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arXiv:0712.0230 · Physical Review A

Experimental test of uncertainty relations for quantum mechanics on a circle

Jaroslav Řeháček, Zdeněk Bouchal, Radek Čelechovský, Z. Hradil, Luis Lorenzo Sánchez-Soto

Published Mar 24, 2008• 33 citations
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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 at both the state preparation and analyzing stages.

DOI
10.1103/physreva.77.032110

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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 at both the state preparation and analyzing stages.
We present an accurate description of the conjugate pair angle-angular momentum in terms of the exponential of the angle instead of the angle itself, which leads to dispersion 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 discuss Gaussian approximations to these optimal states in terms of von Mises distributions. The theory is successfully applied to the spatial degrees of freedom of a photon and verified in an experiment that employs computer-controlled spatial light modulators at both the state preparation and the analyzing stages.
The equality in the uncertainty principle for linear momentum and position is obtained for states which also minimize the uncertainty product. However, in the uncertainty relation for angular momentum and angular position both sides of the inequality are state dependent and therefore the intelligent states, which satisfy the equality, do not necessarily give a minimum for the uncertainty product. In this paper, we highlight the difference between intelligent states and minimum uncertainty states by investigating a class of intelligent states which obey the equality in the angular uncertainty relation while having an arbitrarily large uncertainty product. To develop an understanding for the uncertainties of angle and angular momentum for the large-uncertainty intelligent states we compare exact solutions with analytical approximations in two limiting cases.
The states of linear momentum that satisfy the equality in the Heisenberg uncertainty principle for position and momentum, that is the intelligent states, are also the states that minimize the uncertainty product for position and momentum. The corresponding uncertainty relation for angular momentum and angular position, however, is more complicated and the intelligent states need not be the constrained minimum uncertainty product states. In this paper, we investigate the differences between the intelligent and the constrained minimum uncertainty product states for the angular case by means of instructive approximations, a numerical iterative search and the exact solution. We find that these differences can be quite significant for particular values of angular position uncertainty and indeed may be amenable to experimental measurement with the present technology.
The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator.Instead of the angle variable operator, we introduce the complex position operator Ẑ = x + iŷ and interpret the order parameter µ = Ẑ / Ẑ † Ẑ as a measure of certainty of the angle distribution.We prove the relation between the uncertainty of angular momentum and the angle order parameter.We also prove its generalizations and discuss experimental methods for testing these relations.
We present an experimental demonstration of a form of the angular uncertainty principle for single photons. Producing light from type I down-conversion, we use spatial light modulators to perform measurements on signal and idler photons. By measuring states in the angle and orbital angular momentum basis, we demonstrate the uncertainty relation of Franke-Arnold et al (2004 New J. Phys. 6 103). We consider two manifestations of the uncertainty relation. In the first we herald the presence of a photon by detection of its paired partner and demonstrate the uncertainty relation on this single photon. In the second, we perform orbital angular momentum measurements on one photon and angular measurements on its correlated partner exploring, in this way, the uncertainty relation through non-local measurements.