# Theoretical Investigation of All Optical Switching by Intersystem Crossing
## 3 Discussion and Results  
We start out our calculation from Equation (11) to get the decay rate k needed in Equation (3). As Fe is the core element in the AOS scenario, our four-level system uses some parameters that refer to Fe. According to the typical displacement \(\Delta\)Q of a few tenths of Angstrom [32; 33; 42], the values of \(\lambda\) and \(|\Delta\)E\(|\) are taken as 0.2 eV. In classical description, \(\Delta\)E is interpreted as Gibbs free energy, \(\Delta\)G. If \(\Delta\)G \(<\) 0, the process involved is spontaneous and accompanied with energy releasing. In the case of spin flip, the energy of electrons at triplet states is smaller than that at singlet states, so ISC is an energy releasing process, in which phonons are released, so \(\Delta\)E = \(-0.2\) eV. For SOC, we take a typical atomic Fe value [32], V = 0.05 eV, \(\hbar(\omega)\) = 0.03 eV. Evaluating Equation (11) with all parameters above, we get Figure 2a which shows the relation between temperature and the ISC rate. The initial temperature starts at room temperature, 300 K. The ISC rate dramatically declines with temperature increasing from 300 K to 600 K.  
Because of the sensitivity of the ISC rate to the temperature, the temperature fluctuation heated by the incident light influences the ISC rate. Another well-developed theory, the two-temperature model, successfully describes ultrafast heating process. The details of the two-temperature model, where the 'two-temperature' refers to the temperature of electrons and lattices, are discussed by [PERSON], etc. [43]. The lattice temperature is calculated using this model, and inserted into Equation (11) to represent the temperature of the sample. The two-temperature model is used to reflect the temperature fluctuation  
Figure 2: (**a**) The intersystem crossing (ISC) rate fluctuates with temperature; (**b**) The lattice temperature fluctuation over time; the electron temperature over time is in the box at right. and we ignore the influence of the effect on magnetization. Even though we do not present the details of the two-temperature model, but equations and values can be found in Reference [6]. Taking the time \(T_{r}\) when the electric field of the incident pulse gets its maximum as reference, our simulation starts 200 fs earlier than \(T_{r}\). After substituting the electric field amplitude in Equation (5) and the pump intensity \(P\) in two-temperature model into Equation (3) by \(P\propto E^{2}\), we then plot the temperature fluctuation over time in Figure 2b. Figure 2b shows the electron temperature increases dramatically at the beginning, while the lattice temperature lags much later, that it cannot reach 600 K within the first 0.5 ps. The temperature in Equation (11) corresponds to the lattice temperature because the ISC rate is associated with vibration of nucleus. Constrained by the pulse width of the incident laser, the Rabi oscillation and ISC mainly occur within 0.5 ps. Hence, the ISC rate keeps decreasing within the first 0.5 ps. In the simulation, we assume that all electrons stay at ground state initially. The amplitude of electron field \(E_{0}\) equals to \(10^{9}\) V/m. Since we have the ISC rate, \(W(t)\) already, we then can calculate \(c(t)\) and \(P(t)\). The probability of electrons staying at \({}^{0}\)S, P(\({}^{0}\)S), shown as the blue curve in Figure 3a, declines from 1 dramatically and starts oscillating. This ultrafast laser pulse induced oscillation lasts about 300 fs, during which some electrons secede to triplet state. After the oscillation, there are still electrons staying at \({}^{1}\)S with the probability P(\({}^{1}\)S) equals 0.2. The remaining electrons decay to \({}^{1}\)T corresponding to the smooth decline of the orange curve, because the decay from \({}^{1}\)S to \({}^{0}\)S is neglected, comparing with the two orders lager decay rate. We plot the probability that electrons decay to the triplet state in blue, and the probability of the final spin state in orange in Figure 3b.