Experimental test of uncertainty relations for quantum mechanics on a circle

TL;DR

This paper derives and tests uncertainty relations for angular position and momentum on a circle, using Mathieu functions for optimal states, and verifies the theory experimentally with photon beams.

Contribution

It introduces a new formulation of uncertainty relations on a circle using exponential angle and experimentally verifies the optimal states with photon orbital angular momentum.

Findings

Mathieu wave functions minimize the uncertainty product.

Experimental verification with photon beams confirms theoretical predictions.

Feasible approximations to optimal states are discussed.

Abstract

We rederive uncertainty relations for the angular position and momentum of a particle on a circle by employing the exponential of the angle instead of the angle itself, which leads to circular variance as a natural measure of resolution. Intelligent states minimizing the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum turn out to be given by Mathieu wave functions. We also discuss a number of physically feasible approximations to these optimal states. The theory is applied to the orbital angular momentum of a beam of photons and verified in an experiment that employs computer-controlled spatial light modulators both at the state preparation and analyzing stages.