Periodically driven Rydberg chains with staggered detuning

We study the stroboscopic dynamics of a periodically driven finite Rydberg chain with staggered ($\Delta$) and time-dependent uniform ($\lambda(t)$) detuning terms using exact diagonalization (ED). We show that at intermediate drive frequencies ($\omega_D$), the presence of a finite $\Delta$ results in violation of the eigenstate thermalization hypothesis (ETH) via clustering of Floquet eigenstates. Such clustering is lost at special commensurate drive frequencies for which $\hbar \omega_d=n \Delta$ ($n \in Z$) leading to restoration of ergodicity. The violation of ETH in these driven finite-sized chains is also evident from the dynamical freezing displayed by the density-density correlation function at specific $\omega_D$. Such a correlator exhibits stable oscillations with perfect revivals when driven close to the freezing frequencies for initial all spin-down ($|0\rangle$) or Neel ($|{\mathbb Z}_2\rangle$, with up-spins on even sites) states. The amplitudes of these oscillations vanish at the freezing frequencies and reduces upon increasing $\Delta$; their frequencies, however, remains pinned to $\Delta/\hbar$ in the large $\Delta$ limit. In contrast, for the $|{\bar {\mathbb Z}_2}\rangle$ (time-reversed partner of $|{\mathbb Z}_2\rangle$) initial state, we find complete absence of such oscillations leading to freezing for a range of $\omega_D$; this range increases with $\Delta$. We also study the properties of quantum many-body scars in the Floquet spectrum of the model as a function of $\Delta$ and show the existence of novel mid-spectrum scars at large $\Delta$. We supplement our numerical results with those from an analytic Floquet Hamiltonian computed using Floquet perturbation theory (FPT) and also provide a semi-analytic computation of the quantum scar states within a forward scattering approximation (FSA).