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Orbital angular momentum of photons is an intriguing system for the storageand transmission of quantum information, but it is rapidly degraded byatmospheric turbulence. We explore the ability of adaptive optics to compensatefor this disturbance by measuring and correcting cumulative phase shifts in thewavefront. These shifts can be represented as a sum of Zernike functions; weanalyze the residual errors after correcting up to a certain number of Zernikemodes when an orbital angular momentum state is transmitted through a turbulentatmosphere whose density fluctuations have a Kolmogorov spectrum. Weapproximate the superoperator map that represents these residual errors andfind the solution in closed form. We illustrate with numerical examples howthis perturbation depends on the the number of Zernike modes corrected and theorbital angular momentum state of the light.

Recovering Quantum Information in Orbital Angular Momentum of Photons by Adaptive Optics