## 4 XUV Wave Mixing and Coherent Control
### Four Wave Mixing
Moreover, XCRS is not limited to the study of valence band excitons, since the same concept in principle applies for all kind of excitations, related to any dynamical variable coupled to the field (in some circumstances also to those uncoupled in the linear regime), with energies lower than that of core-resonances. Such class basically includes all kind of modes related to nuclear (phonons, structural relaxations, heat diffusion, etc. ), electronic (excitons, plasmons, etc.) and \"mixed\" (polarons, polaritons, etc.) degrees of freedom. The XCRS approach may hence allow to study, e.g., charge and energy transfer processes among different atoms in molecules, the delocalization and correlations of electronic excitations as well as structural fluctuations, nuclear motions and relaxation processes, other than detecting the dynamics of elementary excitations such as, e.g., phonons, plasmons and polarons. Among the worldwide existing FEL facilities, FERMI possesses unique characteristics for FWM experiments in the XUV/soft X-ray domain. Firstly, most of the photon parameters of the FEL radiation emitted (the longitudinal coherence properties among them) are related to those of the seed laser, which is fully controllable in all relevant parameters, so that it is possible to control the FEL output by simply acting on the seed. Moreover, it is possible to use two (or more) independently controllable seed laser pulses, which result into the radiation of two (or more) FEL-pulses with controllable photon energy, polarization, time delay, etc. ; a further development of such scheme includes the simultaneous use of multiple resonances in the FEL amplifier, which largely extends the separation in the photon energy (i.e., up to several eV's) of the multiple FEL pulses [37]. Finally, the layout of FERMI could be adapted to generate a few fs to sub-fs FEL-pulses with the aforementioned benefits. In a nutshell, today FERMI is the FEL source most similar to a \"multi-color\" conventional laser and, therefore, it could be used to attempt the development of XUV/soft X-ray FWM. A significant endorsement of the latter statement is represented by the recent demonstration of a time-resolved FEL-stimulated FWM response [45] (see Figure 8). In this experiment two time-space coincident ultrafast (time duration \(\sim\)70 fs) FEL pulses (photon energy \(\omega_{\text{XUV}}\)\(\sim\)45 eV) were crossed at the sample (amorphous SiO\({}_{2}\)) position. Figure 8: FEL-based FWM experiment stimulated by transient extreme ultraviolet (XUV) gratings (inset (b) reports the phase matching geometry: \(k_{\text{FEL1}}\), \(k_{\text{FEL2}}\), \(k_{\text{opt}}\) and \(k_{\text{out}}\) are the wavevectors of the two FEL pulses, the optical pulse and the FWM signal, respectively). The signal (XTG) has been registered on a charged coupled detector (CCD). These beams were expected to generate an XUV grating able to scatter off, through FWM processes, a third coherent beam, provided that the latter is send into the sample in phase matching conditions. Indeed, using a third \"phase matched\" optical pulse (time duration \(\sim\) 100 fs, photon energy \(\omega_{\text{opt}}\sim\) 3.1 eV) in the interaction region allowed the observation of an optical beam emerging from the sample along the expected \"phase matched\" direction (see Figure 8 XTG signal).