## 3 Experiments
### Experimental Results
#### 3.3.2 Carrier-Density-Dependent Ultrafast Spin Dynamics  
Having interpreted a microscopic model of photoinduced spin dynamics at a relatively high temperature, the question now is how much the magnitude of the magnetic interaction has changed by the photoexcitation and how is the relationship between the spin dynamics and the carrier density \(n_{\rm c}\). To simplify this investigation, the corresponding dynamics were measured at 10 K where the magnetic moment is fully saturated at 7 \(\mu_{\rm B}\) per Eu\({}^{2+}\) ion. Hence, referring to relation (3), the nonthermal changes of the MSHG intensity is associated to the changes of \(J_{\rm ex}\). Figure 6 shows the time evolution of the photoinduced changes of the MSHG intensity at 10 K for Gd concentrations between 0.013% and 19.5%. The excitation density of the pump pulse is \(\sim\)130 \(\upmu\)J/cm\({}^{2}\) for all traces, which roughly corresponds to \(\Delta n_{\rm c}\sim 1\times 10^{19}\) cm\({}^{-3}\) carriers excited by photoexcitation. A common behavior observed for all samples is the decreasing MSHG intensity on a ns timescale. As mentioned above, this is caused by the thermal destabilization of the magnetic order mediated by the spin-lattice relaxation. A prolonged positive signal (\(\Delta I^{\rm MSHG}/I^{\rm MSHG}>0\)), especially observed for \(x=0.25\%\), is related to the stability of photoenhanced ferromagnetic state, as described below. The MSHG response on the ps timescale strongly depends on the Gd concentration. At the lowest doping (\(x=0.013\%\)), an ultrafast increase of the MSHG signal is observed. It nicely confirms that our nonlinear magneto-optical probe reflects a variation of \(J_{\rm ex}\) rather than that of the already saturated magnetization. By increasing the Gd concentration, the magnitude of this initial increase becomes progressively more pronounced and reaches a maximum near \(x=0.25\%\). Higher Gd doping reduces the magnitude of the initial enhancement, and at \(x>5\%\) an ultrafast decrease of the MSHG signal is observed. In order to illustrate this behavior, in Figure 7a the pump-induced change of the MSHG intensity \(\Delta I^{\rm MSHG}/I^{\rm MSHG}\) at the fixed time delay \(\Delta t=3\) ps is plotted as a function of Gd-dependent carrier density \(n_{\rm c}\). Here, \(n_{\rm c}\) has been derived for each Gd concentration \(x\) from the relation displayed in the inset of Figure 3b. When the photoinduced change in the magnetization \(M\) is neglected, the relationship of \(\Delta I^{\rm MSHG}/I^{\rm MSHG}=2\Delta J_{\rm ex}/J_{\rm ex}\) is established from the relation (3). To obtain insight into this relation, it should be clarified how the photoinduced population of the \(5d/6s\) band affects \(J_{\rm ex}\). Figure 7b shows \(\Delta J_{\rm ex}/J_{\rm ex}\) obtained by increasing the equilibrium carrier density \(n_{\rm c}\) of the \(5d/6s\) band by a fixed value \(\Delta n_{\rm c}\sim 1\times 10^{19}\) cm\({}^{-3}\) which represents the photo-carrier density injected by the pump pulse in this experiment. The relation between \(\Delta J_{\rm ex}/J_{\rm ex}\) and \(\Delta n_{\rm c}\) was derived by the relation between \(T_{\rm C}\) and \(n_{\rm c}\) in Figure 3b, assuming that \(J_{\rm ex}\) is simply proportional to \(T_{\rm C}\) (\(\Delta J_{\rm ex}/J_{\rm ex}=\Delta T_{\rm C}/T_{\rm C}\)). For low  
Figure 6: Change of the MSHG intensity at 10 K in Eu\({}_{1-x}\)Gd\({}_{x}\)O as a function of time delay and Gd concentration (\(0.013\%\leq x\leq 19.5\%\)). The excitation density of the pump pulse is \(\sim\)130 \(\upmu\)J/cm\({}^{2}\) for all traces, which creates a photo-carrier density \(\Delta n_{\rm c}\sim 1\times 10^{19}\) cm\({}^{-3}\). The inset shows a magnified view of the ultrafast increase and decrease of the MSHG intensity for \(x=0.64\%\) and \(x=9.63\%\). values of \(n_{\rm c}\), \(\Delta n_{\rm c}\) simply set the net carrier density to \(1\times 10^{19}\) cm\({}^{-3}\) which results in a constant shift of \(\Delta J_{\rm ex}\).