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.
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