## 3 Coherent \(\pi\)-Electron Angular Momentum and Current of a Molecule Having Two Aromatic Rings
### Quantum Switching of Coherent \(\pi\)-Electron Rotations in a Nonplanar Chiral Aromatic Molecule
#### 3.2.1 Control Scheme for Angular Momentum Switching  
As shown in Figure 12a, four rotational patterns (CC, AA, CA, and AC), in which C (A) means clockwise (anticlockwise) rotation along the \(L\) or \(R\) phenol ring, are possible for \((P)\)-2,2\({}^{\prime}\)-biphenol, depending on two factors: one is the polarization direction of the applied linearly polarized laser pulse and the other is the symmetry of the created coherent state. Let us first clarify how these patterns can be prepared, and then how ultrafast switching between these rotational patterns can be realized. As already shown by Equation (31), the polarization direction is expressed in terms of a unit vector, \(\mathbf{e}_{a\beta}^{(+)}\) or \(\mathbf{e}_{a\beta}^{(-)}\) with \(\alpha\), \(\beta=a\), \(b_{1}\), \(b_{2}\). Here, \(\mathbf{e}_{a\beta}^{(+)}\) and \(\mathbf{e}_{a\beta}^{(-)}\) are defined by \(\mathbf{e}_{a\beta}^{(+)}\cdot\mathbf{\mu}_{ga}=\mathbf{e}_{a\beta}^{(+)}\cdot \mathbf{\mu}_{\beta\beta}\) and \(-\mathbf{e}_{a\beta}^{(-)}\cdot\mathbf{\mu}_{ga}=\mathbf{e}_{a\beta}^{(-)} \cdot\mathbf{\mu}_{\beta\beta}\), respectively. The excitation by laser pulses with \(\mathbf{e}_{a\beta}^{(+)}\) [\(\mathbf{e}_{a\beta}^{(-)}\)] creates a linear combination of two electronic states \(\alpha\) and \(\beta\) with their in-phase (out-of-phase) relation, which is called \(\mathbf{e}_{a\beta}^{(+)}\) [\(\mathbf{e}_{a\beta}^{(-)}\)] excitation hereafter. The schematic drawings on the left-hand side of Figure 12a represent the initial rotational directions of \(\pi\) electrons after the preparation of a superposition of the two excited states, \(b_{1}\) and \(b_{2}\). The upper (lower) drawing shows the rotational direction just after coherent \(\mathbf{e}_{b1b2}^{(+)}\) [\(\mathbf{e}_{b1b2}^{(-)}\)] excitation, which indicates CC (AA) rotation, i.e., \(\pi\) electrons start to rotate along the two phenol rings in the same directions. Figure 11: Bond currents for the three types of the electronic coherence at a time when the maximum electronic coherence is created. Modified from [87] with permission of the American Institute of Physics. Figure 12b shows the resultant angular momentum \(\mathbf{I}=\mathbf{I}_{L}+\mathbf{I}_{R}\). Here, the \(\pi\)-electron angular momentum \(\mathbf{I}_{L}\) (\(\mathbf{I}_{R}\)) is perpendicular to the \(L\) (\(R\)) phenol ring, and the resultant total angular momentum \(\mathbf{I}_{Z}\) (\(\mathbf{I}_{X}\)) is parallel to the \(Z\)- (\(X\)-) axis on the \(ZX\)-plane. The directions of \(\mathbf{I}_{Z}\) indicate that clockwise (anticlockwise) rotation around the \(Z\)-axis is created by \(\mathbf{\mathrm{e}}_{\alpha\beta}^{(+)}\) [\(\mathbf{\mathrm{e}}_{\alpha\beta}^{(-)}\)] excitation. The schematic drawings on the right-hand side of Figure 12a (12b) represent \(\pi\)-electron rotations (the resultant angular momenta) after preparation of a superposition of the two excited states, \(a\) and \(b_{1}\), with different irreducible representations. \(\pi\) Electrons start to rotate along the two phenol rings in opposite directions to each other (CA or AC rotation). Therefore, the total angular momentum parallel to the \(Z\)-axis is zero and the nonzero resultant angular momentum \(\mathbf{I}_{X}\) is generated along the \(X\)-axis on the \(ZX\)-plane. If the two phenol rings are in the same plane without twist, both \(\mathbf{I}_{X}\) and \(\mathbf{I}_{Z}\) vanish. Furthermore, \(\mathbf{\mathrm{e}}_{\alpha\beta 1}^{(+)}\) and \(\mathbf{\mathrm{e}}_{\alpha\beta 1}^{(-)}\) excitations induce CA and AC rotations, respectively. Let us now consider two-dimensional current switching on the basis of the results shown in Figure 12. Here, two-dimensional switching means a  
Figure 12: (**a**) Four patterns of \(\pi\)-electron rotations induced by selective coherent electronic excitations. For example, CA rotation means clockwise and anticlockwise rotations along the phenol rings \(L\) and \(R\), respectively. (**b**) Resultant angular momenta \(\mathbf{I}_{Z}\) and \(\mathbf{I}_{X}\) for the four rotational patterns.