Weak-Value Amplification for Longitudinal Phase Measurements Approaching the Shot-Noise Limit Characterized by Allan Variance

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 $T$ = $0.01-0.1$ s, representing two orders of magnitude variance reduction compared to the $T=300$ s operating point in prior implementations. We demonstrate that the Allan-variance noise floor scales with the inverse of the detected photon number $1/N_r$, confirming shot-noise-limited operation with WVA. Furthermore, this $1/N_r$ 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 ($>10$ Hz).