Singularity of Berry Connections Inhibits the Accuracy of Adiabatic Approximation

Abstract

Adiabatic approximation for quantum evolution is investigated quantitatively with addressing its dependence on the Berry connections. We find that, in the adiabatic limit, the adiabatic fidelity may uniformly converge to unit or diverge manifesting the breakdown of adiabatic approximation, depending on the type of the singularity of the Berry connections as the functions of slowly-varying parameter $R$. When the Berry connections have a singularity of $1/R^\sigma$ type with $\sigma < 1$, the adiabatic fidelity converges to unit in a power-law; whereas when the singularity index $\sigma$ is larger than one, adiabatic approximation breaks down. Two-level models are used to substantiate our theory.