Quantum phase corrections from adiabatic iteration

The phase change γ acquired by a quantum state |ψ(t)> driven by a hamiltonian H0(t), which is taken slowly and smoothly round a cycle, is given by a sequence of approximants γ(k) obtained by a sequence of unitary transformations. The phase sequence is not a perturbation series in the adiabatic parameter ∊ because each γ(k) (except γ(0)) contains ∊ to infinite order. For spin-½ systems the iteration can be described in terms of the geometry of parallel transport round loops Ck on the hamiltonian sphere. Non-adiabatic effects (transitions) must cause the sequence of γ(k) to diverge. For spin systems with analytic H0(t) this happens in a universal way: the loops Ck are sinusoidal spirals which shrink as ∊k until k ~ ∊-1 and then grow as k!; the smallest loop has a size exp{-1/∊}, comparable with the non-adiabaticity.