Coherent excitation of a two-level atom driven by a far-off-resonant classical field: Analytical solutions

We present an analytical treatment of coherent excitation of a two-level atom driven by a far-off-resonant classical field. A class of pulse envelope is obtained for which this problem is exactly solvable. The solutions are given in terms of the Heun function, which is a generalization of the hypergeometric function. Degeneracy of the Heun to a hypergeometric equation can give all the exactly solvable pulse shapes of Gauss hypergeometric form from the generalized pulse shape obtained here. We discuss the application of the results obtained to the generation of soft x-ray and ultraviolet radiations.