Feedback control of Rabi oscillations in circuit QED

We consider the feedback stabilization of Rabi oscillations in a superconducting qubit which is coupled to a microwave readout cavity. The signal is readout by homodyne detection of the in-phase quadrature amplitude of the weak-measurement output. By multiplying the time-delayed Rabi reference, one can extract the signal, with maximum signal-to-noise ratio, from the noise current. We further track and stabilize the Rabi oscillations by using Lyapunov feedback control to properly adjust the input Rabi drives. Theoretical and simulation results illustrate the effectiveness of the proposed control law.

Figure 1

(a) Simplified circuit diagram of measurement and feedback control. A superconducting qubit (brown) is coupled to a microwave readout detector cavity (blue). The amplified output is homodyne detected and the quadrature signal (green) is then extracted from the noise current $I\left(t\right)$ by multiplying the time-delayed Rabi reference. The discrepancy is used to design the feedback-control law to correct Rabi oscillations (red). (b) Schematic of the readout drive to build up the photon population of the cavity and Rabi drive to stabilize the Rabi oscillation.

Figure 2

Evolution of the conditional state [here we only plot $\langle {\sigma }_z\left(t\right)\rangle$] by continuous weak measurements of the open-loop controlled microwave readout cavity with Rabi frequency ${\Omega }_R/2\pi =1$ MHz. The Rabi-drive amplitude ${\epsilon }_r={\Omega }_R\Delta /2g$ and the frequency ${\omega }_r={\omega }_q+\chi$, which makes the Lamb-shifted qubit transition frequency equals zero. The qubit is initially in the excited state and the readout drive amplitudes are ${\epsilon }_d/2\pi =1$ MHz (a), 4 MHz (b), 8 MHz (c), and 20 MHz (d), respectively.

Figure 3

(a) Feedback-controlled ensemble-averaged (over 1,000 realizations) Rabi oscillations, which persist for much longer time than those with open-loop control. The Rabi frequency ${\Omega }_R^0/2\pi =2.5$ MHz and the read-out drive amplitude is ${\epsilon }_d/2\pi =1$ MHz. (b) Power spectral density for the averaged measurement of feedback-controlled Rabi oscillations from (a) (red curve); the blue curve corresponds to the open-loop case with the same parameters of (a).