Spontaneous decay of a soft optical phonon in the relaxor ferroelectric PbMg$_{1/3}$Nb$_{2/3}$O$_{3}$

We report the spontaneous decay of a soft, optical phonon in a solid. Using neutron spectroscopy, we find that specific phonon lifetimes in the relaxor PbMg$_{1/3}$Nb$_{2/3}$O$_{3}$ are anomalously short within well-defined ranges of energy and momentum. This behavior is independent of ferroelectric order and occurs when the optical phonon with a specific energy and momentum can kinematically decay into two acoustic phonons with lower phase velocity. We interpret the well-known relaxor waterfall effect as a form of quasiparticle decay analogous to that previously reported in quantum spin liquids and quantum fluids.