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Infinitesimal propagation equation for atmospheric decoherence with multiphoton correlations

Previously a set of coupled first order differential equations were derived for the decoherence of a pair of spatial mode entangled photons, propagating along different paths through turbulence. Here we extend this analysis to the situation where both photons travel along the same path, which introduces the effect of multiple photon correlations. The resulting equation now contains additional terms that take these multiphoton correlations into account. At the same time, we provide a more thorough formulation of the quantized field, starting from a Lorentz invariant formulation, which is then explicitly broken by the choice of a particular propagation direction. The effect of the latter improvement in the quantization on the form of the final equation is minimal.