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.
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.
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.
Circuit modeling and simulation techniques are necessary to the development of advanced microwave architectures. In particular, microwave circuits used in the quantum regime can produce a variety of single-photon phenomena related to qubits such as unitary gate operations, quantum illumination, and quantum interferometry. However, classical microwave formalism is incomplete to sufficiently describe these systems at this level. Noisy environments, for instance, destroy internal quantum information at the single-photon level, which makes it difficult to reliably control a quantum state. A theoretical infrastructure to account for this problem has been well-established in quantum optics as quantum input-output network (QION) theory. In this work, we consider an extension to QION theory, called SLH theory, to expand quantum stochastic differential equations to multi-port networks. A mathematical object composed of a network’s scattering parameters (S), coupling vector (L), and internal energy (H), known as an SLH triple, can fully describe its dynamical evolution at the quantum level. We show how microwave components can be converted into an SLH framework that retains the mathematical formalism of classical S-parameter techniques at the quantum limit. This makes SLH theory applicable to many interdisciplinary subfields within physics and engineering. We describe how a dynamical circuit model with universal quantum multi-port networks can be devised within SLH formalism. This can be applied to microwave circuit designs in the quantum limit.
The precise engineering of electromagnetic couplings is paramount for constructing scalable and high-fidelity superconducting quantum processors. While essential for orchestrating qubit operations, these couplings also present significant design challenges, including the mitigation of crosstalk and the management of environmental decoherence. A clear and unified theoretical framework is therefore crucial for the design, simulation, and analysis of these complex quantum circuits. This paper presents a comprehensive theoretical treatment of the fundamental electromagnetic coupling mechanisms in superconducting devices. Starting from first principles, we formulate the equations of motion and derive the input-output relations for canonical systems, including a single resonator coupled to a multi-port microwave network, interacting resonators, and coupled transmission lines. We review rigorous definitions for key parameters such as the energy decay rate (κ) and the dimensionless coupling coefficient (ζ) and connect these formalisms to practical methods of parameter extraction from electromagnetic simulations. This work provides a rigorous and pedagogical foundation for understanding and modeling linear electromagnetic interactions, serving as a vital resource for the development of advanced superconducting quantum hardware.
Quantum information processing technologies are gaining interest due to their potential for performance exceeding what is possible with the best classical computers. One of the most promising platforms is the circuit quantum electrodynamics (cQED) architecture that controls superconducting circuit qubits by coupling them to electromagnetic fields in microwave resonators or waveguides. As these devices become increasingly complex, existing analysis and design approaches that often rely on lumped element circuit approximations can become inadequate due to their lack of accuracy and flexibility. This is leading to a growing interest in developing quantum full-wave numerical modeling methods for the high-fidelity analysis of cQED devices. In this work, we focus on a quantum full-wave modeling approach that uses a projector-based quantization approach that allows a problem to be decomposed into distinct subdomains whose fields can be quantized more easily. This leads to a framework that can naturally incorporate microwave ports, which is not straightforward with other formulations. Here, we consider an explicit problem of a coaxial-fed rectangular waveguide cavity to validate parts of our projector-based formalism. Toward this end, we apply this approach to a classical Hamiltonian analysis of this system that can be used to more easily validate the relevant full-wave aspects of the corresponding quantum formulation. In addition to presenting the formulation of the full-wave classical Hamiltonian of the system, we derive equations of motion and discuss an analytical solution procedure with an input-output theory approach. We validate our formulation by comparing the input-output theory results to standard finite element method (FEM) scattering parameter simulations.
Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, e.g. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple (S,L,H). Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.
Quantum feedback networks have been introduced in quantum optics as a framework for constructing arbitrary networks of quantum mechanical systems connected by unidirectional quantum optical fields, and has allowed for a system theoretic approach to open quantum optics systems. Our aim here is to establish a network theory for quantum transport systems where typically the mediating fields between systems are bidirectional. Mathematically, this leads us to study quantum feedback networks where fields arrive at ports in input-output pairs, making it a special case of the unidirectional theory where inputs and outputs are paired. However, it is conceptually important to develop this theory in the context of quantum transport theory-the resulting theory extends traditional approaches which tend to view the components in quantum transport as scatterers for the various fields, in the process allowing us to consider emission and absorption of field quanta by these components. The quantum feedback network theory is applicable to both Bose and Fermi fields, moreover, it applies to nonlinear dynamics for the component systems. We advance the general theory, but study the case of linear passive quantum components in some detail.