## 4 Discussion
The density of photoexcited carriers is proportional to the spectral density at the energy \(\hbar\omega_{\mathrm{pump}}-E_{\mathrm{gap}}\) addressed by the photoexcitation. Therefore, as \(E_{\mathrm{gap}}\) is reduced with increasing Gd doping and, thus, increasing \(n_{\mathrm{c}}\), more states can be populated with photoexcited carriers. The enhanced density of excited carriers contributes to the increase of \(\Delta f_{\mathrm{ex}}/f_{\mathrm{ex}}\) with \(n_{\mathrm{c}}\). In contrast, for \(n_{\mathrm{c}}>10^{20}\) cm\({}^{-3}\), \(\Delta f_{\mathrm{ex}}/f_{\mathrm{ex}}\) and \(\Delta I^{\mathrm{MSHG}}/I^{\mathrm{MSHG}}\) get quenched to negative values for two reasons. First, with further downward shift of the conduction band the energy mismatch \(\hbar\omega_{\mathrm{pump}}-E_{\mathrm{gap}}\), and with it the wavenumber of the photoexcited carriers, increases further. Thus, because of the oscillating dependence of the RKKY coupling on the wavenumber, \(f_{\mathrm{ex}}\) turns over towards weaker ferromagnetic coupling. Second, with the increasingly metallic nature, the electron-electron scattering time gets reduced, leading to ultrafast thermalization and concomitant magnetization quenching, that is, conventional demagnetization according to the 3 TM begins to dominate. This is nicely reflected by the change of the build-up time from \(\sim\)3 ps (manifestation of RKKY interaction) to \(\sim\)100 fs (thermalization according to the 3 TM) in the inset of Figure 6.  
Finally, I will show the continuous tuning of the exchange interaction is possible by changing the excitation density of the pump pulse. Figure 8a shows the excitation density dependence of the change of the MSHG intensity at \(\Delta t=3\) ps and at 10 K for selected Gd concentrations. Apart from the result for \(x=10.2\%\), where the decrease of \(\Delta I^{\mathrm{MSHG}}/I^{\mathrm{MSHG}}\) due to the ultrafast demagnetization is observed, the overall tendencies are roughly consistent with those of \(\Delta f_{\mathrm{ex}}/f_{\mathrm{ex}}\) plotted in Figure 8b as a function of the additional carrier density \(\Delta n_{\mathrm{c}}\). Here, the \(\Delta n_{\mathrm{c}}\) dependence of \(\Delta f_{\mathrm{ex}}/f_{\mathrm{ex}}\) was calculated by increasing the equilibrium carrier density \(n_{\mathrm{c}}\) of the \(5d/6s\) band for each Gd concentration. The largest enhancement of \(\Delta f_{\mathrm{ex}}/f_{\mathrm{ex}}\) is seen for \(x=0.25\%\) because \(f_{\mathrm{ex}}\) is very sensitive to the change of \(n_{\mathrm{c}}\) in this doping region. This result corroborates the above discussion and provides a possibility to control the exchange interaction on the sub-ps timescale in a well-controlled way. Figure 8: Continuous optical tuning of the exchange interaction \(J_{\mathrm{ex}}\) in ferromagnetic semiconductor Eu\({}_{1-x}\)Gd\({}_{x}\)O. (**a**) Excitation density dependence of the change of the MSHG intensity at \(\Delta t=3\) ps and at 10 K for selected Gd concentrations. Solid lines are guides to the eyes. (**b**) Calculated result of the \(\Delta n_{\mathrm{c}}\) dependence of \(\Delta J_{\mathrm{ex}}/f_{\mathrm{ex}}\). Here, \(\Delta J_{\mathrm{ex}}/f_{\mathrm{ex}}\) was calculated using the curve in Figure 3b, assuming that \(J_{\mathrm{ex}}\) is simply proportional to \(T_{\mathrm{C}}\left(\Delta J_{\mathrm{ex}}/J_{\mathrm{ex}}=\Delta T_{\mathrm{C }}/T_{\mathrm{C}}\right)\).