Post-selected von Neumann measurement with Hermite-Gaussian and Laguerre-Gaussian pointer states

TL;DR

This paper explores how Hermite-Gaussian and Laguerre-Gaussian pointer states in post-selected von Neumann measurements can enhance signal-to-noise ratios, offering insights into weak and strong measurement regimes in optical systems.

Contribution

It introduces a theoretical framework for using higher-order Gaussian modes as pointer states, analyzing their advantages and disadvantages in weak measurement scenarios.

Findings

HG and LG modes can improve SNR over fundamental Gaussian modes.

The analysis covers arbitrary interaction strengths and specific system observables.

Potential applications include optical vortex experiments and advanced quantum measurements.

Abstract

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 $\hat{A}$ with $\hat{A}^{2}=\hat{I}$ and $\hat{A}^{2}=\hat{A}$ for an arbitrary interaction strength, where $\hat{I}$ 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 (SNR). 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.