Weyl Heisenberg Research Articles

Irregular Weyl–Heisenberg wave packet frames in [formula omitted

Let {fk} be a frame for a Hilbert H and let {αj,k} be a sequence of scalars. Aldroubi (1995) [1] considered a linear combination of the form gj=∑k=1∞αj,kfk (j∈N) and proved sufficient conditions on {αj,k} such that {gj} constitutes a frame for H. In this paper we discuss linear combination of the irregular Weyl–Heisenberg wave packet frames for L2(R). Necessary and sufficient conditions for a finite sum of irregular Weyl–Heisenberg wave packet frames to be a frame for L2(R) are given.

Deformed Weyl–Heisenberg algebra and quantum decoherence effect

We study the dynamics of a catlike superposition of f-deformed coherent states under dissipative decoherence. For this purpose, we investigate two important categories of f-deformed coherent states: Gazeau–Klauder and displacement-type coherent states. In addition, we consider two deformation functions; one of them describes a harmonic oscillator in an infinite well and another corresponds to a harmonic oscillator in a quantum well with finite depth. The decoherence effects appeared through a dissipative interaction of the environment with the catlike states. In this study, we first show that the Gazeau–Klauder coherent state is more resistant under the decoherence process, in contrast to the displacement-type one, and second, that the potential range of the infinite well and the depth of potential possess a remarkable role in the decoherence process.

Hamiltonian deformations of Gabor frames: First steps

Gabor frames can advantageously be redefined using the Heisenberg–Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies (i.e. arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. We will thereafter discuss possible applications and extensions of our method, which can be viewed – as the title suggests – as the very first steps towards a general deformation theory for Gabor frames.

Orbits of mutually unbiased bases

We express Alltop’s construction of mutually unbiased bases as orbits under the Weyl–Heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4. We reproduce Alltop’s mutually unbiased bases using abelian subgroups of the Clifford group in prime dimensions, in direct analogy to the well-known construction of mutually unbiased bases using abelian subgroups of the Weyl–Heisenberg group. Finally, we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases.

An SU(2)⊗SU(2) Jaynes–Cummings model with a maximum energy level

The SU(2)⊗SU(2) extension of the Jaynes–Cummings model provides a quantum mechanical system with a finite Hilbert space together with a maximum energy level in the spectrum. The model, which remains exactly solvable, is based on the replacement of the bosonic (creation and annihilation) operators by spin operators defined to act within a definite SU(2) irreducible representation, in such a way that the photon field has a finite number of excitations. The usual Heisenberg–Weyl () algebra can be obtained by contraction of the su(2) Lie algebra. We analyze the behavior of both the atomic and field quantum properties, like collapse and revivals, photon antibunching and squeezing, giving special attention to their dependence upon the maximum number of photon excitations.

Integral quantizations with two basic examples

The paper concerns integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical–quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl–Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that a world of quantizations exist, which yield the canonical commutation rule and the usual quantum spectrum of the harmonic oscillator. The second one concerns the affine group of the real line and gives rise to an interesting regularization of the dilation origin in the half-plane viewed as the corresponding phase space.

Generalized Weyl–Heisenberg (GWH) groups

Let $$H$$ be a locally compact group, $$K$$ be a locally compact Abelian (LCA) group, $$\theta :H\rightarrow Aut(K)$$ be a continuous homomorphism, and let $$G_\theta =H\ltimes _\theta K$$ be the semi-direct product of $$H$$ and $$K$$ with respect to the continuous homomorphism $$\theta $$ . In this article, we introduce the Generalized Weyl–Heisenberg (GWH) group $${\mathbb {H}}(G_\theta )$$ associate with the semi-direct product group $$G_\theta $$ . We will study basic properties of $${\mathbb {H}}(G_\theta )$$ from harmonic analysis aspects. Finally, we will illustrate applications of these methods in the case of some well-known semi-direct product groups.

The oscillator model for the Lie superalgebra $\mathfrak {sh}(2|2)$sh(2|2) and Charlier polynomials

We investigate an algebraic model for the quantum oscillator based upon the Lie superalgebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sh}(2|2)$\end{document}sh(2|2), known as the Heisenberg–Weyl superalgebra or “the algebra of supersymmetric quantum mechanics,” and its Fock representation. The model offers some freedom in the choice of a position and a momentum operator, leading to a free model parameter γ. Using the technique of Jacobi matrices, we determine the spectrum of the position operator, and show that its wavefunctions are related to Charlier polynomials Cn with parameter γ2. Some properties of these wavefunctions are discussed, as well as some other properties of the current oscillator model.

New physics in Landau levels

The most general displaced number ‘coherent’ states, based on the Heisenberg, su(2) and su(1, 1) Lie algebras symmetries, are constructed. They depend on two parameters, and can be converted into the well-known photon-added, two variable Glauber coherent states and displaced number states respectively, depending on which of the parameters is equal to zero. The relations of the Weyl–Heisenberg algebra guarantee a corresponding resolution of the identity conditions. A discussion of the statistical properties of these states is included. Significant are their squeezing properties, which can be raised by increasing the energy and angular momentum quantum numbers n and m. The maximum squeezing is obtained for Bext = 0. Depending on the particular choice of parameters in the above scenarios, we are able to determine the status of compliance with Poissonian statistics. In the limiting case, we obtain a major result about the non-classical properties of the Glauber minimum uncertainty coherent states. In other words, in addition to the requirement to minimize uncertainty conditions, they carry non-classical features too. Finally, a theoretical framework is proposed to generate them.

