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

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.

Figure 1

(a) Sketch of the optical setup. The tilted birefringent crystal plates is set between the two polarizers, the angles of which are tuned almost orthogonal. On the screen, we observe the beam position $y$ and we decide whether or not the crystal is birefringent. Also shown are (b) the beam polarization and (c) the probe distribution, in each stage. For the illustration, the angles of the polarizers are $\alpha =\pi /4$ and $\beta =3\pi /4$.

Figure 2

Powers $b_{\mathrm{ps}}$ (solid line) and $b_{\mathrm{nps}}$ (dashed line) plotted with the critical points $c$ as the horizontal axis in the two cases: (b) $\beta =3\pi /4+2.2\times {10}^{-2}$ and (c) $\beta =3\pi /4$. The other parameters are fixed as $\alpha =\pi /4,w_0=55\mu \mathrm{m}$, and $g{\lambda }_{-}=0.32\mu \mathrm{m}$.