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In the physics literature it is common to see the rotating wave approximationandthe adiabatic approximation used "in cascade" to justify the use of chirpedpulses for two-level quantum systems driven by one external field, inparticular when the resonance frequency of the system is not known precisely.Both approximations need relatively long time and are essentially based onaveraging theory of dynamical systems. Unfortunately, the two approximationscannot be done independently since, in a sense, the two time scales interact.The purpose of this paper is to study how the cascade of the two approximationscan be justified and how large becomes the final time as the fidelity goes toone, while preserving the robustness of the adiabatic strategy. Our firstresult, based on high-order averaging techniques, gives a precisequantification of the uncertainty interval of the resonance frequency for whichthe population inversion works. As a byproduct of this result, we prove that itis possible to control an ensemble of spin systems by a single real-valuedcontrol, providing a non-trivial extension of a celebrated result of ensemblecontrollability with two controls by Khaneja and Li.

Ensemble qubit controllability with a single control via adiabatic and rotating wave approximations