For a qubit capacitively coupled to a transmission-line feedline, the photon decay rate into the feedline (the Purcell rate) is set by the qubit frequency fq, the coupling capacitance Cc, the total qubit capacitance CΣ, and the feedline characteristic impedance Z0. The calculator gives the decay rate, the external Q, and the Purcell-limited T1.
The qubit, with total shunt capacitance CΣ, behaves at its transition frequency ωq = 2π fq like an LC oscillator with effective characteristic impedance Zq = 1/(ωq CΣ). The coupling capacitor Cc taps a small fraction of that voltage onto the feedline of impedance Z0; the feedline absorbs the power as a real load.
Treating Cc and Z0 as a series load with admittance Y = jωCc/(1 + jωCcZ0) and expanding for weak coupling ωqCcZ0 << 1:
Re Y ≈ ωq² Cc² Z0.κ = Re Y / CΣ = ωq² Cc² Z0 / CΣ (rad/s).Qext = ωq / κ = CΣ / (ωq Cc² Z0).T1Purcell = 1/κ (this is the T1 floor from the feedline alone; intrinsic / other channels are not included).Cc/CΣ; the rate scales as that ratio squared.The weak-coupling expansion is quantitative while ωq Cc Z0 << 1. For typical transmons (5 GHz, 50 Ω), this means Cc << 1/( 2π·5×109·50) ≈ 0.6 pF ≈ 640 fF — nearly always satisfied. The calculator flags violations explicitly.
Reference: Koch et al., "Charge-insensitive qubit design derived from the Cooper pair box," Phys. Rev. A 76, 042319 (2007); Blais et al., "Circuit quantum electrodynamics," Rev. Mod. Phys. 93, 025005 (2021).