What is the Purcell filter and how does it protect qubit coherence during dispersive readout?

A Purcell filter is a bandpass filter placed between the readout resonator and the external measurement chain that protects qubit coherence during dispersive readout by suppressing the Purcell effect, which is the enhanced spontaneous emission of the qubit into the readout line mediated by the readout resonator. In dispersive readout, the qubit is coupled to a readout resonator that is detuned from the qubit frequency by delta = |f_qubit - f_readout| (typically 1-2 GHz). The qubit can decay by emitting a photon at the qubit frequency through the readout resonator into the external 50-ohm line. The Purcell decay rate is: gamma_Purcell = (g^2 / delta^2) x kappa, where g is the qubit-resonator coupling strength and kappa is the readout resonator decay rate into the external line. Without a Purcell filter: increasing kappa (for faster readout) directly increases gamma_Purcell, creating a trade-off between readout speed and qubit lifetime. The Purcell filter resolves this trade-off by: presenting a high impedance (open circuit) to the readout line at the qubit frequency (blocking the qubit's spontaneous emission), while presenting a matched 50-ohm impedance at the readout resonator frequency (allowing the readout signal to pass freely). This decouples gamma_Purcell from kappa: the qubit cannot emit through the high-impedance filter at its frequency, regardless of how fast kappa is. The Purcell-limited T1 improves from T1_Purcell = delta^2 / (g^2 x kappa) without filter to T1_Purcell = infinity (ideal) or >> 1 ms (practical, limited by filter rejection) with the filter.