Precise coupling terms in adiabatic quantum evolution

TL;DR

This paper constructs superadiabatic representations for multi-level quantum systems, explicitly determines exponentially small coupling terms, and confirms Berry's predictions on adiabatic transitions.

Contribution

It explicitly constructs superadiabatic representations for two-state systems with real-symmetric Hamiltonians and analyzes the asymptotic behavior of small coupling terms.

Findings

Explicit asymptotic behavior of coupling terms determined

First order perturbation describes adiabatic transition dynamics

Confirms and generalizes Berry's predictions

Abstract

It is known that for multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. For a family of two-state systems with real-symmetric Hamiltonian we construct such a superadiabatic representation and explicitly determine the asymptotic behavior of the exponentially small coupling term. First order perturbation theory in the superadiabatic representation then allows us to describe the time-development of exponentially small adiabatic transitions. The latter result rigorously confirms the predictions of Sir Michael Berry for our family of Hamiltonians and slightly generalizes a recent mathematical result of George Hagedorn and Alain Joye.