## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Beyond the Frozen-Nuclei Approximation: Nonadiabatically Coupled Vibronic Dynamics
#### 2.3.3 Angular Momentum Expectation Value and Vibrational Spectrum  
Figure 5 shows \(l_{z}(t)\) for the four types of excitations. Its behavior is analogous to that in Figure 3 while \(P_{X}(t)\simeq P_{Y}(t)\) (\(t<4\) fs) except for its smaller amplitude due to the less efficiency of excitation and the coordinate dependence of \(\lambda(\mathbf{Q})\). The amplitude of \(l_{z}(t)\), however, diminishes gradually, which is a characteristic feature absent in Figure 3. This is ascribed to two factors: decrease of the overlap between the WPs moving on the two excited-state PESs, which occurs even without nonadiabatic couplings, and population transfer due to nonadiabatic couplings shown in Figure 4. Both of the two factors cause the loss of a superposition of \(|X\rangle\) and \(|Y\rangle\); in particular, the latter contributes to the loss dominantly. Indeed, we performed nuclear WP simulations within the Born-Oppenheimer approximation (BOA) [83], where the WPs simply propagated on the individual PESs, and found that the amplitude of \(l_{z}(t)\) exhibits periodic reduction and recovery in turn owing to the temporal change in the WP overlap but never decays monotonically [56; 57; 58]. The curves of \(l_{z}(t)\) for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}_{+1}\) can be approximated by a sinusoidal exponential decay. The lifetime of the decay is \(\sim\)6 fs for \(\mathbf{e}^{(+)}\) and  
Figure 4: Temporal behavior in the populations of the quasi-degenerate excited states for (**a**) \(\mathbf{e}^{(+)}\), (**b**) \(\mathbf{e}^{(-)}\), (**c**) \(\mathbf{e}_{+1}\), and (**d**) \(\mathbf{e}_{-1}\) calculated from nuclear WP simulations. In each panel, the populations of \(|X\rangle\) and \(|Y\rangle\) are denoted by the solid and dotted lines, respectively. The laser pulses fully decay at \(t=7.26\) fs. Reprinted from [56] with permission of the American Chemical Society. \(\sim\)18 fs for \(\mathbf{e}_{+1}\); this difference stems from the different rates of nonadiabatic transition in Figure 4a,c. Meanwhile, the amplitudes of \(l_{z}(t)\) for \(\mathbf{e}^{(-)}\) and \(\mathbf{e}_{-1}\) do not undergo a monotonic decrease but make a small transient recovery [in the time ranges of \(t\sim\) 14-20 fs for \(\mathbf{e}^{(-)}\) and \(t\sim\) 18-24 fs for \(\mathbf{e}_{-1}\)]. This arises from the regeneration of the superposition of \(|X\rangle\) and \(|Y\rangle\) due to the upward population transfer in the respective time ranges in Figure 4b,d. We also noticed that \(l_{z}(t)\) oscillates with slightly different periods among the four types of excitations. The oscillation period of \(l_{z}(t)\) is determined by the energy gap between the two PESs in the region where the WPs run; the region depends on the applied pulse as will be shown in Section 2.3.4. The results in Figure 5 demonstrate that nonadiabatic couplings weaken laser-driven ring currents within tens of femtoseconds. Figure 6 displays the expectation value of the normal coordinates \(\mathbf{Q}(t)\equiv\big{\langle}\Psi(t)\big{|}\hat{\mathbf{Q}}\big{|}\Psi(t) \big{\rangle}\) with \(\hat{\mathbf{Q}}\) being the operator of \(\mathbf{Q}\). As seen in Figure 6a, the behaviors of \(\mathbf{Q}(t)\) for linear polarizations are strongly dependent on the polarization direction: The vibrational amplitude for \(\mathbf{e}^{(-)}\) is more than two-times as large as that for \(\mathbf{e}^{(+)}\). In contrast, in Figure 6b, the trajectories of \(\mathbf{Q}(t)\) for circular polarizations (\(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\)) differ only a little and are located between those for linear ones. The behavior of \(\mathbf{Q}(t)\) hardly depends on laser polarization under the BOA [56; 57; 58], which implies the importance of nonadiabatic couplings.