## 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
#### 2.2.4 Case of Quasi-Degenerate Excited States  
Now, we move on to the quasi-degenerate case of \(\Delta\omega\simeq 0\) for aromatic molecules of low symmetry. It is difficult to analytically integrate Equation (16) with an arbitrary envelope function \(f(t)\). For this reason, \(f(t)\) is restricted to a rectangular function: \(f(t)=1\) for [0, \(t_{f}\)] and otherwise zero. During irradiation we have:  
\[\frac{d\mathbf{D}(t)}{dt}=\frac{i}{2}\left( \begin{array}{ccc}0&\overline{ \Omega}&0\\ \overline{\Omega}&0&2\Delta\omega\\ 0&2\Delta\omega&0\end{array} \right)\mathbf{D}(t) \tag{25}\]and its solution is derived as:  
\[\mathbf{D}(t)=\left( \begin{array}{c}\left(\frac{\overline{\Omega}}{\Omega} \right)^{2}\cos\left(\frac{\Omega}{2}t\right)+\left(\frac{2\Lambda\omega}{ \overline{\Omega}}\right)^{2}\\ i\frac{\overline{\Omega}}{\Omega}\sin\left(\frac{\Omega}{2}t\right)\\ -\frac{\overline{\Omega}(2\Lambda\omega)}{\Omega^{2}}\left[1-\cos\left(\frac {\Omega}{2}t\right)\right]\end{array} \right). \tag{26}\]  
The generalized Rabi frequency \(\Omega\equiv\left[\overline{\Omega}^{2}+\left(2\Lambda\omega\right)^{2} \right]^{1/2}\) is the root mean square of the Rabi frequencies and the detuning frequency \(2\Lambda\omega\). The pulse area of a laser field for a quasi-degenerate system is defined as \(\Omega F(t)\), which is equal to \(\Omega t_{f}\) in the rectangular-envelope case. The ground state \(\left|\hat{G}\right\rangle\) can be fully emptied by a rectangular-envelope pulse that meets \(2|\Delta\omega|\leq\overline{\Omega}\) and has a pulse area of \(2\arccos\left[-\left(2\Lambda\omega/\overline{\Omega}\right)^{2}\right]\). The pulse area of this laser field (termed the full-excitation pulse in this article) is larger than that of the \(\pi\) pulse, that is, \(2\arccos\left[-\left(2\Lambda\omega/\overline{\Omega}\right)^{2}\right]>\pi\) (but not larger than \(2\pi\)); however, in fact, the full-excitation pulse is very close to the \(\pi\) pulse for \(\Delta\omega\simeq 0\). The electronic WP of a quasi-degenerate system is eventually written down as:  
\[|\Psi(t)\rangle=\left[\left(\frac{\overline{\Omega}}{\overline{\Omega}} \right)^{2}\cos\left(\frac{\Omega}{2}t\right)+\left(\frac{2\Lambda\omega}{ \overline{\Omega}}\right)^{2}\right]|G\rangle+i\frac{\Omega_{X}}{\Omega} \alpha(t)e^{-i(\omega t+\varphi)}\Big{\{}|X\rangle+e^{i|\theta-\theta(t)|}|Y \rangle\Big{\}}, \tag{27}\]  
where  
\[\alpha(t)\equiv 2\sin\left(\frac{\Omega}{4}t\right)\left[\cos\left(\frac{ \Omega}{4}t\right)+i\frac{2\Lambda\omega}{\overline{\Omega}}\sin\left(\frac{ \Omega}{4}t\right)\right]=2\sin\left(\frac{\Omega}{4}t\right)\overline{\pi}(t) \tag{28}\]  
and \(\vartheta(t)\equiv 2\arg\overline{\alpha}(t)\). The relative quantum phase between \(|X\rangle\) and \(|Y\rangle\) is \(\theta-\vartheta(t)\) and its temporal behavior is the main subject of discussion in Section 2.2. The populations \(P_{\pm}(t)\) are derived as:  
\[P_{\pm}(t)=\frac{1}{2}\left[\frac{\overline{\Omega}}{\overline{\Omega}}| \alpha(t)|\right]^{2}\left\{1\pm\sin[\theta-\vartheta(t)]\right\} \tag{29}\]  
and the angular-momentum expectation value \(l_{z}(t)\) is  
\[l_{z}(t)=\lambda\left[\frac{\overline{\Omega}}{\overline{\Omega}}|\alpha(t)| \right]^{2}\sin[\theta-\vartheta(t)], \tag{30}\]  
where \(\lambda\equiv\left\langle+\left|\hat{l}_{z}\right|+\right\rangle=-\left\langle -\left|\hat{l}_{z}\right|-\right\rangle\) (\(>0\)). In sharp contrast to Equation (21), the time-dependent phase \(\vartheta(t)\) may change the direction of ring currents during irradiation. For aromatic molecules of low symmetry, where \(|\mid\mu_{X}\mid\mid\
eq|\mid\mu_{Y}\mid\mid\) and \(\mu_{X}\cdot\mu_{Y}\
eq 0\) in general, Equations (22) and (23) do not strictly hold. In this case, the linear polarization vectors \(\mathbf{e}^{(+)}\) (\(\theta=0\)) and \(\mathbf{e}^{(-)}\) (\(\theta=\pi\)) are determined by [55; 57; 60; 61; 62]:  
\[\pm\mu_{X}\cdot\mathbf{e}^{(\pm)}=\mu_{Y}\cdot\mathbf{e}^{(\pm)}, \tag{31}\]  
which fulfills \(\pm\Omega_{X}=\Omega_{Y}\).