Dynamical Crossover in Landau$-$Zener Tunneling in Dissipative Rydberg Lattices
In this work, we investigate the excitation dynamics of a Rabi-coupled dissipative Rydberg lattice with a time-dependent detuning. The system is analyzed using (i) a Lindblad master equation within a mean-field approximation and (ii) an effective non-Hermitian Hamiltonian framework. While the mean-field approach captures the emergence of an antiferromagnetic order in the Rydberg excitation profile, the non-Hermitian description provides direct insight into the complex energy spectrum and its avoided crossings, which govern the Landau$-$Zener dynamics. We identify a regime in which the sublattice population imbalance vanishes near the avoided crossing, resulting in identical Landau$-$Zener probabilities on the two sublattices. Beyond a critical effective blockade strength there is a dynamical crossover to another regime in which the sublattice population imbalance persists through the avoided crossing, giving rise to sublattice-dependent Landau$-$Zener probabilities. Furthermore, Rydberg interactions prolong the lifetime of Landau$-$Zener-induced excitations in the presence of weak dissipation and strong Rabi coupling. In contrast, for weak Rabi coupling, the Rydberg blockade inhibits excitation and suppresses the Landau$-$Zener transition probability.