Pre- and post-selected (PPS) measurement, which was first proposed by Aharonov, Bergmann, and Lebowitz (ABL), is of fundamental significance in quantum measurement.^[1] PPS measurement is performed on a sub-ensemble of a system with chosen pre- and post-selected states. The coupling strength between the system and the meter can be measured with a subsequent projective measurement of the output meter. A typical extension of PPS measurement is the weak PPS measurement (i.e., the so-called weak measurement), which requires an extremely weak coupling strength.^[2–4] Weak measurement has attracted much attention due to the striking amplification effect on some ultra-small physical parameters with an amplification factor of the weak value[5–19]

In this paper, we propose a modulated pre- and post-selected measurement (PPSM) scheme based on the coupling-strength-dependent modulation. With the attainable highest sensitivity and precision, the coupling strength can be further relaxed to an arbitrary magnitude. We theoretically analyze the measurement sensitivity and precision for the case of unbalanced input meter. We also give the optimal pre- and post-selected states, as well as the optimal modulation of the coupling strength that correspond to the highest precision without any approximation. Our numerical comparison demonstrates that the modulated PPSM scheme is more feasible and efficient in precisely measuring an unknown parameter.

Consider a two-level system which is pre-selected at a state of superposition |ѱ_i〉 = cos(θ_i/2)|0〉+sin(θ_i/2)e^iϕ_i|1〉, with |0〉 and |1〉 representing the eigenstates of the observable  = |0〉〈|–|1〉〈|. Besides, the meter is prepared at |Φ_i〉 = ∫ dFΦ(F)|F〉, where Φ(F) represents the wave function with respect to the continuous input meter variable F. Without loss of generality, suppose that Φ(F) is the real Gaussian-shape wave function centered on with a standard deviation ΔF, which is written as The meter then is coupled to the system with a coupling strength γ. As outlined in Section 1, in order to retain both the ultra-high measurement sensitivity and precision while extending the measurable range of the coupling strength, here we consider an additional modulation in the coupling strength, denoted as γ_M. Therefore, the interaction Hamiltonian between the system and the meter is given as

In the following, we investigate the best attainable measurement precision of the coupling strength γ (i.e., the physical parameter to be measured) for the joint state |ѱ′〉. Specifically, we will introduce the Fisher information (FI) metric to represent the measurement precision of the coupling strength. FI is defined based on the maximum likelihood estimation strategy, where the lower-bound of the estimation variance (i.e., the Cramer–Rao bound) corresponds to the maximal FI.^[28] In principle, more FI about the parameter to be measured contained in a joint state means that one can reach a higher measurement precision. Thus, it is worth calculating the maximal FI contained in the pure joint state |ѱ′〉, i.e., the so-called quantum fisher information (QFI)^[28–30]

In the context of reaching the highest possible precision, the pre- and post-selected states of the system are respectively recast as

To specifically reflect the restriction of the magnitude of the coupling strength on the measurement sensitivity and precision in the standard PPSM scheme, and the advantages of the modulated PPSM scheme, we compare the two schemes in measurement sensitivity, precision and the post-selected probability.

For the case of the unbalanced pointer, we take the time delay measurement for example, in which the frequency ω serves as the meter variable. Suppose that a Gaussian-shape wave function is centered on ω₀ = 2400 THz with a standard deviation Δω = 200 THz. The time delay τ serves as the coupling strength to be measured. For the modulated PPS measurement, an added time delay τ_M reconstructs a modulated interaction Hamilton . Moreover, the system is considered to be a two-level system with some alternative variables, such as the polarization of the light, and the which-path of a Sagnac interferometer.

In Figs. 1(a) and 1(c), we plot the spectrum shift and the measurement sensitivity with respect to the time delay to be measured in the standard PPSM scheme, respectively. Unlike the case of balanced input meter in which the pointer shift curve is always centered at the zero-value of the coupling strength, here the spectrum shift curve depends on the post-selected angle ϕ_S and is centered at the time delay τ = ϕ_S/2ω₀. In this case, the measurable range of the time delay is limited at the nonlinear intermediate region |τω₀–ϕ_S/2|≪|τΔω| which shows higher sensitivity, see Fig. 1(c). As shown in Fig. 1(a), the spectrum shift curve shows a walk-off effect along the time delay induced by the increased post-selected angle. Besides, the nonlinear intermediate region tends to be broadened with the increase of the post-selected angle, resulting in the reduction of the measurement sensitivity. Overall, the standard PPSM scheme is impeded in measuring larger time delay due to the sharply decreased sensitivity.

The modulated PPSM scheme shows the superiority with the same high sensitivity retained for an arbitrary time delay, as shown in Fig. 1(d). Concretely, by adding an appropriate time-delay-dependent modulation τ_M and keeping the small post-selected angle ϕ_M unchanged, the spectrum shift curve can be centered on an arbitrary time delay τ = ϕ_M/2ω₀–τ_M, see Fig. 1(c). The walk-off effect vanishes with a fixed post-selected angle. Moreover, the measurable range of the time delay is limited at the modulated nonlinear intermediate region τ′ω₀–ϕ_M/2|≪|τ′Δω| with τ′ = τ+τ_M. Fig. 1.

Spectrum shift and measurement sensitivity with respect to time delay τ. Panels (a) and (b) show spectrum shift with respect to time delay τ in standard PPSM and modulated PPSM schemes, respectively. Panels (c) and (d) show measurement sensitivity with respect to time delay τ in standard PPSM and modulated PPSM schemes, respectively. The solid parts of the curves in panels (a) and (b) represent the corresponding nonlinear intermediate regions which are respectively limited at |τω₀ –ϕ_S/2|≤|τΔω|/10 and |τ′ω₀ –ϕ_M/2|≤|τ′Δω|/10 for the two PPSM schemes.

