Quantum phase corrections from adiabatic iteration

Abstract

The phase change γ acquired by a quantum state | ψ ( t )> driven by a hamiltonian H 0 ( 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 H 0 ( 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.