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

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.