Analytical solutions for two-level systems with damping

A method is proposed to transform any analytic solution of the Bloch equation into an analytic solution of the Landau-Lifshitz-Gilbert equation. This allows for the analytical description of the dynamics of a two level system with damping. This method shows that damping turns the linear Schr\"{o}dinger equation of a two-level system into a nonlinear Schr\"{o}dinger equation. As applications, it is shown that damping has a relatively mild influence on self-induced transparency but destroys dynamical localization.