Dispersive readout works because the qubit state shifts the cavity frequency

When a qubit is detuned from its cavity by $\Delta ={\omega }_r-\Omega \gg g$, the Jaynes-Cummings interaction produces a qubit-state-dependent cavity frequency shift:

${\omega }_r\to {\omega }_r\pm \frac{g^2}{\Delta }$

Ground state pulls the cavity one way; excited state pulls it the other. Measuring the transmitted phase of a probe tone at ${\omega }_r$ reveals the qubit state without exchanging energy — a quantum non-demolition (QND) measurement.

The measurement backaction is controlled by the photon number $\bar{n}$ in the readout pulse. Too many photons ($\bar{n}\gtrsim n_{\text{crit}}={\Delta }^2/4g^2$) break the dispersive approximation and cause measurement-induced transitions. The signal-to-noise ratio improves with integration time and is ultimately limited by the amplifier noise temperature, not quantum backaction.

Routing boundary

Dispersive readout and purcell-protection-via-detuning are two consequences of the same small cavity admixture $g/\Delta$, but they answer different questions:

• Stay here when the useful quantity is the state-dependent cavity pull $\chi \sim g^2/\Delta$ and the question is how strongly or quickly the qubit can be measured.
• Switch to purcell-protection-via-detuning when the harmful quantity is cavity-mediated decay ${\gamma }_{\kappa }\sim \kappa (g/\Delta {)}^2$ and the question is how the readout channel limits $T_1$.
• Read both when choosing detuning: increasing $|\Delta |$ protects lifetime but weakens the readout signal generated by the same hybridization.

This mechanism underpins essentially all modern superconducting qubit readout and was first proposed in blais-2004-circuit-qed.

Related: jaynes-cummings-in-circuits, purcell-protection-via-detuning, circuit-qed, transmon, high-quality-superconducting-cavities-coupled-to-nonlinear