## 2 Model, Concept, Theory and Methods
### Conceptual Background for the New Symmetry Restoration Strategy
 \tag{35}\]  
Some important aspects of the concept are illustrated in Figure 1, with application to the oriented model benzene as an example. Initially (\(t=t_{i}\)), the system is in its electronic ground state \(|\Psi_{g}\rangle\) with symmetry \(D_{6h}\) and \(IRREP_{g}=A_{1g}\). This is documented by the one-electron density labeled \"\(A_{1g}\)\" at the bottom of Figure 1. The first circularly right (+) polarized laser pulse centered at \(t=t_{b}\) breaks symmetry by exciting the ground state (Equation (7)) to the superposition state (Equation (5)) of the ground state and an excited state \(|\Psi_{e}\rangle\) with different \(IRREP_{e}\)--here this is one of the lowest two degenerate excited states with \(IRREP_{e}=E_{1u}\). This excitation is symbolized by the first red arrow in Figure 1. The superposition state (Equation (5)) at the central time \(t_{c}\) is written symbolically as \"\(A_{1g}-iE_{1u}\)\" ( \(=A_{1g}+e^{-i\pi/2}E_{1u}\)) in Figure 1, in accord with the phase condition in Equation (35) for the case of \(N=9\) periods \(T\) between the peaks of the laser pulses at \(t_{b}\) and \(t_{r}\). The symmetry of the superposition state is \(C_{s}\), a sub-group of \(D_{6h}\) that contains nothing but the identity and the reflection at the molecular plane--all other symmetry operations of \(D_{6h}\) are destroyed by the first laser pulse. The superposition state (Equation (5)) represents charge migration as indicated by the two curved arrows in Figure 1. Two snapshots of the one-electron densities during charge migration are also illustrated in Figure 1--the first one corresponds to the superposition \"\(A_{1g}-iE_{1u}\)\" for the time \(t=t_{c}=0\) (or periodically for \(t=T,2T,3T\), etc. ), the second one is for \"\(A_{1g}+iE_{1u}\)\" at time \(t=T/2\) (or for \(t=3T/2,5T/2\), etc.). The second laser pulse excites the component \(c_{g}(t_{c})|\Psi_{g}\rangle\) of the ground state in the superposition state (Equation (5)) to the excited state \(|\Psi_{e}\rangle\) with the same symmetry \(D_{6h}\) as the ground state, but with different \(IRREP_{e}=E_{1u}\) (cf. Equation (6)); one can also say symbolically that it transforms the superposition state \"\(A_{1g}-iE_{1u}\)\" to the excited state \"\(E_{1u}\)\", as illustrated by the second red arrow in Figure 1. The target state \(|\Psi_{e}\rangle\) is illustrated by its one-electron density at the top of Figure 1. One readily recognizes its \(D_{6h}\) symmetry--the same as for \(|\Psi_{g}\rangle\). Close inspection reveals, however, that the density of \(|\Psi_{e}\rangle\) is slightly different from the density of \(|\Psi_{g}\rangle\)--this is associated with the different \(IRREP_{e}\
eq IRREP_{g}\). Successful preparation of the target state \(|\Psi_{e}\rangle\) depends on perfect timing of the second laser pulse, i.e., it must be centered at \(t=t_{r}=-t_{b}=NT/2\), or at \(t=t_{r}+T,t_{r}+2T,t_{r}+3T\), etc., which means it must be fired precisely at \(t_{r}(=9T/2\) in the present application), or precisely after one, two or more full cycles of charge migration (see Equations (33) and (34)). Any other attempts to restore symmetry during non-integer multiples of the period \(T\) of charge migration are useless. Such unsuccessful attempts are indicated by the arrows that are crossed out in Figure 1.  
The \(x\)- and \(y\)-components of the electric fields of the laser pulses are also sketched in Figure 1, together with the products \(\epsilon_{b}s_{b}(t)\) and \(\epsilon_{s}s_{r}(t)\) of the field amplitudes times the shape functions which are modeled as Gaussians,  
\[s_{b}(t) =e^{-(t-t_{b})^{2}/2\tau_{b}^{2}}, \tag{36}\] \[s_{r}(t) =e^{-(t-t_{r})^{2}/2\tau_{r}^{2}}.\]  
Our \"proof-of-principle\" employs the same parameters for the durations,  
\[\tau_{b}=\tau_{r}=\tau, \tag{37}\]  
Figure 1: The concept for symmetry breaking of the electronic ground state and symmetry restoration in an electronic excited state by two laser pulses according to the new strategy.