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

Journal of the Optical Society of America A

Geometry and dynamics in the Fresnel transforms of discrete systems

Kurt Bernardo Wolf, Guillermo Krötzsch

Published Jan 1, 2007• 6 citations
Publisher

Abstract

Free propagation in continuous optical and mechanical systems is generated by the momentum-squared operator and results in a shear of the phase space plane along the position coordinate. We examine three discrete versions of the Fresnel transform in periodic systems through their Wigner function on a toroidal phase space. But since it is topologically impossible to continuously and globally shear a torus, we examine a fourth version of the Fresnel transform on a spherical phase space, in a model based on the Lie algebra of angular momentum, where the corresponding Fresnel transform wrings the sphere.

DOI
10.1364/josaa.24.002568

Related papers

Translations and reflections on the torus: identities for discrete Wigner functions and transforms

Marcos Saraceno, Alfredo M Ozorio de Almeida

Abstract A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.

arXiv:1808.00990•Journal of Physics A Mathematical and Theoretical•Jan 11, 2019•4 citations

Wigner transform for continuous and for discrete finite-dimensional Hilbert spaces

A. Mann, Pier A. Mello, M. Revzen

We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality $N$. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator $P$ and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.

arXiv:1507.03167•Jul 11, 2015•2 citations

Transformations in the angle–angular-momentum phase space

Apostol Vourdas

Squeezing transformations in the discrete ${\mathit{Z}}_{2\mathit{j}+1}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2\mathit{j}+1}$ angle--angular-momentum phase space are shown to be associated with the SL(2,${\mathit{Z}}_{2\mathit{j}+1}$) group and important special cases are explicitly constructed. ``Spherical'' bases are introduced in the direct sum of all the Hilbert spaces ${\mathit{H}}_{2\mathit{j}+1}$ and the corresponding representations are defined. Transformations of these bases, using area preserving diffeomorphisms on a sphere, are studied and potential applications in quantum-optics models are discussed.

Physical Review A•Feb 1, 1991•40 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

Geometry and dynamics of squeezing in finite systems

Kurt Bernardo Wolf, Guillermo Krötzsch

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.

Journal of the Optical Society of America A•Jan 1, 2007•7 citations

Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.

quant-pharXiv:2512.19840•Dec 24, 2025
Free propagation in continuous optical and mechanical systems is generated by the momentum-squared operator and results in a shear of the phase space plane along the position coordinate. We examine three discrete versions of the Fresnel transform in periodic systems through their Wigner function on a toroidal phase space. But since it is topologically impossible to continuously and globally shear a torus, we examine a fourth version of the Fresnel transform on a spherical phase space, in a model based on the Lie algebra of angular momentum, where the corresponding Fresnel transform wrings the sphere.
Abstract A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality NNN. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator PPP and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
Squeezing transformations in the discrete Z2j+1{\mathit{Z}}_{2\mathit{j}+1}Z2j+1​\ifmmode\times\else\texttimes\fi{}Z2j+1{\mathit{Z}}_{2\mathit{j}+1}Z2j+1​ angle--angular-momentum phase space are shown to be associated with the SL(2,Z2j+1{\mathit{Z}}_{2\mathit{j}+1}Z2j+1​) group and important special cases are explicitly constructed. ``Spherical'' bases are introduced in the direct sum of all the Hilbert spaces H2j+1{\mathit{H}}_{2\mathit{j}+1}H2j+1​ and the corresponding representations are defined. Transformations of these bases, using area preserving diffeomorphisms on a sphere, are studied and potential applications in quantum-optics models are discussed.
Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.