Figure 1. Schematic and circuit diagrams of the system.

(a) A coplanar waveguide cavity which consists of a center conductor located between two ground planes. The cavity has been modified by adding a resistance R and capacitance C_c at positions x_r and x_c, respectively. In addition, one or more SQUIDs, each with an inductance L_J(Φ), where Φ is the magnetic flux penetrating the SQUID loop, are positioned evenly into the left cavity. Our approach makes no assumptions about the type of qubit, however, for the sake of concreteness, we assume a transmon qubit that is capacitively coupled to the resonator with a capacitance C_Q at x_q. (b) The cavity may be represented using an infinite number of capacitors (C₀ = cdx) and inductors (L₀ = ℓdx). The horizontal dots represent many sequences of these L₀, C₀ pairs. It is also possible to include series resistors to model the internal losses of the superconductor but these are assumed to be negligible in our setup.