# Theoretical Investigation of All Optical Switching by Intersystem Crossing
## 2 Dynamic Description of Electron in Laser Field
\right] \tag{4}\]  
\(\Omega\) is the scaled projection of the electric dipole moment \(d\) onto the electric field \(E(t)\), as shown in Equation (5). Figure 1: A schematic four-level system for our model. Electrons non-radioactively decay from singlet state \({}^{1}\)S to triplet state \({}^{1}\)T. The displacement \(\Delta\)Q corresponds to different equilibrium geometries in \({}^{1}\)S and \({}^{1}\)T. \(\Delta\)E is the energy gap between the lowest vibrational levels and \(\lambda\) is the electron-phonon self-energy difference. These two parameters are associated with the Franck-Condon factor. In our study, the incident laser pulse is chosen to have a Gaussian envelope with 100 fs pulse width. Comparing with the frequency of the incident laser and the Rabi oscillation, the rotating wave approximation [35] is valid when the intensity of the laser is not high. Therefore, \(E(t)\) in Equation (5) has the envelope of the Gaussian distribution. \(\hbar\) is the reduced Planck constant. The ISC rate can be described by [PERSON]'s rules [36]. In his work, [PERSON] highlighted the importance of SOC and predicted that the transition from \(1(\pi\pi^{\alpha})\) to \((\pi\pi^{\alpha})\) would be faster than \(1(\pi\pi^{\alpha})\) to \(3(\pi\pi^{\alpha})\), as the latter involves the same orbitals. The electron transition rate is calculated by [PERSON]'s golden rule. If we consider SOC with adiabatic states (following [PERSON]'s rule), the ISC rate from \({}^{1}\)S to \({}^{1}\)T can be written as:  
\[k=\frac{2\pi}{\hbar}|<T|H_{SO}|S>|^{2}\delta(E_{1_{S}}-E_{1_{T}}) \tag{6}\]  
S and T are the singlet and triplet states; H\({}_{SO}\) is the Hamiltonian for SOC; \(E_{1_{S}}\) and \(E_{1_{T}}\) are the energy of \({}^{1}\)S and \({}^{1}\)T. However, if vibrations of states are considered, [PERSON]'s rules will be broken and the speed of ISC will be accelerated. ISC can be rewritten by considering a thermal contribution and assuming that SOC has no dependency on vibration [37]:  
\[\mathrm{k=}\frac{2\pi}{\hbar^{2}}|<T|H_{SO}|S>|^{2}\sum_{j,k}e^{-E_{Sj}/k_{B}T }|<v_{Tk}|v_{Sj}>|^{2}\delta(E_{S}-E_{T})\] \[Z=\sum_{j}e^{-E_{j}/k_{B}T} \tag{7}\]  
\(\mathrm{k_{B}}\) is Boltzmann constant; T is temperature; \(j\), k represents vibrational states; \(v_{Tk}\) and \(v_{Sj}\) are vibration states; Z is a canonical partition function for vibrational motion in the singlet state. Comparing Equation (7) with Equation (6), except the thermal related term Z, we can find that vibrational terms contribute to the ISC rate, which ensures ISC within the same orbit. Hence, this formulation is consistent with experimental results that spin angular momentum transfers to lattice but not orbit. Equation (8) can be simplified to the following formulation [38]:  
\[\mathrm{k=}\frac{2\pi}{\hbar}\mathrm{V}^{2}\mathrm{F} \tag{8}\]  
V is the intensity of interaction between levels where spin-obit coupling drives the transition from singlet state to triplet state. To figure out the contribution of vibration to ISC, SOC is treated as a constant V. F is the Franck-Condon (FC) integrals corresponding to the vibration part in Equation (8). [PERSON] theory [39] can be safely applied to present FC integrals [40]:  
\[ \begin{array}{ll}\mathrm{F}&=\exp[-\mathrm{S_{s}(2\overline{v}_{s}+1)-S(2 \overline{v}+1)}]\\ &\times\sum_{m=-\infty}^{\infty}[(\overline{v}_{s}+1)/\overline{v}_{s}]^{P(m) /2}\mathrm{I_{P(m)}}[2\mathrm{S_{s}[\overline{v}_{s}(\overline{v}_{s}+1)]}^{1 /2}\Big{\}}\\ &\times[(\overline{v}+1)/\overline{v}]^{m/2}\mathrm{I_{m}}\Big{[}2S[\overline {v}(\overline{v}+1)]^{1/2}\Big{)}/\hbar\langle\omega_{s}\rangle\end{array}