We demonstrate an intricate combination of strong and weak interactions of a broadband half-cycle pulse with Rydberg atoms. A transition, which is resonant with a frequency that was taken out of the frequency spectrum, can be enhanced. In the experiment, the broadband ultrashort terahertz half-cycle pulse propagates through water vapor, which absorbs on discrete lines in the spectrum, thus removing frequencies from the spectrum. The resulting pulse interacts with a rubidium Rydberg atom, which has a resonant transition with one of the missing frequencies. Under certain conditions, the transition probability is enhanced although the associated frequency was missing in the spectrum.
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