Photon-added nonlinear coherent states

Abstract

The photon-added nonlinear coherent states, defined as ${a}^{\\dagger m}|\\alpha, f>$ up to a normalization constant, are still nonlinear coherent states with the nonlinear function $f(N-m)[1-m/(N+1)]$. Here $% |\\alpha, f>$ is the nonlinear coherent state with the nonlinear function $f(N)$, $N$ is the number operator, $a^{\\dagger}$ is the boson creation operator and $m$ is a non-negative integer. We introduce a new type of nonlinear coherent states with the nonlinear function $f(N+m)[1+m/(N+1)]$ obtained from the photon-added 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.