## 3 Coherent \(\pi\)-Electron Angular Momentum and Current of a Molecule Having Two Aromatic Rings
### Two Ring Components of \(\pi\)-Electron Angular Momentum and Current
#### 3.1.5 Generation of Two-State Electronic Coherence by Linearly Polarized UV Pulses  
We can generate three types of two-state electronic coherence in the present three excited state model, which can be represented as (\(b_{1}\pm b_{2}\)), (\(a\pm b_{1}\)), and (\(a\pm b_{2}\)). The phase of each electronic coherence can be set by applying a linearly polarized UV laser pulse with a properly selected polarization direction. Because of the nonplanar geometrical structure, the angular momentum is two-dimensional and the ring current is on two different surfaces: \(Z\)-directional angular momentum is generated for the created coherent state with A symmetry and corresponding ring current is generated on the \(Z\)-directional angular momentum, while \(X\)-directional angular momentum is generated for the coherent state with B symmetry and ring current is generated on the \(X\)-directional angular momentum.  
Figure 10 shows time-dependent angular momenta calculated for three types of electronic coherence, (\(b_{1}+b_{2}\)), (\(a+b_{1}\)), and (\(a+b_{2}\)), each of which is created by a linearly polarized UV pulse with a properly selected polarization direction. The envelope function of the pulse for each coherence is shown in the right-hand side. The pulse amplitudes for (\(b_{1}+b_{2}\)), (\(a+b_{1}\)), and (\(a+b_{2}\)) coherences are 0.19, 0.83 and 3.32 TW/cm\({}^{2}\), respectively. Angular momentum along the \(Z\)-axis, \(l_{Z}\), is generated for (\(b_{1}+b_{2}\)) electronic coherence, while those along the \(X\)-axis, \(l_{X}\), are generated with \(\pi\)-phase shift for (\(a+b_{1}\)) electronic coherence as well as (\(a+b_{2}\)) electronic coherence. Similarly, for out-of-phase electronic coherences (\(b_{1}-b_{2}\)), (\(a-b_{1}\)), and (\(a-b_{2}\)), each angular momentum can be expressed by that for the corresponding in-phase coherence with \(\pi\)-phase shift. The time-dependent angular momenta in the \(Z\) and \(X\) directions are, respectively, expressed in terms of \(l_{L}(t)\) as \(l_{Z}(t)=2l_{L}(t)\sin(\Theta_{d}/2)\) and \(l_{X}(t)=2l_{L}(t)\cos(\Theta_{d}/2)\). \(l_{Z}(t)\) was calculated using the relation \(l_{L}(t)=l_{R}(t)\) and \(l_{X}(t)\) was calculated using \(l_{L}(t)=-l_{R}(t)\) as shown in Figure 10. The simple sinusoidal time-dependence with a damping originates from creation of two-state electronic coherence, the oscillation period of which corresponds to the frequency difference between the two states. The damping behaviors of the oscillation amplitudes are due to dephasing between the two excited states. Here, dephasing  
Figure 9: (**a**) Three electronic excited states and optical excitations to create three types of the electronic coherence of (\(P\))-2,2′-biphenol (**b**) The geometrical structure and the electronic transition moment vectors. Modified from [87] with permission of the American Institute of Physics.  
constants \(\gamma_{ab1}=\gamma_{ab2}=\gamma_{b1b2}=0.01\) eV were adopted [87]. The oscillations in the angular momenta should be called angular momentum quantum beats, similar to quantum beats originating from vibronic coherence in fluorescence [90]. It should be noted that in angular momentum quantum beats, \(\pi\) electrons rotate in a unidirectional way a few times within the half cycle of the oscillation. The unidirectional \(\pi\)-electron rotation can produce unidirectional ring current and current-induced magnetic flux. This is the basic principle to design ultrafast switching devices made from organic ring molecules.