Analytic solutions to the two-state problem for a class of coupling potentials

A class of pulse functions is found for which analytic solutions to the problem of two levels coupled by these pulse functions is obtained. The hyperbolic-secant coupling pulse is included in this class of functions leading to the Rosen-Zener solution, but all other pulses belonging to the class function are asymmetric. The asymmetric pulses lead to qualitatively new features in the solutions; in general, it is impossible to have a zero-transition probability with such asymmetric pulses.