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arXiv:2310.06605 · Physical Review Applied

High-Precision Multiparameter Weak Measurement with Hermite-Gaussian Pointer

Binke Xia, Jingzheng Huang, Chen Fang, Hongjing Li, Guihua Zeng

Published Mar 9, 2020• 23 citations
arXivHTMLPDF

Abstract

The weak-value amplification technique has been proved to be useful for precision metrology in both theory and experiment. To explore the ultimate performance of weak-value amplification for multiparameter estimation, we investigate a general weak-measurement formalism with the assistance of the high-order Hermite-Gaussian (H-G) pointer and the quantum Fisher information matrix. Theoretical analysis shows that the ultimate precision of our scheme is improved by a factor of the square root of $2n+1$, where $n$ is the order of the Hermite-Gaussian mode. Moreover, the parameters' estimation precision can approach the precision limit with the maximum-likelihood estimation method and the homodyne method. We also present a proof-of-principle experimental setup to validate the H-G--pointer theory and explore its potential applications in precision metrology.

DOI
10.1103/physrevapplied.13.034023

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The weak-value amplification technique has been proved to be useful for precision metrology in both theory and experiment. To explore the ultimate performance of weak-value amplification for multiparameter estimation, we investigate a general weak-measurement formalism with the assistance of the high-order Hermite-Gaussian (H-G) pointer and the quantum Fisher information matrix. Theoretical analysis shows that the ultimate precision of our scheme is improved by a factor of the square root of 2n+12n+12n+1, where nnn is the order of the Hermite-Gaussian mode. Moreover, the parameters' estimation precision can approach the precision limit with the maximum-likelihood estimation method and the homodyne method. We also present a proof-of-principle experimental setup to validate the H-G--pointer theory and explore its potential applications in precision metrology.
Achieving the ultimate precisions for multiple parameters simultaneously is an outstanding challenge in quantum physics, because the optimal measurements for incompatible parameters cannot be performed jointly due to the Heisenberg uncertainty principle. In this work, a criterion proposed for multiparameter estimation provides a possible way to beat this curse. According to this criterion, it is possible to mitigate the influence of incompatibility meanwhile improve the ultimate precisions by increasing the variances of the parameter generators simultaneously. For demonstration, a scheme involving high-order Hermite-Gaussian states as probes is proposed for estimating the spatial displacement and angular tilt of light at the same time, and precisions up to 1.45 nm and 4.08 nrad are achieved in experiment simultaneously. Consequently, our findings provide a deeper insight into the role of Heisenberg uncertainty principle in multiparameter estimation, and contribute in several ways to the applications of quantum metrology.
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.
Exploring high sensitivity on the measurement of angular rotations is an outstanding challenge in optics and metrology. In this work, we employ the m n -order Hermite–Gaussian (HG) beam in the weak measurement scheme with an angular rotation interaction, where the rotation information is taken by another HG mode state completely after the post-selection. By taking a projective measurement on the final light beam, the precision of angular rotation is improved by a factor of 2 m n + m + n . For verification, we perform an optical experiment where the minimum detectable angular rotation improves 15 -fold with HG55 mode over that of HG11 mode, and achieves a sub-microradian scale of the measurement precision. Our theoretical framework and experimental results not only provide a more practical and convenient scheme for ultrasensitive measurement of angular rotations but also contribute to a wide range of applications in quantum metrology.
Weak measurement as a nonperturbative theory has been shown to be powerful in the fundamentals of quantum mechanics and high-precision metrology, in which the pointer is of great importance to parameter estimation and experimental design. In this work, we find that under the conditions of weak coupling and Gaussian meter, the probability distribution after the postselection follows a bimodal function and the peak contrast ratio (PCR) can be applied as a special pointer, which is different from the shift of mean value. To obtain the PCR, only two values corresponding to the two peaks in the distribution function are required. We theoretically present the relation of the PCR and parameters to be estimated, and then demonstrate it in an experiment. Weak measurement with the PCR pointer has the advantages of reducing the requirements of the apparatus and suppressing noise, which also provides a heuristic approach for studying weak measurement when the weak value is very large.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
Weak-value amplification has been successfully used to amplify the sensitivity of small detector signals, which paved the way towards new developments for precise parameter estimation methods. In general, the scheme is based on the idea that, between pre- and postselection, the measured system and the pointer are coupled in such a way that the interaction depends linearly on the coupling constant, the so-called linear regime. In this article, we propose an improvement to this technique that is based on the quadratic regime of interaction to increase the precision of such measurements. We demonstrate that by employing a purely real (imaginary) weak value in this regime, it is possible to decrease considerably the standard deviation of the final pointer observable in the position (momentum) space, when compared to the traditional linear regime, rendering a precision enhancement. By using an all-optical implementation, we obtained experimental data that confirm our theoretical predictions.