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arXiv:2008.12908 · Physical Review A

Quantum dynamics under simultaneous and continuous measurement of noncommutative observables

Chao Jiang, Gentaro Watanabe A

Published Dec 23, 2020• 7 citations
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Abstract

We consider simultaneous and continuous measurement of two noncommutative observables of the system whose commutator is not necessarily a $c$-number. We revisit the Arthurs-Kelly model and generalize it to describe the simultaneous measurement of two arbitrary observables of the system. Using this generalized model, we continuously measure the system by following the scheme proposed by Scott and Milburn [Scott and Milburn, Phys. Rev. A 63, 042101 (2001)]. We find that the unconditioned master equation reduces to the Lindblad form in the continuous limit. In addition, we find that the master equation does not contain a cross term of these two measurements. Finally, we propose a scheme to prepare the state of a two-level system in an external field by feedback control based on the simultaneous, continuous measurement of the two observables.

DOI
10.1103/physreva.102.062216

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We consider simultaneous and continuous measurement of two noncommutative observables of the system whose commutator is not necessarily a ccc-number. We revisit the Arthurs-Kelly model and generalize it to describe the simultaneous measurement of two arbitrary observables of the system. Using this generalized model, we continuously measure the system by following the scheme proposed by Scott and Milburn [Scott and Milburn, Phys. Rev. A 63, 042101 (2001)]. We find that the unconditioned master equation reduces to the Lindblad form in the continuous limit. In addition, we find that the master equation does not contain a cross term of these two measurements. Finally, we propose a scheme to prepare the state of a two-level system in an external field by feedback control based on the simultaneous, continuous measurement of the two observables.
As an application of the simultaneous and continuous measurement of noncommutative observables formulated in our previous paper [C. Jiang and G. Watanabe, Phys. Rev. A 102, 062216 (2020)], we propose a scheme to generate a pure ideal quadrature-squeezed state in a one-dimensional harmonic oscillator system by the feedback control based on such type of measurement of noncommutative quadrature observables. We find that, by appropriately setting the strengths of the measurement and the feedback control, the pure ideal quadrature-squeezed state with arbitrary squeezedness can be produced. This is in contrast to the scheme based on the single-observable measurement and the feedback control, where only nonideal squeezed states are produced. Furthermore, we discuss the transient dynamics of the system and the experimental feasibility of our scheme.
In contrast to a projective quantum measurement, in a weak measurement the system is only weakly perturbed while only partial information on the measured observable is obtained. A simultaneous measurement of non-commuting observables cannot be projective, however the strongest possible such measurement can be defined as providing their values at the smallest uncertainty limit. Starting with the Arthurs and Kelly (AK) protocol for such measurement of position and momentum, we derive a systematic extension to a corresponding weak measurement along three steps: First, a plausible form of the weak measurement operator analogous to the Gaussian Kraus operator, often used to model a weak measurement of a single observable, is obtained by projecting a naïve extension (valid for commuting observable) onto the corresponding Gabor space. Second, we show that the so obtained set of measurement operators satisfies the normalization condition for the probability to obtain given values of the position and momentum in the weak measurement operation, namely that this set constitutes a positive operator valued measure (POVM) in the position-momentum space. Finally, we show that the so-obtained measurement operator corresponds to a generalization of the AK measurement protocol in which the initial detector wavefunctions is suitable broadened.
We derive the stochastic Schr\"odinger equation for the limit of continuous weak measurement where the observables monitored are canonical position and momentum. To this end we extend an argument due to Smolianov and Truman from the von Neumann model of indirect measurement of position to the Arthurs and Kelly model for simultaneous measurement of position and momentum. We require only unbiasedness of the detector states and an integrability condition sufficient to ensure a central limit effect. Despite taking a weak interaction as opposed to a weak measurement limit, the resulting stochastic wave equation is of the same form as that derived in a recent paper by Scott and Milburn for the specific case of joint Gaussian states.
We combine the concept of Bell measurements, in which two systems are projected into a maximally entangled state, with the concept of continuous measurements, which concerns the evolution of a continuously monitored quantum system. For such time-continuous Bell measurements we derive the corresponding stochastic Schrödinger equations, as well as the unconditional feedback master equations. Our results apply to a wide range of physical systems, and are easily adapted to describe an arbitrary number of systems and measurements. Time-continuous Bell measurements therefore provide a versatile tool for the control of complex quantum systems and networks. As examples we show that (i) two two-level systems can be deterministically entangled via homodyne detection, tolerating photon loss up to 50%, and (ii) a quantum state of light can be continuously teleported to a mechanical oscillator, which works under the same conditions as are required for optomechanical ground-state cooling.
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.
Measurement and feedback control are essential features of quantum science, with applications ranging from quantum technology protocols to information-to-work conversion in quantum thermodynamics. Theoretical descriptions of feedback control are typically given in terms of stochastic equations requiring numerical solutions, or are limited to linear feedback protocols. Here we present a formalism for continuous quantum measurement and feedback, both linear and nonlinear. Our main result is a quantum Fokker-Planck master equation describing the joint dynamics of a quantum system and a detector with finite bandwidth. For fast measurements, we derive a Markovian master equation for the system alone, amenable to analytical treatment. We illustrate our formalism by investigating two basic information engines, one quantum and one classical.