## 2 Model, Concept, Theory and Methods
### Model and Basic Theory
The coefficients are in general complex-valued; they can be written as  
\[H_{e}=T_{e}+V_{c} \tag{3}\]0  
with real valued amplitudes \(C_{k}(t)\) and phases \(\eta_{k}(t)\). The matrix elements of the Hamilton matrix \(H(t)\) are  
\[H_{e}=T_{e}+V_{c} \tag{3}\]1  
with \(k\to l\) transition dipole matrix element \(\vec{d}_{kl}=\langle\Psi_{k}|\vec{d}|\Psi_{l}\rangle\). In the case of benzene, the diagonal matrix elements vanish, \(\vec{d}_{kk}=0\), for symmetry reasons. The absolute squares of the coefficients \(c_{k}(t)\) yield the populations, or the probabilities of occupying the eigenstate labeled \(k\) in the superposition (Equation (9)) at time \(t\),  
\[H_{e}=T_{e}+V_{c} \tag{3}\]2  
These probabilities are normalized,  
\[H_{e}=T_{e}+V_{c} \tag{3}\]3In particular, the initial electronic ground state (Equation (7)) implies that \(P_{k}(t_{i})=\delta_{gk}\) (cf. Equation (11)). The populations of the ground and excited states in the superposition (Equation (5)) at time \(t_{c}\) after the first laser pulse are  
\[ \begin{array}{l}P_{g}(t_{c})=|c_{g}(t_{c})|^{2},\\ P_{e}(t_{c})=|c_{e}(t_{c})|^{2},\end{array}  \tag{16}\]  
whereas all other coefficients and probabilities are equal to zero at time \(t_{c}\). The normalization in Equation (15) implies that the coefficients at time \(t_{c}\) can be parameterized as  
\[ \begin{array}{l}c_{g}(t_{c})=\cos\alpha\cdot e^{i\eta_{g}(t_{c})},\\ c_{e}(t_{c})=\sin\alpha\cdot e^{i\eta_{e}(t_{c})}.\end{array}  \tag{17}\]  
The final populations are  
\[P_{k}(t_{f})=\delta_{ek} \tag{18}\]  
in accord with Equation (6). The formal solution of the TDSE (Equation (10)) with the initial condition in Equation (11) is  
\[\mathbf{c}(t)=\mathbf{U}(t,t_{i})\mathbf{c}(t_{i})=\hat{T}e^{-i\int_{t_{i}}^{t }dt^{\prime}\mathbf{H}(t^{\prime})/\hbar}\mathbf{c}(t_{i}), \tag{19}\]  
where \(\mathbf{U}(t,t_{i})\) is the matrix representation of the unitary operator for propagating the electronic wave function \(|\Psi(t)\rangle\) from the initial time \(t_{i}\) to \(t\), and \(\hat{T}\) is the time-ordering operator. In practice, the algebraic TDSE (Equation (10)) is propagated numerically, using the methods which have been developed in Ref. [19]. The coefficients at the central and final times, i.e., after the first and second laser pulses are thus  
\[\mathbf{c}(t_{c})=\mathbf{U}(t_{c},t_{i})\mathbf{c}(t_{i}), \tag{20}\]  
\[\mathbf{c}(t_{f})=\mathbf{U}(t_{f},t_{i})\mathbf{c}(t_{i})=\mathbf{U}_{r}(t_{ f},t_{c})\mathbf{U}_{b}(t_{c},t_{i})\mathbf{c}(t_{i}). \tag{21}\]  
In the second part of Equation (21), we have separated the evolution operator \(\mathbf{U}(t_{f},t_{i})\) into two parts for symmetry breaking and restoration by the first and second laser pulses,  
\[\mathbf{U}(t_{f},t_{i})=\mathbf{U}_{r}(t_{f},t_{c})\mathbf{U}_{b}(t_{c},t_{i}) \tag{22}\]  
where  
\[\mathbf{U}_{b}(t_{c},t_{i})=\mathbf{U}(t_{c},t_{i}) \tag{23}\]  
and  
\[\mathbf{U}_{r}(t_{f},t_{c})=\hat{T}e^{-i\int_{t_{c}}^{t_{f}}dt^{\prime}\mathbf{ H}(t^{\prime})/\hbar}. \tag{24}\]  
If the time delay between the two laser pulses is prolonged by additional time \(t^{\prime}\) for quasi-field-free evolution, as discussed for Equation (1), we obtain by using the Schrodinger picture  
\[\mathbf{U}(t_{f}^{\prime},t_{i})=\mathbf{U}_{r}(t_{f}^{\prime},t^{\prime}) \mathbf{U}_{f}(t^{\prime},t_{c})\mathbf{U}_{b}(t_{c},t_{i})=\mathbf{U}_{r}(t_ {f},t_{c})\mathbf{U}_{f}(t^{\prime},t_{c})\mathbf{U}_{b}(t_{c},t_{i}) \tag{25}\]  
where \(t_{f}^{\prime}=t_{f}+t^{\prime}\), and \(\mathbf{U}_{f}(t^{\prime},t_{c})\) is the propagator for the free evolution. It is diagonal, with matrix elements [19; 20]  
\[U_{f}(t^{\prime},t_{c})_{kl}=e^{-i\hat{T}e_{k}t^{\prime}/\hbar}\delta_{kl}. \tag{26}\]  
In the second part of Equation (25), we exploit the fact that the effect of the second laser pulse with additional time delay \(t^{\prime}-t_{c}\) during the time-shifted period from \(t^{\prime}\) to \(t_{f}^{\prime}\) is the same as the effect of the original second laser pulse without additional time delay during the period from \(t_{c}\) to \(t_{f}\).