The Quantum Adiabatic Approximation and the Geometric Phase

Abstract

A precise definition of an adiabaticity parameter $ν$ 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 $U(τ)=\sum_\ell U^{(\ell)}(τ)$ with $U^{(\ell)}(τ)$ being at least of the order $ν^\ell$. In particular $U^{(0)}(τ)$ corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the $1/τ$-expansion of $U(τ)$. 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.