Course 13 Superconducting qubits and the physics of Josephson junctions

This chapter discusses the physics of the Josephson junction because, being nonlinear, it is the fundamental circuit element that is needed for the appearance of usable qubit states. In contrast, linear circuit elements such as capacitors and inductors can form low-dissipation superconducting resonators, but are unusable for qubits because the energy-level spacings are degenerate. The nonlinearity of the Josephson inductance breaks the degeneracy of the energy level spacings, allowing dynamics of the system to be restricted to only the two qubit states. The Josephson junction is a remarkable nonlinear element because it combines negligible dissipation with extremely large nonlinearity - the change of the qubit state by only one photon in energy can modify the junction inductance by order unity. The chapter discusses the nonlinear Josephson inductance and three types of qubit circuits, and shows how these circuits use this nonlinearity in unique manners. It presents the derivation of the BCS theory, highlighting the appearance of the macroscopic phase parameter. The Josephson equations are derived using standard first and second order perturbation theory that describe quasiparticle and Cooper-pair tunneling. An exact calculation of the Josephson effect then follows using the quasiparticle bound-state theory. This theory is expanded to describe quasiparticle excitations as transitions from the ground to excited bound states from nonadiabatic changes in the bias. The chapter also describes quasiparticle tunneling with AC voltage excitations, as appropriate for the qubit state.