Exact solutions for nonlinear Hamiltonians

We find the eigenvalues and eigenvectors of two nonlinear Hamiltonians describing the interaction between a two-level system and a quantized linear harmonic oscillator. In the first case we obtain exact isolated solutions for the Hamiltonian used as a model of an ion in a harmonic trap and interacting with a laser field, not restricted to the Lamb-Dicke limit. After projecting these eigenstates onto one of the levels of the two-level system the oscillator state is described by a finite superposition of Fock states. In the second case we consider a Hamiltonian, with a squeeze operator in the interaction part. We give perturbation results in the weak-coupling limit and results obtained by numerical diagonalization for the strong coupling limit. Non-classical results are pointed out also in this case.