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arXiv:1805.01893 · Chinese Physics B

Pre- and post-selected measurements with coupling-strength-dependent modulation

Zhaoxue Li, Jiangdong Qiu, Linguo Xie, Lan Luo, Xiong Liu, Zhiyou Zhang, Changliang Ren, JingLei Du

Published Mar 1, 2019• 2 citations
arXivHTMLPDF

Abstract

Pre- and post-selected (PPS) measurement, especially the weak PPS measurement, has been proved to be a useful tool for measuring extremely tiny physical parameters. However, it is difficult to retain both the attainable highest measurement sensitivity and precision with the increase of the parameter to be measured. Here, a modulated PPS measurement scheme based on coupling-strength-dependent modulation is presented with the highest sensitivity and precision retained for an arbitrary coupling strength. This idea is demonstrated by comparing the modulated PPS measurement scheme with the standard PPS measurement scheme in the case of unbalanced input meter. By using the Fisher information metric, we derive the optimal pre- and post-selected states, as well as the optimal coupling-strength-dependent modulation without any restriction on the coupling strength. We also give the specific strategy of performing the modulated PPS measurement scheme, which may promote practical application of this scheme in precision metrology.

DOI
10.1088/1674-1056/28/3/030602

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Pre- and post-selected (PPS) measurement, especially the weak PPS measurement, has been proved to be a useful tool for measuring extremely tiny physical parameters. However, it is difficult to retain both the attainable highest measurement sensitivity and precision with the increase of the parameter to be measured. Here, a modulated PPS measurement scheme based on coupling-strength-dependent modulation is presented with the highest sensitivity and precision retained for an arbitrary coupling strength. This idea is demonstrated by comparing the modulated PPS measurement scheme with the standard PPS measurement scheme in the case of unbalanced input meter. By using the Fisher information metric, we derive the optimal pre- and post-selected states, as well as the optimal coupling-strength-dependent modulation without any restriction on the coupling strength. We also give the specific strategy of performing the modulated PPS measurement scheme, which may promote practical application of this scheme in precision metrology.
We theoretically propose and experimentally demonstrate a modulated weak measurement scheme that permits a precise estimation of the extended phase shift. By choosing the frequency and the total post-selected light intensity as the input and output meter variables, respectively, a linear weak measurement model is established that is especially applicable to precise phase estimation. By performing a time-delay-dependent modulation strategy based on the feedback modulation of the experimental outcome, both the high phase-sensitivity and measurement precision can be retained. We have achieved a precision at the order of ∼10−5 rad that can be effectively retained with an increase in the measured phase.
The weak-value amplification is extensively considered in precision metrology in order to achieve a higher sensitivity. Despite the practical benefits in amplifying small physical quantities, its metrological advantage still arouses a broad debate due to the low postselection probability of success. In this paper, by employing the quantum Fisher information metric, we show that the precision of estimating an unknown parameter can be improved by introducing a precoupling process with properly chosen interaction operators. We point out this result is credible for both real and imaginary weak values. By tracking the meter wave functions, we find this enhancement of estimation precision comes from a precoupling induced modulation of the meter wave function, thus the most sensitive regime with respect to the parameter is reached. In addition, the estimation error is investigated by considering the difference between the theoretical and estimated value. The analysis suggests that this kind of error can be effectively suppressed by averaging the estimations resulted from different initial meter states. These results are finally illustrated by an exact numerical simulation where the advantages of our proposal are displayed by the comparison to the standard weak-value amplification scheme.
Weak measurements, which offer a significant signal enhancement with pre- and post-selections, have proven to be powerful for precision metrology. However, the measured precision of the parameter of interest is always limited by the post-selection-induced probe loss, especially in the presence of technical noise. In this paper, we investigate the optimal pre- and post-selections of weak measurements for precision metrology in the presence of technical noise. While weak measurements apply more generally, we experimentally measure the spin splitting induced by the spin Hall effect of light (SHEL) to approach this issue. The highest measured precision of the spin splitting can be obtained with the post-selected state in the nonlinear intermediate regime. Surprisingly, the pre- and post-selections for the largest weak-value amplification are incapable of estimation of the spin splitting, since the amplified shift is independent of the spin splitting. Our results are not restricted to the SHEL and could be applied to other parameter estimations with weak measurements.
In quantum precision metrology, the famous result of Heisenberg limit scaling as 1/N1/N1/N (with NNN the number of probes) can be surpassed by considering nonlinear coupling measurement. In this work, we consider the most practice-relevant quadratic nonlinear coupling and show that the metrological precision can be enhanced from the 1/N321/N^{\frac{3}{2}}1/N23​ super-Heisenberg scaling to 1/N21/N^21/N2, by simply employing a pre- and post-selection (PPS) technique, but not using any expensive quantum resources such as quantum entangled state of probes.
In optical metrological protocols to measure physical quantities, it is, in principle, always beneficial to increase photon number n to improve measurement precision. However, practical constraints prevent the arbitrary increase of n due to the imperfections of a practical detector, especially when the detector response is dominated by the saturation effect. In this work, we show that a modified weak measurement protocol, namely, biased weak measurement significantly improves the precision of optical metrology in the presence of saturation effect. This method detects an ultra-small fraction of photons while maintains a considerable amount of metrological information. The biased pre-coupling leads to an additional reduction of photons in the post-selection and generates an extinction point in the spectrum distribution, which is extremely sensitive to the estimated parameter and difficult to be saturated. Therefore, the Fisher information can be persistently enhanced by increasing the photon number. In our magnetic-sensing experiment, biased weak measurement achieves precision approximately one order of magnitude better than those of previously used methods. The proposed method can be applied in various optical measurement schemes to remarkably mitigate the detector saturation effect with low-cost apparatuses.
Exact pointer states are obtained for projection operator measurements performed upon pre-selected (PS) and upon pre- and post-selected (PPS) quantum systems. These states are used to provide simple exact expressions for both the pointer spatial probability distribution profiles and the mean values of arbitrary pointer observables associated with PS and PPS projection operator measurements that are valid for any strength of the interaction which couples a measurement pointer to the quantum system. These profiles and mean values are compared in order to identify the effects of post-selection upon projector measurement pointers. As a special case, these mean value results are applied to the weak measurement regime - yielding PS and PPS mean value expressions which are valid for any operator (projector or non-projector). Measurement sensitivities which are useful for estimating weak measurement accuracies for PS and PPS systems are also obtained and discussed.