Perturbative approach to the adiabatic approximation (6 pages)

M. S. Sarandy

Gustavo Rigolin, Gerardo Ortíz, V. H. Ponce

We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-$1∕2$ problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.

Ming-Yong Ye, Xiang-Fa Zhou, Yong-Sheng Zhang +1 more

Marco Frasca

We present an analysis of the adiabatic approximation to understand when it applies, in view of the recent criticisms and studies for the validity of the adiabatic theorem. We point out that this approximation is just the leading order of a perturbation series, that holds in a regime of a perturbation going to infinity, and so the conditions for its validity can be only obtained going to higher orders in the expansion and removing secular terms, that is terms that runs to infinity as the time increases. In this way, it is always possible to get the exact criteria for the approximation to hold.

Yi-Tian Ding

YI Xue-hu

The quantum traditional adiabatic theorem was present and traditional approximation conditions were given.Based on existence of insufficence about quntum traditional adiabatic approximation condition,the new adiabatic theory and approximation condition were proposed and discussed by means of perturbative approach and U(1)-invariant expasion.The properties of quantum geometric potential contained in the new adiabatic condition were analyzed and discussed.