## 2 Numerical Simulations
For larger values of \(\Delta_{3}\) population inversion is basically suppressed, as it is shown in Figure 2f, where the yellow color in the plot represents only a 12% of inversion. For longer pulses the supported detuning is scaled accordingly with the duration of the pulses, as it was also reported in the case of a three-level system interacting with two phase-locked pulses [12]. This is, if a factor \(m\) is multiplied to the duration of the pulse, the same factor \(m\) divides the scale of the supported detuning, which concurs with the fact the detuning should be comparable to the spectral width of the pulses. The optimal pulse duration for a particular application and the corresponding effect of detuning will therefore be dependent basically on the line-broadening factors of the system under consideration and the value of the dipole moments of the transitions, which for ultrafast population inversion have to be high enough not to involve too strong laser fields, as commented above. We next have considered pulse chirp on the three pulses interacting with the four-level system and have computed its effect for different transform limited (TL) pulse durations. In Figure 3 the effect of pulse chirp on the final populations is shown. The populations of the four levels after the pulse interaction are plotted as a function of the spectral chirp parameter \(\varphi^{\prime\prime}\), which has been implemented in the Fourier domain as \(\varphi(\omega)=0.5\varphi^{\prime\prime}(\omega-\omega_{0})^{2}\), with \(\varphi(\omega)\) being the group delay dispersion (GDD) added to the spectral phase of the pulse. We note that, as only the spectral phase of the laser pulse is varied in our study, the frequency content remains as the TL pulse for any value of the spectral chirp \(\varphi^{\prime\prime}\). The rephasing of the spectral components produces a time broadening of the pulse given by \(t_{p_{c}}=\sqrt{t_{p}^{4}+16(\ln 2)^{2}{\varphi^{\prime\prime}}^{2}}/t_{p}\), where \(t_{p_{c}}\) is the full width at half maximum of the chirped pulse intensity profile [22], and therefore we implement the regular effect produced by for instance propagating the laser pulse in a linear dispersive medium. For GDD measurements of atmospheric and other gases with ultrashort pulses see, e.g., [27]. In our study we have considered the same amount of spectral chirp for the three frequency components of the interacting pulse and with central frequencies resonant to their closest transition (\(\omega_{1}=\omega_{12}\), \(\omega_{2}=\omega_{23}\), \(\omega_{3}=\omega_{34}\)). The red lines in Figure 3 correspond to the final population in level \(7p\), the blue lines show the final population in level \(4s\), and the green and black lines (close to zero) are the final populations in levels \(3p\) and \(3s\). It is clear from Figure 3 that the main effect of pulse chirp in the case that we have considered is to bring part of the final population to level \(4s\), while levels \(3p\) and \(3s\) remain basically unpopulated. Note that the \(x\)-axis in Figure 3 have been scaled for the different pulse durations, as it is indicated in the legend of the figure. We observe that the trichromatic \(\pi\)-pulse effect is robust for spectral chirps of the order of ps\({}^{2}\) for TL pulses of 10 ps. In Table 1 we show the GDD (\(\varphi^{\prime\prime}\)) values supported for the pulses with TL durations of \(t_{p}\) = 1 ps, 10 ps and 100 fs for a reduction in the population inversion to values of \(\approx\) 92%, together with the associated pulse broadening (\(t_{p_{c}}\)). Clearly, the shorter \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{t_{p}}\) & \(\varphi^{\prime\prime}\) & \(\mathbf{t_{p_{c}}}\) & **\% Inversion** \\ \hline
10 ps & 100 ps\({}^{2}\) & 29.5 ps & \(\approx\) 92\% \\
1 ps & 0.94 ps\({}^{2}\) & 2.8 ps & \(\approx\) 92\% \\
100 fs & 9765 fs\({}^{2}\) & 289 fs & \(\approx\) 92\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effect of GDD.