Quantum adiabatic theorem in light of the Marzlin-Sanders inconsistency

TL;DR

This paper examines the quantum adiabatic theorem's limitations highlighted by the Marzlin-Sanders inconsistency, identifying conditions under which the theorem fails and clarifying the role of resonant terms in Hamiltonians.

Contribution

It demonstrates that standard assumptions do not hold for dual Hamiltonians in the inconsistency and provides simple criteria to predict when the adiabatic approximation fails.

Findings

Inconsistency arises only with Hamiltonians containing resonant terms with asymptotically vanishing amplitudes.

Key premises of the standard theorem do not hold for the dual Hamiltonian involved.

Two simple conditions can identify systems where the adiabatic approximation fails.

Abstract

A consensus that questions the perfunctory use of the quantum adiabatic theorem has emerged since Marzlin and Sanders [Phys. Rev. Lett. {\bf 93}, 160408 (2004)] showed the existence of an inconsistency in the applicability of the theorem. Further analysis proved that the inconsistency may arise from the existence of resonant terms in the Hamiltonian, but recent work indicates that the debate about the full extent of the problem remains open. Here, we first show that key premises required in the standard demonstration of the theorem do not hold for a dual Hamiltonian involved in the Marzlin-Sanders inconsistency. Also, we show that two simple conditions can identify systems for which the adiabatic approximation fails, in spite of satisfying traditional quantitative conditions that were believed to guarantee its validity. Finally, we prove that the inconsistency only arises for Hamiltonians that contain resonant terms whose amplitudes go asymptotically to zero.