## 2 Model, Concept, Theory and Methods
### Extended Theory for the New Strategy
\begin{array}{rcl}c_{g}(t_{c})&=&\cos\left[\sqrt{\frac{1}{2}}d\epsilon_{b} \int_{t_{i}}^{t_{c}}dt^{\prime}s_{b}(t^{\prime})/\hbar\right]=\cos\left[\sqrt {\frac{\pi}{2}}d\epsilon_{b}\tau_{b}/\hbar\right],\\ c_{e}(t_{c})&=&i\sin\left[\sqrt{\frac{1}{2}}d\epsilon_{b}\int_{t_{i}}^{t}dt^{ \prime}s_{b}(t^{\prime})/\hbar\right]e^{-2\pi i(t_{c}-t_{b})/T}\\ &=&i\sin\left[\sqrt{\pi}d\epsilon_{b}\tau_{b}/\hbar\right]e^{-2\pi iN/2}\\ &=&\begin{cases}i\sin\left[\sqrt{\pi}d\epsilon_{b}\tau_{b}/\hbar\right]&\text{ if }\ N=2,4,6,...(\text{even})\\ -i\sin\left[\sqrt{\pi}d\epsilon_{b}\tau_{b}/\hbar\right]&\text{if }\ N=1,3,5,...(\text{odd}).\end{cases}\end{array}  \tag{54}\]  
Gratifyingly, this is in accord with the general result, Equation (17), with parameters  
\[\alpha=\frac{\sqrt{\pi}d\epsilon_{b}\tau_{b}}{\hbar} \tag{55}\]  
for the amplitudes, and  
\[ \begin{array}{l}\eta_{g}(t_{c})=0,\\ \Delta\eta_{e}(t_{c})=\eta_{e}(t_{c})=\begin{cases}-\pi/2&\text{if }\ N=1,3,5,...(\text{odd})\\ +\pi/2&\text{if }\ N=2,4,6,...(\text{even})\end{cases}\end{array}  \tag{56}\]for the phases and phase differences. The derivation of Equations (54)-(56) employs the conditions in Equations (33) and (34), i.e., the time delay between the two laser pulses must be equal to an integer number \(N\) times the period \(T\) for charge migration. The corresponding unitary matrix \(\mathbf{U}_{bge}(t_{c},t_{i})\), Equation (28), for propagating the initial coefficients \(c(t_{i})\), Equation (11), to \(c(t_{c})\), Equation (54), is  
\[\mathbf{U}_{bge}(t_{c},t_{i})= \begin{pmatrix}\cos\alpha&\pm i\sin\alpha\\ \pm i\sin\alpha&\cos\alpha\end{pmatrix} \;. \tag{57}\]  
The first column of the matrix in Equation (57) is imposed by the coefficients in Equation (54), the second column is due to the unitarity of \(\mathbf{U}_{bge}(t_{c},t_{i})\). Likewise, the final coefficients \(c_{k}(t_{f})=c_{e}(t_{f})\delta_{ke}=e^{i\eta_{e}(t_{f})}\delta_{ke}\equiv e^{ i\eta_{e}t_{f}}\delta_{ke}\) imply the unitary matrix for propagating from \(t_{i}\) to \(t_{f}\) (cf. Equation (49)),  
\[\mathbf{U}_{ge}(t_{f},t_{i})=e^{i\eta_{e}t_{f}} \begin{pmatrix}0&1\\ 1&0\end{pmatrix}  \tag{58}\]  
with irrelevant phase factor \(e^{i\eta_{e}t_{f}}\). The separation in Equation (22) of the evolution operator \(\mathbf{U}_{ge}(t_{f},t_{i})\) into two parts for symmetry breaking and restoration then yields the evolution operator for propagating from \(t_{c}\) to \(t_{f}\),  
\[\mathbf{U}_{rge}(t_{f},t_{c})=e^{i\eta_{e}t_{f}} \begin{pmatrix}\pm i\sin\alpha &\cos\alpha\\ \cos\alpha&\pm i\sin\alpha\end{pmatrix} =\pm ie^{i\eta_{e}t_{f}} \begin{pmatrix} \cos\left(\pi/2-\alpha\right)&\mp i\sin\left(\pi/2-\alpha\right)\\ \mp i\sin\left(\pi/2-\alpha\right)&\cos\left(\pi/2-\alpha\right)\end{pmatrix} \;. \tag{59}\]  
In the second part of Equation (59), we use the symmetry relations \(\sin\alpha=\cos\left(\pi/2-\alpha\right)\) and \(\cos\alpha=\sin\left(\pi/2-\alpha\right)\). The second part of Equation (59) for time propagation due to the second laser pulse thus takes the same form as Equation (57) for the first laser pulse, except for the different but irrelevant phase factor \(\pm ie^{i\eta_{e}t_{f}}\). That means that the effect of the second laser pulse is formally equivalent to the first pulse, but the angle \(\alpha\) must be replaced by \(\pi/2-\alpha\). Equation (52) shows that this substitution is equivalent to replacing \(\epsilon_{b}\tau_{b}\) by \(\epsilon_{\pi}\tau_{\pi}-\epsilon_{r}\tau_{r}\) or, if one uses the same parameters for the laser durations, \(\tau_{b}=\tau_{\pi}=\tau_{r}\) then it is equivalent to replacing \(\epsilon_{b}\) by \(\epsilon_{\pi}-\epsilon_{r}\). As anticipated, this requirement is equivalent to Equations (52) and (53). Now, let us add the evolution operator for quasi-field-free evolution during the additional time from \(t_{c}=0\) to \(t^{\prime}\) between the laser pulses (cf. Equations (25) and (26)),  
\[U_{fge}(t^{\prime},t_{c})= \begin{pmatrix}1&0\\ 0&e^{-i\Delta Et^{\prime}/h}\end{pmatrix} = \begin{pmatrix}1&0\\ 0&e^{-2\pi it^{\prime}/T}\end{pmatrix} \equiv