Histories of adiabatic quantum transitions

Abstract

The way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation, for a 2-state quantum system. By transforming to a series of superadiabatic bases, clinging ever closer to the exact evolving state, it is shown that transition histories renormalize onto a universal one, in which the amplitude grows to its final value as an error function (rather than via large oscillations as in the ordinary adiabatic basis). The time for the universal transition is of order $\surd (\hslash /\delta)$ where $\delta $ is the small adiabatic (slowness) parameter. In perturbation theory the pre-exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value ${\textstyle\frac{1}{3}}\pi $ (for the ordinary adiabatic basis) to the correct value unity. The various histories could be observed in spin experiments.