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arXiv:1703.01632 · Physical Review A

Practical advantages of almost-balanced-weak-value metrological techniques

Julián Martínez-Rincón, Zekai Chen, John C. Howell

Published Jun 5, 2017• 17 citations
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Abstract

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.

DOI
10.1103/physreva.95.063804

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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 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.
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 a recent letter, Brunner and Simon propose an interferometric scheme using imaginary weak values with a frequency-domain analysis to outperform standard interferometry in longitudinal phase shifts [N. Brunner and C. Simon, Phys. Rev. Lett {\bf105} (2010)]. Here we demonstrate an interferometric scheme combined with a time-domain analysis to measure longitudinal velocities. The technique employs the near-destructive interference of non-Fourier limited pulses, one Doppler shifted due to a moving mirror, in a Michelson interferometer. We achieve a velocity measurement of 400 fm/s and show our estimator to be efficient by reaching its Cramer-Rao bound.
In a recent Letter, Brunner and Simon proposed an interferometric scheme using imaginary weak values with a frequency-domain analysis to outperform standard interferometry in longitudinal phase shifts [Phys. Rev. Lett105, 010405 (2010)]. Here we demonstrate an interferometric scheme combined with a time-domain analysis to measure longitudinal velocities. The technique employs the near-destructive interference of non-Fourier limited pulses, one Doppler shifted due to a moving mirror in a Michelson interferometer. We achieve a velocity measurement of 400 fm/s and show our estimator to be efficient by reaching its Cramér-Rao bound.
By choosing more orthogonality between preselection and postselection states, one can significantly improve the sensitivity in general optical quantum metrology based on the weak-value amplification (WVA) approach. However, increasing the orthogonality decreases the probability of detecting photons and makes the weak measurement difficult, especially when the weak measurement is disturbed by strong noise and the pointer is drowned in noise with a negative-decibel signal-to-noise ratio (SNR). In this article we introduce and numerically evaluate a modified weak-measurement protocol with a temporal pointer, namely, the autocorrelative weak-value amplification (AWVA) approach. Specifically, a small longitudinal time delay (tiny phase shift) $\ensuremath{\tau}$ of a Gaussian pulse is measured by implementing two simultaneous autocorrelative weak measurements under Gaussian white noise with different SNRs. The small quantities $\ensuremath{\tau}$ are obtained by measuring the autocorrelation coefficient of the pulses instead of fitting the shift of the mean value of the probe in the standard WVA technique. Simulation results show that the AWVA approach outperforms the standard WVA technique in the time domain with smaller statistical errors, remarkably increasing the precision of weak measurement under a strong noise background.
Achieving higher sensitivity is an earnest purpose for precision metrology. As a response to this goal, the weak-value amplification approach has been developed for measuring ultrasmall physical effects, realizing sensitivity that had never been reached before. Encouraged by the successes, many efforts have been devoted to obtain ultimate sensitivity of weak-value amplification. However, the benefit would be easily compromised in practice, because the cost of significant reduction on signal intensity leads to an ultralow signal-to-noise ratio. In this work, we bridge this gap by proposing an alternative weak-value amplification approach, which provides sensitivity several orders of magnitude higher than the standard approach while being compatible with practical imperfections. In the proof-of-principle experiment of measuring longitudinal phase change in time domain, sensitivity up to $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ attosecond is exemplified. Our approach can be applied to measure other small parameters with extremely high sensitivity, providing a method for future precision metrology.