# Theoretical Investigation of All Optical Switching by Intersystem Crossing
## 2 Dynamic Description of Electron in Laser Field  
We construct our Hamiltonian for AOS as:  
\[H=H_{0}+H_{\text{int}}+H_{\text{SOC}} \tag{1}\]  
The right terms in Equation (1) represent an unperturbed Hamiltonian, interaction between laser and material and SOC. In our model, laser interaction and SOC are considered as perturbations that induce electron transitions. H\({}_{\text{int}}\) describes electron oscillation between ground and excited state; H\({}_{\text{SOC}}\) describes electron transitions from singlet state to triplet state, but in our paper, SOC is associated with vibration. The time-dependent Schrodinger equation with the Hamiltonian in Equation (2) has the following solution:  
\[\Psi(t)=\sum_{n}^{N}c_{n}(t)\cdot\Psi(t) \tag{2}\]  
where \(c_{n}(t)\) describes electron transitions between N eigenstates. The probability of finding an electron on state n at time t is denoted as \(P_{n}(t)\), which can be calculated by \(P_{n}(t)=\abs{c_{n}(t)}^{2}\). The possibility that the final state will take can then be calculated by \(P_{n}(\infty)\). In our study, the state vector is a 4-dimensional one; state vectors \(\Psi_{n}\) represent singlet ground state \({}^{0}\)S, singlet excited state \({}^{1}\)S, triplet state \({}^{1}\)T and ground state G, respectively. As shown in Figure 1, we decompose the spin flip process into three stages as excitation, ISC, and decay. Firstly, electrons are excited from \({}^{0}\)S to \({}^{1}\)S and oscillate between \({}^{0}\)S and \({}^{1}\)S; then some electrons in a singlet state \({}^{1}\)S transfer to triplet state \({}^{1}\)T, obeying the ISC rate, which is mainly determined by SOC and the vibration of nucleus; finally, electrons in \({}^{1}\)T state decay to state G but maintaining the same spin state as \({}^{1}\)T.  
These four states are combined into two systems. The first two-level system is constituted by \({}^{0}\)S and \({}^{1}\)S; the second two-level system is constituted by \({}^{1}\)T and G. The spin orientation in these two systems are opposite of each other. ISC links these two separated systems by breaking the selection rule with a non-radiative radiation. Therefore, the difference of probabilities between electrons being on two systems determines the final orientation of spins. Rabi model [34] corresponds to H\({}_{\rm int}\) that describes electron oscillation. The dynamics of ISC can be incorporated into the framework of the Rabi model by treating it as a decay rate k. We use Equation (3) to describe the dynamic process electrons experiencing in AOS. The first term on the right hand side describes electron oscillation between \({}^{0}\)S and \({}^{1}\)S, which has a fundamental impact on the decay speed from \({}^{1}\)S to \({}^{1}\)T. The second term describes the effect of ISC. \[\frac{d}{dt}\left[ \begin{array}{c}c_{0}(t)\\ c_{1}(t)\end{array} \right]=-iW(t)\left[ \begin{array}{c}c_{0}(t)\\ c_{1}(t)\end{array} \right]+k\left[ \begin{array}{c}0\\ c_{1}(t)\end{array} \right] \tag{3}\]  
\(c_{0}(t)\) and \(c_{1}(t)\) in Equation (3) have the same meaning as \(c_{n}(t)\); \(W(t)\) is a \(2\times 2\) matrix of the Hamiltonian. Its off-diagonal elements describe the perturbation responsible for the change of the probability of electrons, and its diagonal elements are detuning elements that determine the coupling efficiency between energy bands and the incident laser. In Equation (4), \(\Delta(t)\) dominated by the energy difference between the incident laser pulse and the energy gap, determines the amplitude of the Rabi oscillation. For clearly presenting electron transition between \({}^{0}\)S and \({}^{1}\)S, \({}^{0}\)S in Figure 1 is not pictured as a vibrational state. The vibration of both \({}^{0}\)S and \({}^{1}\)S may induce the mismatch between laser and energy gap between \({}^{0}\)S and \({}^{1}\)S, but we assume that \(\Delta(t)\) is detuned to zero at the very beginning of our tentative calculation. \[W(t)=\left[ \begin{array}{cc}0&\frac{\Omega}{2}\\ \frac{\Omega}{2}&\Delta(t)\end{array}