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 its scope, such as the behavior of circuits with strong Josephson junctions carrying substantial currents and the properties of very small superconducting devices. By performing gauge transformations on self-consistent solutions of the Bogoliubov--de Gennes equations, we develop here a formalism that treats Josephson couplings nonperturbatively. We apply the formalism to (a) show that Fermi sea effects can contribute to the effective capacitance of small charge qubits; (b) demonstrate an asymmetry in clockwise and counterclockwise current states in small rf superconducting quantum interference device qubits; and (c) provide a microscopic wave function of superconducting Schr\"odinger cats suitable for computing the number of entangled electrons.