  The spin dynamics in two electron double quantum dots embedded in two
dimensional electron gas at the interface between SrTiO$_3$ and LaAlO$_3$ is
studied by an exact numerical solution of the time-dependent Schr\"odinger
equation, in the context of the electric dipole spin resonance experiment.
Based on the three band model of $3d$-electrons localized at Ti ions on the
square lattice we analyze in details the singlet-triplet transition induced by
the AC electric field, in the magnetic field range close to the avoided
crossing which appears as a result of the spin-orbit coupling. Our calculations
show that for symmetric double quantum dots the single photon spin-flip
transitions is prohibited due to the parity symmetry and the transition can
occur only by the higher order two-photon processes. For a weakly asymmetric
system, when the first order singlet-triplet transitions are released due to
the parity symmetry breaking, the spin-flip transition has a character of the
Rabi oscillations for a low electric field amplitude. As the amplitude is
increased the frequency of the transition is blueshifted (redshifted) for the
magnetic field below (above) the single-triplet avoided crossing.
Interestingly, for a sufficiently high magnetic field and high AC field
amplitude the electric field drives the system across the avoided crossing
inducing the spin-flip by the Landau-Zener-Stueckelberg-Majorana transitions
with 100\% spin flip probability for a slow sweep. Finally, the optimization of
the geometrical parameters of the system with respect to the time of spin-flip
of its fidelity is also presented.
