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Modern Physics Letters A

RADON TRANSFORMATION OF THE WIGNER OPERATOR FOR TWO-MODE CORRELATED SYSTEM IN GENERALIZED ENTANGLED STATE REPRESENTATION

Hong-Yi Fan, Guichuan Yu

Published Mar 7, 2000• 23 citations
Publisher

Abstract

We introduce a generalized entangled state |η,λ1,λ2>, which spans a complete and orthonormal representation. Using the technique of integration within an ordered product of operators we prove that the projection operator |η,λ1,λ2><η,λ1,λ2| is just the Radon transformation of the entangled Wigner operator. The inverse Radon transformation is also derived and the tomography theory for two-mode correlated system is established.

DOI
10.1142/s0217732300000487

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We introduce a generalized entangled state |η,λ1,λ2>, which spans a complete and orthonormal representation. Using the technique of integration within an ordered product of operators we prove that the projection operator |η,λ1,λ2><η,λ1,λ2| is just the Radon transformation of the entangled Wigner operator. The inverse Radon transformation is also derived and the tomography theory for two-mode correlated system is established.
Using the technique of integration within an ordered product of operators, we reveal that a new quantum mechanical representation |x,µ,ν exists, the eigenvector of operator µQ + νP (linear combination of coordinate Q and momentum P) with eigenvalue x, which is inherent to the two-parameter (µ,ν) Radon transformation of the Wigner operator. It turns out that the projection operator |x,µ,νx,µ,ν| is just the Radon transformation of the Wigner operator. The inverse of operator Radon transformation is also derived which indicates tomography in operator version.
This paper introduces the generalized excited pair coherent state (GEPCS). Using the entangled state |η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q , α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state |η 1 , η 2 , τ 1 , τ 2 〉. The entangled states |η〉 and η 1 , η 2 , τ 1 , τ 2 〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.
We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ 1 , ξ 2 〉 C,D C,D 〈ξ 1 , ξ 2 | is a Radon transform of Wigner operator.
The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.