Quantum weak measurement is a measurement technique based on minimal intrusion and the weak value amplification (WVA) effect, which significantly expands the capability of quantum precision measurement. Quantum weak measurements have been applied to tiny-phase measurements and have yielded a series of results. Previous standard weak measurements primarily used spectral or CCD pointers to achieve precise measurements of tiny phases under a single weak interaction (SWI). However, a trade-off between enhancing the amplification factor (known as the anomalous weak value) and the post-selection probability is encountered in SWIs. Because of the difficulty in enhancing both simultaneously, the extraction of the anomalous weak value and the measurement precision are limited. This issue is typically alleviated using quantum resources. However, difficulties in preparing quantum resources restrict their practical applications. Hence, practical quantum weak measurement systems must be further investigated to overcome these limitations.

In this study, an enhanced phase weak measurement method based on multiple weak interactions (MWIs) is proposed and demonstrated. The evolution of the intensity pointer was analyzed theoretically and a formulation for enhanced WVA was derived. A system for enhanced phase weak measurements based on MWIs was constructed using continuous coherent light as the incident source. This setup employs double weak interactions with two sets of half-wave plates (HWPs) as well as an avalanche photodiode (APD) as the intensity pointer. The post-selection angle was fixed, and the HWP was tilted to introduce a weak phase delay for the measurement. Meanwhile, the phase delay was detected precisely by recording changes in the light intensity. By maintaining a constant phase delay and adjusting the post-selection angle, anomalously weak values were extracted with a high signal-to-noise ratio (SNR).

Owing to the double weak interactions, the intensity shift is greater than that resulted under the SWI (Fig. 4). The symmetrical post-selection intensity contrast ratios exceed those under the SWI by approximately two folds (Fig. 5), thus indicating that the double-weak-interaction scheme offers a higher measurement sensitivity. At three post-selection angles of 0.002, 0.004, and 0.007 rad, the intensity uncertainties of the double weak interactions are 0.006, 0.015, and 0.040 mV, respectively (Fig. 6), and the phase precisions are 6.00×10^-7, 7.76×10^-7, and 9.46×10^-7 rad, respectively. Compared with the SWI, the double weak interactions improved the phase precision by approximately an order of magnitude. Under the double weak interactions, an anomalously weak value of 239 is obtained with a high SNR of 18.2 dB for a post-selection angle of 0.0084 rad. Decreasing the post-selection angle to 0.002 rad while the SNR remains at 9.8 dB results in a higher anomalous weak value of 993 (Fig. 7).

In this study, based on the amplification effect of anomalous weak values on the interaction parameters, an enhanced WVA scheme based on MWIs was theoretically and experimentally demonstrated and applied to ultraprecise phase measurements. Experimentally, coherent light with a linewidth of 400 kHz was used as the incident light source and an APD was utilized as the intensity pointer, which is different from previous spectral detection schemes. By developing the MWI theory and experimental scheme as well as realizing double weak interaction (N=2) measurements, the experimental results show that the phase measurement sensitivity is significantly better than that achieved under SWIs. In particular, an optimal measurement precision of 6.0×10^-7 rad is achieved at a post-selection angle of 0.002 rad, which is approximately an order of magnitude higher than that achieved under an SWI. Additionally, based on a high SNR of 9.8 dB at a post-selection angle of 0.002 rad, an enhanced anomalous weak value of 993 is measured, which is approximately an order of magnitude greater than the standard weak measurement. The enhanced WVA scheme based on MWIs was validated via phase measurements, whereas the relationship between the phase and other physical quantities, such as displacement, temperature, and magnetic induction strength, can be established in optical experiments; thus, the scheme provides important technical support for practical enhanced quantum precision sensing.