Algebraic treatment of the time-dependent Jaynes-Cummings Hamiltonian including nonlinear terms

A generalization of the time-dependent Jaynes-Cummings Hamiltonian is considered including nonlinear terms in the field and/or an intensity-dependent coupling term. Using algebraic methods we find the corresponding time evolution operator of the problem as a product of exponential operators. Also, we study the effect of the atomic motion in the cavity on its quantum state. We found that, when the atom is vibrating classically with a small amplitude A(0) (in comparison with the effective length of the cavity L), i.e. A(0) << L, the effects of the atom's motion are negligible but when A(0) similar to L/4, the state of the atom changes drastically. Our study is devoted to a Jaynes-Cummings Hamiltonian with an additional Kerr medium.