  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.
