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Chinese Physics B

Wigner functions and tomograms of the even and odd binomial states

Zhang Xiao-Yan, Ji‐Suo Wang, Meng Xiang-Guo, Jie Su

Published Feb 1, 2009• 0 citations
Publisher

Abstract

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the even and odd binomial states (EOBSs) are obtained. The physical meaning of the Wigner functions for the EOBSs is given by means of their marginal distributions. Moreover, the tomograms of the EOBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.

DOI
10.1088/1674-1056/18/2/036

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Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the even and odd binomial states (EOBSs) are obtained. The physical meaning of the Wigner functions for the EOBSs is given by means of their marginal distributions. Moreover, the tomograms of the EOBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.
Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner functions for the photon-depleted even and odd coherent states (PDEOCSs). Moreover, in terms of the Wigner functions with respect to the complex parameter α the nonclassical properties of the PDEOCSs are discussed. The results show that the nonclassicality for the state |β, m ⟩ o (or |β, m ⟩ e ) is more pronounced when m is even (or odd). According to the marginal distributions of the Wigner functions, the physical meaning of the Wigner functions is given. Further, the tomograms of the PDEOCSs are calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.
Using the coherent state representation of the Wigner operator, the Wigner function for the photon-added even and odd coherent state (PAEOCS) is obtained. In terms of the variations of the Wigner function with the complex parameter α in the phase space, the non-classical properties of the PAEOCS are discussed. It is found that the PAEOCS always exhibits the non-classical properties, and the photon-added even (or odd) coherent state exhibits the non-classical properties more easily when m is odd (or even). By virtue of the marginal distributions of the Wigner function for the PAEOCS, we illuminate the physical meaning of the Wigner function. Finally, based on the intermidate coordinate-momentum presentation the quantum tomogram function for the PAEOCS is obtained.
We show that for tomographic approach there exist two mutual conjugate quantum states | p ,σ,τ⟩ and | x ,μ,ν⟩ (named the intermediate coordinate-momentum representation), and the two Radon transforms of the Wigner operator are just the pure-state density matrices | p ⟩ σ,τσ,τ ⟨ p | and | x ⟩ λ,νλ,ν ⟨ x | respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states' wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.