Binomial States and Negative Binomial States of the Radiation Field and their Excitations are Nonlinear Coherent States

We show that the well-known binomial states and negative binomial states of the radiation field and their excitations are nonlinear coherent states.Excited nonlinear coherent state are still nonlinear coherent states with different nonlinear functions.We finally give exponential form of the nonlinear coherent states. PACS numbers:42.50.Dv,03.65.Db,32.80.Pj,42.50.Vk Typeset using REVTEX ∗email:[email protected] 1 Currently there has been much interest in the study of nonlinear coherent states[14]and their superpositions[5-6]. The so-called nonlinear coherent states are defined as the right-hand eigenstates of the product of the boson annihilation operator a and a nonlinear function of number operator N̂ = aa, f(N̂)a|α, f〉 = α|α, f〉, (1) where f(N̂) is an operator-valued function of the number operator and α is a complex eigenvalue.The ordinary coherent states |α〉 are recovered for the special choice of f(N̂) = 1. A class of nonlinear coherent states can be realized physically as the stationary states of the center-of-mass motion of a trapped ion[1].These nonlinear coherent states can exhibit various nonclassical features like squeezing and self-splitting. In a recent paper,Sivakumar[7] pointed out that the excited coherent states[8-10] |α,m〉 = a †m|α〉 〈α|ama†m|α〉 (2) are nonlinear coherent states and gave an explicit form of the corresponding nonlinear function f(N̂) = 1 − m/(1 + N̂). Here m is nonnegative integer. The work brings us to think the following question: if there exist other quantum states as nonlinear coherent states? The answer is affirmative. We find that the well-known binomial states[11-16] and negative binomial states[17-22] are nonlinear coherent states. We also show that the excited nonlinear coherent states are still nonlinear coherent states with different nonlinear function f(N̂). Finally,the exponential form of the nonlinear coherent states is given. Expanding |α, f〉 in terms of Fock states |n〉 |α, f〉 = ∞