We study the time-dependent response of a cold atom cloud illuminated by a laser beam immediately after the light is switched on experimentally and theoretically. We show that cooperative effects, which have been previously investigated in the decay dynamics after the laser is switched off, also give rise to characteristic features in this configuration. In particular, we show that collective Rabi oscillations exhibit a superradiant damping. We first consider an experiment that is performed in the linear-optics regime and well described by a linear coupled-dipole theory. We then show that this linear-optics model breaks down when increasing the saturation parameter, and that the experimental results are then well described by a nonlinear mean-field theory.
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