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arXiv:2611.00521

Optimal Estimation in Weak Value Amplification with a Position Pointer

Jing-Hui Huang, Jing Zhang, Jian Ge, Lichao Liu, Wang Luo, Xiang-Yun Hu

quant-phSubmitted Oct 31, 2026• Announced Nov 3, 2026
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Abstract

We experimentally evaluate five estimators for weak-value amplification (WVA) using a position pointer to measure birefringence time delays. Two general approaches are considered: measuring the position shift $Δx$ and measuring the probability change $ΔP_w$ of successful postselection. Specifically, five estimators are investigated: CCD-detected fringe shift $Δx$, quadrant photodiode (PDQ) transversal voltage difference (XDIFF), and lock-in amplifier output phase $η$ (for $Δx$ measurement), as well as PDQ total output voltage (SUM) and CCD-detected average photon number $N_r$ (for $ΔP_w$ measurement). The measurement precision is characterized by the Allan deviation over a range of input laser powers $P_0$. Generally, measuring the position shift $Δx$ yields higher precision than measuring the probability change $ΔP_w$. At $P_0=0.5$~mW, corresponding to a laser-power-limited regime, direct measurement of the fringe displacement with the CCD achieves a minimum Allan deviation of $(5.30\pm0.10)\times10^{-3}$~as, approaching the corresponding photon-statistical Cramér--Rao bound (CRB). At $P_0=50$~mW, corresponding to a detector-limited regime, PDQ detection combined with lock-in demodulation provides the best precision, reaching $(1.80\pm0.06)\times10^{-3}$~as and remaining close to the corresponding CRB. These results demonstrate that the optimal readout depends on the available optical power, detector dynamic range, and technical noise, providing practical guidance for selecting readout strategies in high-precision WVA-based metrology.

