The Quantum Adiabatic Approximation and the Geometric Phase

TL;DR

This paper introduces a precise adiabaticity parameter and a novel perturbation series expansion for quantum evolution, clarifying the relationship between adiabatic and non-adiabatic dynamics and their connection to the geometric phase.

Contribution

It proposes a new definition of the adiabaticity parameter and a variation of perturbation theory that expands the evolution operator in powers of this parameter, distinct from traditional methods.

Findings

Series expansion of the evolution operator in powers of the adiabaticity parameter.

The zeroth order corresponds to the adiabatic approximation and Berry's phase.

Non-adiabatic effects are generated by a transformed, off-diagonal Hamiltonian.

Abstract

A precise definition of an adiabaticity parameter $\nu$ of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator $U(\tau)=\sum_\ell U^{(\ell)}(\tau)$ with $U^{(\ell)}(\tau)$ being at least of the order $\nu^\ell$. In particular $U^{(0)}(\tau)$ corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the $1/\tau$-expansion of $U(\tau)$. It is also shown that the non-adiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. Some related issues concerning the geometric phase are also discussed.