## 3 Coherent \(\pi\)-Electron Angular Momentum and Current of a Molecule Having Two Aromatic Rings
### Quantum Localization of Coherent \(\pi\)-Electron Angular Momentum
#### 3.3.1 Optimal Control Theory  
Localization of \(\pi\)-electron angular momentum can be intuitively understood from the coherent character of the four patterns of coherent \(\pi\)-electron rotations as shown in Figure 14. Here, each pattern is generated by rotations of \(\pi\) electrons localized at the two benzene rings with a definite rotational phase. We now outline optimal control procedure [92; 93; 94]. Quantum dynamics of \(\pi\) electrons induced by the electric field of a laser pulse \(\mathbf{E}(t)\) is described by the Hamiltonian:  
\[\hat{H}(t)=\hat{H}_{0}-\hat{\mathbf{u}}\cdot\mathbf{E}(t) \tag{46}\]  
Here, \(\hat{H}_{0}\) is the \(\pi\)-electronic Hamiltonian of (\(P\))-2,2\({}^{\prime}\)-biphenol in the free field. For simplicity, effects of molecular vibrations on electron dynamics are omitted. Let us take an objective functional defined as:  
\[J[\mathbf{E}]=\
u\Big{\langle}\Psi\Big{(}t_{f}\Big{)}\big{|}\hat{O}\big{|}\Psi \Big{(}t_{f}\Big{)}\Big{\rangle}+\frac{1-\
u}{t_{f}}\int_{0}^{t_{f}}dt\big{\langle} \Psi(t)\big{|}\hat{O}\big{|}\Psi(t)\big{\rangle}-\int_{0}^{t_{f}}dt\frac{ \mathbf{E}(t)^{2}}{\hbar\alpha(t)}. \tag{47}\]  
where, \(\hat{O}\) is a projection operator for the target state of interest, \(t_{f}\) is the final (control) time, \(\alpha(t)\) is a time-dependent penalty factor to suppress the intensity of the optimal field, and \(\
u\) (= 0 or 1) is a parameter to switch the objective functionals. For \(\
u\) = 1, the target population is maximized at the final time, and on the other hand, for \(\
u\) = 0 the time-integrated population of the target state is maximized. Equation (47) is to be maximized with a constraint of the time-dependent Schrodinger equation:  
\[i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\hat{H}_{0}|\Psi(t)\rangle, \tag{48}\]  
with the initial condition \(|\Psi(0)\rangle=|G\rangle\). Here, \(|G\rangle\) denotes the ground state. After taking the variation in Equation (47), we obtain the optimal electric field as:  
\[\mathbf{E}(t)=-\alpha(t)\text{Im}\langle\xi(t)|\hat{\mathbf{n}}|\Psi(t)\rangle, \tag{49}\]  
Here, \(\xi(t)\) is time-dependent Lagrange multiplier, which satisfies a time-dependent equation:  
\[\bigg{[}i\hbar\frac{\partial}{\partial t}-\hat{H}(t)\bigg{]}|\xi(t)\rangle=( \
u-1)\frac{i\hbar}{t_{f}}\hat{O}|\Psi(t)\rangle, \tag{50}\]with the final condition \(\left|\xi\left(t_{f}\right)\right\rangle=\upsilon\hat{O}\left|\Psi\left(t_{f} \right)\right\rangle\).  
In order to carry out quantum control simulation for localization of the \(\pi\)-electron angular momentum of (\(P\))-2,2\({}^{\prime}\)-biphenol, we need a set of angular momentum eigenstates defined in each aromatic ring. The localized angular momentum eigenstates (\(W_{K}^{A}\) and \(W_{K}^{C}\) with \(K=L\), \(R\)) can be expressed in terms of the three excited states (\(\Phi_{a}\), \(\Phi_{b1}\), \(\Phi_{b2}\)) as [91]:  
\[\left(\begin{array}{c}W_{L}^{A}\\ W_{L}^{C}\end{array}\right)=\left(\begin{array}{ccc}0.56&0.71i&-0.43\\ 0.56&-0.71i&-0.43\end{array}\right)\left(\begin{array}{c}\Phi_{a}\\ \Phi_{b1}\\ \Phi_{b2}\end{array}\right), \tag{51a}\]  
\[\left(\begin{array}{c}W_{R}^{A}\\ W_{R}^{C}\end{array}\right)=\left(\begin{array}{ccc}-0.56&0.71i&-0.43\\ -0.56&-0.71i&-0.43\end{array}\right)\left(\begin{array}{c}\Phi_{a}\\ \Phi_{b1}\\ \Phi_{b2}\end{array}\right). \tag{51b}\]  
The transformation matrices in the above equations were calculated by using the electronic angular momentum operator. The target states for the optimal control are, for example, expressed as follows: \(\hat{O}=\left|W_{L}^{A}\right\rangle\left\langle W_{L}^{A}\right|\) for the [AO] localization and \(\hat{O}=\left|W_{R}^{A}\right\rangle\left\langle W_{R}^{A}\right|\) for the [OA] localization. Here, O means no rotation on the \(L\) or \(R\) ring.