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

Infinitesimal propagation equation for atmospheric decoherence with multiphoton correlations