Lifetime renormalization of driven weakly anharmonic superconducting qubits. II. The readout problem

Recent experiments in superconducting qubit systems have shown an unexpectedly strong dependence of the qubit relaxation rate on the readout drive power. This phenomenon limits the maximum measurement strength and thus the achievable readout speed and fidelity. We address this problem here and provide a plausible mechanism for drive-power dependence of relaxation rates. To this end we introduce a two-parameter perturbative expansion in qubit anharmonicity and the drive amplitude through a unitary transformation technique introduced in Part I. This approach naturally reveals number-nonconserving terms in the Josephson potential as a fundamental mechanism through which applied microwave drives can activate additional relaxation mechanisms. We present our results in terms of an effective master equation with renormalized state- and drive-dependent transition frequency and relaxation rates. Comparison of numerical results from this effective master equation to those obtained from a Lindblad master equation which only includes number-conserving terms (i.e., Kerr interactions) shows that number-nonconserving terms can lead to significant drive-power dependence of the relaxation rates. The systematic expansion technique introduced here is of general applicability to obtaining effective master equations for driven-dissipative quantum systems that contain weakly nonlinear degrees of freedom.

Figure 1

Schematics of the readout setup. A Josephson junction qubit (mode $\widehat{a}$) is capacitively coupled to a waveguide through a readout cavity (mode $\widehat{c}$). A microwave drive (amplitude ${\varepsilon }_{\text{d}}$, frequency ${\omega }_{\text{d}}$) is applied through the waveguide.

Figure 2

Master equation simulations for our model of dispersive readout (see Fig. 1). (a) EME solution: The natural logarithm of the qubit occupation number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ as a function of time, for different values of the drive power. As the drive strength is increased, the relaxation rate of the qubit increases linearly as a function of the cavity steady-state population. Inset: The Kerr-only master equation (43) predicts no drive- or nonlinearity-induced renormalization of the qubit relaxation rate. (b) The drive strength is adjusted such that the cavity has a mean steady-state population ${\overline{n}}_c$. (c) Qubit relaxation rate [Eq. (39)] extracted from a number of numerical simulations: Kerr theory Eq. (43), EME with all terms included [i.e., solving (31) together with the full expression for the collapse operator $\widehat{C}\left({\omega }_a\right)$, Eq. (F19) in Appendix pp6, and its analog for $\widehat{C}\left({\omega }_c\right)]$, as well as EME in which a subset of the terms are included only (see text for complete discussion).

Figure 3

Magnitudes of (a) ${\eta }_x$ as a function of ${\overline{n}}_c$ for the parameters chosen for the EME simulation (see text); (b) for the same parameters, the magnitudes of the most significant terms in the EME.

Figure 4

EME results versus drive frequency, at ${\overline{n}}_c=0.5$ steady-state cavity photons. (a) The relaxation rate obtained from the EME (solid red) exhibits a large renormalization only in the vicinity of the cavity resonance. At large drive-cavity detuning ${\omega }_c-{\omega }_{\text{d}}$ (ten times the cross-Kerr energy scale ${\chi }_{\text{ac}}$) there is almost no correction from the drive-induced terms, as exemplified by a comparison with the EME result for an undriven system (black dashed line). (b) This is consistent with the coefficients of the most resonant contributions in the EME decaying algebraically with the detuning.

Figure 5

An EME is derived here for a transmon qubit (mode $\widehat{a}$) coupled to an infinite waveguide. A flux bias line is used to tune the frequency of the transmon qubit by adjusting the magnetic flux through its SQUID loop. Flux noise is the dominant source of noise for this setup.

Figure 6

Results from EME solution for a qubit coupled to an infinite waveguide. The drive strength is represented on the horizontal axis in units of $\overline{n}$, which represents the steady-state mean photon number for the case of a linear oscillator. The vertical axis represents the relaxation rate ${\kappa }_a^{\text{EME}}$ of the nonlinear oscillator as a function of $\overline{n}$, rescaled by the linear oscillator ${\kappa }_a$. Solving the EME with $\epsilon =0$ amounts to simulating a driven-dissipative linear oscillator, and the relaxation rate remains unchanged when the drive is applied (horizontal black curve). The solid red and purple curves show a renormalization of the nonlinear oscillator relaxation rate, for $\epsilon =0.2$ and $\epsilon =0.15$, respectively. The dashed lines represent an estimate of the relaxation rate obtained from the single-photon relaxation process in Eq. (D18). This is an underestimate of the actual relaxation rate: in fact, additional drive-induced processes in Eqs. (D18), (D20), and (D21), among which we mention single-photon excitation, dephasing, and multiphoton transitions, are responsible for the depletion of the first excited state and, consequently, an enhancement of the relaxation rate.