  Ultrachirped pulses for which the frequency chirp is of the order of the
transition frequency of a two-level atom are examined. When the chirp is large
enough, the resonance may be crossed twice, for positive and negative
quadrature frequencies. In this scenario the analytic signal and quadrature
decompositions of the field into amplitude and phase factors turn out to be
quite different. The corresponding interaction pictures are strictly
equivalent, but only as long as approximations are not applied. The domain of
validity of the formal rotating wave approximation is dramatically enhanced
using the analytic signal representation.
