Quantum phase corrections from adiabatic iteration

Abstract

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 C_k 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 C_k 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.