize the electromagnetic field in a less mathematically and more physically motivated way. Starting from a direct description of what one sees in experiments, we show that the usual expressions of the electric and magnetic field observables follow from Heisenberg's equation of motion. In our treatment, there is no need to invoke the vector potential in a specific gauge and we avoid the commonly used notion of a fictitious cavity that applies boundary conditions to the field.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.