Adiabatic approximation, Gell-Mann and Low theorem and degeneracies: A pedagogical example

TL;DR

This paper analyzes a simple 2x2 quantum system to evaluate the validity of the adiabatic approximation and Gell-Mann and Low theorem, highlighting issues with degeneracies and proposing methods to select initial states to avoid failures.

Contribution

It provides a clear pedagogical example demonstrating when the adiabatic approximation and Gell-Mann and Low theorem succeed or fail, especially in degenerate cases.

Findings

Adiabatic approximation valid for non-degenerate initial states

Gell-Mann and Low formula can track eigenstates despite no limit in evolution operator

Degeneracies cause failure of approximation and formula unless initial states are carefully chosen

Abstract

We study a simple system described by a 2x2 Hamiltonian and the evolution of the quantum states under the influence of a perturbation. More precisely, when the initial Hamiltonian is not degenerate,we check analytically the validity of the adiabatic approximation and verify that, even if the evolution operator has no limit for adiabatic switchings, the Gell-Mann and Low formula allows to follow the evolution of eigenstates. In the degenerate case, for generic initial eigenstates, the adiabatic approximation (obtained by two different limiting procedures) is either useless or wrong, and the Gell-Mann and Low formula does not hold. We show how to select initial states in order to avoid such failures.