k-quantum nonlinear Jaynes－Cummings model in two trapped ions

We have obtained a k-quantum nonlinear Jaynes－Cummings model for two trapped ions interacting with laser beams resonant to the kth red side-band of the centre-of-mass mode, far from the Lamb－Dicke regime. The exact analytic solution shows the existence of quantum collapses and revivals of the occupation of two atoms.