  We study the dynamical generation and storage of spin squeezed states, as
well as more entangled states up to macroscopic superpositions, in a system
composed of a few ultra-cold atoms trapped in a one-dimensional optical
lattice. The system, initially in the superfluid phase with each atom in a
superposition of two internal states, is first dynamically entangled by
atom-atom interactions then adiabatically brought to the Mott-insulator phase
with one atom per site where the quantum correlations are stored. Exact
numerical diagonalization allows us to explore the structure of the stored
states by looking at various correlation functions, on site and between
different sites, both at zero temperature and at finite temperature, as it
could be done in an experiment with a quantum-gas microscope.
