Producing and storing spin-squeezed states and Greenberger-Horne-Zeilinger states in a one-dimensional optical lattice

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 by 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.