Linear dependencies in Weyl–Heisenberg orbits

Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl–Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.

Coherent orthogonal polynomials

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory.We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis {|x〉}, for an alternative countable basis {|n〉}. The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively.Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained.Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L2 functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L2 and, in particular, generalized coherent polynomials are thus obtained.

Matula numbers, Gödel numbering and Fock space

By making use of Matula numbers, which give a 1-1 correspondence between rooted trees and natural numbers, and a Godel type relabelling of quantum states, we construct a bijection between rooted trees and vectors in the Fock space. As a by product of the aforementioned correspondence (rooted trees $$\leftrightarrow $$ Fock space) we show that the fundamental theorem of arithmetic is related to the grafting operator, a basic construction in many Hopf algebras. Also, we introduce the Heisenberg–Weyl algebra built in the vector space of rooted trees rather than the usual Fock space. This work is a cross-fertilization of concepts from combinatorics (Matula numbers), number theory (Godel numbering) and quantum mechanics (Fock space).

COHERENT STATE OF α-DEFORMED WEYL–HEISENBERG ALGEBRA

At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.

Symmetries of the quantum damped harmonic oscillator

For the non-conservative Caldirola–Kanai system, describing a quantum damped harmonic oscillator, a couple of constant-of-motion operators generating the Heisenberg–Weyl algebra can be found. The inclusion of the standard time evolution generator (which is not a symmetry) as a symmetry in this algebra, in a unitary manner, requires a non-trivial extension of this basic algebra and hence of the physical system itself. Surprisingly, this extension leads directly to the so-called Bateman dual system, which now includes a new particle acting as an energy reservoir. In addition, the Caldirola–Kanai dissipative system can be retrieved by imposing constraints. The algebra of symmetries of the dual system is presented, as well as a quantization that implies, in particular, a first-order Schrödinger equation. As opposed to other approaches, where it is claimed that the spectrum of the Bateman Hamiltonian is complex and discrete, we obtain that it is real and continuous, with infinite degeneracy in all regimes.

A characterisation of Weyl–Heisenberg frame operators

We give a simple characterisation of Gabor frame operators in L2(R).

Geometric measure of pairwise quantum discord for superpositions of multipartite generalized coherent states

We give the explicit expressions of the pairwise quantum correlations present in superpositions of multipartite coherent states. A special attention is devoted to the evaluation of the geometric quantum discord. The dynamics of quantum correlations under a dephasing channel is analyzed. A comparison of geometric measure of quantum discord with that of concurrence shows that quantum discord in multipartite coherent states is more resilient to dissipative environments than is quantum entanglement. To illustrate our results, we consider some special superpositions of Weyl–Heisenberg, SU(2) and SU(1,1) coherent states which interpolate between Werner and Greenberger–Horne–Zeilinger states.

Bosonic and k-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras

The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su(1, 1) algebra (for κ > 0), the su(2) algebra (for κ < 0) and the h4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k-fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k-fermionic) variables. Some of the results are applied to su(2), su(1, 1) and the harmonic oscillator (in a truncated or not truncated form).This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Harmonic analysis on rational numbers

Harmonic analysis on Q is studied. The Fourier transforms of complex functions f(y) (y∈Q) in the Schwartz–Bruhat space S, are functions F(u) where u∈AQ/Q. It is shown that the corresponding Heisenberg–Weyl group is a locally compact topological group. Various time–frequency analysis methods (properties of displacement operators, Wigner and Ambiguity functions, etc.) are discussed in this context.

From the generalized uncertainty relations on fuzzy AdS 2 to the Poincaré geometry

The positive and discrete unitary irreps of SU(1,1) are used to construct fuzzy (Euclidean) AdS2. Two different types of uncertainty relation involving the Weyl–Heisenberg and a weaker type are studied. It is shown that there are no generalized coherent states which simultaneously minimize the Weyl–Heisenberg uncertainty relations among three non-commuting embedding coordinates of the fuzzy AdS2. However, generalized squeezed states that simultaneously satisfy the three weaker uncertainty relations do exist, and reproduce some properties of the classical AdS2. Up to a common scaling factor in terms of the irrep label, the expectation values of the non-commuting coordinates over such states are described in the same manner as the classical AdS2, in terms of the Poincare coordinates. The expectation values on the fuzzy AdS2 tend to their corresponding values in the commutative limit.

D-Orthogonal polynomials and [formula omitted

Two families of d-orthogonal polynomials related to su(2) are identified and studied. The algebraic setting allows for their full characterization (explicit expressions, recurrence relations, difference equations, generating functions, etc.). In the limit where su(2) contracts to the Heisenberg–Weyl algebra h1, the polynomials tend to the standard Meixner polynomials and d-Charlier polynomials, respectively.