# Theoretical Investigation of All Optical Switching by Intersystem Crossing
## 3 Discussion and Results
The probability of electrons transferring to \({}^{1}\)T, P(\({}^{1}\)T), can be presented as \(\text{P}(^{1}\text{T})=1-[\text{P}(^{0}\text{S})+\text{P}(^{1}\text{S})]\), which is the equivalence of the probability of electrons at the second two-level state. Electrons decaying to triplet state will be trapped even for a few seconds, sothe transition from \({}^{1}\)T back to \({}^{1}\)S in our system is assumed to be zero and the direct transition between \({}^{0}\)S and G is forbidden. The final spin state is determined by the difference between two sub-systems. Dynamics of electrons in the second two-level system also can be described by Rabi  
Figure 3: **(a)** Blue and orange lines represent probabilities of electrons staying at ground state \({}^{0}\)S and singlet state \({}^{1}\)S, respectively, The Rabi oscillation with decaying to triplet state. The final probability of \({}^{0}\)S steady at 0.4; **(b)** Probabilities of electrons decaying to \({}^{1}\)T and the dynamic of spin state are presented in blue and orange lines. model, but the oscillations do not affect the final state of spins. The final spin state, \(\rm P_{m}\) can be calculated by Equation (12). \[ \begin{array}{ll}\rm P_{m}&=P\!\left({}^{0}S\right)+P\!\left({}^{1}S\right)-P \!\left({}^{1}T\right)\\ &=2\!\left[P\!\left({}^{0}S\right)+P\!\left({}^{1}S\right)\right]-1\end{array}  \tag{12}\]  
For about 300 fs, spin states of the system get the balance between spin up and down. The time scale of the simulation is approximately consistent with the sub-lattice of Fe [14]. Then, the orientation of spins reverses slowly from the initial state. The electron transitions cease at about 1 ps, and the normalized difference between the spin up and down reaches \(-0.2\) from 1 which agrees with experimental results [44; 45]. Now we focus on the impact of amplitude of the electric field \(\rm E_{0}\) of the ultrafast laser pulse on final system states. The impact is shown in Figure 4: the blue curve is the final probability of electrons at the first two-level system, the orange one is the final spin state of whole system over \(\rm E_{0}\). The most impressive element in Figure 4 is the oscillating pattern of blue and orange curves. In the Rabi model, when laser pulse areas equal to \(2\pi\) and the pulses traverse samples without photon absorption during oscillations, the self-induced transparency is achieved and electrons oscillate back to initial state, \({}^{0}\)S. Illustrated in Figure 4, when \(\rm P\!\left({}^{0}S\right)+P\!\left({}^{1}S\right)\) gets its peaks, the first two-level system appears a self-induced transparency. On the contrary, when laser pulses excite electrons to their excited state with an abrupt ending, most electrons at \({}^{1}\)S decay to triplet state, which realizes the near 100% spin flip. The dynamics of the spin state, shown as the orange curve in Figure 4, undergo two phases: a dramatic decrease before \(\rm E_{0}\) reaches \(E_{0}=0.25\times 10^{9}\) V/m followed by oscillations. The first phase corresponds to the situation that the laser with low intensity excites electrons to \({}^{1}\)S; at this time, the ISC rate is faster than the speed of the excited radiation, which in turn lowers the probability from 1 with the increasing intensity of the laser. During the oscillating phase, the final spin state oscillates with up-shifting valleys because of the temperature. As shown in Figure 2, the high intensity of the laser apparently causes an increase in temperature and the high temperature declines the decay rate of the transition from \({}^{1}\)S to \({}^{1}\)T.  
Some non-trivial issues need to be addressed. Firstly, the intensity of the laser causes AOS to range within a relatively narrow window\({}^{3}\), which is similar to the results shown in Figure 4 before \(\rm E_{0}\) reaches \(E_{0}=0.5\times 10^{9}\) V/m. However, the periodic appearances of the window have not been confirmed by other studies. This result is induced by the ultra-high intensity of laser which cannot be treated as perturbation in the Hamiltonian.