Detuning a qubit from its cavity exponentially suppresses spontaneous emission (Purcell effect)

A qubit resonant with a lossy cavity inherits the cavity’s decay rate — the Purcell effect enhances spontaneous emission. But when the qubit is detuned by $\Delta \gg g$, the effective decay rate through the cavity drops to:

${\gamma }_{\kappa }=\kappa {\left(\frac{g}{\Delta }\right)}^2$

For typical circuit QED parameters ($g/2\pi \sim 50\text{MHz}$, $\Delta /2\pi \sim 1\text{GHz}$, $\kappa /2\pi \sim 1\text{MHz}$), this gives ${\gamma }_{\kappa }\sim 2.5\text{kHz}$ — a lifetime contribution of $\sim 400\mu \text{s}$.

This is the central tension in circuit QED design: you want large $g/\Delta$ for strong readout signal, but small $g/\Delta$ for long qubit lifetime. The transmon solved this by operating at $E_J/E_C\gg 1$, which reduces charge dispersion (noise sensitivity) while keeping $g$ large.

Source: blais-2004-circuit-qed Related: circuit-qed, transmon, fluxonium