Evolution laws of multi-level atoms interacting with multi-mode cavity fields

Interaction of multi-level atoms with multi-mode fields is studied. Under rotation wave approximation, one can reveal the evolution laws of an N-level atom interacting with (N - 1) mode cavity fields, and obtain the analytic solution of the corresponding Schrodinger equation in the interaction picture. Furthermore, the evolution expression of a three-level atom interacting with two mode fields and a four-level atom interacting with three mode fields, respectively, are given in detail. When the initial states of atoms are ground states, the probability amplitude of ground states will have the same evolution expression for both three-level atoms and four-level atoms.