## 3 Results and Discussions
We note in passing that the conditions in Equations (63) and (64) imply the phase difference \(\Delta\eta(t_{c})\) mod \(2\pi=\pm\pi/2\) for the coefficients \(c_{e}(t_{c})\) and \(c_{g}(t_{c})\) of the superposition state (Equation (5)) at central time \(t_{c}\); see Equation (35) and the derivation in Equation (54). This result is confirmed in Figure 3d. Third, the field strength \(\epsilon_{r}\) is determined according to the rule in Equation (53),  
\[\epsilon_{r}=\epsilon_{\pi}-\epsilon_{b} \tag{65}\]  
where \(\epsilon_{\pi}\) is the field strength of a circularly right (+) polarized \(\pi\)-pulse with Gaussian shape,  
\[\epsilon_{\pi}=\frac{\sqrt{\pi}\cdot\hbar}{2d\tau}, \tag{66}\]  
in accord with Equation (50) (cf. Ref. [42]). The value of the parameter for the laser duration, \(\tau=0.47\) fs, together with the value of the transition dipole \(d=2.0576 ea_{0}\), yields the field amplitude of the \(\pi\)-laser pulse, \(\epsilon_{\pi}=11.399\times 10^{7}\) V/cm. Subtraction of the field amplitude of the laser pulse for symmetry breaking, \(\epsilon_{b}=4.207\times 10^{7}\) V/cm, then yields the field strength of the laser pulse for symmetry restoration, \(\epsilon_{r}=7.192\times 10^{7}\) V/cm. It is illuminating to compare the population transfers between the electronic ground and excited states, namely 0.30 and 0.70, which are achieved by these pulses, with the values 0.0037 for the weak field strengths \(\epsilon_{b}=\epsilon_{r}=4.42\times 10^{6}\) V/cm that are employed in Ref. [19]. Assuming the weak field limit, the population transfers should increase linearly with intensity, i.e., one would expect the value \(0.0037\times(42.07/4.42)^{2}=0.33\) and \(0.0037\times(42.07/4.42)^{2}=0.98\). The deviations from the values 0.30 and 0.70 show that while the intensity of the pulse for symmetry breaking may still be considered as \"weak\", the required high field strength of the pulse for symmetry restoration is well beyond that limit. The predicted laser pulse for symmetry restoration is shown in Figure 3a as second pulse, together with the laser pulse for symmetry breaking as first laser pulse. As anticipated already in Figure 1, the rules in Equations (53) and (65) imply that the second pulse is a bit stronger than the first one. Step (iii): As decisive effect of the laser pulse for the symmetry restoration designed in Section 2.2, Figure 3b documents the laser driven evolution of the population of the excited state \(P_{e}(t)\) in the superposition state (Equation (5)), from the value \(P_{e}(t_{c})=0.3\) to \(P_{e}(t_{f})=1\). This is a key result: it implies complete and exclusive population of the excited state with \(D_{6h}\) symmetry and \(IRREP_{e}=E_{1u}\). This provides the proof-of-principle, i.e., the well-designed (see Section 2.2) second laser pulse (Figure 3a) restores \(D_{6h}\) symmetry by exciting the model benzene from the superposition state (Equation (5)) to the target state with \(IRREP_{e}\
eq IRREP_{g}\). At the same time, it stops charge migration because the target state is an electronic eigenstate, i.e., it is stationary. The one-electron density of the target state is shown in Figure 3g. Its \(D_{6h}\) symmetry is obvious. There are, however, tiny differences between the initial (Figure 3e) and the final (Figure 3g) electron densities--this is a consequence of the different \(IRREP_{s}\)\(A_{1g}\) and \(E_{1u}\) of the initial and final states, respectively. The Requirement of Attosecond Precision for the Proper Time Delay Between the Laser Pulses for Electronic Structure Symmetry Breaking and Restoration  
The proof-of-principle (Section 3.1) employs the perfectly synchronized second laser pulse for symmetry restoration which has been designed according to the recipe derived in Section 2. In particular, its time delay with respect to the first laser pulse that breaks symmetry satisfies the condition in Equation (63), in accord with the general rule in Equation (33). In this subsection, we investigate the consequences of violations of the condition in Equation (63).