## 2 Model, Concept, Theory and Methods
### Extended Theory for the New Strategy
Implicitly, this approximation makes use of the so-called rotating wave approximation (RWA) that excludes transitions from \(|\Psi_{g}\rangle\) to the degenerate excited state \(|\Psi_{e-}\rangle=|\Psi_{E_{1u-}}\rangle\). The validity of the RWA has been tested by excellent agreement of numerically accurate solutions of the TDSE (Equation (10)) with and without the RWA [19]. Moreover, the durations \(\tau=\tau_{b}=\tau_{r}\) of the laser pulses imply sufficiently narrow widths \(\Gamma=0.5\hbar/\tau\) that exclude dipole-allowed one-photon transitions to any other excited states. We assume that multi-photon processes such as ladder climbing to more excited states are negligible--the test of this assumption is left to future accurate model simulations. Using the two-state approximation with transition dipole matrix elements in Equation (41) for selective excitation from the ground state \(|\Psi_{g}\rangle\) to the target state \(|\Psi_{e+}\rangle=|\Psi_{E_{1u+}}\rangle\), we obtain the specific form of the \(2\times 2\) Hamilton matrix in Equation (13),  
\[H(t)= \begin{pmatrix}E_{g}&-\left[\epsilon_{b}s_{b}(t)+\epsilon_{r}s_{r}(t) \right]e^{i\omega(t-t_{b})}d/\sqrt{2}\\ -\left[\epsilon_{b}s_{b}(t)+\epsilon_{r}s_{r}(t)\right]e^{-i\omega(t-t_{b})} d/\sqrt{2}&E_{e+}\end{pmatrix} \,. \tag{46}\]  
The solution in Equation (19) of the TDSE (Equation (10)) with the initial condition in Equation (11) yields the coefficients [42]  
\[ \begin{array}{rcl}c_{g}(t)&=&\cos\left\{d\left[\epsilon_{b}\int_{t_{i}}^{t}dt ^{\prime}s_{b}(t^{\prime})+\epsilon_{r}\int_{t_{i}}^{t}dt^{\prime}s_{r}(t^{ \prime})\right]/\sqrt{2}\hbar\right\}e^{-iE_{g}(t-t_{b})/\hbar},\\ c_{e}(t)&=&i\sin\left\{d\left[\epsilon_{b}\int_{t_{i}}^{t}dt^{\prime}s_{b}(t^{ \prime})+\epsilon_{r}\int_{t_{i}}^{t}dt^{\prime}s_{r}(t^{\prime})\right]/ \sqrt{2}\hbar\right\}e^{-iE_{e}(t-t_{b})/\hbar}\\ &=&\sin\left\{d\left[\epsilon_{b}\int_{t_{i}}^{t}dt^{\prime}s_{b}(t^{\prime})+ \epsilon_{r}\int_{t_{i}}^{t}dt^{\prime}s_{r}(t^{\prime})\right]/\sqrt{2}\hbar \right\}e^{-iE_{e}(t-t_{b})/\hbar+i\pi/2}.\end{array}  \tag{47}\]  
In the following, we set \(E_{g}=0\), for convenience. Equation (45) then simplifies to  
\[ \begin{array}{rcl}c_{g}(t)&=&\cos\left\{d\left[\epsilon_{b}\int_{t_{i}}^{t} dt^{\prime}s_{b}(t^{\prime})+\epsilon_{r}\int_{t_{i}}^{t}dt^{\prime}s_{r}(t^{ \prime})\right]/\sqrt{2}\hbar\right\},\\ c_{e}(t)&=&i\sin\left\{d\left[\epsilon_{b}\int_{t_{i}}^{t}dt^{\prime}s_{b}(t^{ \prime})+\epsilon_{r}\int_{t_{i}}^{t}dt^{\prime}s_{r}(t^{\prime})\right]/ \sqrt{2}\hbar\right\}e^{-i\Delta E(t-t_{b})/\hbar}\\ &=&i\sin\left\{d\left[\epsilon_{b}\int_{t_{i}}^{t}dt^{\prime}s_{b}(t^{\prime})+ \epsilon_{r}\int_{t_{i}}^{t}dt^{\prime}s_{r}(t^{\prime})\right]/\sqrt{2}\hbar \right\}e^{-2\pi i(t-t_{b})/T}.\end{array}  \tag{48}\]  
The new strategy aims at complete \(g\to e\) population transfer by the two laser pulses, with exclusive population of the excited target state, Equation (18). Hence,  
\[|c_{e}(t_{f})|^{2}=1 \tag{49}\]  
or equivalently  
\[ \begin{array}{rcl}&d\left[\epsilon_{b}\int_{t_{i}}^{t_{f}}dt^{\prime}s_{b} (t^{\prime})+\epsilon_{r}\int_{t_{i}}^{t_{f}}dt^{\prime}s_{r}(t^{\prime}) \right]/\sqrt{2}\hbar\\ &=&\sqrt{\pi}d(\epsilon_{b}\tau_{b}+\epsilon_{r}\tau_{r})/\hbar\\ &=&\sqrt{\pi}d\epsilon_{r}\tau_{r}/\hbar\\ &=&\pi/2,\end{array}  \tag{50}\]  
(cf. Equations (14) and (18)). In the first part of Equation (50), we use the result  
\[ \begin{array}{rcl}&\int_{-\infty}^{+\infty}dt^{\prime}s_{b}(t^{\prime})= \sqrt{2\pi}\tau_{b}\\ &\int_{-\infty}^{+\infty}dt^{\prime}s_{r}(t^{\prime})=\sqrt{2\pi}\tau_{r}\end{array}  \tag{51}\]  
for Gaussian shape functions (Equation (36)) with parameters for the durations \(\tau_{b}\) and \(\tau_{r}\), respectively.