## 2 Numerical Simulations  
We have solved numerically the Liouville equations for a four-level ladder system coherently interacting with a tricolor pulse without the assumptions of the rotating-wave approximation (RWA). Avoiding the RWA in the density-matrix formalism allows to account for the interaction of all the Fourier components of the pulses with all the permitted transitions in the system, and therefore in our study we can evaluate the effect of large detuning and chirp of ultrashort pulses with a broad range of intensities. The density matrix equations that we numerically solve are as follows [24]. based on a 4-5 order Runge-Kutta solver which scans the areas of the three pulse components \(E_{1}(t)\), \(E_{2}(t)\) and \(E_{3}(t)\) between 0 to 4\(\pi\) in steps of 0.01\(\pi\). Starting with the system in the ground \(3s\) state the goal is to obtain complete population transfer to the \(7p\) state. We first consider a trichromatic pulse with a duration of 10 ps composed of three resonant components with central wavelengths of 589 nm, 1141 nm and 781 nm (see Figure 1). We obtain that the only solution for total population transfer from the ground state \(3s\) to the highest state \(7p\) is the set of pulse areas \(\sqrt{3}\pi\), \(2\pi\), and \(\sqrt{3}\pi\), as it is illustrated in Figure 1. As it is shown in Figure 1, the intermediate levels \(3p\) and \(4s\) are temporally occupied during the interaction process, but at the end of the interaction all the population is efficiently transferred to the \(7p\) state. By varying the area of pulses \(E_{1}(t)\) and \(E_{3}(t)\) in a 4% we obtain that the population transfer to the \(7p\) state is reduced to roughly 98%, which gives an estimation of the accuracy of our result. In the inset of Figure 1 we show how doubling the area of each pulse the \(2\pi\)-pulse effect is retrieved, with all the population being back to the ground state at the end of the interaction. The solution that we report can be scaled to other pulse durations, which is also described in the case of a three-level ladder system [12]. Indeed, we find that complete population inversion is reproduced for pulses longer than 10 ps and also for shorter pulses. The limitation of the result is determined, on the one hand, on the relaxation times and line-broadening factors of the system, which prevent the usability of long pulses, and on the other hand, on the value of the transition dipole moments combined with the duration of the pulse, since weak coupling dipole moments with ultrashort pulses  
Figure 1: Time evolution of the populations of the four levels, as indicated, for a “\(\pi\)-pulse” transition produced by a tricolor resonant 10 ps Gaussian pulse of central angular frequencies \(\omega_{1}\), \(\omega_{2}\) and \(\omega_{3}\). The inset on the left shows a scheme of the four-level ladder system with the pulse areas that completely invert the population. Atomic sodium has been considered in our study as an example, with transition energies such as \(\omega_{12}=\omega_{1}=16,978\) cm\({}^{-1}\) (589 nm for \(3s\)–\(3p\)), \(\omega_{23}=\omega_{2}=8762\) cm\({}^{-1}\) (1141 nm for \(3p\)–\(4s\)), and \(\omega_{34}=\omega_{3}=12,801\) cm\({}^{-1}\) (781 nm for \(4s\)–\(7p\)). The inset on the right shows a complete oscillation of a “\(2\pi\)-pulse”, which is produced with the corresponding double pulse areas \(2\sqrt{3}\pi\), \(4\pi\), and \(2\sqrt{3}\pi\). result in large peak intensities involving carrier field dynamics [26] and ac-Stark shifts that preclude the population inversion effect.