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We synthesize sub-THz longitudinal quasi-monochromatic acoustic phonons in aSrTiO$_3$ single crystal using a SrRuO$_3$/SrTiO$_3$ superlattice as anoptical-acoustic transducer. The generated acoustic phonon spectrum isdetermined using ultrafast X-ray diffraction. The analysis of the generatedphonon spectrum in the time domain reveals a k-vector dependent phononlifetime. It is observed that even at sub-THz frequencies the phonon lifetimeagrees with the 1/$\omega^2$ power law known from Akhiezer's model for hypersound attenuation. The observed shift of the synthesized spectrum to the higher$q$ is discussed in the framework of non-linear effects appearing due to thehigh amplitude of the synthesized phonons.

Direct time domain sampling of sub-THz coherent acoustic phonon spectra in SrTiO$_3$ using ultrafast X-ray diffraction