## 2 Model, Concept, Theory and Methods
### Conceptual Background for the New Symmetry Restoration Strategy  
After the first laser pulse has broken the symmetry of the initial electronic ground state (Equation (7)) by exciting it to the superposition state (Equation (5)), our new strategy suggests that the second laser pulse should restore symmetry by transferring the superposition state (Equation (5)) to the excited state (Equation (6)). The net result of the combined first and second laser pulses is thus to excite the ground state (Equation (7)) to the target state (Equation (6)). It is well known that complete population transfer can be achieved by a resonant \(\pi\)-pulse [41; 42]. This suggests the concept which can be written symbolically as  
\[\text{first laser pulse}\ +\ \text{second laser pulse}=\text{resonant }\pi-\text{pulse}. \tag{29}\]  
Equation (29) should be read as condition for the second laser pulse: It should be designed such that the sum of the two pulses is a resonant \(\pi\)-pulse. This condition holds irrespective of the polarizations of the laser pulses, i.e., they may be linearly \(x\)- or \(y\)-polarized, or circularly right (\(+\)) or left (\(-\)) polarized. The expressions for linearly [41] and circularly [42] polarized resonant \(\pi\)-pulses are similar. They all have in common that the \(x\)- and/or \(y\)-components of the electric field of the pulse that propagates along \(\vec{\varepsilon}_{z}\) are written as amplitude \(e_{0}\) ( = the maximum field strength) times a shape function \(s(t)\) (where max \(s(t)=1\)), times cosinusoidal or sinusoidal functions \(\cos\left[\omega(t-t_{ref})+\eta_{CEP}\right]\) or \(\sin\left[\omega(t-t_{ref})+\eta_{CEP}\right]\) where \(\omega\) is the carrier frequency, \(t_{ref}\) is a reference time, and \(\eta_{CEP}\) is the carrier envelope phase (CEP); additional information on the consequences of different laser polarizations is provided at the end of this subsection. The resonance condition means that the photon energy \(\hbar\omega\) matches the energy gap between the levels of the excited and ground states,  
\[\hbar\omega=E_{e}-E_{g}=h/T. \tag{30}\]  
Here, we must emphasize that the reference time and the CEP of resonant \(\pi\)-pulses are time-independent,  
\[t_{ref}=\text{const},\ \ \eta_{CEP}=\text{const}. \tag{31}\]  
That means that the first and second laser pulses in Equation (29) must have the same reference time and the same CEP. Mathematical formulations of the two conditions in Equations (29) and (31) are worked out below. There are various ways of realizing the conditions in Equations (29) and (31). Below, we demonstrate a rather simple example that serves as proof-of-principle, and also as a reference for extensions of the present approach. Specifically, we consider the case of circularly right (+) polarized laser pulses with  
\[t_{ref}=t_{b},\ \ \eta_{CEP}=0. \tag{32}\]  
For our proof-of-principle, the second laser pulse is designed such that its electric field at maximum intensity points along \(\vec{e}_{x}\), just like the first laser pulse. We show below that the condition in Equation (32) is then equivalent to the requirement  
\[t_{d}=t_{r}-t_{b}=NT,N=1,2,3,... \tag{33}\]  
or  
\[t_{r}=-t_{b}=NT/2,N=1,2,3,... \tag{34}\]  
This means that during the delay time \(t_{d}\) between the peaks of the laser pulses, the system undergoes an integer number \(N\) of periodic cycles of charge migration. Our specific example employs \(N=9\) periods \(T\) between the laser pulses. We also derive an equivalent alternative condition, namely the difference between the phases of the coefficients \(c_{g}(t_{c})=C_{g}(t_{c})e^{i\eta_{g}(t_{c})}\) and \(c_{e}(t_{c})=C_{e}(t_{c})e^{i\eta_{e}(t_{c})}\) of the superposition state, Equations (5), (12) and (17) at the central time \(t_{c}=(t_{b}+t_{r})/2=0\) should satisfy  
\[\Delta\eta(t_{c})\ \ \text{mod}\ \ 2\pi=(\eta_{e}(t_{c})-\eta_{g}(t_{c}))\ \ \text{mod}\ \ 2\pi= \begin{cases}-\pi/2\ \ \text{if}\ \ N=1,3,5,...(\text{odd})\\ +\pi/2\ \ \text{if}\ \ N=2,4,6,...(\text{even}).\end{cases}