Adaptive weak-value amplification with adjustable postselection

Weak-value amplification (WVA) has recently become an important technique for parameter estimation, owing to its ability to enhance the signal-to-noise ratio by amplifying extremely small signals with proper postselection strategies. In this paper, we propose an adaptive WVA scheme to achieve the highest Fisher information when using an unbalanced pointer. Different from previous schemes, the adaptive WVA scheme is associated with a real-time update on the postselection states with the help of feedback information from the outcomes, and the “extremely small” condition set on the parameter of interest is relaxed. By applying this scheme to a time-delay measurement scenario, we show by numerical simulation that the precision achieved in our scheme is several times higher than the standard WVA scheme. Our result might open a path for improving the WVA technique in a more flexible and robust way.

Figure 1

Schematic diagram of the SWVA and the AWVA scheme. (a) AWVA scheme with feedback control. Pre- and postselection in each measurement is adjusted according to an adaptive method. (b) SWVA scheme with fixed pre- and postselection. Pre- and postselection is fixed without any adaptivity. The estimative value of the coupling parameter $g$ can be calculated from the detection outcome.

Figure 2

(a) The Fisher information (arb. units) with respect to the imaginary part of weak value $\text{Im}{A}_w$ and the parameter $g$. The peak of the curved surface, which denotes the maximum Fisher information, shows that the optimal weak value varies with the parameter $g$. (b) is captured from this three-dimensional surface. (b) The Fisher information with respect to the parameter $g$. The three dotted curves denote that $\text{Im}{A}_w$ takes three different but constant values. For the three different weak values, denoted by the three dotted curves, the parameter that can achieve the maximum Fisher information is different. For an alterable weak value, denoted by the solid curve, it can always lead to the maximum Fisher information by adjusting the weak value to be optimal.

Figure 3

(a) Spectrum shift with respect to postelection angle $\epsilon$ and time delay $\tau$. (b) is captured from this curved surface. (b) Spectrum shift with respect to $\epsilon$. The three curves denote that when the time delay takes three different values, a relevance is shown between the spectrum shift and the time delay. Note that the sign of the spectrum shift is opposite on different sides of the zero point.

Figure 4

The red continuous curve and the blue dashed curve represent the minimum errors predicted by the Fisher information of the SWVA and AWVA schemes with respect to the photon number $N$, respectively. The black dots are the numerical simulation of the error of the AWVA scheme with the optimal postselection angle for increasing photon number. The pre- and postselection are $|{\phi }_i\rangle =\frac{|H\rangle +|V\rangle }{\sqrt{2}}$ and $|{\phi }_f\rangle =\frac{|H\rangle -e^{i\epsilon }|V\rangle }{\sqrt{2}}$, respectively. The postselection angle in SWVA is fixed at $\epsilon =0.03$ in every run while in AWVA it is nearly adjusted to reach the optimal value via the adaptive method. The time delay takes the value ${\tau }_0=8\text{as}$, the mean frequency ${\omega }_0$ is 2400 THz, and the width $\mathrm{\Delta }$ is 55 THz. The numerical simulation error is defined as $\mathrm{\Delta }\tau =|\widehat{\tau }-{\tau }_0|$ and the error limit is defined as $1/\sqrt{NI}$. The increment in the numerical simulation is $\delta \epsilon ={10}^{-6}$.