## 3 Main Results Obtained from the Simulations  
In order to resolve the dependence on the central input pulse frequency and the amount of chirp on the transfer of population and the creation of vibrational coherences in the system, the central wavenumber of the input pulse is considered from \(\widetilde{v}=\) 15,000 cm\({}^{-1}\) to 25,000 cm\({}^{-1}\) (see Figure 1), in steps of 100 cm\({}^{-1}\), and the values of the spectral phase chirp have been taken from \(\phi^{\prime\prime}=-10^{3}\) fs\({}^{2}\) to \(10^{3}\) fs\({}^{2}\), in steps of 10 fs\({}^{2}\). The spectral chirp parameter \(\phi^{\prime\prime}\) has been implemented in the Fourier domain and is defined as \(\phi(\omega)=0.5\)\(\phi^{\prime\prime}\)\(\omega^{2}\), with \(\phi(\omega)\) being the group delay dispersion (GDD) added to the spectral phase of the pulse. Therefore, since only the spectral phase of the laser pulse is varied in our study, the frequency content and the power remains as in the TL pulse for any value of the spectral chirp \(\phi^{\prime\prime}\), as it is usually the case in the experiments. The rephasing of the spectral components produces a time broadening of the pulse given by \(t_{p_{c}}=\sqrt{t_{p}^{4}+16(ln2)^{2}{\phi^{\prime\prime}}^{2}}/t_{p}\), where \(t_{p_{c}}\) is the full width at half maximun (FWHM) of the chirped pulse intensity profile and \(t_{p}\) is the duration corresponding to the TL pulse [66]. Figure 2 shows the results for the enhancement/suppression factor of the population in the ground state (Figure 2a), the vibrational coherence in the ground state (Figure 2b), the population in the upper state (Figure 2c) and the vibrational coherence in the upper state (Figure 2d), sufficiently long after the interaction of the system with the laser pulse. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Transitional dipole coupling coefficient (\(\upmu_{ij}\)) & \(4.22\times 10^{-29}\) Cm \\ \hline Peak amplitude of field, \(E\) & \(8.7\times 10^{7}\) V/m \\ \hline Vibrational Frequency & \\ (\(\widetilde{\upsilon}_{12}\) and \(\widetilde{\upsilon}_{34}\))\(\widetilde{\upsilon}_{12}=\widetilde{\upsilon}_{34}=600\) cm\({}^{-1}\) & 600 cm\({}^{-1}\) \\ \hline Wave numbers of the different transitions. & \\ \(\widetilde{\upsilon}_{13}\) & \(|1\rangle\rightarrow|3\rangle\) & 19,400 cm\({}^{-1}\) \\ \(\widetilde{\upsilon}_{14}\) & \(|1\rangle\rightarrow|4\rangle\) & 20,000 cm\({}^{-1}\) \\ \(\widetilde{\upsilon}_{23}\) & \(|2\rangle\rightarrow|3\rangle\) & 18,800 cm\({}^{-1}\) \\ \hline Spectral chirp \(\phi^{\prime\prime}\) & \(-1000\) fs\({}^{2}\) to 1000 fs\({}^{2}\) \\ \hline Purely dephasing rates (\(\Gamma_{\widetilde{\eta}}\)) & \(\Gamma_{12}^{-1}=\Gamma_{34}^{-1}=1.7\) ps & \\  & \(\Gamma_{13}^{-1}=\Gamma_{14}^{-1}=\Gamma_{23}^{-1}=\Gamma_{24}^{-1}=100\) fs \\ \hline Decay rates of the transitions, \(i-j\) (\(\gamma_{ij}\)) & \(\gamma_{ij}^{-1}=1\) ns \\ \hline Duration of the Gaussian transform-limited (TL) pulses & 30 fs, 17 fs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used in the simulations. As is clear from Figure 2, the interplay between chirp and excitation detuning is crucial for the coherent control of the population transfer and the enhancement or suppression of the vibrational coherences. We can observe regions with more than a 2% enhancement/suppression in the case of the population in the excited state (Figure 2c), and enhancement factors as high as 4 for the vibrations in this state (Figure 2d). In order to understand the origin of the processes involved in the calculations of Figure 2, the different time-scales involved in the interaction need to be considered. On the one hand, the period of the vibration both in the ground and the excited states, which in our simulations is \(T_{\rm vib}=1/(\tilde{\upsilon}\tilde{\upsilon})\simeq 60\) fs. On the other hand, we note that the duration of the 30 fs laser pulse with \(\phi^{\prime\prime}=\pm 10^{3}\) fs\({}^{2}\) is \(\simeq 94\) fs.