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Quantum Bayesian approach to circuit QED measurement

Alexander N. Korotkov

Published Jun 12, 2014• 22 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780199681181.003.0017

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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.
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.
We analyze the backaction of homodyne detection and photodetection on superconducting qubits in circuit quantum electrodynamics. Although both measurement schemes give rise to backaction in the form of stochastic phase rotations, which leads to dephasing, we show that this can be perfectly undone provided that the measurement signal is fully accounted for. This result improves on an earlier one [Phys. Rev. A 82, 012329 (2010)], showing that the method suggested can be made to realize a perfect two-qubit parity measurement. We propose a benchmarking experiment on a single qubit to demonstrate the method using homodyne detection. By analyzing the limited measurement efficiency of the detector and bandwidth of the amplifier, we show that the parameter values necessary to see the effect are within the limits of existing technology.
We theoretically investigate the backaction of a sensor quantum dot with strong local Coulomb repulsion on the transient dynamics of a qubit that is probed capacitively. We show that the measurement backaction induced by the noise of electron cotunneling through the sensor is surprisingly mitigated by the recently identified coherent backaction [M. Hell, M. R. Wegewijs, and D. P. DiVincenzo, Phys. Rev. B 89, 195405 (2014)] arising from quantum fluctuations. This indicates that a sensor with quantized states may be switched off better than naively expected. This renormalization effect is missing in semiclassical stochastic fluctuator models and typically also in Born-Markov approaches, which try to avoid the calculation of the nonstationary, nonequilibrium state of the qubit plus sensor. Technically, we integrate out the current-carrying electrodes to obtain kinetic equations for the joint, nonequilibrium detector-qubit dynamics. We show that the sensor current response, level renormalization, cotunneling broadening, and leading non-Markovian corrections always appear together and cannot be turned off individually in an experiment or ignored theoretically. We analyze the backaction on the reduced qubit state---capturing the full non-Markovian effects imposed by the sensor quantum dot on the qubit---by applying a Liouville-space decomposition into quasistationary and rapidly decaying modes. Importantly, the sensor cannot be eliminated completely even in the simplest high-temperature, weak-measurement limit since the qubit state experiences an initial slip depending on the initial preparation of qubit plus sensor quantum dot. The slip persists over many qubit cycles, i.e., also on the time scale of the qubit decoherence induced by the backaction. A quantum-dot sensor can thus not be modeled as usual as a ``black box'' without accounting for its dynamical variables; it is part of the quantum circuit. We furthermore find that the Bloch vector relaxes (rate 1/T11/{T}_{1}1/T1​) along an axis that is not orthogonal to the plane in which the Bloch vector dephases (rate 1/T21/{T}_{2}1/T2​), blurring the notions of relaxation and dephasing times. Moreover, the precessional motion of the Bloch vector is distorted into an ellipse in the tilted dephasing plane.
The purpose of a phase-preserving linear amplifier is to make a small signal larger, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise. Quantum mechanics limits how well this can be done: the noise added by the amplifier, referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we provide the quantum constraints on the entire distribution of added noise: any phasepreserving linear amplifier is equivalent to a parametric amplifier with a physical state σ for the ancillary mode; σ determines the properties of the added noise.
Measuring a quantum system can randomly perturb its state. The strength and nature of this back-action depend on the quantity that is measured. In a partial measurement performed by an ideal apparatus, quantum physics predicts that the system remains in a pure state whose evolution can be tracked perfectly from the measurement record. We demonstrated this property using a superconducting qubit dispersively coupled to a cavity traversed by a microwave signal. The back-action on the qubit state of a single measurement of both signal quadratures was observed and shown to produce a stochastic operation whose action is determined by the measurement result. This accurate monitoring of a qubit state is an essential prerequisite for measurement-based feedback control of quantum systems.