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arXiv:1809.04222 · Physical Review Letters

Correlators Exceeding One in Continuous Measurements of Superconducting Qubits

Juan Atalaya, Shay Hacohen-Gourgy, Irfan A. Siddiqi, Alexander N. Korotkov

Published Jun 7, 2019• 7 citations
arXivHTMLPDF

Abstract

We consider the effect of phase backaction on the correlator ⟨I(t)I(t+τ)⟩ for the output signal I(t) from continuous measurement of a qubit. We demonstrate that the interplay between informational and phase backactions in the presence of Rabi oscillations can lead to the correlator becoming larger than 1, even though |⟨I⟩|≤1. The correlators can be calculated using the generalized "collapse recipe," which we validate using the quantum Bayesian formalism. The recipe can be further generalized to the case of multitime correlators and arbitrary number of detectors, measuring non-commuting qubit observables. The theory agrees well with experimental results for continuous measurement of a transmon qubit. The experimental correlator exceeds the bound of 1 for a sufficiently large angle between the amplified and informational quadratures, causing the phase backaction. The demonstrated effect can be used to calibrate the quadrature misalignment.

DOI
10.1103/physrevlett.122.223603

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In quantum physics, measurements give random results and yield a corresponding random backaction on the state of the system subject to measurement. If a quantum system is probed continuously over time, its state evolves along a stochastic quantum trajectory. To investigate the characteristic properties of such dynamics, we perform weak continuous measurements on a superconducting qubit that is driven to undergo Rabi oscillations. From the data we observe a number of striking temporal correlations within the time dependent signals and the quantum trajectories of the qubit, and we discuss their explanation in terms of quantum measurement and photodetection theory.
Abstract One of the hallmarks of quantum mechanics is the impossibility of simultaneous measurement of non-commuting observables with projective measurements. This, however, can be circumvented by using continuous quantum measurements. Here we investigate the temporal correlations of the output signals of detectors continuously and simultaneously measuring the qubit observables σz and σz cosφ + σx sinφ, for various angles φ. Using the quantum Bayesian formalism, we obtain analytical expressions for the correlators, which we find to be in good agreement with those obtained from experimentally measured output signals. The agreement is particularly good for cross-correlators, even at times shorter than the cavity modes decay time. We further discuss how the correlators can be applied for parameter estimation, and use them to infer a small residual qubit Hamiltonian arising from calibration inaccuracy in the experimental data. Our work opens up new possibilities to perform quantum metrology based on temporal correlations of measured data.
We consider the temporal correlations of the quantum state of a qubit subject to simultaneous continuous measurement of two noncommuting qubit observables. Such qubit state correlators are defined for an ensemble of qubit trajectories, which has the same fixed initial state and can also be optionally constrained by a fixed final state. We develop a stochastic path integral description for the continuous quantum measurement and use it to calculate the considered correlators. Exact analytic results are possible in the case of ideal measurements of equal strength and are also shown to agree with solutions obtained using the Fokker-Planck equation. For a more general case with decoherence effects and inefficiency, we use a diagrammatic approach to find the correlators perturbatively in the quantum efficiency. We also calculate the state correlators for the quantum trajectories which are extracted from readout signals measured in a transmon qubit experiment, by means of the quantum Bayesian state update. We find an excellent agreement between the correlators based on the experimental data and those obtained from our analytical and numerical results.
The Bayesian formalism for a continuous measurement of solid-state qubits is extended to a model which takes into account several factors of the detector nonideality. In particular, we consider additional classical output and backaction noises (with finite correlation), together with quantum-limited output and backaction noises, and take into account possible asymmetry of the detector coupling. The formalism is first derived for a single qubit and then generalized to the measurement of entangled qubits.
Designing high-precision and efficient protocols is of crucial importance for quantum parameter estimation in practice. Estimation based on continuous quantum measurement is one possible type of this, which also appears to be the most natural choice for continuous dynamical processes. In this work we consider the state-of-the-art superconducting circuit quantum-electrodynamics (QED) systems, where high-quality continuous measurements have been extensively performed in the past decade. Within the framework of Bayesian estimation and particularly using the quantum Bayesian rule in circuit QED, we numerically simulate the likelihood function as an estimator for the Rabi frequency of qubit oscillations. We find that, by proper design of the interaction strength of measurement, the estimate precision can scale with the measurement time beyond the standard quantum limit, which is usually assumed for this type of continuous measurement. This unexpected result is supported by the simulated Fisher information and can be understood as a consequence of the quantum correlation between the output signals by simulating the effect of quantum efficiency of measurement.