  We present the origin of the Abraham-Minkowski controversy of light-matter
wave interacting system, which is a special case of the centaury-old
Abraham-Minkowski controversy. We solve the controversy of laser-atom
interacting case and find that for systems with perfect atomic spatial
coherence, the systems prefer to show Minkowski momentum and canonical momentum
for the atoms and light, respectively; while for the systems where the atoms
are spatially incoherent, the momenta of light and atoms would choose the
Abraham and kinetic forms. The provement of our solution can be realized with
current techniques, using three-dimensional optical lattices and
electromagnetically-induced absorption (EIA) to distinguish the kinetic and
canonical recoil momentum of ultra-cold atoms.
