Abstract
The technical merits of weak-value-amplification techniques are analyzed. We consider models of several different types of technical noise in an optical context and show that weak-value-amplification techniques (which only use a small fraction of the photons) compare favorably with standard techniques (which use all of them). Using the Fisher-information metric, we demonstrate that weak-value techniques can put all of the Fisher information about the detected parameter into a small portion of the events and show how this fact alone gives technical advantages. We go on to consider a time-correlated noise model and find that a Fisher-information analysis indicates that the standard method can have much larger information about the detected parameter than the postselected technique. However, the estimator needed to gather the information is technically difficult to implement, showing that the inefficient (but practical) signal-to-noise estimation of the parameter is usually superior. We also describe other technical advantages unique to imaginary weak-value-amplification techniques, focusing on beam-deflection measurements. In this case, we discuss combined noise types (such as detector transverse jitter, angular beam jitter before the interferometer, and turbulence) for which the interferometric weak-value technique gives higher Fisher information over conventional methods. We go on to calculate the Fisher information of the recently proposed photon-recycling scheme for beam-deflection measurements and show it further boosts the Fisher information by the inverse postselection probability relative to the standard measurement case.
- Received 19 September 2013
DOI:https://doi.org/10.1103/PhysRevX.4.011031
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Published by the American Physical Society
Popular Summary
A central problem in the theory of precision measurement is how to extract the value of an unknown parameter from a collection of data that depends both on the parameter and a random variable.
It is intuitively obvious that if you choose to ignore some of that data, the statistical uncertainty of your estimate will increase, or at best remain the same. This intuition is quantified with the concept of Fisher information, which sets the minimal possible statistical uncertainty about the parameter of interest.
Recent experiments have successfully used “weak-value amplification” as a metrological technique to detect a parameter more accurately. Weak-value amplification is an interference effect that was introduced quantum mechanically, but it can also be realized using classical electromagnetic waves. Surprisingly, this technique uses only a small fraction of the available events to make this precise measurement. How can this be? We show how this technique funnels nearly all the information into a small fraction of the events. While the theory applies generally, we focus on optical interference experiments. In that case, the funneling occurs because the interference results in an amplified signal for one detector output having few photons and a suppressed signal for the detector output having many photons. Almost all the Fisher information about the parameter can be extracted from the amplified signal giving approximately the same precision as a standard method.
We further show there are cases where, in the presence of certain types of technical noise, the weak-value amplification technique has greater Fisher information than a comparable direct technique. Therefore, even when using optimal statistical estimators (those which achieve the minimum statistical uncertainty), the weak-value amplification technique can give a metrological advantage over standard measurement techniques.
