What is the phonon/lattice interaction? To my understanding, phonons in solids are an excited state of a lattice, i.e., a lattice of atoms oscillate at a certain frequency. Phonons are quasipartices.
But what should the phonon/lattice interaction be? 
I found this term in the abstract of Jie Chen et al., http://www.pnas.org/content/pnas/108/47/18887.full.pdf.
 A: I believe he is talking about this section:

Low  Laser  Fluence:  Periodic  Expansion  and  Contraction. Coherent
  phonons could be excited and induce lattice vibrations (17) dur-
  ing the propagation of the acoustic wave and blast wave, which
  result in lattice expansion and contraction.
Conclusion. We conducted an experimental and computer simula-
  tion study of a single-crystal metal that monitored and detected
  directly the changes in the structure during femtosecond heating,
  as a function of time starting with photon/electron and electron/
  phonon coupling through phonon/lattice interaction, contraction,
  melting, nucleation, recrystallization, and mosaic crystal forma-
  tion. 

I believe that he is talking about when coherent phonons get excited, and they induce lattice vibrations. This is in this case an acoustic wave or blast wave, that is traveling in the material, and it causes phonons to become excited, and cause expansion and contraction in the lattice. Now this is similar to a gravitational wave, that causes the same expansion and contraction in the fabric of spacetime. In your case this fabric is the material, and the waves are expansions and contractions in the lattice.
This is why he is talking about an interaction between phohons and the lattice itself. The phonons are themselves are quanta of energy or quasiparticles associated with a compressional wave as a sound wave or vibration of the crystal lattice. These phonons get excited, and create expansions and contractions in the lattice.
Just as gravitons are quanta of gravitation, they somehow cause gravitational expansion and contraction (GWs) of the fabric of spacetime itself.
