  We study the population dynamics in a two-atom setup in which each atom is
driven independently by different light fields, but coupling the same Rydberg
state. In particular, we look at how an offset in the Rabi frequencies between
two atoms influences the dynamics. We find novel features such as amplifying
the Rabi frequency of one atom, together with strong Rydberg-Rydberg
interactions freezes the dynamics in the second atom. We characterize the
Rydberg-biased freezing phenomenon in detail, with effective Hamiltonians
obtained for various limits of the system parameters. In the absence of
Rabi-offset, the doubly excited state population exhibits a Lorentzian profile
as a function of interaction, whereas for very small offsets it shows splitting
and thus peaks. Using an effective Hamiltonian as well as the perturbation
theory for weak interactions, we show that the peak arises from a competition
between Rabi-offset and Rydberg-Rydberg interactions when both are sufficiently
small, together with the Rydberg blockade at large interactions. The effective
Hamiltonians provide us with analytical results which are in an excellent
agreement with full numerical solutions. Also, we analyze the growth and the
dynamics of quantum correlations such as entanglement entropy and quantum
discord for the coherent dynamics. We extend our studies to the dissipative
case in which the spontaneous emission from the Rydberg state is taken into
account and in particular, we look at the purity and quantum discord of the
steady states. To conclude, our studies reveal that the local manipulation of
an atom using Rabi-offset can be an ideal tool to control the quantum
correlations and in general, quantum states of the composite two-qubit systems.
