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arXiv:1606.07162 · Physical Review A

Quantum Bayesian approach to circuit QED measurement with moderate bandwidth

Alexander N. Korotkov

Published Oct 18, 2016• 47 citations
arXivHTMLPDF

Abstract

We consider continuous quantum measurement of a superconducting qubit in the circuit QED setup with a moderate bandwidth of the measurement resonator, i.e., when the ``bad-cavity'' limit is not applicable. The goal is a simple description of the quantum evolution due to measurement, i.e., the measurement backaction. Extending the quantum Bayesian approach previously developed for the bad-cavity regime, we show that the evolution equations remain the same, but now they should be applied to the entangled qubit-resonator state, instead of the qubit state alone. The derivation uses only elementary quantum mechanics and basic properties of coherent states, thus being accessible to nonexperts.

DOI
10.1103/physreva.94.042326

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•Jun 12, 2014•22 citations
We consider continuous quantum measurement of a superconducting qubit in the circuit QED setup with a moderate bandwidth of the measurement resonator, i.e., when the ``bad-cavity'' limit is not applicable. The goal is a simple description of the quantum evolution due to measurement, i.e., the measurement backaction. Extending the quantum Bayesian approach previously developed for the bad-cavity regime, we show that the evolution equations remain the same, but now they should be applied to the entangled qubit-resonator state, instead of the qubit state alone. The derivation uses only elementary quantum mechanics and basic properties of coherent states, thus being accessible to nonexperts.
Compared with the quantum trajectory equation (QTE), the quantum Bayesian approach has the advantage of being more efficient to infer a quantum state under monitoring, based on the integrated output of measurements. For weak measurement of qubits in circuit quantum electrodynamics (cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest. Elegant work towards this task was carried out by Korotkov in 'bad-cavity' and weak-response limits (Korotkov 2011 Quantum Bayesian approach to circuit QED measurement (arXiv: 1111.4016 )). In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of Korotkov to more general system parameters. The obtained Bayesian rule is in full agreement with Korotkovʼs result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.
Using a superconducting circuit QED device, we present a theoretical study of real-time quantum state estimation via the quantum Bayesian approach. Suitable conditions under which the quantum Bayesian approach can accurately update the density matrix of the qubit are analyzed. We also consider the correlation between some basic and physically meaningful parameters of the circuit QED and the performance of the Bayesian approach. Our results advance the understanding of the quantum Bayesian approach.
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.
Abstract This chapter introduces a puzzling property of the monitoring of a qubit by a coherent signal traversing a cavity dispersively coupled to the qubit. If the measurement of the signal is done by a fully efficient phase-sensitive amplifier, no noise is added by the amplifier and the backaction of the amplifier on the qubit induces the Bloch vector of the latter to diffuse along a line of longitude. When the measurement of the signal is done by a fully efficient phase-preserving amplifier, extra noise of quantum origin is added by the amplifier owing to the presence of the idler port, but the state of the qubit remains pure, the backaction-induced diffusion of the Bloch vector now occurring on the entire Bloch sphere.