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arXiv:1505.07199 · Physical Review A

Physical description of statistical hypothesis testing for a weak-value-amplification experiment using a birefringent crystal

Yuki Susa

Published Aug 20, 2015• 3 citations
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Abstract

We investigate the weak measurement experiment demonstrated by Ritchie et al. [N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (1991)] from the viewpoint of the statistical hypothesis testing for the weak-value amplification proposed by Susa and Tanaka [Y. Susa and S. Tanaka, Phys. Rev. A 92, 012112 (2015)]. We conclude that the weak-value amplification is a better method to determine whether the crystal used in the experiment is birefringent than the measurement without postselection, when the angles of two polarizers are almost orthogonal. This result gives a physical description and intuition of the hypothesis testing and supports the experimental usefulness of the weak-value amplification.

DOI
10.1103/physreva.92.022118

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We study the detection capability of the weak-value amplification on the basis of the statistical hypothesis testing. We propose a reasonable testing method in the physical and statistical senses to find that the weak measurement with the large weak value has the advantage to increase the detection power and to reduce the possibility of missing the presence of interaction. We enhance the physical understanding of the weak value and mathematically establish the significance of the weak-value amplification. Our present work overcomes the critical dilemma of the weak-value amplification that the larger the amplification is, the smaller the number of data becomes, because the statistical hypothesis testing works even for a small number of data. This is contrasted with the parameter estimation by the weak-value amplification in the literature which requires a large number of data.
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.
How to realize the detection of tiny birefringence is an important but challenging issue for investigations and applications in optics. In weak measurements, weak-value amplification has been widely applied to the precise estimation of time delay. Here, we show that the linear amplification in weak measurements has the potential to outperform standard weak-value amplification for ultra-small parameter characterization. The higher measurement accuracy in the linear amplification enables us to achieve a measurement of stress-induced birefringence with a high resolution of 5×10−10. Compared with existing methods, the accuracy based on weak measurements for birefringence detection can improve two orders of magnitude. Our method may have important applications in a variety of fields involving the precise measurement of time delay or birefringence.
Interpretations of an experiment on the backaction in a weak measurement in [Iinuma et al., New J. Phys. 13, 033041 (2011)] are revisited. We show two different but essentially equivalent interpretations for this experiment along the original scenario of weak measurements proposed by Aharonov, Albert, and Vaidman [Phys. Rev. Lett. 60, 1351 (1988)]. To do this, we introduce the notion of extended weak values, which is associated not only with the states of the system but also the state of the measuring device. We also evaluate fluctuations in this experiment, and found that an optimal measurement strength exists for a fixed polarization angle prepared as an initial state, at which fluctuations in measurement results vanish. The consistency of this evaluation with the experimental results is discussed.
We present an expanded mathematical model of the optical weak value amplification device introduced by Duck, Stevenson, and Sudarshan in 1989 [ Phys. Rev. D 40 , 2112 ( 1989 ) PRVDAQ 0556-2821 10.1103/PhysRevD.40.2112 ]. We model the pointer behavior in and around the weak value and inverse weak value amplification regions with respect to the incidence angle of the birefringent crystal. We present laboratory measurements that support the validity of this model. We measure a tilt sensitivity of ∼.58m/rad, which represents an amplification of 390 times over nonamplified refraction. This corresponds to an increase in SNR of ∼110 times with respect to shot noise. This paper demonstrates the novel utility of a birefringence-based WVA device as a sensitive tiltmeter, to our knowledge.