  We study the dynamics of the periodically driven Rydberg chain starting from
the state with zero Rydberg excitations (vacuum state denoted by $|0\rangle$)
using a square pulse protocol in the high drive amplitude limit. We show, using
exact diagonalization for finite system sizes ($L\le 26$), that the Floquet
Hamiltonian of the system, within a range of drive frequencies which we chart
out, hosts a set of quantum scars which have large overlap with the $|0\rangle$
state. These scars are distinct from their counterparts having high overlap
with the maximal Rydberg excitation state ($|\mathbb{Z}_2\rangle$); they
coexist with the latter class of scars and lead to persistent coherent
oscillations of the density-density correlator starting from the $|0\rangle$
state. We also identify special drive frequencies at which the system undergoes
perfect dynamic freezing and provide an analytic explanation for this
phenomenon. Finally, we demonstrate that for a wide range of drive frequencies,
the system reaches a steady state with sub-thermal values of the
density-density correlator. The presence of such sub-thermal steady states,
which are absent for dynamics starting from the $|\mathbb{Z}_2\rangle$ state,
imply a weak violation of the eigenstate thermalization hypothesis in finite
sized Rydberg chains distinct from that due to the scar-induced persistent
oscillations reported earlier. We conjecture that in the thermodynamic limit
such states may exist as pre-thermal steady states that show anomalously slow
relaxation. We supplement our numerical results by deriving an analytic
expression for the Floquet Hamiltonian using a Floquet perturbation theory in
the high amplitude limit which provides an analytic, albeit qualitative,
understanding of these phenomena at arbitrary drive frequencies. We discuss
experiments which can test our theory.
