Beyond the quantum adiabatic approximation: Adiabatic perturbation theory

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.