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arXiv:0911.0471 · New Journal of Physics

Weak value measurement with an incoherent measuring device

Young‐Wook Cho, Hyang-Tag Lim, Young-Sik Ra, Yoon-Ho Kim

Published Feb 24, 2010• 30 citations
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Abstract

In the Aharonov–Albert–Vaidman (AAV) weak measurement, it is assumed that the measuring device or the pointer is in a quantum mechanical pure state. In reality, however, it is often not the case. In this paper, we generalize the AAV weak measurement scheme to include more generalized situations in which the measuring device is in a mixed state. We also report an optical implementation of the weak value measurement in which the incoherent pointer is realized with the pseudo-thermal light. The theoretical and experimental results show that the measuring device under the influence of partial decoherence could still be used for amplified detection of minute physical changes and is applicable for implementing the weak value measurement for massive particles.

DOI
10.1088/1367-2630/12/2/023036

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In the Aharonov–Albert–Vaidman (AAV) weak measurement, it is assumed that the measuring device or the pointer is in a quantum mechanical pure state. In reality, however, it is often not the case. In this paper, we generalize the AAV weak measurement scheme to include more generalized situations in which the measuring device is in a mixed state. We also report an optical implementation of the weak value measurement in which the incoherent pointer is realized with the pseudo-thermal light. The theoretical and experimental results show that the measuring device under the influence of partial decoherence could still be used for amplified detection of minute physical changes and is applicable for implementing the weak value measurement for massive particles.
In the Aharonov-Albert-Vaidman (AAV) weak measurement, it is assumed that the measuring device or the pointer is in a quantum mechanical pure state. In reality, however, it is often not the case. In this paper, we generalize the AAV weak measurement scheme to include more generalized situations in which the measuring device is in a mixed state. We also report an optical implementation of the weak value measurement in which the incoherent pointer is realized with the pseudo-thermal light. The theoretical and experimental results show that the measuring device under the influence of partial decoherence could still be used for amplified detection of minute physical changes and are applicable for implementing the weak value measurement for massive particles.
The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al. (Phys. Rev A 41,11 (1990)), hereafter referred to as AAV. We have performed an experiment similar to one proposed by Knight et al. (1990) and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity. We suggest that preselection and postselection measurement techniques could be useful for study of physical phenomena that might be otherwise unobservable.
Weak values as introduced by Aharonov, Albert, and Vaidman (AAV) are ensemble-average values for the results of weak measurements. They are interesting when the ensemble is preselected on a particular initial state and postselected on a particular final measurement result. It is shown that weak values arise naturally in quantum optics, as weak measurements occur whenever an open system is monitored (as by a photodetector). The quantum-trajectory theory is used to derive a generalization of AAV's formula to include (a) mixed initial conditions, (b) nonunitary evolution, (c) a generalized (nonprojective) final measurement, and (d) a non-back-action-evading weak measurement. This theory is applied to the recent cavity-QED experiment demonstrating wave particle duality [G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, Phys. Rev. Lett. 85, 3149 (2000)]. It is shown that the ``fractional-order'' correlation function measured in that experiment can be recast as a weak value in a form as simple as that introduced by AAV.
The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al., hereafter referred to as AAV. Their analysis led to the controversial result that the weak values for an observable can lie well outside its associated eigenvalue spectrum. We have performed an experiment similar to one proposed by Knight et al. and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity defined by Duck et al. It is shown that these properties are produced by interference phenomena that result from the loss of ‘welcher Weg’ information. We have also identified an intrinsic ‘weak energy’ which appears in the equation of motion for the weak value of an observable and have shown that it can be non–vanishing only for systems which are both preselected and postselected. It is noted that such a ‘weak spin–rotation coupling energy’ can be associated with our experiment and that it is manifested in our experimental data. We suggest that preselection and postselection measurement techniques could be useful for the creation and study of physical phenomena that might be otherwise unobservable.
We present the first realization of a measurement of a ``weak value,'' a concept recently introduced by Aharonov, Albert, and Vaidman (AAV). Our experiment uses a birefringent crystal to separate the two linear-polarization components of a laser beam by a distance small compared to the laser-beam waist. This ``weak measurement'' is followed by a strong measurement which translates the centroid of the beam by a distance far larger than the birefringence-induced separation. In addition, we present data corresponding to orthogonal initial and final states, for which the weak value is not defined. This interference effect may have application in the amplification and detection of weak effects.
For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory- equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the "very" weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be important when the AAV's WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful result for homodyne measurement in circuit-QED system, which allows for measuring the real and imaginary parts of the AAV's WV by simply tuning the phase of the local oscillator. This advantage can be exploited as efficient state-tomography technique.