Evolution equation for classical and quantum light in turbulence

Abstract

Recently, an infinitesimal propagation equation was derived for the evolution of orbital angular momentum entangled photonic quantum states through turbulence. The authors will discuss its derivation and application within both classical and quantum contexts. While quantum information science promises significant advances in information technology, such as enhanced security in communication, it is often based on quantum entanglement, which is a fragile resource. In particular, free-space quantum communication systems that are based on the entanglement of the spatial modes of photons, are adversely affected by the loss of entanglement due to turbulence in the atmosphere. For this reason it became important to understand how entanglement in the spatial degrees of freedom of photons decays in turbulence. First, a single phase screen approach [1] was proposed to compute the evolution of spatial modes in turbulence, but it assumes weak scintillation conditions. To overcome this limitation, an equation was proposed [2,3] for the evolution of a photonic quantum state, propagating in arbitrary atmospheric conditions. Originally, this infinitesimal propagation equation (IPE) was presented directly in terms of the Laguerre-Gaussian (LG) modes, which is an orbital angular momentum (OAM) basis. As such, the IPE consists of an infinite set of coupled first order differential equations. To solve this IPE one needs to truncate the set, which introduces errors that render the solutions inaccurate [4]. So far the IPE in this form has only been solved for small numbers of dimensions: 2 [2] and 3 [5]. The truncation problem is solved by expressing the IPE in the plane wave basis (spatial Fourier domain), instead of the LG basis. The result is a single differential equation that can be solved without truncation. For a single photon, its expression is given by [2, 3].