Experimental Demonstration of Higher Precision Weak-Value-Based Metrology Using Power Recycling

The weak-value-based metrology is very promising and has attracted a lot of attention in recent years because of its remarkable ability in signal amplification. However, it is suggested that the upper limit of the precision of this metrology cannot exceed that of classical metrology because of the low sample size caused by the probe loss during postselection. Nevertheless, a recent proposal shows that this probe loss can be reduced by the power-recycling technique, and thus enhance the precision of weak-value-based metrology. Here we experimentally realize the power-recycled interferometric weak-value-based beam-deflection measurement and obtain the amplitude of the detected signal and white noise by discrete Fourier transform. Our results show that the detected signal can be strengthened by power recycling, and the power-recycled weak-value-based signal-to-noise ratio can surpass the upper limit of the classical scheme, corresponding to the shot-noise limit. This work sheds light on higher precision metrology and explores the real advantage of the weak-value-based metrology over classical metrology.

Figure 1

The schematic diagram of three metrological protocols. The classical metrology is constituted by three steps: the probe preparation, the interaction between the probe and the target system (in this work, we focus on the weak-interaction situation), and the probe detection. For the weak-value-based metrologies, an additional step of probe postselection is inserted. The probe that is not postselected is discarded in the SWV metrology, and is, however, recycled in the PRWV metrology. The steps in the dashed box can be regarded as the measurement devices, which output the results of the probe coupling to the target system, which are then detected by the detector.

Figure 2

Experimental setup of the power-recycled weak-value-based beam-deflection measurement. This setup is constituted by four parts: the probe source, the measurement device, the splitting detector, and the PDH system used to stabilize the cavity length. The probe source is the beam attenuated by the neutral-density filter (NDF). This beam is focused onto the cavity mirror and sent into the measurement device. The measurement device consists of a cavity mirror and a Sagnac interferometer. The beam deflection is imposed via a pizeo-driven $50:50$ BS and is coupled to the transverse beam position when exiting from the dark port. The splitting detector is realized using a KEP, two SPADs, and a time-to-digital converter (TDC), which is used to detect the beam-position shift. The PDH system contains the reference beam modulated by the chopper, the transimpedance amplifier (TIA) used to detect the power of this beam, the following signal-processing circuits, and the servo to stabilize the cavity mirror. LPF, low-pass filter; M1, mirror 1; DP, dove prism; FR, Faraday rotator; EOM, electro-optic modulator; PBS, polarizing beam splitter; VA, variable attenuator; QWP, quarter-wave plate; HWP, halfwave plate.

Figure 3

The relations between the average difference signal $\overline{s_0}$ (of 260 sample sequences) and the beam deflection in the PRWV and SWV situations. The red solid and blue dashed lines are the linear-fit results of the corresponding experimental results. The slopes indicate that the difference signal is amplified by a factor of $2.40\pm 0.12$ after combining the power-recycling technique. The error bars correspond to the standard error of the results.

Figure 4

(a) The relation between the white-noise-related SNR and the input probe photon number per integration time for the classical, SWV, and PRWV protocols. The orange solid and blue dashed lines are the shot-noise-limited SNR of the classical and SWV protocols, respectively. The light orange region is the confidence interval of the classical shot-noise-limited SNR under the level of 0.999. These results indicate that the SNR of the PRWV protocol surpasses the classical limit under the confidence level of 0.999. (b),(c) The relation between the white noise and the input probe photon number in the (b) SWV and (c) classical protocols. The dots represent the experimental data, and the lines show the shot noises, which are calculated from the square roots of the detected photon numbers per integration time. The error bars correspond to the standard error of the results.