  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.
