  Neutral atom arrays driven into Rydberg states constitute a promising
approach for realizing programmable quantum systems. Enabled by strong
interactions associated with Rydberg blockade, they allow for simulation of
complex spin models and quantum dynamics. We introduce a new Floquet
engineering technique for systems in the blockade regime that provides control
over novel forms of interactions and entanglement dynamics in such systems. Our
approach is based on time-dependent control of Rydberg laser detuning and
leverages perturbations around periodic many-body trajectories as resources for
operator spreading. These time-evolved operators are utilized as a basis for
engineering interactions in the effective Hamiltonian describing the
stroboscopic evolution. As an example, we show how our method can be used to
engineer strong spin exchange, consistent with the blockade, in a
one-dimensional chain, enabling the exploration of gapless Luttinger liquid
phases. In addition, we demonstrate that combining gapless excitations with
Rydberg blockade can lead to dynamic generation of large-scale multi-partite
entanglement. Experimental feasibility and possible generalizations are
discussed.
