Population difference of two-level atomic system due to a running pulsed field

The method of multiple scales is used to derive an expression for the population difference in an absorbing atomic system due to a running pulsed field. The main contribution to this expression comes from a quasi-steady-state part which has the same functional form as the hole produced by a continuous running field, except that the saturation parameter contains the time dependence of the field. The expression includes also an oscillatory term and a quasisteady term, which decay with a rate that is equal to the inverse of the lifetime of the levels.