  We study the nonclassical properties and algebraic characteristics of the
negative binomial states introduced by Barnett recently. The ladder operator
formalism and displacement operator formalism of the negative binomial states
are found and the algebra involved turns out to be the SU(1,1) Lie algebra via
the generalized Holstein-Primarkoff realization. These states are essentially
Peremolov's SU(1,1) coherent states. We reveal their connection with the
geometric states and find that they are excited geometric states. As
intermediate states, they interpolate between the number states and geometric
states. We also point out that they can be recognized as the nonlinear coherent
states. Their nonclassical properties, such as sub-Poissonian distribution and
squeezing effect are discussed. The quasiprobability distributions in phase
space, namely the Q and Wigner functions, are studied in detail. We also
propose two methods of generation of the negative binomial states.
