## 3 Discussion
### Spin State Transition  
\(\omega_{0}\) is the intrinsic property of materials, but \(\Delta\omega\) is determined by the intensity of laser. By controlling \(\Delta\omega\) and the difference between the center frequencies of femtosecond laser \(\omega\) and \(\omega_{0}\), the distributions of final spin states are various. Here, Figure 3a shows a specific relation between \(\omega\), \(\omega_{0}\) and \(\Delta\omega\) (\(\omega_{12}=\omega_{0}-\Delta\omega\), \(\omega_{23}=\omega_{0}+\Delta\omega\)) and the electric field of laser, E\({}_{12}\) = 2E\({}_{23}\), where \(\omega_{12}\) equals to the center frequency of the incident laser. We ignore the difference of electric dipoles d\({}_{12}\) = d\({}_{23}\), because \(\Delta\omega\) is much less than \(\omega_{0}\); decay rates \(\gamma_{1}=\gamma_{3}=0.01\), which can be neglected during the interaction between laser and materials. The pick intensity of electric field, E\({}_{12}\), is \(\sim\)10\({}^{9}\) V/m. The pulse width \(\tau\) of the simulation in Figure 3 is 100 fs and the peak of the laser is reached at 2\(\tau\). In this model, the effective field induced by the [PERSON] effect breaks the transition forbidden with spin-flip, but the influence to the transition between 12\(>\) and 13\(>\) is not discussed. Instead, the magnitude of electric filed directly determines the speed of the transition. Figure 3a shows the relationship between the incident laser and the energy gap of \(\omega_{12}\) and \(\omega_{23}\). The full line describes the distribution of frequency versus the electric field. The intersections of vertical imaginary lines and the profile of the laser are corresponding to the magnitude of electric field for the interaction of laser and material. In Figure 3b, the intensity of electric dipole transition, \(\omega_{12}\) and \(\omega_{23}\) are 1.4 eV and 0.7 eV, respectively. \(\rho_{1}\) and \(\rho_{3}\) are the probabilities of spin state at 1\(>\) and 13\(>\), respectively; initially, \(\rho_{1}=\rho_{3}=0.5\). The results in Figure 3b can be qualitatively explained by the fact that more electrons at state 1\(>\) are excited by higher intensity of laser than those at state  
Figure 2: Two-temperature model for electrons and lattice. It can be seen that the temperature of electrons dramatically rises to T\({}_{\rm c}\) of 500 K from room temperature of 300K within 150 fs. \(\mid\)3\(>\), which maintains the population density at state 12\(>\), while more electrons stimulated radiate to state 13\(>\). The separate dynamics of the three states are ignored, but the process of magnetization is shown in Figure 3b, because the difference of the probability of spin-up and -down, \(\Delta\rho=\rho_{3}-\rho_{1}\), reflects the magnetization of the sample. The final state of \(\Delta\rho\) is steady at about 0.11, which indicates a determinate all-optical magnetic switching. Because of the equality of \(\gamma_{1}\) and \(\gamma_{3}\) which are assigned previously, electrons decay to 1\(>\) and 13\(>\) with the same speed. \(\Delta\rho\) keeps constant after about 300 fs when there are electrons that remain staying at the virtual state. However, the decay rate should be considered with the surrounding areas. The direction of the magnetization of material after the interaction of laser affects the tendency to decaying to spin-up or -down. \(\Delta\rho\) should be greater than the result displayed in Figure 3b, but we still assume that \(\gamma_{1}\) equals to \(\gamma_{3}\) in the following simulations, which does not significantly affect the final probability \(\Delta\rho\). We further investigate the effect of the variation of the pulse width and the intensity of laser to the final state of \(\Delta\rho\). In Figure 4, the intensity variation is reflected by \(\omega_{12}\). The summation of \(\omega_{12}\) and \(\omega_{23}\) is constant to be 2.1 eV and \(\omega_{12}\) varies from 1.05 to 1.6 eV. \(\tau\) represents the pulse width of laser; the peak of the intensity of laser is reached at 2\(\tau\); the simulation time is set as 6\(\tau\).