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.