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Completeness of photon-added squeezed vacuum and one-photon-states and of photon-added coherent states on a circle

Quesne, C

Published Aug 29, 2001• 10 citations
OpenAlex

Abstract

A discrete completeness relation and a continuous one with a positive measure are found for the photon-added squeezed vacuum states. Extension to the photon-added squeezed one-photon states is considered. Photon-added coherent states on a circle are introduced. Their normalization and unity resolution relation are given.

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We demonstrate explicitly the completeness of photon-added coherent states (PACSs), introduced by Agarwal and Tara (Agarwal G S and Tara K 1991 Phys. Rev. A 43 492) and defined, up to normalization, by (â † ) M | z ⟩, M = 0,1,2,...,, where â † is the boson creation operator and | z ⟩ are conventional Glauber-Klauder coherent states. We find the analytical form of the positive weight function in their resolution of unity by solving the associated Stieltjes power-moment problem. We furnish an example of generation of another set of PACSs which are complete.
We show that the photon-added squeezed vacuum a m S ( z )|0 and the photon-added squeezed one-photon state a m S ( z )|1 , where m is a non-negative integer, may be regarded as even and odd nonlinear coherent states, respectively. To achieve this, we derive an operator-valued function f ( N , m ) of the number operator N = a a such that a m S ( z )| i ( i = 0,1) are eigenstates of f ( N , m ) a 2 . Based on this, and using the unified method developed by Shanta et al , we find that a m S ( z )| i can be equivalently expressed in exponential forms. The eigenstates of f ( N ,- m ) a 2 are also constructed and their nonclassical features are studied in detail.
In this paper, we construct the m-photon-added and m-photon-subtracted coherent states on a sphere. These states are shown to satisfy the usual conditions of continuity in the label, normalizability, and the resolution of identity. The preparation of the constructed states, as the states of the radiation field, is considered. We examine and analyze the nonclassical properties of these states, including the photon mean number, Mandel parameter, and quadrature squeezing. We find that these states are sub-Poissonian in nature, whereas the degree of squeezing is reduced (enhanced) by increasing m for the photon-added (photon-subtracted) coherent states on a sphere. The results also exhibit that the curvature of the sphere contributes to the enhancement of nonclassical behavior of the photon-added and photon-subtracted coherent states on the sphere.
Photon-added squeezed vacuum states were fabricated by act inverse operator of Boson annihilation operator ~(-1) and Boson creation operator ~+ on the squeezed vacuum states first.Then the squeezing properties of the photon-added squeezed vacuum states are discussed.The results show that only the photon-added squeezed state which was acted on ~+ can appear squeezing effects,but was acted on ~(-1) did not.The squeezing range have relation with the squeezing parameter η.So the squeezing properties of the photon-added squeezed vacuum states have closed relation with ~(-1) and ~+.
From the photon-added one-photon nonlinear coherent states a†m|α, ƒ〉, we introduce a new type of nonlinear coherent states with negative values of m. The nonlinear coherent states corresponding to the positive and negative values of m are shown to be the result of nonunitarily deforming the number states |m〉 and |0〉, respectively. As an example, we study the sub-Poissonian statistics and squeezing effects of the photon-added geometric states with negative values of m in detail. Finally, we investigate the photon-added two-photon nonlinear coherent states and find that they are still the two-photon nonlinear coherent states with certain nonlinear functions. PACS Nos.: 42.50Dv, 03.65Fd