Towards Microscopic Understanding of the Phonon Bottleneck
  The problem of the phonon bottleneck in the relaxation of two-level systems (spins) to a narrow group of resonant phonons via emission-absorption processes is investigated from the first principles. It is shown that the kinetic approach based on the Pauli master equation is invalid because of the narrow distribution of the phonons exchanging their energy with the spins. This results in a long-memory effect that can be best taken into account by introducing an additional dynamical variable corresponding to the nondiagonal matrix elements responsible for spin-phonon correlation. The resulting system of dynamical equations describes the phonon-bottleneck plateau in the spin excitation, as well as a gap in the spin-phonon spectrum for any finite concentration of spins. On the other hand, it does not accurately render the lineshape of emitted phonons and still needs improving. 
D. A. Garanin