Hyperpolarizability effect of Rydberg states in rubidium and cesium atoms

The hyperpolarizabilities of Rydberg states in rubidium and cesium atoms play a crucial role in understanding higher-order Stark shifts and illustrating high precision experiments. We calculate the dc hyperpolarizabilities of rubidium and cesium atoms in the ns1/2, np1/2, np3/2, nd3/2, and nd5/2 Rydberg states with principal quantum numbers $n=40-90$ using the sum-over-states method and provide analysis of uncertainties arising from {perturbation theory} in dc electric field amplitude measurements. We also find that for several Rydberg states in rubidium atoms, the Stark shifts are dominated by hyperpolarizability contributions, leading to a characteristic quartic dependence on the electric field amplitude {once the field amplitude exceeds approximately 0.3~V/cm}. These findings give more comprehensive understanding of Stark shifts in Rydberg atoms, and may contribute to improve sensitivity for weak electric field sensing.