  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.
