## 3 Results and Discussions  
This section has two parts. First, we apply the theory of Section 2 to demonstrate successful quantum control of breaking and restoring the \(D_{6h}\) symmetry of the electronic structure of the model benzene in its electronic ground and excited states with different \(IRREP_{s}\)\(A_{1g}\) and \(E_{1u}\). This part consists of three steps: (i) the symmetry breaking by a circularly right (+) polarized laser pulse and the resulting periodic charge migration; (ii) the design of the laser pulse for symmetry restoration; and (iii) the quantum dynamics simulation of the laser driven symmetry restoration. The purpose of this first part is to provide a proof-of-principle of the new strategy that has been developed in Section 2, for the example of the model benzene. The second part presents an in-depth-investigation of the condition in Equation (33) for the time delay between the two laser pulses, \(t_{d}=NT\), i.e., the time delay must be equal to an integer number of period \(T\) of charge migration. We show that this condition must be satisfied with precision of a few attoseconds. The results for both parts are documented in Figure 3. The layout of Figure 3 is the same as corresponding figures in Refs. [19; 20; 21], but the results are of course entirely different, i.e., Figure 3 documents a new mechanism of symmetry restoration, different from those of Refs. [19; 20; 21]. The values of the parameters of the laser pulses that are applied for the proof-of-principle are listed in the caption of Figure 3.  
Figure 3: Symmetry breaking of the electronic ground state of benzene labeled \(A_{1g}\) (symmetry \(D_{6h}\), irreducible representation \(IRREP_{g}=A_{1g}\)) and symmetry restoration in an electronic excited labeled \(E_{1u}\) (symmetry \(D_{6h}\), irreducible representation \(IRREP_{e}=E_{1u}\)) by two laser pulses according to the new strategy. (**a**) Gaussian envelopes (dashed lines) and the \(x\)- and \(y\)-components (red and green continuous  
lines) of the circularly right (+) polarized laser pulses centered at \(t_{b}=-4.5T\) and \(t_{r}=+4.5T\) with period \(T=504\) as of charge migration. The parameter of the circularly right (+) polarized laser pulses (Equation (40)) are \(\epsilon_{b}=4.207\times 10^{7}\) V/cm, \(\epsilon_{r}=7.192\times 10^{7}\) V/cm, \(\omega=2\pi/T,T=504\) as, \(\tau_{b}=\tau_{r}=0.47\) fs, \(t_{b}=-4.5T,t_{r}=+4.5T\). **(b)** Time evolution of the population of the excited state due to the first and second laser pulses shown in (**a**) for the case \(t_{r}=+4.5T=2.267\) fs. The results for sixteen different times \(t_{r}^{\prime}=t_{r}+t^{\prime}\) where \(t^{\prime}=kT/16,k=1,2,...,16\) are also shown. (**c**) Numerical results (continuous blue line) and analytical result (dotted red line, Equation (62)) for the final populations \(P_{e}(t_{f}^{\prime})\) of the excited state at time \(t_{f}^{\prime}=t_{f}+t^{\prime}\) versus delay time \(t_{d}^{\prime}=t_{r}^{\prime}-t_{b}\), in units of the period \(T\) (top abscissa) or fs (bottom abscissa, as in (**d**)). The results coincide within graphical resolution. (**d**) Phase difference \(\Delta\eta(t_{c}^{\prime})=\eta_{e}(t_{c}^{\prime})-\eta_{g}(t_{c}^{\prime})\) of the wave functions in electronic excited and ground states at the central time \(t_{c}^{\prime}=(t_{r}^{\prime}+t_{b})/2\). (**e**) One-electron density of the electronic ground state of benzene labeled \(A_{1g}\). (**f**) Five snapshots of the one-electron density during periodic charge migration of the superposition of the ground state labeled \(A_{1g}\) and the excited state labeled \(E_{1u}\) during one period, from \(t=t_{c}=0\) to \(T\). (**g**) One-electron density of the excited target state labeled \(E_{1u}\). All densities were created using detCI@ORBKKIT [43; 44; 45] and plotted using Matplotlib [46].