Landau-Zener transition in a multilevel system. An exact result

TL;DR

This paper derives an exact formula for transition amplitudes in a multilevel Landau-Zener problem by extending quantum evolution to complex time, confirming previous conjectures and covering both slow and fast regimes.

Contribution

It provides an exact solution for multilevel Landau-Zener transitions, extending the classic two-level problem and confirming the Brundobler-Elser conjecture.

Findings

Exact transition amplitude matches the sequential pairwise crossing approximation.

The method applies to both adiabatic and diabatic regimes.

Confirms the conjecture of Brundobler and Elser.

Abstract

We study the S-matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schroedinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic regime (slow evolution.) and the diabatic regime (fast evolution). The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser.