Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0605137 · Physical Review Letters

Minimum Uncertainty Measurements of Angle and Angular Momentum

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

Published Dec 15, 2006• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.97.243601

Related papers

Experimental test of uncertainty relations for quantum mechanics on a circle

Jaroslav Řeháček, Zdeněk Bouchal, Radek Čelechovský +2 more

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.

arXiv:0712.0230•Physical Review A•Mar 24, 2008•33 citations

Minimum uncertainty states of angular momentum and angular position

David Thomas Pegg, Stephen Mark Barnett, Roberta Zambrini +2 more

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.

New Journal of Physics•Feb 17, 2005•41 citations

Large-uncertainty intelligent states for angular momentum and angle

Jörg B. Götte, Roberta Zambrini, Sonja Franke‐Arnold +1 more

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.

Journal of Optics B Quantum and Semiclassical Optics•Nov 15, 2005•7 citations

Angular minimum uncertainty states with large uncertainties

Jörg B. Götte, PAUL M. RADMORE, Roberta Zambrini +1 more

The uncertainty relation for angle and angular momentum has a lower bound which depends on the form of the state. Surprisingly, this lower bound can be very large. We derive the states which have the lowest possible uncertainty product for a given uncertainty in the angle or in the angular momentum. We show that, if the given angle uncertainty is close to its maximum value, the lowest possible uncertainty product tends to infinity.

arXiv:quant-ph/0512077•Journal of Physics B Atomic Molecular and Optical Physics•May 30, 2006•5 citations

Fourier relationship between the angle and angular momentum of entangled photons

Anand K. Jha, Barry Jack, E. Yao +6 more

We study the Fourier relationship between angle and orbital angular momentum of entangled photons. Spatial light modulators allow us to define and control the spatial mode measurement state. We observe strong quantum correlations, establishing that angular position and momentum distributions between the photons are related as conjugate Fourier pairs.

Physical Review A•Oct 8, 2008•70 citations

Minimum uncertainty states of angular momentum and angular position

Roberta Zambrini, D.T. Pegg, S.M. Barnett +2 more

In this paper, the much less known uncertainty relation for angular position and momentum to find the system at the edge of the phase interval is considered. States that minimize the uncertainty product with a constraint given either by the angular variance or by the angular momentum variance is examined.

European Quantum Electronics Conference (EQEC)•Jan 18, 2006•3 citations
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.
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.
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 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.