As for the precision difference between the standard and the modulated PPSM schemes, we calculate the classical FI as follows:^[29,31] The normalized probability distribution with respect to time delay τ after a measurement on frequency is expressed as with the post-selected probability pd = {1–exp[–(Δω)^2_τ2]cos(2ω₀τ–ϕ)}/2.

As can be seen in Fig. 2(a), the classical FI curve in the standard PPSM scheme reaches the maximum I_max at the peak time delay τ = ϕ_S/2ω₀ for a fixed post-selected angle ϕ_S. The peak time delay exactly falls within the nonlinear intermediate region. Unlike the measurement sensitivity as discussed above, it seems that I_max (approximate to ) can always be retained for different time delay with the corresponding post-selected angle. In fact, I_max tends to slightly reduced with the further increase of the post-selected angle (see Appendix B and Fig. B1), which is hardly shown in the most measurement scenarios. Furthermore, the classical FI at the time delay τ except for the peak time delay is increased significantly with the increase of the post-selected angle. That is to say, the measurable range of the time delay around τ = ϕ_S/2ω₀ can be increased with an increased post-selected angle from the point of view of the measurement precision.

Fig. 2.

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	Fig. 2. Classical Fisher information as a function of the time delay τ in (a) standard PPSM scheme and (b) modulated PPSM scheme.

For contrast, it seems that in the most scenarios, the modulated PPSM scheme shows no improvement in precision, see Fig. 2(b). Combined with the sensitivity curves (see Figs. 1(c) and 1(d)), we can find that the sensitivity in PPS measurements can be improved at the sacrifice of the precision corresponding to the time delay except for the peak time delay τ = ϕ_M/2ω₀ – τ_M. However, it is more crucial for one to reach the attainable highest measurement sensitivity and precision. In this regard, the modulated PPSM scheme becomes the dominate scheme to reach both the highest sensitivity and precision.

In Figs. 3(a) and 3(b), we compare the post-selected probability between the standard and modulated PPSM schemes. For a fixed post-selected angle, there always exists the minimum of the post-selected probability which corresponds to the time delay τ to be measured. In the standard PPSM scheme (see Fig. 3(a)), we can see that the post-selected probability tends to be increased with the increase of the post-selected angle ϕ_S. More precisely, the standard PPSM scheme shows a larger post-selected probability for a larger time delay. However, in the modulated PPSM scheme, the low probability is retained for an arbitrary time delay when keeping the small post-selected angle ϕ_M (corresponding to a narrow modulated intermediate region) unchanged, see Fig. 3(b).

Fig. 3.

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	Fig. 3. Post-selected probability as a function of the time delay τ in (a) standard PPSM scheme and (b) modulated PPSM scheme.

The standard PPSM scheme shows extremely high sensitivity and precision for an extremely small coupling strength, but has little feasibility to attain the high sensitivity or precision with adjustable post-selection in practice. The schematic diagram of the standard PPSM scheme is shown in Fig. 4(a). For the case of the unbalanced pointer, the walk-off effect of the pointer shift curve hinders one from detecting the rough magnitude of the coupling strength to be measured. For contrast, the modulated PPSM scheme has the ability to attain both the highest sensitivity and precision for an arbitrary coupling strength. In this case, the specific strategy of performing the modulated PPSM scheme becomes much crucial.

Suppose an arbitrary coupling strength (i.e., the parameter to be measured) is expressed as γ = γ₀ + Δ γ with γ₀ and Δγ representing the expected value of measurement outcomes and the uncertainty, respectively. More precisely, the uncertainty Δ γ can be regarded as a small fluctuation of the measured coupling strength which is not detectable in general. The modulated PPSM scheme shows its significance in measuring Δγ with high-precision, with a schematic diagram shown in Fig. 4(b). Concretely, the modulated PPSM scheme in the case of unbalanced pointer is performed in terms of three steps: first, the post-selected angle is chosen as ϕ_M = π/4 (which is analogous to the balanced homodyne detection^[32]) and the coupling-strength-dependent modulation is set as , the coupling strength is roughly estimated as γ₀; then, the coupling-strength-dependent modulation is determined as and the post-selected angle is modulated based on as a function of the feedback information of the rough estimation, i.e., Δγ is located within the modulated nonlinear intermediate region ; finally, the uncertainty Δγ is precisely estimated. Fig. 4.

Schematic diagrams of (a) standard PPSM scheme and (b) modulated PPSM scheme. In the standard PPSM scheme, the post-selection is adjustable based on the feedback outcomes. In the modulated PPSM scheme, the post-selected angle is initially set as ϕ_M = π/4; then, the coupling-strength-dependent modulation is introduced and the post-selection is modulated based on the feedback outcomes.

By including appropriate modulation in the coupling strength and the post-selection state, we are able to reach the attainable highest sensitivity and precision in the modulated PPSM scheme. We have given the optimal modulation corresponding to the optimal sensitivity and precision, which allows a precise measurement for an arbitrary coupling strength without any approximation.

In this work, we have compared the modulated PPSM scheme with the standard PPSM scheme in the case of unbalanced input meter. Both the highest sensitivity and precision may be reached only via the modulated PPSM scheme, which inevitably requires a low post-selected probability. As for the implement of the modulated PPSM scheme in practical experiments, we have given the specific strategy of performing the modulated PPSM scheme.