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In this paper, we investigate the propagation of two-mode spatiallyGaussian-entangled quantum light fields passing through the turbulenceatmosphere. From the propagation formula of the two-mode wave function in theposition representation, we have derived the analytical expressions for thefidelity, purity and logarithmic negativity (entanglement) of the resultingquantum state after the long-distance atmospheric transportation. Based on thederived formulae, the effects of the atmospheric turbulences on the evolutionsof quantum properties of the resulting two-mode quantum state are discussed indetail under different input parameters of the initial two-mode quantum state.The results show that the maximal distributing distance of quantum entanglementis strongly dependent on the atmospheric conditions: when the atmosphericturbulence becomes stronger and stronger, the maximal distance becomes shorterand shorter, and both the fidelity and purity decrease quicker and quicker asfunctions of propagating distances. Under a certain atmospheric condition, withthe increasing of the input entanglement of the initial two-mode spatiallyGaussian-entangled quantum state, the maximal distributing distance forpreserving the entanglement gradually increases and always has a saturated(upper) limitation, and both the evolutions of the fidelity and purity areaffected by the input parameters of the initial two-mode quantum state, Finallythe optimal parameters of the input two-mode quantum state with the fixed inputentanglement are discussed in order to obtain the optimal transfer distributionof the quantum entanglement over a long distance under a certain atmosphere.Our theoretical results are very helpful for building the distribution of thequantum entanglement via free-space atmosphere link.

Fidelity, purity and entanglement of two-mode spatially Gaussian-entangled light fields in Turbulence Atmosphere