Doubly resonant ultrachirped pulses

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.