Perturbative Analysis of Possible Failures in the Traditional Adiabatic Conditions

Recently, Marzlin and Sanders (2004) demonstrated an inconsistency when the adiabatic approximation was applied to specific, "inverse" time-evolving systems. Following that, Tong et al. (2005) showed that the widely used traditional adiabatic conditions are insufficient to guarantee the validity of the adiabatic approximation for this class of systems. In this article we explore the origin of these observations by a perturbative approach and find that in first order approximation certain nonzero terms appear in the solution which gives rise to the breakdown of the adiabatic approximation (despite the fact that the traditional adiabatic conditions are satisfied). We argue that in this case the Hamiltonian of Marzlin and Sanders' inverse time evolving system cannot be written in terms of t/T, where T denotes the total evolution time. It is further demonstrated that the new qualitative adiabatic condition of Ye et al. (2005) performs well in some cases when the traditional conditions fail to describe properly non-adiabatic evolution.