Time Development of Exponentially Small Non-Adiabatic Transitions

TL;DR

This paper rigorously analyzes the time evolution of exponentially small non-adiabatic transitions in quantum systems, confirming the universality of the switching function predicted by Sir Michael Berry.

Contribution

It provides a rigorous derivation of the leading order non-adiabatic corrections for a specific family of two-level Hamiltonians, validating Berry's universal switching function.

Findings

Confirmed the form of the switching function as predicted by Berry

Derived the leading order non-adiabatic corrections explicitly

Validated the universality of the switching function for this Hamiltonian family

Abstract

Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non--adiabatic corrections to these approximations for a particular family of two--level analytic Hamiltonian functions. Our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function. Our results confirm the physics predictions of Sir Michael Berry in the sense that the switching function for this family of Hamiltonians has the form that he argues is universal.