  Atomic spin-wave (SW) memory is a building block for quantum repeater.
However, due to decoherence caused by atomic motions and magnetic-field
gradients, SW retrieval efficiency decays with storage time. Slow
magnetic-field fluctuations (SMFFs) lead to shot-to-shot phase noises of SWs,
but have not been considered in previous repeater protocols. Here, we develop a
SW model including a homogeneous phase noise caused by SMFFs, and then reveal
that such phase noise induces decoherence of single-excitation entanglement
between two atomic ensembles in a repeater link. For verify this conclusion, we
experimentally prepare single-excitation entanglement between two SWs in a cold
atomic ensemble and observe entanglement decoherence induced by SMFFs. Limited
by SMFFs, the evaluated lifetime of the single-excitation entanglement over a
repeater link is ~135 ms, even if the link uses optical-lattice atoms as nodes.
We then present an approach that may overcome such limitation and extend this
lifetime to ~1.7 s.
