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Journal of Physics A Mathematical and General

Laguerre moments and generalized functions

Salomon Sylvain Mizrahi, D. Galetti

Published Apr 8, 2002• 4 citations
Publisher

Abstract

Here we explore the link between the moments of the Laguerre polynomials or Laguerre moments and the generalized functions (as the Dirac delta-function and its derivatives), presenting several interesting relations. A useful application is related to a procedure for calculating mean values in quantum optics that makes use of the so-called quasi-probabilities. One of them, the P-distribution , can be represented by a sum over Laguerre moments when the electromagnetic field is in a photon-number state. Consequently, the P-distribution can be expressed in terms of Dirac delta-function and derivatives. More specifically, we found a direct relation between P-distributions and the Laguerre factorial moments .

DOI
10.1088/0305-4470/35/15/312

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Here we explore the link between the moments of the Laguerre polynomials or Laguerre moments and the generalized functions (as the Dirac delta-function and its derivatives), presenting several interesting relations. A useful application is related to a procedure for calculating mean values in quantum optics that makes use of the so-called quasi-probabilities. One of them, the P-distribution , can be represented by a sum over Laguerre moments when the electromagnetic field is in a photon-number state. Consequently, the P-distribution can be expressed in terms of Dirac delta-function and derivatives. More specifically, we found a direct relation between P-distributions and the Laguerre factorial moments .
Laguerre 2D polynomials are defined and their properties are investigated. The Laguerre 2D functions, introduced in [1, 2] are related to the Laguerre 2D polynomials in such a way that they also include the weight function for the orthonormalization of the Laguerre 2D polynomials. A one-parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied, in particular, to Hermite polynomials and to Laguerre 2D polynomials. These transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments. They contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for Hermite and Laguerre 2D polynomials. Applications of the Laguerre 2D polynomials and 2D functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the Fock-state basis and by normally and otherwise ordered moments are considered. The inversion of these representations is obtained in all cases. A restricted design of quasiprobabilities should become possible.
A set of orthonormalized and complete functions of two real variables in complex representation involving Laguerre polynomials as a substantial part is introduced and is referred to as the set of Laguerre two-dimensional (2D) functions. The properties of the set of Laguerre 2D-functions are discussed and the Fourier and Radon transforms of these functions are calculated. The Laguerre 2D-functions form a basis for a realization of the five-dimensional Lie algebra to the Heisenberg-Weyl group for a two-mode system. Real representation of this set of functions by a sum over products of Hermite functions involving Jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between Laguerre and Hermite polynomials in both directions. The set of Laguerre 2D-functions is the most appropriate set of functions for the Fock-state representation of quasi-probabilities in quantum optics. The Wigner quasi-probability in Fock-state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding Laguerre 2D-function. The properties of orthonormality and completeness of the Laguerre 2D-functions provide the Fock-state matrix elements of the density operator directly from the quasi-probabilities. The Perina-Mista representation of the Glauber-Sudarshan quasi-probability can be represented with advantage by the Laguerre 2D-functions. The Fock-state matrix elements of the displacement operator and the scalar product of displaced Fock states are closely related to Laguerre 2D-functions.
The practical aspect of the moment problem of the P function is developed; i.e., a simple explicit expression for the P function in terms of its moments and the derivatives of the Dirac \ensuremath{\delta} function in the complex domain is derived. Its connection with the concept of the nonclassical depth of a quantum state introduced recently is explored. The basic formulas derived are used in the calculations of several specific examples that are popular in the quantum-optics community, such as photon-number state, binomial state, two-photon thermal state, squeezed-vacuum state, etc.
The Dirac delta function δ(x) is widely used in many areas of physics and mathematics. Here we consider the generalization of a Dirac delta function to allow the use of complex arguments. We show that the properties of a generalized delta function are very different from those of a Dirac delta function and that they behave more like a pole in the complex plane. We use the generalized delta function to derive the Glauber-Sudarshan P-function, P(α), for a Schrödinger cat state in a surprisingly simple form. Aside from their potential applications in classical electromagnetism and quantum optics, these results provide insight into the ability of the diagonal P-function to describe density operators with off-diagonal elements.
Very recently, an exponential probability distribution with parameter has been used to calculate the decoherence factors for quantum states. We derive all moments of this distribution systematically in three different ways, presenting the results in terms of binomial coefficients or Pochhammer symbols, and Stirling numbers of the first and second kinds. We show how generalized harmonic numbers or polygamma functions provide another representation for the moments. Extensions of the approach are briefly mentioned.
Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics.For the coherent states we show that their Fourier transforms are again coherent states.We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries.It is not well suited for the calculation of expectation values.A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new.We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult.The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values.By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics.Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.