Relaxation of a qubit measured by a driven Duffing oscillator

We investigate the relaxation of a superconducting qubit for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The qubit relaxation rates are different in different states. They can display strong dependence on the qubit frequency and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven oscillator changes the effective temperature of the qubit.

Figure 1

(Left panel) Squared scaled attractor radii $r_a^2$ as functions of the dimensionless field intensity $\beta$ for the dimensionless friction $\kappa /\left|\delta \omega \right|=0.3$. (Right panel) The effective frequencies ${\nu }_a/\left|\delta \omega \right|$ for the same $\kappa /\left|\delta \omega \right|$. Curves 1 and 2 refer to small- and large-amplitude attractors, respectively.

Figure 2

The scaled decay-rate factors for the excited and ground states (curves 1 and 2, respectively) as functions of scaled difference between the qubit frequency and twice the modulation frequency; ${\Gamma }_0=\hslash C_{\Gamma }{r}_a^2/6{\gamma }_S$. Left and right panels refer to the small- and large-amplitude attractors, with the values of $\beta$ being 0.14 and 0.12, respectively. Other parameters are $\kappa /|\delta \omega =0.3$, $\mathrm{n̄}=0.5$.

Figure 3

The effective scaled qubit temperature $T_{\mathrm{eff}}^{*}=k_B{T}_{\mathrm{eff}}/\hslash {\omega }_q$ as function of the scaled field strength $\beta$ in the region of bistability for the small- and large-amplitude attractors (left and right panels, respectively); $({\omega }_q-2{\omega }_F)/\left|\delta \omega \right|=-0.2$ and 0.1 in the left and right panels, respectively; other parameters are the same as in Fig. 2.