Inconsistencies of the Adiabatic Theorem and the Berry Phase

TL;DR

This paper investigates fundamental inconsistencies in the quantum adiabatic theorem and Berry phase, resolving apparent violations of unitarity and clarifying conditions under which Berry phases may vanish, thereby strengthening the theorem's validity.

Contribution

The authors resolve known inconsistencies in the adiabatic theorem and Berry phase, introducing a unitary operator decomposition method to support the theorem's validity.

Findings

Resolved the Marzlin-Sanders inconsistency within the quantum adiabatic theorem.

Showed that Berry phases can vanish under strict adiabatic conditions.

Developed a unitary operator decomposition method to validate the adiabatic approximation.

Abstract

The adiabatic theorem states that if we prepare a quantum system in one of the instantaneous eigenstates then the quantum number is an adiabatic invariant and the state at a later time is equivalent to the instantaneous eigenstate at that time apart from phase factors. Recently, Marzlin and Sanders have pointed out that this could lead to apparent violation of unitarity. We resolve the Marzlin-Sanders inconsistency within the quantum adiabatic theorem. Yet, our resolution points to another inconsistency, namely, that the cyclic as well as non-cyclic adiabatic Berry phases may vanish under strict adiabatic condition. We resolve this inconsistency and develop an unitary operator decomposition method to argue for the validity of the adiabatic approximation.