Nonlinear optical tomography of q-deformed entangled pair coherent states

We introduce entangled pair coherent states of a noncommutative harmonic oscillator operators associated with a q-deformed oscillator algebra. The definition of two-mode q-deformed operator and the decomposition of its eigenstates in terms of the q-deformed Fock states are obtained. The general solution of the recurrence relation of the first nondeformed quadrature is obtained. The asymptotic behavior of the general solution is studied, a complete expansion is derived, and also representations of the general solution in terms of the Gauss hypergeometric function are given. Also, we prove that the general solution is related to Meixner-Pollaczek orthogonal polynomial and calculate the weight function. The photons number distribution, the second order correlation function and the optical tomogram of the entangled pair coherent states are discussed. The recurrence relation for the first q-deformed quadrature is inferred. We evaluate the eigenstate of the q-deformed quadrature operator and give a numerical study for the optical tomogram of the q-deformed of the entangled pair coherent states. Finally, we show the change of the parameters in the entangled pair coherent states that connected with physical quantities and explain how they affect the format of optimal tomography.