Landau-Zener Formula in a "Non-Adiabatic" regime for avoided crossings

Abstract

We study a two-level transition probability for a finite number of avoided crossings with a small interaction. Landau-Zener formula, which gives the transition probability for one avoided crossing as $e^{-\pi \frac{{\epsilon }^2}{h}}$, implies that the parameter $h$ and the interaction $\epsilon$ play an opposite role when both tend to $0$. The exact WKB method produces a generalization of that formula under the optimal regime $\frac{h}{{\epsilon }^2}$ tends to~0. In this paper, we investigate the case $\frac{{\epsilon }^2}{h}$ tends to 0, called "non-adiabatic" regime. This is done by reducing the associated Hamiltonian to a microlocal branching model which gives us the asymptotic expansions of the local transfer matrices.

52 pages, 4 figures, accepted in Analysis and Mathematical Physics on 26 February 2021