Precise coupling terms in adiabatic quantum evolution: The generic case

TL;DR

This paper explicitly determines the asymptotic behavior of exponentially small coupling terms in multi-level quantum systems, confirming the universal form of adiabatic transition histories predicted by Berry.

Contribution

It provides a detailed analysis of superadiabatic coupling terms for generic two-state systems, extending previous results to a broader class of Hamiltonians.

Findings

Coupling terms are exponentially small and depend on singularities of the coupling function.

The superadiabatic coupling has a universal form.

First order perturbation theory describes adiabatic transitions accurately.

Abstract

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. Based on results from [BeTe1] for special Hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. As shown in [BeTe1], first order perturbation theory in the superadiabatic representation then allows to describe the time-development of exponentially small adiabatic transitions and thus to rigorously confirm Michael Berry's [Ber] predictions on the universal form of adiabatic transition histories.