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Annalen der Physik

Reformulation of the Optical Equivalence Theorem in Terms of the LAGUERRE Polynomials

Jan Peřina, Ladislav Mišta

Published Jan 1, 1969• 10 citations
Publisher

Abstract

Abstract The “diagonal” weighting function Φ(α) of the density matrix expressed in the coherent states is examined being decomposed in the base of the Laguerre polynomials. A criterion of the convergence in L2‐space is obtained. This decomposition is also studied as a generalized function. The equivalence of the FOCK and SUDARSHAN representations of the density matrix is proved using present formulae. Some theorems from the moment theory are applied and general assumptions are given under which the Φ‐representation is an ordinary non‐negative function or a generalized function whose support is composed of a finite number of points. The results are demonstrated on the examples of the thermal and coherent fields, their superposition and on the Fock state of the field. The decomposition of the Φ‐function in terms of the LAGUERRE polynomials is useful for approximative determination of the Φ‐function from a finite number of moments.

DOI
10.1002/andp.19694770707

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Journal of Physics A Mathematical and General•Jan 1, 1999•52 citations
Abstract The “diagonal” weighting function Φ(α) of the density matrix expressed in the coherent states is examined being decomposed in the base of the Laguerre polynomials. A criterion of the convergence in L2‐space is obtained. This decomposition is also studied as a generalized function. The equivalence of the FOCK and SUDARSHAN representations of the density matrix is proved using present formulae. Some theorems from the moment theory are applied and general assumptions are given under which the Φ‐representation is an ordinary non‐negative function or a generalized function whose support is composed of a finite number of points. The results are demonstrated on the examples of the thermal and coherent fields, their superposition and on the Fock state of the field. The decomposition of the Φ‐function in terms of the LAGUERRE polynomials is useful for approximative determination of the Φ‐function from a finite number of moments.
The optical equivalence theorem is examined for the particular class of the electromagnetic fields. It is shown that the distribution function can be taken as a non-negative function, if the density operator describing the states of the electromagnetic field is diagonal in the occupation number representation. Hence the equivalence is not only formal but also essential for this class of the electromagnetic fields, so long as the description of the optical coherence theory is concerned. As a typical example the “diagonal” representation of the density operator for blackbody radiation is studied.
The invariance properties of the double-integral and diagonal representations of operators in terms of the coherent states are examined. It is shown that a unique diagonal representation always exists for bounded operators, hence for every density operator, and for unbounded operators which are polynomials in the boson creation and annihilation operators. The associated weight function $P(\ensuremath{\alpha})$ is a generalized function in the space ${Z}^{\ensuremath{'}}$. The physical significance of this result is discussed, with particular emphasis on the diagonal representation of the density operator of arbitrary radiation fields. A general formula for the weight function $P(\ensuremath{\alpha})$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.
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.
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.