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

Abstract

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).

Figures (5)

• Figure 1: Study of optimal weak-value amplification via Allan variance analysis. (a) Schematic of experimental realization of a WVA-based two-slits interferometer for measuring few-attosecond time delays. Two weak measurements with the same preselection and weak interaction but different postselections are conducted inside a two-slit interferometer. Few-attosecond time delays are controlled by two perpendicular true zero-order HWPs, with the second HWP tilted at an angle $\theta$ around the $y$ axis. The two pointers are post-selected using two D-shaped HWPs at different angles followed by a PBS. Components: polarizing beam splitter (PBS), half-wave plate (HWP), D-shaped half-wave plate (D-HWP), Charge-Coupled Device (CCD). (b) The principle of the Allan variance analysis in a sampled system with a sampling period $\delta t$, where the number of time-series data $\tau_j$ is finite and is constrained to be an averaging interval ${T=n \times \delta t}$ (here $n=5$).
• Figure 2: Short-term and long-term precision for a $\tau = 1.69$ as time delay measurement.(a) Allan variance $\sigma_{e}^{2}(T)$ versus averaging interval $T$ (log-log scale). (b) Power spectral density (PSD) of the measured time delay under different conditions. Data were acquired at sampling rates of $f_s = 100$ Hz and $f_s = 0.1$ Hz, with post-selection angles $\beta^{u,d} = \pm1.6^{\circ}$ (weak-value amplification) and $\beta^{u,d} = \pm45^{\circ}$ (traditional measurement without WVA). The standard quantum limit (SQL) $\sigma_{f}(T)$ (dashed line) is given by the Cramér-Rao bound, and is calculated based on Eq. (\ref{['Eq:define_FisherInformation']}). The SQL lines are calculated from the raw (non-averaged) sampling data and represent the fundamental shot-noise limit at the single-measurement level. The red triangle marks the minimum experimental variance achieved at $T = 300$ s in Ref. PhysRevLett.134.080802. The detected photon number for the WVA condition ($\beta^{u,d} = \pm1.6^{\circ}$) was measured to be $N_r = 3.6 \times 10^{4}$ photons per measurement. This value is consistent with the experimental parameters used in the previous work PhysRevLett.134.080802, confirming a comparable post-selection efficiency and facilitating a direct comparison of the shot-noise-limited performance.
• Figure 3: Dependence of the measured variance $\sigma_{e,f}^2$ on the detected photon number $N_r$. Each data point represents the average of 200 independent measurements collected at a sampling rate $f_s = 100$ Hz. The experimental data obtained under WVA with $\beta^{u,d}=\pm1.6^{\circ}$ follows the SQL, confirming the predicted $\sigma^2 \propto 1/N_r$ scaling.
• Figure 4: Dependence of the measured Allan curve $\sigma_{e,f}^2(T)$ on the detected photon number $N_r$. Data were acquired at sampling rates of $f_s = 100$ Hz with post-selection angles $\beta^{u,d} = \pm1.6^{\circ}$ (WVA).
• Figure 5: Dependence of the measured Allan variance $\sigma_{e,f}^2$ on the magnitude of the time delay $\tau$ being estimated. Data were acquired at sampling rates of $f_s = 100$ Hz with post-selection angles $\beta^{u,d} = \pm1.6^{\circ}$ (WVA).