Phonon elastic scattering on two-level systems in cryogenic piezoelectric crystals

We demonstrate that two-level systems (TLSs), which are caused by impurities or defects, lead to a kind of phonon elastic scattering in cryogenic piezoelectric crystals. We apply a model in which a TLS is coupled to the strain field in disordered crystals. Due to this coupling, we propose that phonons can be elastically scattered by TLSs when their frequency is much lower than the transition frequency of TLSs. Based on the second-order perturbation theory, we then obtain an attenuation formula. Consequently, we show how frequency and temperature influence the attenuation, which gives a theoretical explanation for the recent experiment (GORYACHEV M. et al., Phys. Rev. Lett., 111 (2013) 085502). This work provides some help for understanding the loss mechanism and designing low-loss cryogenic resonant cavities made of piezoelectric quartz.