We derive a first-order differential equation for the decoherence of an orbital angular momentum entangled biphoton state propagating through a turbulent atmosphere. The derivation is based on the distortion that orbital angular momentum states experience due to propagation through a thin sheet of turbulent atmosphere. This distortion is treated as an infinitesimal transformation leading to a first-order differential equation, which we call an infinitesimal propagation equation. The equation is applied to a simple qubit case to show how the entanglement decays.
Instead of considering the transformation properties of irreducible tensors under finite rotations, we will now consider transformations under infinitesimal rotations. Those rotations can be characterized by choosing a definite axis and a rotation around it by an infinitesimal angle δϕ . Consider rotations around the z-axis ( z′ = z ). Under a finite anticlockwise rotation of the frame of reference by an angle ϕ , the coordinates x′, y′ are given in terms of x, y by ( 6.1 ) x ′ = x cos ϕ + y sin ϕ x = x ′ cos ϕ − y ′ sin ϕ y ′ = − x sin ϕ + y cos ϕ y = x ′ sin ϕ + y ′ cos ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq268.tif"/>
The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.