## 1 Introduction
In the applications to \({}^{87}Rb\) and benzene (where \(T=0.992\) fs and \(0.504\) fs, respectively, see above), Condition (ii) thus requires \"perfect\" time delays, with precision of a few attoseconds. In Ref. [19], we developed robust numerical techniques for quantum dynamics simulations that fulfill this condition, with applications to the two model systems. The experimental feasibility is also presented in Ref. [19], by means of high-contrast Ramsey interferometry of the \({}^{87}Rb\) atom. In retrospect, one can find rather early examples in the literature of works involving electronic structure symmetry breaking and restoration, in particular in early work on Ramsey fringes [27; 28], but the authors did not discuss the phenomenon. A more recent example comes from high harmonic generation (HHG). According to [PERSON]'s \"simple man's three step mechanism\" [29], an atom or a molecule in the electronic ground state is first ionized by an intensive ultrashort laser pulse. In the second step, the same laser pulse accelerates the electron that was ejected in the first step, first by driving it away and then back to the ion. In the third step, the electron re-collides with the ion in a highly excited electronic state that decays back to the electronic ground state by spontaneous emission of HHG, see, e.g., Refs. [29; 30; 31; 32; 33]. In retrospective, this correspond to a kind of _spontaneous_ symmetry restoration, although it was never recognized as such. The example of HHG is rewarding because it shows that restoration of electronic structure symmetry can be exploited for useful application. Laser control of electronic symmetry may also be used for chemical reactions [34; 35; 36; 37] and for charge migration [10; 17]. The efficiency of electronic symmetry restoration in HHG is rather low, however, because the probabilities of re-collision and spontaneous emission in the third step are rather small. Here, we aim at designing laser pulses that achieve close to 100% restoration of electronic structure symmetry. In a general perspective, the present development may thus be considered as contribution to quantum control [38; 39; 40]. The goal of this article is to develop a new strategy for restoration of electronic structure symmetry by means of the second laser pulse, after symmetry has been broken by a first pulse. For convenience, we employ the same mechanism of symmetry breaking by the first laser pulse as previously [19; 20; 21], namely by exciting the system from its electronic ground state to a superposition of the ground state and an excited state with \(IRREP_{e}\) different from \(IRREP_{g}\). The purpose and the mechanism of the second laser pulse is, however, quite different from the previous de-excitation of the superposition state back to the ground state [19; 20]: It excites the component of the ground state in the superposition state to the excited state, which means it prepares the excited state as a pure eigenstate with the same electronic symmetry as the original one. Its \(IRREP_{e}\) is, however, different from the original \(IRREP_{g}\) (e.g., in the case of benzene with symmetry \(D_{6h}\), the excited target state may have \(IRREP_{e}=E_{1u}\), which is different from \(IRREP_{g}=A_{1g}\)). A special case of this strategy is already presented in Ref. [21]. There, the discovery is motivated by Equation (1), which yields \(P_{e}(t^{\prime}_{f})=1\) if \(P_{g}(t_{c})=P_{e}(t_{c})=0.5\) and \(t^{\prime}=T/2\) or \(3T/2,5T/2\), etc. (Note that the work in Ref. [21] employs alternative definitions of time delays.) The requirement of equal populations \(P_{g}(t_{c})=P_{e}(t_{c})=0.5\) called for the special design of the first laser pulse as so-called \(\pi/2\) laser pulse [41]--in Ref. [21], this is a circularly polarized \(\pi/2\) pulse as in Ref. [11; 17].