## 4 Discussion  
The observed carrier-density-dependent ultrafast spin dynamics can be explained more qualitatively as follows. In the low-doped Eu\({}_{1-x}\)Gd\({}_{x}\)O the carrier thermalization time is much longer than the one in dense itinerant ferromagnets like Ni or Gd. Charge thermalization is slow because it occurs by electron-electron scattering, the rate of which is proportional to carrier concentration. In low-doped Eu\({}_{1-x}\)Gd\({}_{x}\)O, the carrier concentration is by a factor of about 100 lower than that in pure Gd. The carrier thermalization time in pure Gd equals 50-100 fs [56]. With this, the factor of 100 leads to a prolonged carrier thermalization time of low-doped Eu\({}_{1-x}\)Gd\({}_{x}\)O of 5-10 ps. It is therefore concluded that the dynamics for the first few ps is dominated by a non-equilibrium charge carrier distribution, that is, the three-temperature model (3 TM) [57], where the coupled electron, spin and lattice subsystems are each assumed to be in equilibrium at their respective temperatures, is not applicable. Hence, for the first few ps after the photoexcitation, the 4\(f\) electron population, which is photoexcited into the state with 5\(d\) electron character, contributes to a stronger magnetic coupling between the Eu 4\(f\) spins. This coupling mechanism is in agreement with the experimental observations on the low-doped samples. On the other hand, as shown in Figure 7b, it does not explain the photoinduced decrease of the MSHG intensity in the region where the carrier density is sufficiently high (\(n_{\rm c}\sim 1\times 10^{21}\) cm\({}^{-3}\)). In the picture of the 3 TM, the optical pump pulse would, in addition to exciting the localized 4\(f\) electrons and creating the 4\(f^{6}5d\)(\(t_{2g}\)) magnetic excitons, lead to the excitation of the chemically doped carriers in the 5\(d\)/6\(s\) conduction band and to their thermalization at the elevated temperature corresponding to the deposited energy. This would invariably reduce \(J_{\rm ex}\) since the ferromagnetic coupling with electrons in the energetically higher band states becomes weaker because of their  
Figure 7: Optical tuning of the exchange interaction \(J_{\rm ex}\) in carrier-density-controlled ferromagnetic semiconductor Eu\({}_{1-x}\)Gd\({}_{x}\)O. (**a**) Relation between \(n_{\rm c}\) and the photoinduced change of the MSHG intensity at \(\Delta t=3\) ps and at 10 K, extracted from the experiments in Figure 6. (**b**) Calculated result of \(\Delta J_{\rm ex}\)/\(J_{\rm ex}\) by adding a photoinduced carrier density \(\Delta n_{\rm c}=1\times 10^{19}\) cm\({}^{-3}\) to the Gd-dependent carrier density \(n_{\rm c}\) of the conduction band. Here, \(\Delta J_{\rm ex}\)/\(J_{\rm ex}\) was calculated using the curve in Figure 3b, assuming that \(J_{\rm ex}\) is simply proportional to \(T_{\rm C}\) (\(\Delta J_{\rm ex}\)/\(J_{\rm ex}\) = \(\Delta T_{\rm C}\)/\(T_{\rm C}\) ). (**c**) Theoretical calculation of \(\Delta J_{\rm ex}\)/\(J_{\rm ex}\) based on the modification of the RKKY interaction by the photoinduced non-equilibrium population of the 5\(d\)/6\(s\) conduction band. increased wavenumber. Furthermore, the increased temperature would directly contribute to the demagnetization. As shown in Ref. [52], an increase in the carrier density in the conduction band by chemical or photo-doping strengthens its interaction with the Gd impurity band. This entails an energetic downshift of the conduction band toward the Gd impurity band and, hence, a reduction of the \(4f\)-\(5d/6s\) gap energy. This shift affects the \(4f\)-\(4f\) RKKY exchange. Using the many-body renormalization theory introduced in Ref. [52], the RKKY-like coupling for the non-equilibrium electron distribution was calculated. The resultant \(\Delta f_{\mathrm{ex}}/f_{\mathrm{ex}}\) is plotted in Figure 7c as a function of the carrier density \(n_{\mathrm{c}}\). According to the theory, the non-monotonic behaviors in Figure 7a,c can be understood as follows.