Approximate analytical solutions for a general two-level periodic Hamiltonian

Approximate analytical solutions are derived for a two-level system with a periodic Hamiltonian. Cases are considered where there are 0, 1 or 2 crossings of the energy levels in one complete period. The solutions are derived by applying Lie algebra techniques and by considering two regions of the problem separately. One around the crossing point where a parameter dα/dt is significant and the other where the same is negligible. The solution is sufficient to describe the analytical behaviour for all parameters. It is also found that dα/dt can be ignored throughout if the interaction potential is large.