Perturbative solution to the time-dependent two-level problem and the validity of the Rosen-Zener conjecture

We consider the time-dependent two-level problem of quantum mechanics, where the levels are coupled by a radio-frequency pulse with an arbitrary time-dependent envelope $V(t)$. We derive an approximate solution for the system's transition amplitude P(∞) which is correct to the third order of perturbation theory, and which applies to all pulses $V(t)$ with finite first and second moments which obey the following: lim t³V(t)=0, as t→∞. Our form of solution for P(∞) provides a criterion for judging the validity of a solution previously conjectured by Rosen and Zener, and it is generally useful for providing line-shape details in many cases of practical interest.