While the atoms are strongly localized in the initial state (upmost row) in the deep lattice, correlations develop during the crossing of the phase transition induced by lowering the lattice depth.
In the final weak lattice (lowest row), the correlations would in principle span the whole lattice, if the phase transition could be crossed infinitely slow. In reality, however, the finite ramp speed limits the correlation length to a finite value.

Graphic: Quantum Optics Group, LMU

While the atoms are strongly localized in the initial state (upmost row) in the deep lattice, correlations develop during the crossing of the phase transition induced by lowering the lattice depth.
In the final weak lattice (lowest row), the correlations would in principle span the whole lattice, if the phase transition could be crossed infinitely slow. In reality, however, the finite ramp speed limits the correlation length to a finite value.

Graphic: Quantum Optics Group, LMU