Single and Double-click High-Rate Entanglement Generation Between Distant Ions Using Multiplexed Atomic Ensembles

In an accompanying paper [arxiv:2511.04488], we introduced an approach to interface trapped-ion quantum processors with ensemble-based quantum memories by matching a spontaneous parametric down conversion source to both the ions and the memories. This enables rapid entanglement generation between single trapped ions separated by distances of hundreds of kilometers. In this article, we extend the protocol and provide additional details of the analysis. Particularly, we compare a double-click and single-click approaches for the ion edge nodes. The double-click approach relaxes the phase stability requirement but is strongly affected by finite efficiencies. Choosing the optimal protocol thus depends on the access to the phase stabilization as well as the efficiencies of the interfaces of the ions and ensemble-based memories.