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

Optics Letters

Wigner representation and geometric transformations of optical orbital angular momentum spatial modes

Gabriel F. Calvo

Published May 15, 2005• 64 citations
Publisher

Abstract

An exact Wigner representation of optical spatial modes carrying orbital angular momentum is found in closed form by exploiting the underlying SU(2) Lie-group algebra of their associated Poincaré sphere. Orthogonality relations and observables of these states are obtained within the phase space picture. Development of geometric phases on mode transformations is also elucidated.

DOI
10.1364/ol.30.001207

Related papers

Geometric representation of higher-order optical modes

Claire Marie Cisowski

An octant representation of higher-order optical modes that includes Laguerre-Gaussian and Hermite-Gaussian modes is presented. The octant picture captures the high-dimensional nature of three-state optical systems and beyond, with standard Poincaré spheres for orbital angular momentum forming subspaces of the entire state space. This representation enables intuitive manipulation of both classical modes and optical qudits and provides a framework for extending Berry phases and topological invariants to high dimensions.

arXiv:2602.21991•Physical Review A•Jun 3, 2026•0 citations

Lie algebra of Wigner operators of angular momentum

D. Sourias, A. D. Jannussis, A. Streclas

In this paper we study the Wigner operators of angular momentum in the phase space, which are obtained from the well-known operators of angular momentum by virtue of Wigner representation. In the sequel we construct the Lie algebra of these operators, as well as the corresponding Lie group, and we examine two representations of this group. We find also the eigenfunction and the eigenvalues of these operators.

Hadronic J.; (United States)•Aug 1, 1980•0 citations

Understanding Squeezed States of Light Through Wigner's Phase-Space

Sibel Baskal, Marilyn E. Noz

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.

quant-pharXiv:2606.23102•Mathematics•Jun 23, 2026•1 citation

Two-dimensional signal representation on the angular Poincaré sphere

Tatiana Alieva, MJ Martin Bastiaans

It is common to use the Poincaré sphere for the representation of the polarization1 (or, in other words, the spin components of the angular momentum) of a radiation beam. This approach has recently been generalized2–4 to the case of Gaussian-type modes which possess orbital angular momentum. In the present paper we propose to use this formalism for the description of an arbitrary scalar twodimensional signal, which might be deterministic or stochastic (coherent or partially coherent).\nA point on the Poincar´e sphere represents – as we may call it – a certain ‘direction’ of the orbital angular momentum, whose expectation value can be found from the ten second-order moments of the signal’s Wigner distribution,5, 6 arranged in a 4 × 4 matrix M. Transforming the signal into its canonical form, corresponding to a diagonal moment matrix,7 we associate this state with the intersection of the main meridian with the equator, where Cartesian coordinates are more appropriate for signal description. Applying orthosymplectic transformations8 (which, in particular, include the antisymmetric fractional Fourier transformation, the rotation transformation, and their cascades) to this state, the entire sphere can be populated, leading to a generalized canonical form. At the poles, we thus have the states with a possible z-component of the orbital angular momentum, where polar coordinates are the best choice for signal analysis. The way in which the signal is transformed into its canonical form defines principal axes (including relative coordinate scaling) for signal representation in phase space. The proposed approach is useful for different applications: beam characterization, adaptive filtering, signal analysis and synthesis, etc.

•Jan 1, 2008•1 citation

Orbital Angular Momentum

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials

•Sep 7, 2023•1 citation

Laguerre-Gaussian modes: entangled state representation and generalized Wigner transform in quantum optics

Liyun Hu, Hong-Yi Fan

By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2ϕ between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.

arXiv:1008.4846•Aug 28, 2010•0 citations
An exact Wigner representation of optical spatial modes carrying orbital angular momentum is found in closed form by exploiting the underlying SU(2) Lie-group algebra of their associated Poincaré sphere. Orthogonality relations and observables of these states are obtained within the phase space picture. Development of geometric phases on mode transformations is also elucidated.
An octant representation of higher-order optical modes that includes Laguerre-Gaussian and Hermite-Gaussian modes is presented. The octant picture captures the high-dimensional nature of three-state optical systems and beyond, with standard Poincaré spheres for orbital angular momentum forming subspaces of the entire state space. This representation enables intuitive manipulation of both classical modes and optical qudits and provides a framework for extending Berry phases and topological invariants to high dimensions.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
It is common to use the Poincaré sphere for the representation of the polarization1 (or, in other words, the spin components of the angular momentum) of a radiation beam. This approach has recently been generalized2–4 to the case of Gaussian-type modes which possess orbital angular momentum. In the present paper we propose to use this formalism for the description of an arbitrary scalar twodimensional signal, which might be deterministic or stochastic (coherent or partially coherent).\nA point on the Poincar´e sphere represents – as we may call it – a certain ‘direction’ of the orbital angular momentum, whose expectation value can be found from the ten second-order moments of the signal’s Wigner distribution,5, 6 arranged in a 4 × 4 matrix M. Transforming the signal into its canonical form, corresponding to a diagonal moment matrix,7 we associate this state with the intersection of the main meridian with the equator, where Cartesian coordinates are more appropriate for signal description. Applying orthosymplectic transformations8 (which, in particular, include the antisymmetric fractional Fourier transformation, the rotation transformation, and their cascades) to this state, the entire sphere can be populated, leading to a generalized canonical form. At the poles, we thus have the states with a possible z-component of the orbital angular momentum, where polar coordinates are the best choice for signal analysis. The way in which the signal is transformed into its canonical form defines principal axes (including relative coordinate scaling) for signal representation in phase space. The proposed approach is useful for different applications: beam characterization, adaptive filtering, signal analysis and synthesis, etc.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2ϕ between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.