Comments
5 pages, 3 figures

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We experimentally evaluate five estimators for weak-value amplification (WVA) using a position pointer to measure birefringence time delays. Two general approaches are considered: measuring the position shift ΔxΔxΔx and measuring the probability change ΔPwΔP_wΔPw​ of successful postselection. Specifically, five estimators are investigated: CCD-detected fringe shift ΔxΔxΔx, quadrant photodiode (PDQ) transversal voltage difference (XDIFF), and lock-in amplifier output phase ηηη (for ΔxΔxΔx measurement), as well as PDQ total output voltage (SUM) and CCD-detected average photon number NrN_rNr​ (for ΔPwΔP_wΔPw​ measurement). The measurement precision is characterized by the Allan deviation over a range of input laser powers P0P_0P0​. Generally, measuring the position shift ΔxΔxΔx yields higher precision than measuring the probability change ΔPwΔP_wΔPw​. At P0=0.5P_0=0.5P0​=0.5~mW, corresponding to a laser-power-limited regime, direct measurement of the fringe displacement with the CCD achieves a minimum Allan deviation of (5.30±0.10)×10−3(5.30\pm0.10)\times10^{-3}(5.30±0.10)×10−3~as, approaching the corresponding photon-statistical Cramér--Rao bound (CRB). At P0=50P_0=50P0​=50~mW, corresponding to a detector-limited regime, PDQ detection combined with lock-in demodulation provides the best precision, reaching (1.80±0.06)×10−3(1.80\pm0.06)\times10^{-3}(1.80±0.06)×10−3~as and remaining close to the corresponding CRB. These results demonstrate that the optimal readout depends on the available optical power, detector dynamic range, and technical noise, providing practical guidance for selecting readout strategies in high-precision WVA-based metrology.
We report a quantitative evaluation of weak-value amplification (WVA) for longitudinal phase measurements using Allan variance analysis. Building on a recent double-slit interferometry experiment with real weak values [Phys. Rev. Lett. 134, 080802 (2025)], our Allan variance analysis demonstrates measurement of a few attosecond time delay approaching the shot noise limit at short averaging intervals of TTT = 0.01−0.10.01-0.10.01−0.1 s, representing two orders of magnitude variance reduction compared to the T=300T=300T=300 s operating point in prior implementations. We demonstrate that the Allan-variance noise floor scales with the inverse of the detected photon number 1/Nr1/N_r1/Nr​, confirming shot-noise-limited operation with WVA. Furthermore, this 1/Nr1/N_r1/Nr​ scaling experimentally validates that WVA can outperform conventional measurement under fixed detected photon number and detector saturation, in the presence of technical noise, as theoretically predicted [Phys. Rev. Lett. 118, 070802 (2017)]. Our results provide rigorous, quantitative evidence of the near-optimal noise performance achievable with WVA, establishing a new benchmark for precision optical metrology. This advancement is particularly relevant to applications such as gravitational-wave detection, where signals predominantly occupy the high-frequency regime (>10>10>10 Hz).
By choosing more orthogonality between preselection and postselection states, one can significantly improve the sensitivity in general optical quantum metrology based on the weak-value amplification (WVA) approach. However, increasing the orthogonality decreases the probability of detecting photons and makes the weak measurement difficult, especially when the weak measurement is disturbed by strong noise and the pointer is drowned in noise with a negative-decibel signal-to-noise ratio (SNR). In this article we introduce and numerically evaluate a modified weak-measurement protocol with a temporal pointer, namely, the autocorrelative weak-value amplification (AWVA) approach. Specifically, a small longitudinal time delay (tiny phase shift) $\ensuremath{\tau}$ of a Gaussian pulse is measured by implementing two simultaneous autocorrelative weak measurements under Gaussian white noise with different SNRs. The small quantities $\ensuremath{\tau}$ are obtained by measuring the autocorrelation coefficient of the pulses instead of fitting the shift of the mean value of the probe in the standard WVA technique. Simulation results show that the AWVA approach outperforms the standard WVA technique in the time domain with smaller statistical errors, remarkably increasing the precision of weak measurement under a strong noise background.
We introduce an ultrasensitive interferometric protocol that combines weak value amplification (WVA) with traditional interferometry. This protocol WVA+interferometry uses weak value amplification of the relative delay between two paths to enhance interferometric sensitivity. As an example, we demonstrate a proof-of-principle experiment that achieves few-attosecond timing resolution (few nanometer path length resolution) with a double-slit interferometer using only common optical components. Since our example uses only the spatial shift of double-slit interference fringes, its precision is not limited by the timing resolution of the detectors but is instead limited by the fundamental shot noise associated with classical light and the diminished technical noise. We experimentally demonstrate that the signal-to-noise ratio can be improved by one to two orders of magnitude relative to a measurement that does not use WVA. Two key conclusions are drawn: (i) Most conventional interferometric techniques primarily rely on determining the path difference (time delay or longitudinal phase), with their precision constrained by technical noise. Our protocol offers a robust solution for minimizing the technical noise in traditional interferometry, with precision in principle approaching the shot-noise limit. (ii) Although WVA has achieved significant advancements in ultrasensitive longitudinal phase measurement, its applicability is constrained by the need for broad spectral bandwidths and high-resolution spectrometers. Contrary to previous assumptions, we demonstrate that quantum-limited WVA time delay measurements are achievable with narrow band light using real weak values. Thus, the cost-effectiveness and practicality of the proposed WVA+interferometry protocol using narrow band light broaden the scope of WVA applications. This protocol holds potential for broad applications in optical metrology, quantum optics and quantum information, biomedical imaging, and interferometric telescopes for astrophysics.
Abstract Most studies for postselected weak measurement in optomechanical system focus on using a single photon as a measured system. However, we find that using weak coherent light instead of a single photon can also amplify the mirror’s position displacement of one photon. In the weak value amplification (WVA) regime, the weak value of one photon can lie outside the eigenvalue spectrum, proportional to the difference of the mirror’s position displacement between the successful and failed postselection and its successful postselection probability is dependent of the mean photon number of the coherent light and improved by adjusting it accordingly. Outside the WVA regime, the amplification limit can reach the level of the vacuum fluctuations, and when the mean photon number and the optomechanical coupling parameter are selected appropriately, its successful postselection probability becomes higher, which is beneficial to observe the maximum amplification value under the current experimental conditions. This result breaks the constraint that it is difficult to detect outside the WVA regime. This opens up a new regime for the study of a single photon nonlinearity in optomechanical system.
The weak-value-amplification (WVA) technique has been extensively considered and debated in the field of quantum precision measurement, largely owing to the reduced Fisher information caused by the low probability of successful postselection. In this work we show that, rather than the transverse-spatial-wave-function meter typically considered, using the optical coherent state as a meter, the WVA measurement can outperform the conventional measurement not involving the strategy of postselection. We also show that the postselection procedure involved in the WVA scheme can make a mixture of coherent states work better than a pure coherent state with identical average photon numbers. This is in contrast to the main result proved in the absence of postselection. The postselection strategy can also result in the precision of Heisenberg scaling with the photon numbers without using expensive quantum resources. The present work may stimulate further interest in postselection strategy for quantum precision measurements.
Weak-value amplification (WVA), a postselection-based technique that amplifies weak physical signals by preparing nearly orthogonal pre- and postselected quantum states, is intrinsically limited by various kinds of technical noise, which distorts amplified weak values, especially when discarding photons in postselection. While prior work established the efficacy of autocorrelative weak-value amplification (AWVA) under Gaussian noise, practical implementations face challenges from band-limited laser-source noise and detection noise (including shot noise and electrical noise). Here, we demonstrate that the AWVA protocol robustly suppresses both laser-power fluctuations and detection noise. Numerical experiments in Simulink further reveal AWVA's dual advantage: under high-power conditions, the noise-reduction superiority of AWVA over WVA becomes in- creasingly pronounced as input laser power increases, whereas in detection-limited regimes AWVA achieves an order-of-magnitude lower uncertainty, closely approaching the Cram\'er-Rao bound. Crucially, this work demonstrates that AWVA improves precision in both high-power (laser-noise-dominated) and photon-starved (detector-noise-dominated) regimes, thereby bridging these operating extremes and advancing precision in applications from gravitational-wave detection to hybrid quantum systems.