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arXiv:1504.04777 · Physical Review A

Weak-value amplification beyond the standard quantum limit in position measurements

A. Nishizawa

Published Sep 21, 2015• 9 citations
arXivHTMLPDF

Abstract

In a weak measurement with postselection, a measurement value, called the weak value, can be amplified beyond the eigenvalues of an observable. However, there are some controversies whether weak-value amplification is practically useful in increasing the sensitivity of the measurement, in which fundamental quantum noise dominates. In this paper, we investigate the sensitivity limit of an optical interferometer when weak-value amplification is implemented, properly accounting for quantum shot noise and radiation-pressure noise. To do so, we formulate weak-value amplification in the Heisenberg picture instead of in the Schr\"odinger picture, which is conventionally used. This formulation enables us to understand intuitively what happens when the measurement outcome is postselected and the weak value is amplified. As a result, we find that the sensitivity limit is given by the standard quantum limit that is the same as in a standard interferometry. We also discuss a way to circumvent the standard quantum limit.

DOI
10.1103/physreva.92.032123

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In a weak measurement with postselection, a measurement value, called the weak value, can be amplified beyond the eigenvalues of an observable. However, there are some controversies whether weak-value amplification is practically useful in increasing the sensitivity of the measurement, in which fundamental quantum noise dominates. In this paper, we investigate the sensitivity limit of an optical interferometer when weak-value amplification is implemented, properly accounting for quantum shot noise and radiation-pressure noise. To do so, we formulate weak-value amplification in the Heisenberg picture instead of in the Schr"odinger picture, which is conventionally used. This formulation enables us to understand intuitively what happens when the measurement outcome is postselected and the weak value is amplified. As a result, we find that the sensitivity limit is given by the standard quantum limit that is the same as in a standard interferometry. We also discuss a way to circumvent the standard quantum limit.
We study the weak-value amplification (WVA) in a phase measurement with an optical interferometer in which shot noise limits the sensitivity. We compute the signal and the shot noise including the full-order interaction terms of the WVA, and show that the shot-noise contribution to a phase shift in a pointer variable is always larger than the final variance of the pointer variable. This yields the difference in estimating the noise level up to a factor of 1.5. To clarify an advantage for practical uses of the WVA, we discuss the signal-to-noise ratio and its optimization in the presence of the shot noise.
In a quantum-noise limited system, weak-value amplification using postselection normally does not produce more sensitive measurements than standard methods for ideal detectors: the increased weak value is compensated by the reduced power due to the small postselection probability. Here, we experimentally demonstrate recycled weak-value measurements using a pulsed light source and optical switch to enable nearly deterministic weak-value amplification of a mirror tilt. Using photon counting detectors, we demonstrate a signal improvement by a factor of 4.4±0.2 and a signal-to-noise ratio improvement of 2.10±0.06, compared to a single-pass weak-value experiment, and also compared to a conventional direct measurement of the tilt. The signal-to-noise ratio improvement could reach around six for the parameters of this experiment, assuming lower loss elements.
Abstract Most studies for postselected weak measurement in optomechanical system focus on using a single photon as a measured system. However, we find that using weak coherent light instead of a single photon can also amplify the mirror’s position displacement of one photon. In the weak value amplification (WVA) regime, the weak value of one photon can lie outside the eigenvalue spectrum, proportional to the difference of the mirror’s position displacement between the successful and failed postselection and its successful postselection probability is dependent of the mean photon number of the coherent light and improved by adjusting it accordingly. Outside the WVA regime, the amplification limit can reach the level of the vacuum fluctuations, and when the mean photon number and the optomechanical coupling parameter are selected appropriately, its successful postselection probability becomes higher, which is beneficial to observe the maximum amplification value under the current experimental conditions. This result breaks the constraint that it is difficult to detect outside the WVA regime. This opens up a new regime for the study of a single photon nonlinearity in optomechanical system.
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.
We show that weak measurement can be used to "amplify" optical nonlinearities at the single-photon level, such that the effect of one properly postselected photon on a classical beam may be as large as that of many unpostselected photons. We find that "weak-value amplification" offers a marked improvement in the signal-to-noise ratio in the presence of technical noise with long correlation times. Unlike previous weak-measurement experiments, our proposed scheme has no classical equivalent.
Weak-value amplification (WVA) is a metrological protocol that amplifies ultrasmall physical effects. However, the amplified outcomes necessarily occur with highly suppressed probabilities, leading to the extensive debate on whether the overall measurement precision is improved in comparison to that of conventional measurement (CM). Here, we experimentally demonstrate the unambiguous advantages of WVA that overcome practical limitations including noise and saturation of photodetection and maintain a shot-noise-scaling precision for a large range of input light intensity well beyond the dynamic range of the photodetector. The precision achieved by WVA is 6 times higher than that of CM in our setup. Our results clear the way for the widespread use of WVA in applications involving the measurement of small signals including precision metrology and commercial sensors.