  We consider spin rotations in single- and two-electron quantum dots that are
driven by external AC electric field with two mechanisms that couple the
electron spatial motion and the spin degree of freedom: the spin-orbit
interaction and a random fluctuation of the Overhauser field due to nuclear
spin bath. We perform a systematic numerical simulation of the driven system
using a finite difference approach with an exact account taken for the
electron-electron correlation. The simulation demonstrates that the electron
oscillation in fluctuating nuclear field is translated into an effective
magnetic field during the electron wave packet motion. The effective magnetic
field drives the spin transitions according to the electric-dipole spin
resonance mechanism. We find distinct signatures of selection rules for direct
and higher-order transitions in terms of the spin-orbital symmetries of the
wave functions. The selection rules are violated by the random fluctuation of
the Overhauser field.
