## 2 Numerical Simulations
As an example, for values of the transition dipole moments in sodium as \(\mu_{3s-3p}=2.11\times 10^{-29}\) Cm, \(\mu_{3p-4s}=2.09\times 10^{-29}\) Cm, and \(\mu_{4s-7p}=1.32\times 10^{-30}\) Cm, and considering 10 ps laser pulses, the peak intensity of the corresponding laser pulses for \(\sqrt{3}\pi\), \(2\pi\), and \(\sqrt{3}\pi\) illumination is \(I_{1}=4.3\times 10^{5}\) W/cm\({}^{2}\), \(I_{2}=5.8\times 10^{5}\) W/cm\({}^{2}\), \(I_{3}=1.1\times 10^{8}\) W/cm\({}^{2}\), which results in a population inversion to the \(7p\) state of 99.3%. This population inversion is reduced to 98% for 5 ps pulses and to 84% for 1 ps pulses, which require pulse peak intensities of two more orders of magnitude. Dephasing times of the considered transitions in sodium are in general of the order of ns and therefore they do not influence the results obtained in the present study, which considers ps and fs pulses, but as commented above the complete density matrix equations including dephasing rates need to be computed for longer pulses. We have investigated the robustness of the full \(\pi\)-pulse trichromatic scheme against detuning and chirp of the laser pulses by considering different pulse durations. In order to study the effect of field detuning we define the following detuning parameters: \(\Delta_{1}=\omega_{12}-\omega_{1}\), \(\Delta_{2}=\omega_{12}+\omega_{23}-\omega_{1}-\omega_{2}\) and \(\Delta_{3}=\omega_{12}+\omega_{23}+\omega_{34}-\omega_{1}-\omega_{2}-\omega_{3}\), with \(\omega_{ij}\) corresponding to the energy transition of the \(i,j\) levels (\(1\equiv 3s\), \(2\equiv 3p\), \(3\equiv 4s\), \(4\equiv 7p\), see Figure 1). \(\Delta_{3}=0\) corresponds to all field detunings that keep the three-photon resonance between levels \(7p\) and \(3s\), \(\Delta_{2}=0\) means that the two-photon resonance between \(3s\) and \(4s\) is maintained, and \(\Delta_{1}=0\) means that the \(E_{1}\) field is in resonance with the \(3s\)-\(3p\) transition. Figure 2 shows how complete population inversion from level \(3s\) to level \(7p\) is affected by detuning in the case of using 100 fs pulses. Figure 2a shows the results obtained when the three-photon resonance between \(3s\) and \(7p\) is kept (\(\Delta_{3}=0\)). In the case shown in Figure 2b the two-photon resonance between \(3s\) and \(7p\) is kept (\(\Delta_{3}=0\)). In the case shown in Figure 2c the two-photon resonance between \(3s\) and \(7p\) is kept (\(\Delta_{3}=0\)). In the case shown in Figure 2d the two-photon resonance between \(3s\) and \(7p\) is kept (\(\Delta_{3}=0\)). In the case shown in Figure 2d the two-photon resonance between \(3s\) and \(7p\) is kept (\(\Delta_{3}=0\)). In the case shown in Figure 2e the two-photon resonance between \(3s\) and \(7p\) is kept (\(\Delta_{3}=0\)). and \(4s\) is maintained while the frequency of the three pulses is varied (\(\Delta_{2}=0\)). In Figure 2c the results corresponding to having resonance in the lowest transition \(3s\)-\(3p\) are shown (\(\Delta_{1}=0\)). In these three first cases we observe a roughly symmetrical geometry around the center of the figure where the detuning is zero (\(\Delta_{1}=\Delta_{2}=\Delta_{3}=0\)), which shows the sensitivity of the complete inversion effect. We observe that for inversion to stay at values larger than 95% the system supports detuning of the order of about 100 cm\({}^{-1}\), which roughly corresponds to the width of the Gaussian 100 fs pulses used (147 cm\({}^{-1}\)). In Figure 2d-f the three detuning parameters have been varied simultaneously. Figure 2d shows that for \(\Delta_{3}=75\) cm\({}^{-1}\) a comparable detuning \(\Delta_{2}\) produces a population inversion as high as 80%. The combined effect of \(\Delta_{2}\) and \(\Delta_{3}\) is further shown in Figure 2e for a higher value of \(\Delta_{3}=150\) cm\({}^{-1}\), where a corresponding higher value of \(\Delta_{2}\) reduces the inversion to about 40%.