  Orbital angular momentum of photons is an intriguing system for the storage
and transmission of quantum information, but it is rapidly degraded by
atmospheric turbulence. We explore the ability of adaptive optics to compensate
for this disturbance by measuring and correcting cumulative phase shifts in the
wavefront. These shifts can be represented as a sum of Zernike functions; we
analyze the residual errors after correcting up to a certain number of Zernike
modes when an orbital angular momentum state is transmitted through a turbulent
atmosphere whose density fluctuations have a Kolmogorov spectrum. We
approximate the superoperator map that represents these residual errors and
find the solution in closed form. We illustrate with numerical examples how
this perturbation depends on the the number of Zernike modes corrected and the
orbital angular momentum state of the light.
