The Quantum Adiabatic Approximation and the Geometric Phase

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.

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