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Devices built using circuit quantum electrodynamics architectures are one ofthe most popular approaches currently being pursued to develop quantuminformation processing hardware. Although significant progress has been madeover the previous two decades, there remain many technical issues limiting theperformance of fabricated systems. Addressing these issues is made difficult bythe absence of rigorous numerical modeling approaches. This work begins toaddress this issue by providing a new mathematical description of one of themost commonly used circuit quantum electrodynamics systems, a transmon qubitcoupled to microwave transmission lines. Expressed in terms ofthree-dimensional vector fields, our new model is better suited to developingnumerical solvers than the circuit element descriptions commonly used in theliterature. We present details on the quantization of our new model, and derivequantum equations of motion for the coupled field-transmon system. Theseresults can be used in developing full-wave numerical solvers in the future. Tomake this work more accessible to the engineering community, we assume only alimited amount of training in quantum physics and provide many backgrounddetails throughout derivations.

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