Weak Value Amplification Can Outperform Conventional Measurement in the Presence of Detector Saturation

Weak value amplification (WVA) is a technique by which one can magnify the apparent strength of a measurement signal. Some have claimed that WVA can outperform more conventional measurement schemes in parameter estimation. Nonetheless, a significant body of theoretical work has challenged this perspective, suggesting WVA to be fundamentally suboptimal. Optimal measurements may not be practical, however. Two practical considerations that have been conjectured to afford a benefit to WVA over conventional measurement are certain types of noise and detector saturation. Here, we report a theoretical study of the role of saturation and pixel noise in WVA-based measurement, in which we carry out a Bayesian analysis of the Fisher information available using a saturable, pixelated, digitized, and/or noisy detector. We draw two conclusions: first, that saturation alone does not confer an advantage to the WVA approach over conventional measurement, and second, that WVA can outperform conventional measurement when saturation is combined with intrinsic pixel noise and/or digitization.

Figure 1

Saturation is necessary to afford an advantage to weak value amplification over conventional measurement. (a) Beam distortion produced by saturation for both CM and WVA with $g|A_w|=0.364$, for a beam with waist $w=0.8$. Orange and green lines show the beam profiles impinging on the camera, while the corresponding beam measured beam profiles obtained from a noisy and saturable camera are indicated by the shaded orange and green profiles. Measured beam profiles obtained via WVA (shaded magenta and blue areas) are dimmer, though more robust against saturation effects. (b) FI obtained from a camera with $k_{\max }=256$, $\sigma =k_{\max }/20$, and $N_{\text{sat}}=500$, for an incident beam with $w=1$ and $g=0.01$, carrying a range of average photon numbers. FI obtained for CM (solid blue line) are plotted alongside FI for indicated WVA schemes (dotted-dashed purple lines). The plot shows that more FI can be recovered using WVA than using CM when saturation effects become important, meaning that WVA can outperform CM in this regime. (c) FI obtained for a beam and camera identical to that simulated in (b), except that $N_{\text{sat}}=10^5$, in order to render saturation negligible. Without saturation, but with intrinsic pixel noise, pixelation, and digitization, WVA is outperformed by CM for all $\overline{n}$ plotted. We note that the FI values obtained in (c) are lower than those in part (b) of the figure, due to the effective decrease in the photoelectron response’s sensitivity (and therefore, increase in digitization effects) that accompanies the large value of $N_{\text{sat}}$.

Figure 2

Saturation is not sufficient to afford an advantage to weak value amplification. (a) FI for a saturable ($N_{\text{sat}}=500$), noiseless, 100-pixel camera subject to negligible digitization ($k_{\max }=4096$), for a beam identical to that simulated in Fig. 1, for a range of $\overline{n}$ values. Despite the presence of saturation, CM measurement (blue solid line) outperforms WVA with $|A_w|=1.8$ (dotted purple line), 2.4 (dotted-dashed purple line), and 3.2 (dashed purple line) for all $\overline{n}$. (b) Simulation identical to (a), with the exception that digitization has been introduced by setting $k_{\max }=256$. Although saturation alone fails to confer an advantage to WVA over CM [part (a)], WVA can outperform CM when digitization is introduced [part (b)].

Figure 3

Optimal post selection with weak value amplification. (a) Performance of CM (blue line) and WVA schemes with amplification factors of 1.2 (lightest purple line), 1.3, 1.5, 1.6, 1.8, 2.1, 2.4, 2.7, 3.2, and 3.9 (darkest purple line) for a 100-pixel camera with $N_{\text{sat}}=500$, $k_{\max }=256$, and $\sigma =k_{\max }/100$ for indicated incident beam brightnesses. (b) FI for the camera simulated in (a), for an incident beam with $\overline{n}=11500$ [dotted line in (a)], showing that an optimal amplification $|A_w|$ exists, which elicits the maximum possible FI from the camera. The dashed line indicates the performance of CM.