Metrology with Weak Value Amplification and Related Topics : Final

This report summarizes our accomplishments in the past 3+1 years concerning our investigation into the advantages of weak value approaches to precision measurements, as well as other investigations into weak measurements and weak values. The report is divided into two sections: The first focuses on a detailed discussion of results concerning applications and advantages of weak value amplification techniques to precision measurement. Here, we discuss four separate but connected projects. The first concerns the possibility of reusing the rejected photons in the postselection portion of the weak value amplification. We examine a sequence of polarized laser pulses effectively trapped inside an interferometer using a Pockels cell and polarization optics. In principle, all photons can be post-selected, which will improve the measurement sensitivity. The main advantage of such a recycling scheme is an effective power increase, while maintaining an amplified deflection. The second project concerns the technical merits of weak value amplification. 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 uses all of them). Using the Fisher information metric, we demonstrate that weak value techniques give much higher information over conventional methods in some cases, showing a fundamental advantage. We identify other cases where the standard technique has higher Fisher information, but the statistical estimator needed to uncover that information is technically difficult to implement, demonstrating technical advantages. We also discuss a third project using our interferometric weak value technique to make precision frequency measurements on an optical beam, reporting a resolution of 129 +/7 kHz per root Hz with only 2 mW of continuous-wave power. The fourth project deals with a weak value inspired phase amplification technique. We monitor the relative phase between two paths of a slightly misaligned interferometer by measuring the average position of a split-Gaussian mode in the dark port. Although we monitor only the dark port, we show that the signal varies linearly with phase and that we can obtain similar sensitivity to balanced homodyne detection. We relate the effect to an inverse weak value. The second main part of the report concerns results related to weak measurements and weak values, but not directly related to the amplification issue. We give a brief survey of our results in this area. We have investigated weak measurements as a test of quantum mechanics, by deriving generalized Leggett-Garg inequalities for multiple parties and violating them with entangled photons. We have made a systematic analysis of coupled electronic Mach-Zehnder interferometers, where one is used as a detector for the other; this set-up can also be used for parameter estimation. We have extensively investigated a new approach to generalized measurement we call contextual values, which gives a precise connection between an indirectly measured system and the outcomes of the detector by assigning a set of generalized eigenvalues to each detector outcome. This assignment permits a principled definition of a conditioned average, which limits to the weak value under certain conditions that are usually satisfied in the laboratory. We have also discussed the significance of the imaginary part of the weak value. While the imaginary weak value is very useful in amplification (parameter estimation) experiments, we show it has no direct connection any conditioned average of the operator itself. Often the weak value is restricted to the weak limit, where the measurement interaction is infinitesimally small. We have shown that the general structure of the weak value is actually universal for any interaction strength and any meter wavefunction, so long as the form of the interaction is of the von Neumann type. We have also investigated the concept of the quantum instrument, and how it directly deals with observable events in the lab, without using states or observables. (a) Papers published in peer-reviewed journals (N/A for none) Enter List of papers submitted or published that acknowledge ARO support from the start of the project to the date of this printing. List the papers, including journal references, in the following categories: