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Optics Communications

Simplified algebraic description of weak measurements with Hermite–Gaussian and Laguerre–Gaussian pointer states

Bertúlio de Lima Bernardo, Sérgio Azevedo, Alexandre Rosas

Published Jun 20, 2014• 19 citations
Publisher
DOI
10.1016/j.optcom.2014.06.008

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Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes as pointer states and calculate the average shift of the pointer states of the post-selected von Neumann measurement by assuming the system observable with and for an arbitrary interaction strength, where represents the identity operator. Our results show that the HG and LG pointer states for a given coupling direction have advantages and disadvantages over the fundamental Gaussian mode in improving the signal-to-noise ratio. We expect that our general treatment of the weak values will be helpful for understanding the connection between weak- and strong-measurement regimes and may be used to propose new experimental setups with higher-order Gaussian beams to investigate further the applications of weak measurement in optical systems such as the optical vortex.
We investigated an orbital angular momentum (OAM) pointer within the framework of von Neumann measurements and discovered its significant impact on optimizing superpositions of Gaussian and Laguerre-Gaussian (LG) states. Calculations of the quadrature squeezing, the second-order cross-correlation function, the Wigner function, and the signal-to-noise ratio (SNR) support our findings. Specifically, by carefully selecting the anomalous weak value and the coupling strength between the measured system and the pointer, we demonstrated that the initial Gaussian state transforms into a non-Gaussian state after postselection. This transition highlights the potential of OAM pointers in enhancing the performance of quantum systems by tailoring state properties for specific applications.
The weak-value amplification technique has been proved to be useful for precision metrology in both theory and experiment. To explore the ultimate performance of weak-value amplification for multiparameter estimation, we investigate a general weak-measurement formalism with the assistance of the high-order Hermite-Gaussian (H-G) pointer and the quantum Fisher information matrix. Theoretical analysis shows that the ultimate precision of our scheme is improved by a factor of the square root of 2n+12n+12n+1, where nnn is the order of the Hermite-Gaussian mode. Moreover, the parameters' estimation precision can approach the precision limit with the maximum-likelihood estimation method and the homodyne method. We also present a proof-of-principle experimental setup to validate the H-G--pointer theory and explore its potential applications in precision metrology.