  We present optimal control protocols to prepare different many-body quantum
states of Rydberg atoms in optical lattices. Specifically, we show how to
prepare highly ordered many-body ground states, GHZ states as well as some
superposition of symmetric excitation number Fock states, that inherit the
translational symmetry from the Hamiltonian, within sufficiently short
excitation times minimizing detrimental decoherence effects. For the GHZ
states, we propose a two-step detection protocol to experimentally verify the
optimal preparation of the target state based only on standard measurement
techniques. Realistic experimental constraints and imperfections are taken into
account by our optimization procedure making it applicable to ongoing
experiments.
