Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback

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The act of measurement bridges the quantum and classical worlds by projecting a superposition of possible states into a single (probabilistic) outcome. The timescale of this ‘instantaneous’ process can be stretched using weak measurements1, 2, such that it takes the form of a gradual random walk towards a final state. Remarkably, the interim measurement record is sufficient to continuously track and steer the quantum state using feedback3, 4, 5, 6, 7, 8. Here we implement quantum feedback control in a solid-state system, namely a superconducting quantum bit (qubit) coupled to a microwave cavity9. A weak measurement of the qubit is implemented by probing the cavity with microwave photons, maintaining its average occupation at less than one photon. These photons are then directed to a high-bandwidth, quantum-noise-limited amplifier10, 11, which allows real-time monitoring of the state of the cavity (and, hence, that of the qubit) with high fidelity. We demonstrate quantum feedback control by inhibiting the decay of Rabi oscillations, allowing them to persist indefinitely12. Such an ability permits the active suppression of decoherence and enables a method of quantum error correction based on weak continuous measurements13, 14. Other applications include quantum state stabilization4, 7, 15, entanglement generation using measurement16, state purification17 and adaptive measurements18, 19.

At a glance


  1. Experimental set-up.
    Figure 1: Experimental set-up.

    a, Signal generation set-up. One generator provides the Rabi drive at the a.c. Stark-shifted qubit frequency ( ), and the output of another generator at 7.2749GHz is split to create the measurement signal, paramp drive and local oscillator. The relative amplitudes and phases of these three signals are controlled by variable attenuators and phase shifters (not shown). I, in-phase component; Q, quadrature component; LO, local oscillator; RF, radio frequency. b, Simplified version of the cryogenic part of the experiment; all components are at 30mK (except for the high-electron-mobility transistor (HEMT) amplifier, which is at 4K). The combined qubit and measurement signals enter the weakly coupled cavity port, interact with the qubit and leave from the strongly coupled port. The output passes through two isolators (which protect the qubit from the strong paramp drive), is amplified and then continues to the demodulation set-up. The coherent state at the output of the cavity for the ground and excited states is shown schematically before and after parametric amplification. c, d, The amplified signal is homodyne-detected and the two quadratures are digitized (c). The amplified quadrature (Q) is split off and sent to the feedback circuit (d), where it is multiplied with the Rabi reference signal. The product is low-pass-filtered and fed back to the IQ mixer in a to modulate the Rabi drive amplitude.

  2. Rabi oscillations and feedback.
    Figure 2: Rabi oscillations and feedback.

    a, We average 104 measurement traces using weak continuous measurement and simultaneous Rabi driving to obtain ensemble-averaged Rabi oscillations that decay in time as a result of ensemble dephasing. b, Averaged Fourier transforms of the individual measurement traces from a. The spectrum shows a peak at the Rabi frequency (blue trace) with a full-width at half-maximum of Γ/2π. The grey trace shows an identically prepared spectrum for the squeezed quadrature (multiplied by 20 for clarity), which contains no qubit state information. c, Γ/2π plotted as a function of cavity photon occupation, (measurement strength), showing the expected linear dependence. The vertical offset is dominated by pure environmental dephasing, Γenv/2π, but has contributions from qubit relaxation (T1) and thermal excitation into higher qubit levels. d, Feedback-stabilized, ensemble-averaged Rabi oscillations, which persist for much longer times than those without feedback (a). The corresponding spectrum, shown in b, has a needle-like peak at the Rabi reference frequency (red trace). The slowly changing mean level in the Rabi oscillation traces in a and d is due to the thermal transfer of population into the second excited state of the qubit. See Supplementary Information, section IV(C), for more details.

  3. Tomography and feedback efficiency.
    Figure 3: Tomography and feedback efficiency.

    a, Quantum state tomography of the feedback-stabilized state. We plot left fenceσXright fence, left fenceσYright fence and left fenceσZright fence for different time points τtomo in one full Rabi oscillation of the qubit. The solid lines are sinusoidal fits. The magnitude of these sinusoidal oscillations is approximately equal to the feedback efficiency, D = 0.45. b, D plotted as a function of the dimensionless feedback gain, F. Solid red squares show experimental data with a maximum value of D = 0.45. The dashed black line is a plot of equation(1) with η = 0.40 and Γ/2π = 0.154MHz ( , Γenv/2π = 0.020MHz), whereas the solid black line is obtained from full numerical simulations of the Bayesian equations including finite loop delay (250ns) and feedback bandwidth (10MHz).


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Author information


  1. Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA

    • R. Vijay,
    • C. Macklin,
    • D. H. Slichter,
    • S. J. Weber,
    • K. W. Murch,
    • R. Naik &
    • I. Siddiqi
  2. Department of Electrical Engineering, University of California, Riverside, California 92521, USA

    • A. N. Korotkov
  3. Present address: Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA.

    • D. H. Slichter


R.V., C.M. and D.H.S. performed the experiment, which is based on a proposal by A.N.K. R.V. analysed the data, performed numerical simulations and wrote the manuscript. S.J.W. and K.W.M. fabricated the qubit and cavity. R.N. helped with cavity design by performing electromagnetic simulations. A.N.K. provided theoretical support and helped with numerical simulations. All authors helped in editing the manuscript. All work was carried out under the supervision of I.S.

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The authors declare no competing financial interests.

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  1. Supplementary Information (293K)

    This file contains Supplementary Text and Data, Supplementary Figures 1-3 and Supplementary References.

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