Even and Odd Deformed Photon Added Nonlinear Coherent States

Abstract

In this paper, we introduce even and odd deformed photon added nonlinear coherent states which in a special case lead to the even and odd photon-added coherent states |z, m〉_±. After choosing a particular nonlinearity function corresponding to the Pöschl-Teller potential, we show that they satisfy the over-completeness relation and thus are coherent states in actual concept. Meanwhile, by investigating their non-classical features, we find that in the superposing process the quadrature squeezing disappears. Based on the interaction of a trapped ion with traveling wave light fields, we give a scheme for how to prepare even and odd deformed excited nonlinear coherent states. Finally, we compare their non-classicality aspects with |z, m〉_±.

Appendix

The mean values of the relevant operators over the state |z, m, f〉_±, needed for our numerical calculations, can be obtained as follows:

$$\begin{array}{@{}rcl@{}} &&\langle \hat{a}\rangle_{+}=\overline{\langle \hat{a}^{\dagger}}\rangle_{+}=0, \end{array} $$

(26a)

$$\begin{array}{@{}rcl@{}} &&\langle \hat{a}^{2}\rangle_{+}=\overline{\langle \hat{a}^{\dagger\,2}}\rangle_{+}=z^{2}(\mathcal{N}^{\,(m)}_{+})^{2}\sum\limits_{n=0}^{\infty}\frac{|z|^{4n}}{(2n)!(2n+2)!}\\&&\qquad\frac{[f(2n+m)]![f(2n+m+2)]!(2n+m+2)!}{[f^{2}(2n)]![f^{2}(2n+2)]!}, \end{array} $$

(26b)

$$\begin{array}{@{}rcl@{}} &&\langle \hat{a}^{\dagger}\hat{a}\rangle_{+}=(\mathcal{N}^{\,(m)}_{+})^{2}\sum\limits_{n=0}^{\infty}\frac{|z|^{4n}}{((2n)!)^{2}}\frac{[f^{2}(2n+m)]!(2n+m)!(2n+m)}{[f^{4}(2n)]!}, \end{array} $$

(26c)

$$\begin{array}{@{}rcl@{}} &&\langle \hat{a}\rangle_{-}=\overline{\langle \hat{a}^{\dagger}}\rangle_{-}=0, \end{array} $$

(26d)

$$\begin{array}{@{}rcl@{}} &&\langle \hat{a}^{2}\rangle_{-}=\overline{\langle \hat{a}^{\dagger\,2}\rangle}_{-}=z^{2}(\mathcal{N}^{\,(m)}_{-})^{2}\sum\limits_{n=0}^{\infty}\frac{|z|^{4n+2}}{(2n+1)!(2n+3)!}\\&&\qquad\qquad\frac{[f^{2}(2n+m+1)]![f(2n+m+3)]!(2n+m+3)!}{[f^{2}(2n+1)]![f^{2}(2n+3)]!} \end{array} $$

(26e)

$$\begin{array}{@{}rcl@{}} &&\langle \hat{a}^{\dagger}\hat{a}\rangle_{-}=(\mathcal{N}^{\,(m)}_{-})^{2}\sum\limits_{n=0}^{\infty}\frac{|z|^{4n+2}}{((2n+1)!)^{2}}\frac{[f^{2}(2n+m+1)]!(2n+m+1)!(2n+m+1)}{[f^{4}(2n+1)]!}. \end{array} $$

(26f)