## 3 Results and Discussions
The Proof-of-Principle for Quantum Control of Symmetry Breaking and Restoration of Molecules in Electronic Ground and Excited States with Different \(1 RREP_{s}\)  
Our proof-of-principle uses the concept and theory that are developed in Section 2, with application to the model benzene as described in Sections 1 and 2. The proof consists of three steps that implement various results of Section 2. The presentation below documents the key results in a straightforward manner, with references to several key expressions, but without repetition of the derivation. The reader is expected to be familiar with the context developed in Section 2. Step (i): The electronic structure of benzene in its initial ground state (\(IRREP_{g}=A_{1g}\)) is illustrated by a contour plot of its one-electron density in the molecular plane in Figure 3e. One readily recognizes its \(D_{6h}\) symmetry. The \(x\)- and \(y\)-components and the envelope of the electric field of the first circularly right (+) polarized laser pulse (Equation (42)) that breaks the \(D_{6h}\) symmetry are shown in the left of Figure 3a. This pulse is centered at \(t_{b}=-4.5T=-2.267\) fs. Its parameters are chosen rather arbitrarily, within the frame of the general rules explained in Sections 1 and 2. The pulse thus serves as \"general\" example, quite different from the special \(\pi/2\) pulse that has been employed in Ref. [21]. Consequently, its effect of symmetry breaking and launching charge migration should also be considered as \"general\", providing a \"generic example\" for the challenge of designing a second laser pulse that stops charge migration and restores symmetry. In other words, the present example lends itself for a rather general proof-of-principle. The first laser pulse breaks \(D_{6h}\) symmetry by preparing the superposition (Equation (5)) of the electronic ground and excited states with \(IRREP_{s}\)\(A_{1g}\) and \(E_{1u}\), here with probabilities \(P_{g}(t_{c}=0)=0.7\) and \(P_{e}(t_{c}=0)=0.3\) at central time \(t_{c}=0\) (cf. Equation (16)). The laser-induced increase of \(P_{e}(t)\) from the initial value zero to \(P_{e}(t_{c}=0)=0.3\) is documented in Figure 3b. The one-electron density of the superposition state (Equation (5)) at \(t_{c}\) is illustrated in the first panel of Figure 3e. Symmetry breaking is obvious, i.e., the superposition state (Equation (5)) can no longer be assigned to the original \(D_{6h}\) symmetry. Since it is not an eigenstate, it is non-stationary. This gives rise to periodic charge migration with period \(T\), as illustrated in Figure 3f by five snapshots of the one-electron density at times \(t=0,T/4,T/2,3T/4,T\). On first glance, these snapshots may suggest that the superposition (Equation (5)) may have \(C_{2v}\) symmetry, but additional snapshots at arbitrary times \(t\in[0,T]\) reveal that the superposition (Equation (5)) evolves with \(C_{s}\) symmetry: the first laser pulse breaks all symmetry elements except the molecular plane as mirror plane. The challenge is thus to design the second laser pulse that stops charge migration and restores symmetry by transferring the superposition state (Equation (5)) to the excited target state (Equation (6)). Step (ii): The second laser pulse for symmetry restoration is constructed according to the recipe derived in Sections 2.2 and 2.3. First, it should be circularly right (+) polarized, similar to the first laser pulse, Equation (42). The laser parameters of the two pulses are the same, with two exceptions. Second, the pulse is centered at \(t_{r}=-t_{b}\), and the time delay \(t_{d}\) between the two laser pulses must be equal to an integer number \(N\) of period \(T\) of charge migration. In the present case, we choose  
\[t_{d}=t_{r}-t_{b}=9T, \tag{63}\]  
that means  
\[t_{r}=-t_{b}=4.5T; \tag{64}\]  
compare with Equations (33) and (34).