Diagram rules for a chain of entangled ultracold atoms in optical lattices

The creation of highly entangled state for neutral atoms trapped in optical lattices was reported in a recent experiment by Mandel et al. [Nature 425 (2003) 937]. In the present work, simple diagram rules are given to get the final expression of the multi-particle entangled state created by performing certain multi-particle operations in this experiment. These diagram rules may has potential application in the further experimental test for the nonlocal property of the entangled cluster state.