Extracting joint weak values from two-dimensional spatial displacements

TL;DR

This paper extends a method to extract joint weak values of quantum operators from two-dimensional spatial displacements of Laguerre-Gauss states, avoiding the need for momentum data and enabling direct measurement.

Contribution

It provides an analytical extension for |l|>1 Laguerre-Gauss states to retrieve joint weak values solely from spatial displacements, improving efficiency.

Findings

Method applicable to higher azimuthal indices |l|>1

Joint weak values can be obtained directly from spatial data

Enhanced statistical efficiency by omitting momentum measurements

Abstract

The joint weak value is a counterfactual quantity related to quantum correlations and quantum dynamics, which can be retrieved via weak measurements, as initiated by Aharonov and colleagues. In this Rapid Communication, we provide a full analytical extension of the method described by Puentes et al. [Phys. Rev. Lett. 109, 040401 (2012)], to extract the joint weak values of single-particle operators from two-dimensional spatial displacements of Laguerre-Gauss probe states, for the case of azimuthal index |l|>1. This method has a statistical advantage over previous ones since information about the conjugate observable, i.e., the momentum displacement of the probe, is not required. Moreover, we demonstrate that under certain conditions, the joint weak value can be extracted directly from spatial displacements without any additional data processing.