## 3 Coherent \(\pi\)-Electron Angular Momentum and Current of a Molecule Having Two Aromatic Rings
### Quantum Localization of Coherent \(\pi\)-Electron Angular Momentum
#### 3.3.2 Simulation Results of Angular Momentum Localization  
Figure 15 shows the quantum control results of the first step for the localization of angular momentum. The control time was set to \(t_{f}=30\) fs, which is the inverse of the energy differences between two electronic excited states. In Figure 15a, the expectation value at \(t_{f}=30\) fs, \(l_{L}(t_{f})\) [\(l_{R}(t_{f})\)], reaches the maximum value for the [AO] ([OA]) target, while that of \(l_{R}(t_{f})\) [\(l_{L}(t_{f})\)] is suppressed to zero. This demonstrates that the \(\pi\)-electron localization on one benzene ring is successfully achieved. Figure 15b represents the time-dependent electronic coherences, i.e., imaginary parts of \(\rho_{ab1}(t)\), \(\rho_{b1b2}(t)\), and \(\rho_{ab2}(t)\). The imaginary part of \(\rho_{b1b2}(t)\) for the [AO] and that for [OA] targets provide the same time dependence, while the imaginary parts of \(\rho_{ab1}(t)\) and \(\rho_{ab2}(t)\) for the [AO] target are shifted by \(\pi\) compared to those for the [OA] target. These phase relations originate from that of the target operator in Equation (51). Figure 15c shows Re \(\textbf{E}(\omega)\), real parts of the optimal electric fields in the frequency domain, for the [AO] target (solid line) and the [OA] target (dashed line). The vertical dotted lines refer to the energies of the three eigenstates. The peaks of \(E_{X}(\omega)\), \(E_{Y}(\omega)\), and \(E_{Z}(\omega)\) appeared at the  
Figure 14: Superposition of coherent \(\pi\)-electron rotation patterns. The arrow with positive (negative) sign specifies an in-phase (out-of-phase) superposition of [AA] and [AC] rotation patterns. The in-phase superposition creates [AO] rotation pattern, which means anticlockwise rotation localized on the \(L\) ring with no rotation on the \(R\) ring, while the out-of-phase superposition creates [OA] rotation pattern. Here, O means no rotation. Reprinted from [91] with permission of the American Chemical Society.  
resonant positions of \(b_{2}\), \(b_{1}\), and \(a\) states, respectively. The phases of the \(z\)-component are shifted by \(\pi\) between [AO] and [OA] targets, while phase shifts for the other two components, \(E_{X}(\omega)\) and \(E_{Y}(\omega)\), are zero. This indicates that the \(\pi\)-phase shift of the electric field \(E_{Z}(\omega)\) is the key to control \(\pi\) electron rotations for the [AO] and [OA] localization. It should be noted that \(E_{Z}(\omega)\) excites only the \(a\) state. The difference in the target coefficients between \(W_{L}^{A}\) and \(W_{R}^{A}\) is that of \(\Phi_{a}\). This is the reason why the \(\pi\)-phase shift appeared only in the \(z\)-component. The phase shift of \(E_{Z}(\omega)\) results in \(\pi\)-phase shift in coefficients of \(\Phi_{a}\), \(\Phi_{b1}\), and \(\Phi_{b2}\), and finally the [AO] and [OA] localizations are realized at \(t=30\) fs. The maximum amplitude of the optimal field is about 1 GVm\({}^{-1}\) (0.13 TWcm\({}^{-2}\)), which is much lower than the so-called threshold of multiphoton processes [91].  
Figure 15: Results of quantum localization of \(\pi\)-electron rotations for the target [AO] and for the target [OA]. (**a**) Calculated time-dependent \(\pi\)-electron angular momenta \(l_{L}(t)\) (blue solid line) and \(l_{R}(t)\) (purple dashed line). (**b**) Imaginary parts of electronic coherences, \(\rho_{ab1}(t)\), \(\rho_{b1b2}(t)\), and \(\rho_{ab2}(t)\). (**c**) Real parts of the optimal electric field in the frequency domain. Results of the [AO] ([OA]) target are shown in solid (dashed) lines. Reprinted from [91] with permission of the American Chemical Society.