## 3 Results and Discussions
For this purpose, let us assume that the second laser pulse is centered at  
\[t_{r}^{\prime}=t_{r}+t^{\prime} \tag{67}\]  
with additional time \(t^{\prime}\) between the laser pulses, that means with additional time for the molecule to evolve in quasi-field-free environment, from \(t_{c}=0\) to \(t^{\prime}\). The corresponding time delay is  
\[t_{d}^{\prime}=t_{d}+t^{\prime}, \tag{68}\]  
the new final time is  
\[t_{f}^{\prime}=t_{f}+t^{\prime}, \tag{69}\]  
and the new \"central time\" is  
Figure 3.0  
Figure 3b shows the resulting time evolutions of the populations \(P_{e}(t)\) of the excited state in the superposition state (Equation (5)) for altogether sixteen different values of additional time, \(t^{\prime}=kT/16\), \(k=1,2,...,16\). The results of this systematic investigation are obtained by solving the TDSE (Equation (10)) by the numerical techniques developed in Ref. [19]. Apparently, the vast majority of these examples yield final populations  
Figure 3.1  
That means the molecule is left in a superposition state such as Equation (5), not only after the first pulse that means at \(t_{c}^{\prime}\), but also after the second pulse at \(t_{f}^{\prime}\). Hence, the final state is not a pure state, but it is \"contaminated\" by both the ground and excited states with different \(IRREPs\), i.e., the second laser pulse was unable to restore symmetry. Figure 3b also shows one exceptional case (\(k=16\)) where \(t^{\prime}=T\). In this case, the final population of the excited state is  
Figure 3.2  
implying symmetry restoration. The corresponding time delay  
Figure 3.3  
is again an integer multiple of the period \(T\) for charge migration. The results in Equations (71)-(73) documented in Figure 3b thus confirm the condition in Equation (33), i.e., to achieve symmetry restoration, the time delay between the two laser pulses must satisfy the requirement \(t_{d}=NT\) or more generally  
Figure 3.4  
The sixteen results for the final populations \(P_{e}(t_{f}^{\prime})\) obtained in Figure 3b are plotted versus time delay \(t_{d}^{\prime}\), Equation (65), in Figure 3c, together with corresponding results for a broader range of time delays. Figure 3c also shows a comparison of these results, which are obtained by quantum dynamics simulations, with the analytical result in Equation (62). The agreement is seen to be almost perfect, the deviations are smaller than the graphical resolution. We consider this agreement as double triumph, both for the robustness of the numerical method for quantum dynamics simulation that has been developed in Ref. [19], and also for the analytical theory derived in Section 2.3. According to Figure 3c and the analytical result in Equation (62), perfect symmetry restoration requires perfect time delays, Equations (33) and (74). In practice, one may allow small deviations from the ideal value \(P_{e}(t_{f}^{\prime})=1\), for example one may accept results \(P_{e}(t_{f}^{\prime})>0.99\) as \"good enough\" for \"almost perfect symmetry restoration\". The limit \(P_{e}(t_{f}^{\prime})>0.99\) would allow small deviations \(\Delta t_{d}^{\prime}<0.035T\) from the ideal time delays. The present value of the period \(T=504\) as thus implies that the time delay between the laser pulses should obey the condition in Equation (33) and (74) with accuracy of few attoseconds. Indeed, this time resolution was achieved experimentally by high-contrast Ramsay interferometry, which was thereby shown to be suitable for symmetry restoration [19]. Finally, Figure 3d documents the validity of the equivalent conditions in Equations (35) and (56) for the phase difference \(\Delta\eta(t_{c}^{\prime})\) of the coefficients of the superposition state (Equation (5)) at the central time \(t_{c}^{\prime}\), Equation (70). As anticipated, the values of \(\Delta\eta(t_{c}^{\prime})\) mod \(2\pi\) at the central times \(t_{c}^{\prime}=T/2,T,3T/2,2T\), etc. and the corresponding time delays \(t_{d}^{\prime}=9T,10T,11T,12T\), etc. alternate according to \(-\pi/2,+\pi/2,-\pi/2,+\pi/2\), etc.