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IEEE Conference on Decision and Control (CDC)

Strong measurement and quantum feedback for persistent Rabi oscillations in circuit QED experiments

Mazyar Mirrahimi, Benjamin Huard, Michel Devoret

Published Dec 1, 2012• 3 citations

Abstract

We investigate the stabilization of the dynamical state of a superconducting qubit. In a series of papers, A. Korotkov and his co-workers suggested that continuous weak measurement of the state of a qubit and applying an appropriate feedback on the amplitude of a Rabi drive, should maintain the coherence of the Rabi oscillations for arbitrary time. Here, in the aim of addressing a metrological application of these persistent Rabi oscillations, we explore a new variant of such strategies. This variant is based on performing strong measurements in a discrete manner and using the measurement record to correct the phase of the Rabi oscillations. Noting that such persistent Rabi oscillations can be viewed as an amplitude-to-frequency convertor (converting the amplitude of the Rabi microwave drive to a precise frequency), we propose another feedback layer consisting of a simple analog phase locked loop to compensate the low frequency deviations in the amplitude of the Rabi drive.

DOI: 10.1109/cdc.2012.6426309

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We investigate the stabilization of the dynamical state of a superconducting qubit. In a series of papers, A. Korotkov and his co-workers suggested that continuous weak measurement of the state of a qubit and applying an appropriate feedback on the amplitude of a Rabi drive, should maintain the coherence of the Rabi oscillations for arbitrary time. Here, in the aim of addressing a metrological application of these persistent Rabi oscillations, we explore a new variant of such strategies. This variant is based on performing strong measurements in a discrete manner and using the measurement record to correct the phase of the Rabi oscillations. Noting that such persistent Rabi oscillations can be viewed as an amplitude-to-frequency convertor (converting the amplitude of the Rabi microwave drive to a precise frequency), we propose another feedback layer consisting of a simple analog phase locked loop to compensate the low frequency deviations in the amplitude of the Rabi drive.

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.

The qubit Rabi oscillations are known to be nondecaying (though with a fluctuating phase) if the qubit is continuously monitored in the weak-coupling regime. In this Rapid Communication we propose an experiment to demonstrate these persistent Rabi oscillations via low-frequency noise correlation. The idea is to measure a qubit by two detectors, biased stroboscopically at the Rabi frequency. The low-frequency noise depends on the relative phase between the two combs of biasing pulses, with a strong increase of telegraph noise in both detectors for the in-phase or antiphase combs. This happens because of self-synchronization between the persistent Rabi oscillations and measurement pulses. Almost perfect correlation of the noise in the two detectors for the in-phase regime and almost perfect anticorrelation for the antiphase regime indicates a presence of synchronized persistent Rabi oscillations. The experiment can be realized with semiconductor or superconductor qubits.

This thesis presents a series of experiments highlighting measurement back action and decoherence in a basic open quantum system, the superconducting qubit. These observations are enabled by recent advances in amplification close to the quantum limit using Josephson circuits. The information extracted from the system can then be used as input in quantum feedback. A stroboscopic projective readout is performed and a feedback loop is used to correct for detected errors, thus stabilizing an arbitrary predetermined state of the qubit. When monitoring continuously the environment of the qubit by heterodyne detection of its fluorescence, we reconstruct individual quantum trajectories during relaxation. Conditioning this detection to the outcome of a following projective measurement, we access the weak values of the fluorescence signal. Included in a continuous feedback loop, this detection is also used to stabilize an arbitrary state of the qubit. Finally, a last experiment witnesses quantum Zeno dynamics of a resonant microwave mode, entailed by its coupling to the qubit.

We propose a scheme of stabilizing the persistent-current Rabi oscillation based on the flux qubit-resonator-atom hybrid structure. The low-

Q

LC

resonator weakly interacts with the flux qubit and maps the persistent-current Rabi oscillation of the flux qubit onto the intraresonator electric field. This oscillating electric field is further coupled to a Rydberg-Rydberg transition of the

^{87}\mathrm{Rb}

atoms. The Rabi-frequency fluctuation of the flux qubit is deduced from measuring the atomic population via the fluorescence detection and stabilized by feedback controlling the external flux bias. Our numerical simulation indicates that the feedback-control method can efficiently suppress the background fluctuations in the flux qubit, especially in the low-frequency limit. This technique may be extensively applicable to different types of superconducting circuits, paving a way to long-term-coherence superconducting quantum information processing.

We analyze the newly proposed quantum feedback loop for a solid-state qubit, based on monitoring the quadrature components of the current from a weakly coupled detector, which continuously measures the qubit. Similar to the earlier proposal of the "Bayesian" feedback, the feedback loop is used to maintain the coherent (Rabi)oscillations in a qubit for an arbitrarily long time; however, this is done in a significantly simpler way, which requires much smaller bandwidth of the control circuitry. The price for simplicity is a less-than-ideal operation: the fidelity is limited to about 95%. The feedback loop operation can be experimentally verified by appearance of a positive in-phase component of the detector current relative to an external oscillating signal used for synchronization. The quadrature-based quantum feedback seems to be within the reach of the present-day technology.