Berry-like phases in structured atoms and molecules

Quantum mechanical phases arising from a periodically varying Hamiltonian are considered. These phases are derived from the eigenvalues of a stationary, ``dressed'' Hamiltonian that is able to treat internal atomic or molecular structure in addition to the time variation. In the limit of an adiabatic time variation, the usual Berry phase is recovered. For more rapid variation, non-adiabatic corrections to the Berry phase are recovered in perturbation theory, and their explicit dependence on internal structure emerges. Simple demonstrations of this formalism are given, to particles containing interacting spins, and to molecules in electric fields.