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/Nᵣ, confirming shot-noise-limited operation with WVA. Furthermore, this 1/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 Hz).