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 $Z_{load}(α)$, 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.
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.
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.
Classical microwave circuit theory is incapable of representing some phenomena at the quantum level. To include quantum statistical effects, various theoretical treatments can be employed. Quantum input-output network (QION) theory is one such treatment. Another formalism, called SLH theory, incorporates scattering matrices (S), coupling vectors (L), and system Hamiltonians (H). These theoretical treatments require a reformulation of classical microwave theory. To make these topics comprehensible to an electrical engineer, we demonstrate some underpinnings of microwave quantum optics in terms of microwave engineering. For instance, we equate traveling-wave phasors in a transmission line (V0+) directly to bosonic field operators. Furthermore, we extend QION to include a state-space representation and a transfer function for a single port quantum network. This serves as a case study to highlight how microwave methodologies can be applied in open quantum systems. Although the same conclusion could be found from a full SLH theory treatment, our method was derived directly from first principles of QION.
This manuscript establishes the absolute operator-theoretic closure of quantum electricity. It formally defines quantum electricity strictly as the gauge-covariant conserved charge transport generated by a self-adjoint matter Hamiltonian under electromagnetic minimal coupling, rather than treating it as a supplementary physical ontology. The paper rigorously derives the exact one-particle current, its operator-valued local density, its Heisenberg continuity law, and its variational identity as the functional derivative of the Hamiltonian with respect to the vector potential. The framework is then systematically extended to spin-1/2 Pauli matter—proving the exact decomposition of the conserved current into distinct transport and divergence-free magnetization sectors—and to second quantization, establishing the many-body continuity law. Furthermore, the manuscript demonstrates that classical electrical source fields emerge exactly as the macroscopic expectation values of the quantum current operators. This provides the rigorous source-side completion for Maxwell field evolution. The theory is also generalized to relativistic quantum electrodynamics (identifying the current as the Noether current of local U(1) gauge invariance), expressed through spectral representation, and applied to open-system transport via scattering theory and wave operators. Ultimately, the manuscript proves a unified closure: all admissible electrical behavior is completely exhausted by the combination of gauge-covariant matter current, Maxwell field evolution, constitutive response, and boundary data.
Classical microwave circuit theory is incomplete in that it is incapable of fully modeling some phenomena at the quantum level. Various theoretical frameworks can be employed to incorporate single-photon statistical effects in the treatment of microwave networks. Such methods include quantum input-output network (QION) theory and SLH theory. A synthesis of the quantum and classical circuit treatments requires a description of second quantization within classical microwave theory. In order to make these topics understandable to an electrical engineer, we demonstrate some underpinnings of quantum optics in terms of microwave engineering. For example, we relate traveling-wave phasors for transmission-lines, such as voltage and current, to bosonic field operators. The second quantization and necessary components for a quantum treatment of microwave circuit theory are summarized in a table that maps microwave scattering parameters to bosonic operators in a transmission-line. To illustrate the need for second quantization, we use the results of second quantization along with first principles of quantum input-output network theory to determine a state-space representation and a transfer function of a single port quantum network. The same results could be obtained from SLH theory, and this serves as a case study for applying microwave theory to open quantum systems. The results of this work could be used to treat NISQ and other superconducting hardware. Additionally, these results could be incorporated into an applied curriculum, assisting in the engineering education of the future quantum workforce.