Taming the divergent terms that occur during adiabatic switching in perturbation theory

TL;DR

This paper addresses divergence issues in adiabatic switching within perturbation theory, demonstrating that divergences can be absorbed into a removable phase factor, ensuring well-defined evolved states for systems with multiple states.

Contribution

The paper shows that divergences in adiabatic switching are confined to a time-independent phase factor, extending previous results to systems with many states.

Findings

Divergent terms are absorbed into a phase factor.

Evolved states remain well-defined despite divergences.

Results apply to systems with arbitrary number of states.

Abstract

A potential problem with adiabatic switching in perturbation theory is that divergent terms appear in the series solution. An example of this was presented by C. Brouder et al [4] for a simple 2 state system where the evolution of system in the presence of a time dependent perturbation was considered. One of their results is that the evolution operator has no well-defined limit for adiabatic switching. We will rework this problem to show that for adiabatic switching the evolved states are well-defined with any divergences being absorbed in a time independent phase factor which can be removed. These results will then be applied to the more general problem of a system with an arbitrary number of states. It will be shown that for this case, also, the potentially divergent terms all appear in a time-independent phase factor.