  We introduce a generalized class of states called K-quantum nonlinear
coherent states. Each K-state has K j-components corresponding to one and the
same eigenvalue. Each Kj-component can be composed of K K=1-states in a
correlated manner. The introduced states are shown to be realized in the
long-term behavior of the vibrational motion of an ion properly trapped and
laser-driven. Nonclassical properties of the states are studied in detail.
