Heisenberg-Limited Metrology via Weak-Value Amplification without Using Entangled Resources

Weak-value amplification (WVA) provides a way for amplified detection of a tiny physical signal at the expense of a lower detection probability. Despite this trade-off, due to its robustness against certain types of noise, WVA has advantages over conventional measurements in precision metrology. Moreover, it has been shown that WVA-based metrology can reach the Heisenberg limit using entangled resources, but preparing macroscopic entangled resources remains challenging. Here, we demonstrate a novel WVA scheme based on iterative interactions, achieving the Heisenberg-limited precision scaling without resorting to entanglement. This indicates that the perceived advantages of the entanglement-assisted WVA are in fact due to iterative interactions between each particle of an entangled system and a meter, rather than coming from the entanglement itself. Our work opens a practical pathway for achieving the Heisenberg-limited WVA without using fragile and experimentally demanding entangled resources.

Figure 1

Three weak-value amplification (WVA) scenarios for estimating an interaction strength $\gamma$ with a total of $N$ interactions between a system and a meter. (a) Typical WVA: the meter state is measured independently $N$ times after a single interaction with a system particle and the postselection of the system. (b) Entanglement-assisted WVA: the meter state is measured once after $N$ consecutive interactions with each particle of an $N$-partite entangled system and the postselection of the system. (c) WVA with iterative interactions: the meter state is measured once after $N$ iterative interactions with a system particle and the postselection of the system.

Figure 2

Experimental schematic to estimate a path-dependent polarization change via WVA based on iterative interactions. The polarization mode and the path mode of a single photon are employed as the system and the meter, respectively, and they are initialized using the first beam displacer (BD) and half-wave plate (HWP). The path-dependent polarization change of $\gamma$ is implemented by rotating the next HWP in each path in opposite directions, and the $N$ iterative interactions are introduced by setting the HWP angles at $N\gamma /2$ and $-N\gamma /2$, respectively. The system (polarization mode) is then projected onto $|\varphi {⟩}_s$ by postselecting the photon coming from the lower path of the second BD, and the postselection state is determined by the HWP angle $\varphi$. The operation of the second BD converts the path mode state into the polarization mode state, so the meter readout is carried out using the polarizing beam splitter (PBS).

Figure 3

The amplified meter readouts $⟨{\widehat{\sigma }}_z{⟩}_m\approx 2\gamma \mathrm{Im}[⟨N{\widehat{\sigma }}_y{⟩}_w]$ and the detection probabilities $P_{\mathrm{iter}}^{(N)}$ for the WVA measurements of $\gamma =1°$ with (a) a single interaction and (b) four iterative interactions. As the postselection state approaches the completely orthogonal state of the initial system state, $\varphi =45°$, the amplification effect on the meter expectation value $⟨{\widehat{\sigma }}_z{⟩}_m$ is enhanced while the detection probability decreases. The solid lines show the exact theoretical curves, and the error bars represent one standard deviation of Poissonian counting statistics.

Figure 4

The detection probabilities $P_{\mathrm{iter}}^{(N)}$ of WVA based on $N$ iterative interactions. At each $N$, we adjust the postselection state $|\varphi {⟩}_s$ in Eq. (17) according to Eq. (11) in order to fix the amplification factor to $|⟨N{\widehat{\sigma }}_y{⟩}_w|=11.5$. For $N=1$, 2, 3, 4, the amplified meter expectation values for $\gamma =1°$ are, respectively, measured as $⟨{\widehat{\sigma }}_z{⟩}_m=0.41\pm 0.16$, $0.52\pm 0.08$, $0.37\pm 0.06$, $0.35\pm 0.04$ with $\varphi =47.5°$, 50.0°, 52.3°, 54.6°. The red bars represent the experimental data, and the dashed bars represent the ideally expected $P_{\mathrm{iter}}^{(N)}$. The corresponding Heisenberg-limited scaling of $P_H^{(N)}\propto N^2$ is shown in the beige bars. The detection probabilities $P_{\mathrm{iter}}^{(N)}$ have slight discrepancy with $P_H^{(N)}$ since the condition of Eq. (6), $N\ll |⟨N{\widehat{\sigma }}_y{⟩}_w|=11.5$, does not hold as $N$ increases. This discrepancy can be eliminated by increasing the amplification factor. The error bars represent one standard deviation of Poissonian counting statistics.