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

Abstract

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 $P(\ensuremath{\infty})$ 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: $\mathrm{lim} {t}^{3}V(t)=0$, as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Our form of solution for $P(\ensuremath{\infty})$ 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.