## 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
Once \(\mu_{X}\) and \(\mu_{Y}\) are given, the values of \(\delta\) for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) can be readily evaluated from Equation (31). It is straightforward to find a pair of \(\beta\) and \(\delta\) for the other values of \(\theta\) from Equation (12). Circular polarizations do not exactly satisfy Equation (12) unless the magnitudes of \(\mu_{X}\) and \(\mu_{Y}\) happen to be equal. If \(\Delta\omega\simeq 0\), the relationship among \(\beta\), \(\delta\), and \(\theta\) should be close to that for degenerate systems; for example, \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) should be almost perpendicular to each other, and we approximately have \(\pm i\Omega_{X}\simeq\Omega_{Y}\) (\(\theta\simeq\pm\pi/2\)) for \(\mathbf{e}_{\pm 1}\). The time-dependent part of the relative quantum phase, i.e., \(\vartheta(t)\) obeys the following from its definition:  
\[\tan\left[\frac{\vartheta(t)}{2}\right]=\frac{2\Delta\omega}{\Omega}\tan\left( \frac{\Omega}{4}t\right) \tag{32}\]  
when \(\Delta\omega>0\), \(\vartheta(t)\) grows as \(0\rightarrow\pi/2\rightarrow\pi\to 3\pi/2\to 2\pi\rightarrow\dots\) with the progression of time, \(0\to T_{\mathrm{R}}\)\(-\)\(\zeta\to T_{\mathrm{R}}\to T_{\mathrm{R}}+\zeta\to 2T_{\mathrm{R}}\rightarrow\dots\), where \(T_{\mathrm{R}}\equiv 2\pi/\Omega\) and \(\zeta\equiv 4\mathrm{arctan}(2\,|\,\Delta\omega\,|\,/\Omega)/\Omega\); when \(\Delta\omega<0\), \(\vartheta(t)\) evolves in the negative direction. Since \(2\,|\,\Delta\omega\,|\,\leq\Omega\), \(T_{\mathrm{R}}\) is not larger than the period \(T\) of the field-free electronic state change and \(\zeta\leq T_{\mathrm{R}}/2\). The relative quantum phase changes from its initial value \(\theta\) in the negative direction by \(\vartheta(t)\); if the full-excitation pulse is employed, \(\,|\,\vartheta(t_{f})\,|\,\leq\pi\) from the inequality \(T_{\mathrm{R}}/2\leq 2\mathrm{arccos}\left[-\left(2\Delta\omega/\overline{ \Omega}\right)^{2}\right]/\Omega\leq T_{\mathrm{R}}\). The populations \(P_{\pm}(t)\) and the expectation value \(l_{z}(t)\) vary in time according to the relative quantum phase \(\theta\,-\,\vartheta(t)\) as shown in Equations (29) and (30), respectively. The electronic WP propagates freely after a laser pulse is turned off at \(t=t_{f}\). Similarly to Equation (3), the relative quantum phase in the free propagation is \(\theta\,-\,\vartheta(t_{f})\,-\,2\Delta\omega(t\,-\,t_{f})\) and thereby the coherent superposition of the quasi-degenerate states oscillates with the period \(T\). The approximate angular momentum eigenstates \(|+\rangle\) and \(|-\rangle\) are alternately created as predicted in Section 2.1. As a consequence, the direction of ring currents switches between clockwise and anticlockwise in marked distinction from the degenerate case, in which the current direction is unchanged and \(l_{z}(t)\) is constant after the end of the laser pulse. It should be emphasized that the initial relative phase \(\theta\) of the superposed quasi-degenerate states (i.e., the initial direction of ring currents) can be controlled by the ellipticity angle \(\beta\) and orientation angle \(\delta\) of the applied laser. Generation of polarization-shaped femtosecond UV laser pulses has become experimentally realizable [63; 64; 65; 66]. The efficient scheme to produce circularly polarized attosecond XUV laser pulses has been theoretically proposed [67]. In Section 2.2.5, we will compare this model analysis in the rectangular-envelope case with numerical electronic WP simulations for pulse excitation with smooth rise and decay.