  This paper is a proposal for the generation of many-body entangled state in
atomic and mechanical systems. Application of strong Rydberg dressing
interaction and fast bifurcation scheme in an ultra-stable Cs lattice clock in
this proposal results to a breakthrough in the formation of large cat states.
By detailed study of the de-coherence effects using Quantum Jump Monte Carlo
approach and considering obstacles in dense ensembles namely level mixing,
molecule formation and collective decoherence this proposal predicts the
formation of 700 atoms cat state. Dipole coupling between Rydberg atoms and
charged cantilevers is proposed for the transfer of the generated superposition
to far separated mechanical oscillators.
