Weak measurement with the peak-contrast-ratio pointer

Weak measurement as a nonperturbative theory has been shown to be powerful in the fundamentals of quantum mechanics and high-precision metrology, in which the pointer is of great importance to parameter estimation and experimental design. In this work, we find that under the conditions of weak coupling and Gaussian meter, the probability distribution after the postselection follows a bimodal function and the peak contrast ratio (PCR) can be applied as a special pointer, which is different from the shift of mean value. To obtain the PCR, only two values corresponding to the two peaks in the distribution function are required. We theoretically present the relation of the PCR and parameters to be estimated, and then demonstrate it in an experiment. Weak measurement with the PCR pointer has the advantages of reducing the requirements of the apparatus and suppressing noise, which also provides a heuristic approach for studying weak measurement when the weak value is very large.

Figure 1

Schematic diagram of weak measurement with the PCR pointer. S denotes the system and M denotes the meter. The single and double lines correspond to the quantum and classical processes, respectively.

Figure 2

Experimental setup. A white laser as the light source; GF: Gaussian filter with the central wavelength 633 nm and the FWHM (full width at half maximum) 10 nm; P1 and P2: Polarizers with optical axes set at ${45}^{\circ }$ and $\phi -{45}^{\circ }$, respectively; HWP1 and HWP2: Half-half wave-plate combination with optical axes perpendicular to each other, the latter one of which can be rotated around its vertical axis by a certain angle to introduce a time delay; QWP: Quarter wave plate with the optical axis set at $-{45}^{\circ }$. A spectrometer is used as the detector.

Figure 3

The PCR $R_{\omega }$ in a function of the phase difference $\xi$. The red squares correspond to the experimental results and the black solid line corresponds to the theoretical prediction. The black dotted line shows the linear relation $R_{\omega }\simeq 2\sqrt{2}{\left(\mathrm{\Delta }\omega \right)}^{-1}v$ when $\left|v\right|\ll \mathrm{\Delta }\omega$.

Figure 4

(a) The experimental uncertainty of the PCR. (b) The experimental uncertainty of the phase difference.