## 4 XUV Wave Mixing and Coherent Control
### Four Wave Mixing  
Among wave-mixing processes, four wave mixing (FWM) processes have a special place, since they are at the basis of most experimental methods, besides being the lowest order non-linear processes that are not vanishing by reason of symmetry. FWM arise from the 3 rd-order non-linear interactions of three coherent electromagnetic fields (\(E_{1,2,3}\)) that might have different frequencies (\(\omega_{1,2,3}\)), wavevectors (\(k_{1,2,3}\)), polarizations, bandwidth, time delays, etc. (see Figure 7). Such interacting fields drive the radiation of a fourth (signal) field, whose photon parameters (frequency, \(\omega_{4}\), wavevector, \(k_{4}\), polarization, etc.) may differ from those of the input fields. The possibility to experimentally control the input field parameters and to determine the output field parameters turns into the capability to stimulate and detect different FWM processes, which often carry out distinct and complementary information on the sample under study. Such a high degree of selectivity and richness of information makes FWM an extremely versatile and informative tool. For instance, through a FWM process termed impulsive stimulated scattering (ISS) it is possible to determine in real time and with wavevector selectivity the dynamics of both ultrafast (sub-picosecond) molecular vibrations or slow (millisecond) thermal and structural relaxations, depending on the bandwidth and time delays of the interacting fields [41]. Information on the dynamics of higher energy (i.e., faster) sample excitations can be achieved, e.g., by coherent Raman scattering (CRS) and coherent multi-dimensional spectroscopy; methods also able to provide insights into the correlations between different dynamical variables, such as those related to vibrational and electronic modes [42]. FWM approaches are also bringing relevant technological advances, ranging from sub-wavelength microscopy [43] to wavefunction tomography during chemical reactions [44]. The hard-limit of optical FWM, as any other optical spectroscopy, is inherently related to the long wavelength (some 100's of nm or more) and low photon energy (a few eV's or less) of the optical radiation itself. This prevents to probe matter at atomic and molecular length scales (a few nm or less) as well as to study excitations with energies larger than a few eV's, a limitation that basically confines optical CRS to the study of vibrational dynamics and that of low-energy electronic excitations. Furthermore, optical photons cannot exploit core-level electronic resonances, typically located in the 10's eV to 10's keV range, adding elemental selectivity to the FWM approach. The latter point combined with the possibility to detect high energy excitation (e.g., valence band excitons in the 1-10 eV energy range) on a Q-range comparable with the inverse molecular size (-nm\({}^{-1}\)) are the main advantages expected from the XUV/X-ray analogue of CRS (XCRS). By tuning the frequencies of the input beams involved in the  
Figure 7: Sketch of the four wave mixing (FWM) experimental scheme. After the full stop: The non-linear interaction of the three photon pulses (i.e., coherent electromagnetic fields \(E_{1,2,3}\)) that may have different frequencies and wavevectors (1,2,3) is represented in the figure. The pulses may have as well different polarization, bandwidth and time delays. In the inset is depicted the phase matching condition. XCRS process to core resonance of distinct atoms it would be possible to determine where a given electronic wave-packet is created and where it is probed, as well as to follow in real time the dynamics of such charge-transfer between the selected atoms. This unique capability arises from the multi-wave nature of XCRS, which permits to overcome the basic limitation of any linear X-ray method, where the light-matter interaction occurs in correspondence of a single atomic site and prevents the detection of real-time dynamics between distinct atoms.