## 4 XUV Wave Mixing and Coherent Control
### Perspective for the XLV/X-ray FWM  
Further steps can be undertaken to make XUV/X-ray FWM experiments feasible on routine basis. One of the most useful would be to replace the optical pulse with a XUV/X-ray. This will permit to probe, via ISS-type FWM methods, low energy modes (e.g., acoustic modes) in a wavevector range (\(\sim\)0.1-1 nm\({}^{-1}\)) hardly accessible by both optical methods and linear X-ray spectroscopy [46]. Such a wavevector range matches the characteristic length-scales of heterogeneities in the local structures of several classes of materials (e.g., block copolymer, relaxor ferroelectrics, glasses etc.), the incommensurate dimensions of many crystalline phases showing super-lattices of different natures, as well as the characteristic dimensions and periodicities of many nanostructures. Among these applications we mention the study of acoustic dynamics in glasses, which is largely believed to be the origin of the anomalous thermal properties of glasses, still a lively debated issue [47]. A debate that essentially comes from the impossibility to fully access such a \(\sim\)0.1-1 nm\({}^{-1}\) wavevector range and from the stark disagreement often observed between the extrapolations of the trends found at lower and larger wavevectors. Also, couplings between acoustic modes and local vibrations in nm-sized elastic domains, inherently connected to the amorphous local structure, may also have a relevant role; such a small length-scale is potentially in the range of XUV/X-ray FWM. Another relevant step to extend the range of applications of FEL-based FWM is to exploit the two-color emission scheme in order to demonstrate FEL-based CRS-type experiments. This would be a major step forward towards the realization of XCRS with atomic selectivity (see Figure 9), which in many respects may be considered as a main target application of XUV/X-ray FWM.  
Figure 9: (**a**) Sketch of the FEL-based X-ray analogue of coherent Raman scattering (XCRS) experiment; (**b**,**c**) are the excitation processes and level scheme for a XUV/X-ray CRS experiment involving core transitions in both the excitation and probing process; (**d**) Phase matching diagram.  
Indeed, the potential of XCRS to follow charge and energy flows between constituent atoms in materials would be of the greatest importance to address some fundamental scientific issues, such as: (a) the study of intramolecular relaxation dynamics in metal complexes, which are the doorway to photo-induced charge separation (the high selectivity of XCRS would be exploited to understand and disentangle processes, such as intramolecular vibrational redistribution, internal conversion and intersystem crossing, occurring upon photo-excitation; those are processes of fundamental interest in devising efficient molecular systems for applications as diverse as solar energy conversion, biology or data storage [48]); and (b) the dynamics of charge injection and transport in photocatalytic reactions taking place in metal oxides nanoparticles, such as TiO\({}_{2}\), that are key materials for renewable energy; for instance, the key event occurring in devices for solar energy conversion is the generation of a charge-separated state through ultrafast electron injection from an excited metal-complex, adsorbed on a nanoporous metal oxide substrate, to the conduction band of the substrate [49]. In this context the atomic selectivity of XCRS may allow us to understand whether such ultrafast electronic excited states have an O or Ti character.