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

TL;DR

This work analyzes two heralded entanglement protocols for linking distant trapped ions via multiplexed SPDC sources and multimode memories (SPDC+M). By modeling realistic detectors, memory losses, dark counts, and time-bin multiplexing, it derives the edge-node and backbone states, swaps, and rate expressions for both two-single-click and double-click schemes, including a purification step for the former. The results show that multiplexed hybrid nodes can dramatically speed up remote ion entanglement relative to direct ion–ion links, with the single-click protocol favored when phase stability is achievable and memory losses are moderate, while the double-click protocol benefits when phase stability is harder to maintain or memory efficiency is high. The findings underscore the potential of hybrid architectures to enable high-rate quantum networking over hundreds of kilometers, and highlight memory efficiency and phase-stability as critical design levers for practical implementations.

Abstract

In an accompanying paper [1], 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 efficiency of interface of the ions and ensemble-based memories.

Figures (6)

• Figure 1: Sketch of the fundamental steps of the single-click protocol. In (a), we show the optical setup of an edge node (EN) to interface an ion with a single rail SPDC+M node. In this step, wavelength conversion to match the central frequency combined with the multi-mode nature of the photonic wave function before the detection is used to match the SPDC photon field to the ion. (b) The backbone (BB) optical setup to generate long-distance entanglement between remote memories using a central heralding station. In both (a) and (b) a click of detector $+$ or $-$ heralds the generation of entanglement. (c) An optical setup analogous to (a) and (b) implements entanglement swapping between memories of EN and BB.
• Figure 2: Sketch of the entanglement generation protocols, including a multimode repeater node at the center. The single-click protocol (a) is visualized using three steps: (i) entangled states are generated between memories in the BB (spanning the long distance $L/2$) and between a multimode memory and an ion in the ENs. (ii) The range of the ion-photon entanglement is extended to a remote memory using an optical entanglement swap, and finally (iii) another entanglement swap heralds entanglement between the ions. (b) Generating entanglement between two ion pairs enables purification using a node-local CNOT gate, followed by read-out of the controlled bits. Detecting both in $\mathinner{|{1}\rangle}$ heralds an entangled state of higher fidelity of the remaining ion pair. The double-click protocol (c) uses (i) two clicks to entangle the ions with two rails and then proceeds with similar steps (ii) and (iii) as for the single-click protocol.
• Figure 3: Sketch of the edge node (EN) setup in the double-click protocol. The setup is similar to the single-click EN sketched in Fig. \ref{['fig:modules']}(a), but the ion emits photons with two different polarizations entangled with internal states of the ion. The polarizations are split using a polarizing beamsplitter (light gray slashed square) and thereby connected to SPDC+M nodes of individual rails.
• Figure 4: (a,c) Average preparation duration for remote ion-ion entanglement and (b,d) average worst case storage duration within the multimode memory as functions of ion-ion distance. The color encodes the protocol, where blue (orange) is the multiplexed two-single-click (double-click) protocol using the hybrid edge nodes and green (pink) is the direct single-click (double-click) between ions, see legend. Additionally, we compare a repeater-less setup (dashed lines) with the use of a central repeater node (solid lines). We take the central repeater to be an SPDC+M node for the multiplexed protocols and an ion repeater for the direct protocols. We use the simplified parametrization introduced in the main text with a target fidelity $F_T=0.9$, global efficiency $\eta_I = 0.8$, edge node duty cycle $O_{\text{EN}}=0.1$ (to account for cooling and initialization of the ions), $N_{\text{BB}}=1000$ multiplexing channels, dark count rate of $10^{-3}\,\text{s}^{-1}$, and an ion-photon duration of $T_a = 10\,$µs, which we match with $N=10$ SPDC time-bins. Finally we use a memory efficiency $\eta_m =0.5$ (a,b) or $\eta_m =0.8$ (c,d).
• Figure 5: Sketch of the optical setup for single-click entanglement generation. Two channels $a$ and $b$ are combined using a beamsplitter (gray) whoose output ports terminate in detectors $d_\pm$. Losses in the channels and detectors are modeled using BS (light pink) with additional vacuum input $\mathinner{|{\emptyset}\rangle}$ and output into loss channels $\cdot_L$.