## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Beyond the Frozen-Nuclei Approximation: Nonadiabatically Coupled Vibronic Dynamics
#### 2.3.4 Laser Control of Wave Packet Interference  
Figure 8 shows the propagations of the adiabatic WPs on the two excited-state PESs. The probability densities \(|\psi_{X}(\mathbf{Q},t)|^{2}\) and \(|\psi_{Y}(\mathbf{Q},t)|^{2}\) at \(t\sim 5\) fs resemble that of the initial WP in \(|G\rangle\) and then the WPs start to move along the gradient of each PES. As they approach the avoided crossing, the nonadiabatic effects on vibronic dynamics appear differently for the four types of excitations. For \(\mathbf{e}^{(+)}\) (Figure 8a), the WP in \(|Y\rangle\) is diminished by nonadiabatic transition at \(t\sim 12\) fs and the contour map of \(|\psi_{X}(\mathbf{Q},t)|^{2}\) clearly displays the node arising from the interference; thereafter, the WPs on the two PESs are distorted significantly. In contrast, for \(\mathbf{e}^{(-)}\) (Figure 8b), the WPs maintain a Gaussian-like form even after the nonadiabatic transition already in progress at \(t\sim 8\) fs. For circular polarizations, the WPs behave as expected from the behavior of the populations in Figure 4c,d. In the case of \(\mathbf{e}_{+1}\) (Figure 8c), only a small fraction of \(\psi_{Y}(\mathbf{Q},t)\) has been transferred to \(|X\rangle\) until \(t\sim 12\) fs and no interference patterns are observed in the WPs. For \(\mathbf{e}_{-1}\) (Figure 8d), the WPs exhibit intermediate features between those for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\): No clear interference is observed in the WPs but the shape of \(|\psi_{X}(\mathbf{Q},t)|^{2}\) is  
Figure 7: The frequency spectra of \(\psi_{X}(\mathbf{Q},t)\), \(\sigma_{X}(\omega)\), defined by Equation (34) for DCP irradiated by (**a**) linearly and (**b**) circularly polarized VUV laser pulses. In panel (**a**), the spectra for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) are denoted by the solid and dotted lines, respectively; in panel (**b**), those for \(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\) are denoted by the solid and dotted lines, respectively. In each case, the values of \(\sigma_{X}(\omega)\) were scaled so that the maximum value is unity. Reprinted from [56] with permission of the American Chemical Society. deformed from a Gaussian especially at \(t\sim 12\) fs. As a result of nonadiabatic couplings, the center of \(\psi_{X}(\mathbf{Q},t)\), which contributes dominantly to \(\mathbf{Q}(t)\) in Figure 6, runs in a low-energy (high-energy) region on the PES for \(\mathbf{e}^{(+)}\) [\(\mathbf{e}^{(-)}\)], whereas the motions of the center for circular polarizations are similar to each other. The polarization effects on the expectation values \(l_{z}(t)\) and \(\mathbf{Q}(t)\) as well as on the populations \(P_{X}(t)\) and \(P_{Y}(t)\) can be explained in terms of WP interference. As defined in Section 2.2, a linearly polarized laser pulse with \(\mathbf{e}^{(-)}\) initially produces a superposition of \(\psi_{X}(\mathbf{Q},t)\) and \(\psi_{Y}(\mathbf{Q},t)\) out of phase (\(\theta=\pi\)). Their relative quantum phase evolves during irradiation [by \(-\theta(t)\) in the rectangular-envelope cases] and also gains a dynamical phase associated with the shape of each PES as the WPs propagate. In nonadiabatic transition, an additional phase shift is further imposed on the WP created by nonadiabatic couplings, which interferes with that on the other PES. We do not quantify the additional phase in DCP, but the downward population transfer around \(t\sim 5\)-10 fs in Figure 4b implies opposite interferences on the two PESs: The WPs are almost in phase and interfere constructively on the lower PES, while those on the higher PES are out of phase with destructive interference. The constructive interference on the lower PES works particularly on high vibrational states in \(\psi_{X}(\mathbf{Q},t)\). The direction of the population transfer flips as the relative quantum phase grows. For \(\mathbf{e}^{(+)}\), the WPs prepared on the two PESs are in phase (\(\theta=0\)).