  We have numerically solved the Hamiltonian of an electron in a semiconductor
double ring subjected to the magnetic flux and Rashba spin-orbit interaction.
It is found that the Aharonov-Bohm energy spectrum reveals multi-zigzag
periodic structures. The investigations of spin-dependent electron dynamics via
Rabi oscillations in two-level and three-level systems demonstrate the
possibility of manipulating quantum states. Our results show that the optimal
control of photon-assisted inter-ring transitions can be achieved by employing
cascade-type and $\Lambda$-type transition mechanisms. Under chirped pulse
impulsions, a robust and complete transfer of an electron to the final state is
shown to coincide with the estimation of the Landau-Zener formula.
