Time of Zener tunneling

The time of Zener tunneling, ${\tau }_z$, is calculated for a general two-level quantum-mechanical system by two methods. In the first we determine the width of the transition profile in time. In the second we apply an oscillating perturbation and examine how the final transition probability depends upon the perturbation frequency and phase. Both methods show, given that the coupling energy between the diabatic levels is Δ and the time rate of change of the energy is α, that in the adiabatic limit ${\tau }_z$ scales as Δ/α and in the sudden limit ${\tau }_z$ scales as (ħ/α${)}^{1/2}$. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.