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\section{Supplementary Information}

A nonlinear response as a consequence of Rabi oscillations for
short pulses ($\tau\ll\Gamma$ ) can also be seen for exponential
pulses. Figure~\ref{short_pulses_rabi_osc}(a) shows the atomic response for
$\tau_e= 5$\,ns and $\langle N\rangle\,\approx\,1000$. During the rise, the
Rabi frequency increases with time, leading to a more pronounced oscillatory
behaviour in the population than in Figure 4 of the main article before the
pulse turns off and the population decays exponentially.

\begin{figure}[here]
  \begin{center}
%    \includegraphics[width=\columnwidth]{./figures/pulses/exp_sq_5ns_1028_980_mult.eps}
    \includegraphics[width=\columnwidth]{figure7.eps}
    \caption{(a) Excitation dynamics for an exponential pulse with
      $\tau_e=5$\,ns. (b) For reference, the response to a square pulse with
      $\tau_{\text{sq}}=5$\,ns. Respective average photon numbers are $\langle
      N_e\rangle = \langle N_s\rangle\,\approx\,1000$. 
      The solid lines shows predictions from the theoretical
      model.
     \label{short_pulses_rabi_osc}}
\end{center}
\end{figure}

The deviation from the theoretical model for the square pulse for this short
pulse time is significant because rising and falling slopes make up a large
fraction of the whole pulse.
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