Theory of superconducting qubits beyond the lumped element approximation

In the design and investigation of superconducting qubits and related devices, a lumped element circuit model is the standard theoretical approach. However, many important physical questions lie beyond the scope of this approach, such as the consequences of very strong or otherwise unconventional Josephson junctions, the properties of small qubit devices, and the number of entangled electrons in superconducting Schrodinger cats. By performing gauge transformations on self-consistent solutions of the Bogoliubov-de Gennes equations, we develop here a formalism that is capable of addressing these questions. We then apply the formalism to a charge qubit and to an RF squid qubit. This theory provides a promising tool to accompany the remarkable experimental achievements driving superconducting qubits forward.