  We describe in this paper how the nonlinear Josephson inductance is the
crucial circuit element for all Josephson qubits. We discuss the three types of
qubit circuits, and show how these circuits use this nonlinearity in unique
manners. We give a brief 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, and then expand
upon this theory to describe quasiparticle excitations as transitions from the
ground to excited bound states from nonadiabatic changes in the bias. Although
quasiparticle current is typically calculated only for a constant DC voltage,
the advantage to this approach is seen where we qualitatively describe
quasiparticle tunneling with AC voltage excitations, as appropriate for the
qubit state. This section describes how the Josephson qubit is typically
insensitive to quasiparticle damping, even to the extent that a phase qubit can
be constructed from microbridge junctions.
