Statistical Properties and Algebraic Characteristics of Quantum Superpositions of Negative Binomial States

We introduce new kinds of states of quantized radiation fields, which are the superpositions of negative binomial states. They exhibit remarkable non-classical properties and reduce to Schr\"odinger cat states in a certain limit. The algebras involved in the even and odd negative binomial states turn out to be generally deformed oscillator algebras. It is found that the even and odd negative binomial states satisfy a same eigenvalue equation with a same eigenvalue and they can be viewed as two-photon nonlinear coherent states. Two methods of generating such states are proposed.