Quantum theory of a bandpass Purcell filter for qubit readout

The measurement fidelity of superconducting transmon and Xmon qubits is partially limited by the qubit energy relaxation through the resonator into the transmission line, which is also known as the Purcell effect. One way to suppress this energy relaxation is to employ a filter which impedes microwave propagation at the qubit frequency. We present semiclassical and quantum analyses for the bandpass Purcell filter realized by E. Jeffrey et al. [Phys. Rev. Lett. 112, 190504 (2014)]. For typical experimental parameters, the bandpass filter suppresses the qubit relaxation rate by up to two orders of magnitude while maintaining the same measurement rate. We also show that in the presence of a microwave drive the qubit relaxation rate further decreases with increasing drive strength.

Figure 1

Schematic of a standard circuit QED qubit readout setup. The qubit state slightly changes the resonator frequency ${\omega }_{\mathrm{r}}$ (due to qubit-resonator interaction with strength $g$), and this is sensed by passing the microwave through (or reflecting from) the resonator. The amplified outgoing microwave is combined with the local oscillator at the mixer, whose output is measured to discriminate the qubit states. The energy decay $\kappa$ of the resonator is mainly due to its coupling with the transmission line.

Figure 2

Qubit measurement schematic with the bandpass Purcell filter of Ref. [17]. The readout resonator with frequency ${\omega }_{\mathrm{r}}$ (which depends on the qubit state) is coupled (coupling $\mathcal{G}$) with a filter resonator of frequency ${\omega }_{\mathrm{f}}$, which decays into the transmission line with the rate ${\kappa }_{\mathrm{f}}$. The further processing of the outgoing microwave (not shown) is the same as in Fig. 1. The microwave drive can be applied either to the readout (${\varepsilon }_{\mathrm{r}}$) or to the filter (${\varepsilon }_{\mathrm{f}}$) resonators. Coupling with the decaying filter resonator produces an effective decay rate ${\kappa }_{\mathrm{eff}}$ of the readout resonator, which depends on the drive frequency. As a result, for the measurement microwave ${\kappa }_{\mathrm{eff}}={\kappa }_{\mathrm{r}}$, while the qubit sees a much smaller value ${\kappa }_{\mathrm{eff}}={\kappa }_{\mathrm{q}}$, thus leading to a suppression of the qubit Purcell decay by a factor ${\kappa }_{\mathrm{q}}/{\kappa }_{\mathrm{r}}$.

Figure 3

Phase-space transient evolution of qubit-state-dependent coherent states in the filter resonator for driving either the readout resonator (red curves, left) or the filter (blue curves, right), with the drive amplitudes related via Eq. (18). Solid curves are for the qubit state $|e\rangle$, dashed curves are for the state $|g\rangle$. See text for the assumed parameters. For (a) we choose the drive frequency ${\omega }_{\mathrm{d}}$ symmetrically for the readout resonator, so that ${\overline{n}}_{\mathrm{r}}^{|g\rangle }={\overline{n}}_{\mathrm{r}}^{|e\rangle }=50$. For (b) we choose ${\omega }_{\mathrm{d}}$ symmetrically for the filter resonator, so that ${\overline{n}}_{\mathrm{f}}^{|g\rangle }={\overline{n}}_{\mathrm{f}}^{|e\rangle }$ when driving the filter; we choose ${\overline{n}}_{\mathrm{r}}^{|e\rangle }=50$. The black dots indicate time moments in the evolution every 10 ns until 100 ns, then every 50 ns. Circles illustrate the coherent state error circles in the steady state.

Figure 4

The Purcell relaxation rate $\mathrm{\Gamma }\left(\overline{n}\right)$ with a microwave drive, normalized by the no-drive rate $\mathrm{\Gamma }\left(0\right)$, with the filter (red solid curve) and without the filter (blue solid curve), as functions of the mean number of photons $\overline{n}$ in the readout resonator. The numerical simulations used the two-level approximation for the qubit, for which $\overline{n}/|{\mathrm{\Delta }}_{\mathrm{rq}}/\chi \left(0\right)|=\overline{n}/4n_{\mathrm{crit}}$. The dashed lines show the values expected from the model based on the ac Stark shift: $\mathrm{\Gamma }\left(\overline{n}\right)/\mathrm{\Gamma }\left(0\right)={(1+\overline{n}/n_{\mathrm{crit}})}^{-2}$ with the filter and ${(1+\overline{n}/n_{\mathrm{crit}})}^{-1}$ without the filter. The parameters used in the simulations are given in the text.