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arXiv:2104.06996

Circuit quantum electrodynamics: A new look toward developing full-wave numerical models

Thomas E. Roth, Weng Cho Chew

Published Apr 14, 2021• 1 citation
arXivHTMLPDF

Abstract

Devices built using circuit quantum electrodynamics architectures are one of the most popular approaches currently being pursued to develop quantum information processing hardware. Although significant progress has been made over the previous two decades, there remain many technical issues limiting the performance of fabricated systems. Addressing these issues is made difficult by the absence of rigorous numerical modeling approaches. This work begins to address this issue by providing a new mathematical description of one of the most commonly used circuit quantum electrodynamics systems, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our new model is better suited to developing numerical solvers than the circuit element descriptions commonly used in the literature. We present details on the quantization of our new model, and derive quantum equations of motion for the coupled field-transmon system. These results can be used in developing full-wave numerical solvers in the future. To make this work more accessible to the engineering community, we assume only a limited amount of training in quantum physics and provide many background details throughout derivations.

DOI
10.48550/arxiv.2104.06996

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Devices built using circuit quantum electrodynamics architectures are one of the most popular approaches currently being pursued to develop quantum information processing hardware. Although significant progress has been made over the previous two decades, there remain many technical issues limiting the performance of fabricated systems. Addressing these issues is made difficult by the absence of rigorous numerical modeling approaches. This work begins to address this issue by providing a new mathematical description of one of the most commonly used circuit quantum electrodynamics systems, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our new model is better suited to developing numerical solvers than the circuit element descriptions commonly used in the literature. We present details on the quantization of our new model, and derive quantum equations of motion for the coupled field-transmon system. These results can be used in developing full-wave numerical solvers in the future. To make this work more accessible to the engineering community, we assume only a limited amount of training in quantum physics and provide many background details throughout derivations.
Devices built using superconducting circuits are a popular approach being pursued to develop quantum information processing hardware. These systems involve the quantized interactions of electromagnetic fields with large coherent collections of charges in superconductors, and so are typically referred to as macroscopic circuit quantum electrodynamics (QED) devices or simply circuit QED devices. Although significant progress has been made recently, substantial engineering efforts are required to improve performance and scale these devices so they can address problems of practical interest. Performing these engineering efforts is difficult due to the absence of rigorous numerical modeling approaches. This work begins to address this by providing a new mathematical description of a commonly used circuit QED system, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our model is better suited to developing rigorous numerical solvers than approximate lumped element circuit descriptions used in the literature. We present details on the quantization of our model, and derive equations of motion for the coupled field-transmon system. These results can be used in developing full-wave solvers in the future. To make this work more accessible, we assume a limited amount of training in quantum physics and provide many background details.
Currently, circuit quantum electrodynamics architectures have emerged as one of the most popular approaches to implement practical quantum information processing hardware. Although significant progress has been made, many technical issues remain that limit the performance of fabricated devices. One approach to accelerate progress in the engineering design of these devices is to develop improved numerical modeling methods. Current modeling methods generally rely on approximate lumped element circuit models to describe the complex network of microwave transmission lines required to operate a circuit quantum electrodynamics device. This reduction in complexity in the theoretical model is valuable for gaining insight into the operation of a device, but does limit the opportunity for using these models to optimize the performance of practical devices. To develop rigorous numerical modeling methods, it is necessary to move toward theoretical descriptions of circuit quantum electrodynamics devices that retain the full details of the three-dimensional vector electromagnetic fields that exist in these systems. In this work, we present details on such a theoretical model for one of the most commonly used circuit quantum electrodynamics systems, a transmon qubit coupled to microwave transmission lines. We discuss the quantization of our new model and show that by adopting relevant approximations our model can be reduced to the same lumped element descriptions commonly used in the literature. We also discuss the derivation of quantum equations of motion for the coupled field-transmon system, which can be used as the starting point for developing full-wave numerical solvers for these circuit quantum electrodynamics systems in the future.
One of the most popular approaches being pursued to achieve a quantum advantage with practical hardware is the use of superconducting circuit devices. Although significant progress has been made over the previous two decades, substantial engineering efforts are required to scale these devices so they can be used to solve many problems of interest. Unfortunately, much of this exciting field is described using technical jargon and concepts from physics that are unfamiliar to a classically trained electromagnetics engineer. As a result, this work is often difficult for engineers to become engaged in.
In the last two decades, the field of circuit quantum electrodynamics, that studies the interaction between superconducting qubits and 1-dimensional waveguides, has been of great interest. It provides a great potential to build quantum devices, which are important for quantum computing, quantum communication and quantum information. The restriction to one dimension decreases losses and information can be transferred efficiently. Superconducting qubits are artificial atoms that consist of a non-linear Josephson element and work in the microwave regime. These superconducting qubits make on-chip tunable quantum experiments possible.In the appended paper, we investigate the spontaneous emission of an initially excited artificial atom (superconducting transmon qubit) which is capacitively coupled to a semi-infinite transmission line (atom in front of a mirror). We can choose the distance to the mirror arbitrarily so the interaction with the reflected field is delayed if the qubit is far away from the mirror and we have to take time-delay effects into account. We derive equations of motion for the transmon by circuit quantization and solve them semi-classically. In this thesis we give an introduction to circuit quantization, transmission lines and superconducting qubits. Then we discuss the methods and results of the appended paper which are based on the topics introduced.
Circuit quantum electrodynamics is one of the most promising platforms for building quantum information processors. To meet the stringent performance requirements, high-fidelity general-purpose numerical methods are increasingly needed to improve designs. Currently, there exist a limited number of methods for analyzing these systems and they suffer from problematic efficiency limitations. One challenge in developing more efficient methods is the lack of available reference data that can be generated to validate them. Here, we develop an analytical quantum full-wave solution for a transmon qubit in a 3D cavity that can validate numerical methods in the future. We start by reviewing the basics of the field-based quantization method used before showing how classical electromagnetic theories can be used to analytically evaluate all field-based parameters in our formulation. We validate our analytical solution by calculating experimentally-relevant system parameters and comparing against approaches that use numerical EM eigenmodes in their solution.
As superconducting circuits emerge as a leading platform for scalable quantum information processing, building comprehensive bridges from the foundational principles of macroscopic quantum phenomena to the architecture of modern quantum devices is increasingly essential for introducing new researchers to the field. This tutorial provides a self-contained, pedagogical introduction to superconducting quantum circuits at the undergraduate level. Beginning with an overview of superconductivity and the Josephson effect, the tutorial systematically develops the quantization of microwave circuits into the framework of circuit quantum electrodynamics (cQED). The transmon qubit is then introduced as a state-of-the-art application, with a detailed derivation of its Hamiltonian and its interaction with control and readout circuitry. The theoretical formalism is consolidated through a numerical simulation of vacuum Rabi oscillations in a driven transmon-resonator system, a canonical experiment that demonstrates the coherent energy exchange characteristic of the strong coupling regime. This work serves as a foundational guide and first point of contact, equipping students and researchers with the conceptual and mathematical tools necessary to understand and engineer superconducting quantum hardware.