Adiabatic theorems for linear and nonlinear Hamiltonians

TL;DR

This paper analyzes the conditions under which the quantum adiabatic approximation holds, providing a simple criterion for linear Hamiltonians and extending the theorem to nonlinear cases.

Contribution

It introduces a simple, general sufficient condition for the validity of the adiabatic approximation for linear Hamiltonians and generalizes the theorem to nonlinear Hamiltonians.

Findings

Derived a simple sufficient condition valid for arbitrary spectra and time variations.

Showed that in some cases, the condition is both necessary and sufficient.

Extended the adiabatic theorem to nonlinear Hamiltonians.

Abstract

Conditions for the validity of the quantum adiabatic approximation are analyzed. For the case of linear Hamiltonians, a simple and general sufficient condition is derived, which is valid for arbitrary spectra and any kind of time variation. It is shown that in some cases the found condition is necessary and sufficient. The adiabatic theorem is generalized for the case of nonlinear Hamiltonians.