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arXiv:2107.14395 · Physical Review A

Modified weak-value-amplification technique for measuring a mirror's velocity based on the Vernier effect

Jing-Hui Huang, Feifan He, Xue-Ying Duan, Guang-Jun Wang, Xiangyun Hu

Published Jan 27, 2022• 15 citations
arXivHTMLPDF

Abstract

A modified weak-value-amplification (MWVA) technique to measure a mirror's velocity based on the Vernier effect is proposed. To demonstrate its enhanced sensitivity and higher signal-to-noise ratio (SNR), we use two cascaded Michelson interferometers with similar optical structures. One has a fixed mirror and acts as a fixed part of the Vernier scale, while the other, with a moving mirror, acts as a sliding part of the Vernier scale for velocity sensing. The envelope of the cascaded interferometers shifts much more than a single one with a certain enhancement factor, which is related to the free space range difference between them. In addition, we calculate the SNR based on the Fisher information with both the MWVA and traditional weak-value-amplification (TWVA) techniques. The results show that both the SNR and the sensitivity with our MWVA technique is greater than that of the TWVA technique within the range of our time measurement window. In particular, MWVA can present a viable and effective alternative to the TWVA technique out of the limit of resolution. Furthermore, by using the principles of the Vernier effect, it is applicable and convenient to improve the sensitivity and SNR in measuring other quantities with the TWVA technique.

DOI
10.1103/physreva.105.013718

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A modified weak-value-amplification (MWVA) technique to measure a mirror's velocity based on the Vernier effect is proposed. To demonstrate its enhanced sensitivity and higher signal-to-noise ratio (SNR), we use two cascaded Michelson interferometers with similar optical structures. One has a fixed mirror and acts as a fixed part of the Vernier scale, while the other, with a moving mirror, acts as a sliding part of the Vernier scale for velocity sensing. The envelope of the cascaded interferometers shifts much more than a single one with a certain enhancement factor, which is related to the free space range difference between them. In addition, we calculate the SNR based on the Fisher information with both the MWVA and traditional weak-value-amplification (TWVA) techniques. The results show that both the SNR and the sensitivity with our MWVA technique is greater than that of the TWVA technique within the range of our time measurement window. In particular, MWVA can present a viable and effective alternative to the TWVA technique out of the limit of resolution. Furthermore, by using the principles of the Vernier effect, it is applicable and convenient to improve the sensitivity and SNR in measuring other quantities with the TWVA technique.
In this paper, we propose a scheme for angular rotation Ω measurement in an optic-fiber-based Sagnac interferometer based on cascaded weak-value amplification (CWVA). CWVA is a modified standard weak-value amplification (SWVA) technique for further enhancing temporal shifts based on the principle of the Vernier effect. By choosing the appropriate CWVA parameters and the repetition time intervals of the Vernier scale, the temporal shifts in SWVA can be further amplified by measuring the envelope shifts in CWVA. Our comprehensive simulations demonstrate that CWVA enables the detection of minuscule rotations in the range of 1.0×10 −9 rad/s≤Ω≤10×10 −9 rad/s, with significantly improved sensitivity and superior signal-to-noise ratios compared to SWVA. The enhancement with a larger detection band may have great influence on physics and related sciences, such as rotational seismology and gravitational sensing.
Precision measurements of ultrasmall linear velocities of one of the mirrors in a Michelson interferometer are performed using two different weak-value techniques. We show that the technique of almost-balanced weak values (ABWV) offers practical advantages over the technique of weak-value amplification, resulting in larger signal-to-noise ratios and the possibility of longer integration times due to robustness to slow drifts. As an example of the performance of the ABWV protocol we report a velocity sensitivity of 60 fm/s after 40 h of integration time. The sensitivity of the Doppler shift due to the moving mirror is 150 nHz.
We introduce an ultrasensitive interferometric protocol that combines weak value amplification (WVA) with traditional interferometry. This protocol WVA+interferometry uses weak value amplification of the relative delay between two paths to enhance interferometric sensitivity. As an example, we demonstrate a proof-of-principle experiment that achieves few-attosecond timing resolution (few nanometer path length resolution) with a double-slit interferometer using only common optical components. Since our example uses only the spatial shift of double-slit interference fringes, its precision is not limited by the timing resolution of the detectors but is instead limited by the fundamental shot noise associated with classical light and the diminished technical noise. We experimentally demonstrate that the signal-to-noise ratio can be improved by one to two orders of magnitude relative to a measurement that does not use WVA. Two key conclusions are drawn: (i) Most conventional interferometric techniques primarily rely on determining the path difference (time delay or longitudinal phase), with their precision constrained by technical noise. Our protocol offers a robust solution for minimizing the technical noise in traditional interferometry, with precision in principle approaching the shot-noise limit. (ii) Although WVA has achieved significant advancements in ultrasensitive longitudinal phase measurement, its applicability is constrained by the need for broad spectral bandwidths and high-resolution spectrometers. Contrary to previous assumptions, we demonstrate that quantum-limited WVA time delay measurements are achievable with narrow band light using real weak values. Thus, the cost-effectiveness and practicality of the proposed WVA+interferometry protocol using narrow band light broaden the scope of WVA applications. This protocol holds potential for broad applications in optical metrology, quantum optics and quantum information, biomedical imaging, and interferometric telescopes for astrophysics.
We propose precision measurements of ultra-small angular velocities of a mirror within a modified Sagnac interferometer, where the counter-propagating beams are spatially separated, using the recently proposed technique of almost-balanced weak values amplification (ABWV) [Phys. Rev. Lett.116, 100803 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.100803]. The separation between the two beams provides additional amplification with respect to using collinear beams in a Sagnac interferometer. Within the same setup, the weak-value amplification technique is also performed for comparison. Much higher amplification factors can be obtained using the almost-balanced weak values technique, with the best one achieved in our experiments being as high as 1.2×107. In addition, the amplification factor monotonically increases with decreasing of the post-selection phase for the ABWV case in our experiments, which is not the case for weak-value amplification (WVA) at small post-selection phases. Both techniques consist of measuring the angular velocity. The sensitivity of the ABWV technique is ∼38 nrad/s per averaged pulse for a repetition rate of 1 Hz and ∼33 nrad/s per averaged pulse for the WVA technique.
Weak-value-amplification has been proven to be advantageous in the detection of small physical effects. However, most of the data are discarded due to the rare post-selection, resulting in a limitation of precision. Here we introduce a dual-recycling cavity based on the interferometric weak-value-based velocity measurement. By properly inserting two partially transmitting mirrors, both the power and signal-to-noise ratio (SNR) are improved, and the peak of the improvement corresponds to two identical matched mirrors. The dual recycling approach also provides a wider optimal range compared to that of the power-recycling scheme. In addition, the practical consideration is discussed to stabilize the lengths of dual-recycling cavity.
We study the amplification of a weak magnetic field with the weak-value amplification (WVA) technique based on the Faraday magneto-optic effect. By using a different scheme to perform the Mach–Zehnder interferometer with the probe in momentum space, we have demonstrated a new, to the best of our knowledge, weak measure protocol to detect the small weak magnetic field by amplifying the phase shift of the Faraday magneto-optic effect. Our numerical results show that our scheme with the WVA technique has a higher signal-to-noise ratio than the traditional optical fiber scheme with the open-loop modulation technique. Besides, considering the instrument limitations of our scheme, we have demonstrated the magnetic field resolution at the 10 − 13 T level.