Remote entanglement of trapped ions through interference of high-dimensional time-bin-encoded photons*

Trapped ions are a leading platform for quantum technologies, including computing, sensing and communication. 

They serve as nearly perfect and replicable quantum memories as well as interfaces to pristine single photons for quantum networking. 

In this work, we harness the multi-level structure of $^{138}\textrm{Ba}^+$ ions to implement high-dimensional (HD) quantum memories, or qudits, with dimensionality up to $d=4$. 

These memories are entangled with HD time-bin-encoded single photons, allowing the heralded entanglement between two spatially-separated qudit memories, through the interference of the two HD single photons. 

The successful detection of both photons generates a maximally entangled Bell state with a higher success probability compared to conventional qubit-based schemes.

This type of remote qudit entanglement may allow new HD quantum communication protocols between multi-level quantum memories.