How do I design a coplanar waveguide resonator for coupling to a transmon qubit?

Designing a coplanar waveguide (CPW) resonator for coupling to a transmon qubit creates the readout and bus resonator used in circuit quantum electrodynamics (cQED), the standard architecture for superconducting quantum computing. The CPW resonator is a planar transmission line consisting of a center conductor strip separated from two ground planes by gaps, patterned on a low-loss substrate (typically silicon or sapphire). The design parameters are: resonance frequency (determined by the resonator length: f_r = c_eff / (2L) for a half-wave resonator, where c_eff = c / sqrt(epsilon_eff) is the effective speed of light on the substrate; for silicon (epsilon_r = 11.7): epsilon_eff approximately 6.35, c_eff approximately 1.19 × 10^8 m/s; for f_r = 7 GHz: L approximately 8.5 mm), characteristic impedance (Z_0: determined by the center conductor width (w) and gap (g); typical: w = 10 um, g = 6 um → Z_0 approximately 50 ohms on silicon; Z_0 = (30pi / sqrt(epsilon_eff)) × K'(k)/K(k), where K is the elliptic integral and k = w/(w+2g)), coupling to the qubit (the CPW resonator is capacitively coupled to the transmon qubit at one end; the coupling capacitance C_c (typically 1-10 fF) determines the coupling strength g = (C_c / sqrt(C_r × C_q)) × sqrt(omega_r × omega_q)/2; target g/2pi approximately 50-200 MHz for readout, 5-30 MHz for bus resonators), and external coupling Q_c (the resonator is coupled to the external feedline through a coupling capacitor; Q_c = pi × Z_0 / (omega × Z_ext × C_ext^2); typical Q_c = 10^3-10^5 depending on the readout speed requirement).