## 1 Introduction
This created a superposition of the electronic ground and excited states that cannot be assigned to any IRREP (e.g., neither to \(IRREP_{g}\) nor to \(IRREP_{e}\)) of the original symmetry groups (\(S_{0}(3)\) and \(D_{6h}\), respectively), but only to one of their subgroups. For example, the electronic structure of the \({}^{87}Rb\) atom in the superposition state was anisotropic [19], and the electronic wave function of the superposition state of benzene had \(C_{s}\)[19; 21] or \(C_{2v}\)[20] symmetry, different from the initial isotropic \(S_{0}(3)\) or \(D_{6h}\) symmetries, respectively. Since the superposition state is not an eigenstate, it is of course non-stationary. Since the nuclear wave functions that are associated with the electronic ground and excited states overlap, the non-stationary superposition state represents charge migration in the applications of Refs. [19; 20; 21]. The period \(T=h/\Delta E\) of charge migration depends on the energy gap \(\Delta E=E_{e}-E_{g}\) between the levels \(E_{g}\) and \(E_{e}\) of the electronic ground and excited states. Typical values of \(\Delta E\) are in the domain of several eV. The periods \(T\) of charge migration are, therefore, in the time domain from sub-fs to several fs. Specifically, for our applications to \({}^{87}Rb\) and benzene, \(T=0.992\) fs and \(0.504\) fs, respectively. The second laser pulse de-excited the superposition state back to the ground state [19; 20]. Consequently, charge migration was stopped, and finally (at time \(t_{f}=-t_{i}\)) the systems were back to their initial symmetry groups--specifically, they were back to \(S_{0}(3)\) and \(D_{6h}\), respectively. Moreover, the previous approaches [19; 20] also re-establish the original \(IRREP_{g}(=D^{(0)}\) and \(A_{1g}\), respectively). Successful applications depended on two conditions: (i) The second laser pulse was designed as a circularly polarized copy [19], or as time-reversed linearly polarized copy [20] of the first pulse for symmetry breaking. By \"copies\" we imply that the two laser pulses must have the same, or time-reversed properties. For example, they have the same carrier frequencies \(\omega_{b}=\omega_{r}\), the same maximum field strengths \(\epsilon_{b}=\epsilon_{r}\) and corresponding maximum intensities \(I_{b}=0.5\epsilon_{0}c_{0}\epsilon_{b}^{2}=I_{r}\) (where \(\epsilon_{0}\) and \(c_{0}\) denote the permittivity and speed of light in vacuo), the same ( e.g., Gaussian) shape functions with the same parameters for the durations \(\tau_{b}=\tau_{r}\), the same carrier envelope phases \(\eta_{b}=\eta_{r}\) (usually set equal to zero), etc. (ii) The time delay \(t_{d}\) had to be chosen \"perfectly\" such that at the central time \(t_{c}=0\), the two components of the superposition states had the same or opposite phases. Ifthis condition is not satisfied, i.e., if the second pulse is centered at \(t^{\prime}_{r}=t_{r}+t^{\prime}\) with time \(t^{\prime}\) added to the \"perfect\" time \(t_{r}\), then at the final time \(t^{\prime}_{f}=t_{f}+t^{\prime}\), the systems are left in the asymmetric superposition state, with non-zero populations \(P_{g}(t^{\prime}_{f})\) and \(P_{e}(t^{\prime}_{f})=1-P_{g}(t^{\prime}_{f})\) of the two components for the ground and excited states. An important analytical result is [19; 20]  
\[P_{e}(t^{\prime}_{f})=4P_{g}(t_{c})P_{e}(t_{c})\left[\frac{1}{2}-\frac{1}{2} \cos\left(\frac{2\pi t^{\prime}}{T}\right)\right], \tag{1}\]  
where \(P_{g}(t_{c})\) and \(P_{e}(t_{c})\) are the populations of the ground and excited states at the central time, \(t_{c}=0\). Symmetry restoration in the ground state requires that \(P_{g}(t^{\prime}_{f})=1\) and \(P_{e}(t^{\prime}_{f})=0\). Obviously, Condition (ii) is satisfied if and only if \(t^{\prime}=0\) or \(t^{\prime}=T,2T,3T\), etc. In practice, one may allow small deviations from this ideal requirement, by the order of few percents of the period \(T\).