Landau-Zener transition between two quantum dots coupled by resonant tunneling

We consider the transition of electron between two quantum dots in which the discrete levels are swept past each other with a constant velocity. If a direct tunneling between the dot levels was allowed, an electron will be transferred between the dots when the levels cross. This transfer is described in terms of the conventional Landau-Zener theory. We assume that direct tunneling between the dots is forbidden. Rather, the transfer is due to the resonant tunneling via a discrete impurity level separating the dots. Then the description of the electron transfer reduces to a three-state (two dots plus impurity) Landau-Zener transition. Transition probability depends on the relative positions of the resonant level and the energy at which the levels cross. It also depends on the left-right asymmetry of tunneling between the impurity and the left(right) dots. We calculate the transition probability in different limits of the horizontal (in space) and vertical (in energy) impurity level positions. It is big horizontal or big vertical asymmetry that allows to get the analytical expression for the transition probability.