Evaluation of weak measurements to all orders

Some exact formulas of the expectation values and probability densities in a weak measurement for an operator $\mathbf{A}$ that satisfies the property ${\mathbf{A}}^2=1$ are derived. These formulas include all-order effects of the unitary evolution due to the von Neumann interaction. These are valid not only in the weak-measurement regime, but also in the strong-measurement regime and tell us the connection between these two regimes. Using these formulas, arguments of the optimization of the signal amplification and the signal-to-noise ratio are developed in two typical experimental setups.

Figure 1

${\langle p_z\rangle }^{\prime }/g$ is shown as a function of the coupling $s=2g^2\langle z^2\rangle$ and the preselection angle $\alpha$ (rad) in various ranges of $s$. (a) $s\in (0,0.1)$, (b) $s\in (0,1)$, (c) $s\in (0,10)$. From these figures, we can see that, for given coupling parameter $s$, there is an optimal preselection angle $\alpha$ such that the expectation value of $p_z$ is maximized.

Figure 2

The optimal expectation-value line (4.13) (top panel), the expectation value ${\langle p_z\rangle }^{\prime }/g$ [Eq. (4.14)], and the fluctuation $\Delta p_z$ [Eq. (4.15)] in $p_z$ after the postselection on the optimal line (4.13) (lower panel) are shown as functions of the coupling parameter $s$. We also show that $({\langle p_z\rangle }^{\prime }\pm \Delta p_z)/g$ in this lower panel. [The red line is $({\langle p_z\rangle }^{\prime }+\Delta p_z)/g$ and the blue line is $({\langle p_z\rangle }^{\prime }-\Delta p_z)/g$.] This figure shows that if we choose the small parameter of $s\ll 1$, we can accomplish the large expectation value of ${\langle p_z\rangle }^{\prime }/g\sim 1/\sqrt{2s}$, but the fluctuations in $p_z$ also amplified as $\Delta p_z/g\sim 1/\sqrt{2s}$.

Figure 3

Probability distribution functions (4.16) in $p_z/g$ space on the optimal expectation-value line (4.13) with some coupling parameters $s$ are shown by the thick lines (red lines). We also plot the initial probability distribution $\sqrt{\frac{s}{\pi }}{e}^{-s{(p_z/g)}^2}$ with the same coupling parameter $s$ by the thin lines (blue lines). The $s=0.1$ case corresponds to the weak measurement on the optimal expectation-value line (4.13). This shows that the peak of the probability distribution slightly deviates from the weak value $A_w$. The $s=1.0$ case is still essentially the same as the $s=0.1$ case. The $s=10$ and 1000 cases correspond to the strong-measurement case. These behaviors of the probability distribution (4.16) also well describe the strong-measurement regime $s\gg 1$.

Figure 4

The signal-to-noise ratio (SNR) (4.17) is shown as a function of the coupling parameter $s$ and the preselection angle $\alpha$ (rad): (a) $s\in (0,1)$ and $\alpha \in (\pi /2,\pi )$; (b) $s\in (0,10)$ and $\alpha \in (\pi ,0)$. There is the ridge of the SNR from $(s,\alpha )=(0,\pi )$ to $(s,\alpha )=(+\infty ,\pi /2)$. (We note that the direction of $s$ axes are opposite to those in Fig. 1.) The ridge around $\alpha =\pi /2$ is due to the strong-measurement regime $s\gg 1$, which is due to the fact that $\alpha =\pi /2$ corresponds to the eigenstate ${|\uparrow }_z\rangle$ of the operator ${\sigma }_z$ with the eigenvalue $+1$. Together with Fig. 3, this shows the behavior of the SNR between the weak-measurement regime $s\ll 1$ and the strong-measurement regime $s\gg 1$.

Figure 5

The optimal SNR line (upper panel) (4.19) on the $(s,\alpha )$ plane and the optimally preselected SNR (4.20) (lower panel) are shown. The optimally preselected SNR is a monotonically increasing function of $s$. In the strong-measurement regime $s\gg 1$, this SNR increases due to the fact that the preselected state $\alpha =\pi /2$ is the eigenstate ${|\uparrow }_z\rangle$ of the operator ${\sigma }_z$. In the weak-measurement regime $s\ll 1$, the SNR can not be larger than that in the strong-measurement regime, but has the minimum value ${\langle p_z\rangle }^{\prime }/\Delta p_z\sim 1.07046$.

Figure 6

$-k{\langle x\rangle }^{\prime }$ [Eq. (5.9)] is shown as a function of the coupling $s=2k^2\langle x^2\rangle$ and the preselection angle $\phi$ (rad) in two ranges of $s$. (a) $s\in (0,10)$, (b) $s\in (0,2)$. For large $s$, the expectation value $-k{\langle x\rangle }^{\prime }$ decays exponentially. From these figures, we can see that, for a given coupling parameter $s$, there is a preselection angle $\phi$ such that the expectation value of $x$ is maximized.

Figure 7

The optimal expectation-value line (5.14) (upper panel) and the expectation value of $-k{\langle x\rangle }^{\prime }$ (5.15) (lower panel) on this optimal line are shown as a function of the coupling parameter $s$. In the limit $s\to \infty$, the optimal preselection angle for expectation value approaches to $\phi \to \pi /2$. Further, on the optimal expectation-value line, the expectation value $-k{\langle x\rangle }^{\prime }$ has the maximum value at $s\sim 0.8$.

Figure 8

Probability distribution functions (5.17) in $x$ space on the optimal expectation-value line (5.14) with some coupling parameters $s$ are shown by the thick line (red line). We also plot the initial probability distribution $\langle x|{\rho }_d\rangle /k$ in Eq. (5.13) with the same coupling parameter $s$ by the thin lines (blue lines). The $s=0.1$ case corresponds to the weak measurement on the optimal expectation-value line (5.14). This shows that the peak of the probability distribution slightly deviates from the linear result $s\text{Im}{A}_w$. The $s=1.0$ case is still essentially the same as the $s=0.1$ case. The maximal expectation value ($s=s_m\sim 0.8$) is obtained around this parameter. The $s=10$ and 1000 cases correspond to the strong-measurement case.

Figure 9

The signal-to-noise ratio (SNR) (5.20) in the simplified DSJH setup is shown as a function of the coupling parameter $s$ and the preselection angle $\phi$ (rad). This shows that the SNR in this setup has the peak only in the weak-measurement regime $s<1$.

Figure 10

The optimal SNR preselection angle ${\phi }_{\mathrm{opt}}$ [the solution to Eq. (5.22)] (upper panel) and the optimized SNR (5.23) (lower panel) in the simplified DSJH setup are shown as a function of the coupling parameter $s$. This shows that the SNR in this setup has the peak only in the weak-measurement regime $s<1$. The maximum SNR is of the order of unity in the single-photon case.