  A quantum simulator is proposed for nucleation and growth dynamics using an
out-of equilibrium optical lattice. We calculate the density of neutral atoms
in the lattice and we establish the connection with the
Kolmogorov-Mehl-Johnson-Avrami model. Here we show that a Avrami equation can
describe most of the evolution in time of the population growth in the lattice,
coherence between neutral atoms leads a complex growth rate.
