Two-level systems: Exact solutions and underlying pseudo-supersymmetry

Chains of first-order SUSY transformations for the spin equation are studied in detail. It is shown that the transformation chains are related with a polynomial pseudo-supersymmetry of the system. Simple determinant formulas for the final Hamiltonian of a chain and for solutions of the spin equation are derived. Applications are intended for a two-level atom in an electromagnetic field with a possible time-dependence of the field frequency. For a specific form of this dependence, the time oscillations of the probability to populate the excited level disappear. Under certain conditions this probability becomes a function tending monotonously to a constant value which can exceed 1/2.