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Acta Physica Sinica

Wigner functions for the photon-added and photon-depleted even and odd coherent states

Qingyi Yang, Jingwen Sun, Lian-Fu Wei, Ding Liang-En, (1)华东师范大学物理系光谱学与波谱学教育部重点实验室，上海 200062; (2)华东师范大学物理系光谱学与波谱学教育部重点实验室，上海 200062;广西大学物理系，南宁 530004; (3)上海交通大学物理系，上海 200030

Published Jan 1, 2005• 8 citations
Publisher

Abstract

By using the definition of Wigner function in Fock states representation，Wigner functions for the photon_added and photon_depleted even and odd coherent states is reconstvucted. The results show that photon_added even and odd coherent stat es always demonstrate non_classical characteristics but photon_depleted even and odd coherent states demonstrate non_classical characteristics only when k t ake odd and even number respectively.

DOI
10.7498/aps.54.2704

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By using the definition of Wigner function in Fock states representation，Wigner functions for the photon_added and photon_depleted even and odd coherent states is reconstvucted. The results show that photon_added even and odd coherent stat es always demonstrate non_classical characteristics but photon_depleted even and odd coherent states demonstrate non_classical characteristics only when k t ake odd and even number respectively.
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.
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.
Wigner functions in phase space are reconstructed for the excited coherent states，which are generated by applying the inverse of k-Boson operators repeatedly to the coherent states. The non-classicality of these states is discussed by calculating their Wigner functions in Fock-state space. Numerical results show that these excited coherent states always reveal non-classical characteristics，no matter whether the excited number k is even or odd，and also the corresponding non-classicality is more obvious with the number k increasing.
Based on the expressions of Wigner functions in Fock-spaces,we reconstruct the Wigner functions of photon-added and-subtracted states of squeezed vacuum states.Negativities of these Wigner functions numerically indicate that these states are really non-classical quantum states.We also discuss how these negativities depend on the photon number k added to and subtraced from the squeezed vacuum states.The negativities are greater when k is odd numbers than those when k is even numbers.The result provides a theoretical foundation for measuring these states.
A measure of non-classicality of even and odd coherent states is studied. We first calculate the Wigner functions of the even and odd coherent states, which consists of two terms: the positive-definite Gaussian term and the wave term with negativity, and then calculate the integrated value ∊ max of the wave term of the Wigner functions of the even and odd coherent states in their area with negativity, and use ∊ max to measure non-classicality of the even and odd coherent states. For the even and odd coherent states with certain photon count, it is very convenient for us to use ∊ max to measure their non-classicality. The methods of our definition and calculation for ∊ max have theoretical reference value.