## 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
#### 2.2.2 Three-Level Model Analysis of Optical Excitation  
We formulate the polarization effects on the optical excitation in aromatic molecules of low symmetry. The time-dependent Schrodinger equation for a molecule in a laser field \(\mathbf{E}(t)\) is given in the length gauge by:  
\[i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\big{[}\hat{P}_{0}-\mathbf{ \hat{u}}\cdot\mathbf{E}(t)\big{]}|\Psi(t)\rangle. \tag{7}\]  
The system is initially in its ground state \(|G\rangle\). We assume that only the ground and quasi-degenerate excited states are involved in the excitation process. The energy of \(|G\rangle\) is defined to be zero and  
Figure 1: Orientation of the polarization ellipse. The laser field propagates in the \(z\) direction and oscillates in the \(xy\) plane. The orientation angle of the major axis of the polarization ellipse with respect to the \(x\) axis is represented by \(\delta\). The minor-to-major axial ratio is \(\mid\)tan \((\pi/4\,-\,\beta)\mid\). Modified from [56] with permission of the American Chemical Society. the central frequency \(\omega\) of the laser field is set on resonance with the average energy of \(|X\rangle\) and \(|Y\rangle\): \(\omega=\omega_{X}+\Delta\omega=\omega_{Y}-\Delta\omega\). The state vector \(|\Psi(t)\rangle\) is then expanded as:  
\[|\Psi(t)\rangle=c_{G}(t)|G\rangle+c_{X}(t)e^{-i(\omega-\Delta\omega)t}|X\rangle +c_{Y}(t)e^{-i(\omega+\Delta\omega)t}|Y\rangle. \tag{8}\]  
All diagonal elements of \(\hat{\mathbf{u}}\) are assumed to be zero for simplicity, although this assumption is not necessary for our model analysis to be valid. For off-diagonal elements, we denote \(\mathbf{u}_{n}\equiv\langle G|\hat{\mathbf{u}}|n\rangle=\langle n|\hat{ \mathbf{u}}|G\rangle\
eq 0\) (\(n=X\) and \(Y\)), while \(\langle X|\hat{\mathbf{u}}|Y\rangle=\langle Y|\hat{\mathbf{u}}|X\rangle=0\). As defined in Section 2.1, the \(z\) axis is perpendicular to the ring plane, which coincides with the \(xy\) plane. Because the excitations to \(|X\rangle\) and \(|Y\rangle\) correspond to \(\pi\pi^{*}\) transitions, both \(\mathbf{\mu}_{X}\) and \(\mathbf{\mu}_{Y}\) lie in the \(xy\) plane. Substituting Equations (4) and (8) into Equation (7) yields the equation of motion for the expansion coefficient vector \(\mathbf{C}(t)\equiv\left(c_{G}(t),c_{X}(t),c_{Y}(t)\right)^{\mathrm{T}}\):  
\[\frac{d\mathbf{C}(t)}{dt}=i\left( \begin{array}{ccc}0&g_{X}(t)&g_{Y}(t)\\ g_{X}(t)^{*}&0&0\\ g_{Y}(t)^{*}&0&0\end{array} \right)\mathbf{C}(t), \tag{9}\]  
where  
\[g_{X}(t)=\frac{f(t)}{2}\Big{\{}\Omega_{X}e^{-i[(2\omega-\Delta\omega)t+\varphi ]}+\Omega_{X}^{*}e^{i(\Delta\omega t+\varphi)}\Big{\}}, \tag{10a}\] \[g_{Y}(t)=\frac{f(t)}{2}\Big{\{}\Omega_{Y}e^{-i[(2\omega+\Delta\omega)t+\varphi ]}+\Omega_{Y}^{*}e^{-i(\Delta\omega t-\varphi)}\Big{\}}, \tag{10b}\]  
and \(\Omega_{n}\equiv\left(E^{0}/h\right)\mathbf{\mu}_{n}\cdot\mathbf{e}\) (\(n=X\) and \(Y\)) are the complex Rabi frequencies. The initial condition is \(\mathbf{C}(0)=(1,0,0)^{\mathrm{T}}\). To simplify Equation (10a,b), we apply the rotating-wave approximation (RWA) [59]: Near resonance (i.e., \(\omega>>|\Delta\omega|\)) is assumed so that the contribution of the first terms in Equation (10a,b), which are rapidly oscillating exponentials, to the time evolution of \(\mathbf{C}(t)\) is averaged out and removed. Under the RWA, Equation (9) is transformed into:  
\[\frac{d\mathbf{C}(t)}{dt}=i\frac{f(t)}{2}\left( \begin{array}{ccc}0&\Omega_{ X}^{*}e^{i(\Delta\omega t+\varphi)}&\Omega_{Y}^{*}e^{-i(\Delta\omega t-\varphi)} \\ \Omega_{X}e^{-i(\Delta\omega t+\varphi)}&0&0\\ \Omega_{Y}e^{i(\Delta\omega t-\varphi)}&0&0\end{array} \right)\mathbf{C}(t). \tag{11}\]  
In order to efficiently prepare an approximate angular momentum eigenstate \(|+\rangle\) or \(|-\rangle\) defined in Equation (2), the quasi-degenerate states \(|X\rangle\) and \(|Y\rangle\) should be populated equally. The laser field is therefore assumed to satisfy \(|\Omega_{X}|=|\Omega_{Y}|\) or, at least, \(|\Omega_{X}|\simeq|\Omega_{Y}|\).