## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Laser-Polarization Effects on Ring Currents within the Frozen-Nuclei Approximation
#### 2.2.5 Numerical Demonstration for 2,5-Dichloropyrazine
The timing of the initial increase in the magnitude of \(l_{z}(t)\) and its sign [i.e., the phase in the oscillation of \(l_{z}(t)\)] manifestly depend on laser polarization. In Figure 3a, the temporal changes in \(l_{z}(t)\) for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) are identical with each other except for opposite signs: For the former (latter), \(\pi\) electrons start to circulate clockwise (anticlockwise), while in both cases the amplitude of \(l_{z}(t)\) reaches almost \(\hbar\) at \(t=5.4\) fs designated by the vertical broken line in Figure 3a. This is consistent with the difference in the initial relative phase \(\theta\) by \(\pi\) between \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) predicted in the model analysis. The magnitudes of \(l_{z}(t)\) for circular polarizations in Figure 3b grow earlier than for linear ones. Besides, a peak of \(l_{z}(t)\) for \(\mathbf{e}_{+1}\) (\(\mathbf{e}_{-1}\)) indicated by the vertical broken lines in Figure 3b appears at \(t=7.0\) fs (4.4 fs), when a laser pulse is almost off (near the maximum intensity). The peak positions of \(l_{z}(t)\) in Figure 3b, which oscillate with different phases from those for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\), are fairly consistent with \(\theta=\pm\left(\eta_{Y}-\eta_{X}\right)=\pm\) 0.35\(\pi\) for \(\mathbf{e}_{\pm 1}\); the 1.6-fs difference for \(\mathbf{e}_{+1}\) from the peak positions at \(t=5.4\) fs in Figure 3a agrees perfectly with that in the field-free condition, 0.35\(\pi/(2\Delta\omega)=1.6\) fs, while the difference for \(\mathbf{e}_{-1}\) is shorter (1.0 fs) because of the strong laser intensity. These agreements with the model analysis confirm the controllability of the initial relative phase \(\theta\) of the superposed quasi-degenerate states by light polarization. As anticipated from Equation (30), the magnitude of \(l_{z}(t)\) turns to decrease or even the direction of ring currents is reversed before the laser pulse ceases at \(t=t_{f}=7.26\) fs. Finally, for all the four types of excitations, the total population in \(|X\rangle\) and \(|Y\rangle\), \(P_{X}(t_{f})+P_{Y}(t_{f})\), reaches 0.995 and thus the amplitude of \(l_{z}(t)\) achieves 0.97\(\hbar\). This indicates that a smoothly rising and decaying pulse with a pulse area of \(2\arccos\left[-\left(2\Delta\omega/\overline{\Omega}\right)^{2}\right]\) is able to excite almost all the population in \(|G\rangle\). At \(t>7.26\) fs, \(l_{z}(t)\) oscillates with the period of \(T=\pi/\Delta\omega\) = 9.4 fs as expected. The oscillation of \(l_{z}(t)\) can be prevented to achieve unidirectional ring currents  
Figure 2: (**a**) Molecular formula of 2,5-dichloropyrazine (DCP). (**b**) Directions of transition moments \(\mu_{X}\) and \(\mu_{Y}\) at the ground-state optimized geometry of DCP as well as those of linear polarization vectors \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) defined by Equation (31). The angle between \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) is 0.48\(\pi\). Normal mode vectors of the (**c**) breathing and (**d**) distortion modes are represented by thick arrows. Modified from [56] with permission of the American Chemical Society. by a pump-dump control method [55; 57; 60]. The application of this method to a molecule with two aromatic rings will be presented in Section 3.2. Such a strong VUV laser pulse as used in Figure 3 may induce multiphoton excitations to higher excited states or ionizations but the contribution of these additional processes is not large [55; 57; 60] The conclusions of this article are unaffected by the use of weaker laser pulses, although a smaller population is transferred to the quasi-degenerate excited states.