Quantum repeaters based on single trapped ions

We analyze the performance of a quantum repeater protocol based on single trapped ions. At each node, single trapped ions embedded into high-finesse cavities emit single photons whose polarization is entangled with the ion state. A specific detection of two photons at a central station located half-way between two nodes heralds the entanglement of two remote ions. Entanglement can be extended to long distances by applying successive entanglement swapping operations based on two-ion gate operations that have already been demonstrated experimentally with high precision. Our calculation shows that the distribution rate of entanglement achievable with such an ion-based quantum repeater protocol is higher by orders of magnitude than the rates that are achievable with the best known schemes based on atomic-ensemble memories and linear optics. The main reason is that for trapped ions the entanglement swapping operations are performed deterministically, in contrast to success probabilities below 50% per swapping with linear optics. The scheme requires efficient collection of the emitted photons, which can be achieved with cavities, and efficient conversion of their wavelength, which can be done via stimulated parametric down-conversion. We also suggest how to realize temporal multiplexing, which offers additional significant speed ups in entanglement distribution, with trapped ions.

Figure 1

(Color online) Setup for entanglement creation based on two-photon detection of remote ions (brown dots) embedded into cavities. Each ion emits a photon, whose polarization is entangled with the atomic state, leading to the state $|{\Psi }^{\text{A}}⟩\otimes |{\Psi }^{\text{B}}⟩$ of Eq. (1). The two photons, one coming from location A and the other one from B, are combined on a polarizing beam splitter to be further detected in a polarization basis rotated by $45°$ with respect to the $H\text{−}V$ basis. The coincident detection of two photons in the corresponding modes $d_{+}$ and ${\tilde{d}}_{+}$, for example, projects the two ions into the entangled state $|{\psi }_{+}^{\text{AB}}⟩$ of Eq. (2).

Figure 2

(Color online) Performance of quantum repeaters based on single ions versus atomic ensembles. The quantity shown is the average time for the distribution of one entangled pair for the given distance. Curve A: as a reference, the time required using direct transmission of photons through optical fibers, with losses of 0.2 dB/km, corresponding to the best available telecom fibers at a wavelength of $1.5\mu \text{m}$ and a pair generation rate of 10 GHz. Curve B: protocol based on atomic ensembles in Ref. [6]. High-fidelity entangled pairs are generated locally, and entanglement generation and swapping operations are based on two-photon detections. We have assumed memory and detector efficiencies of 90%. We imposed a maximum number of 16 links in the repeater chain (see [6] for details). This approach leads to a repeater protocol that, as far as we know, achieves the highest entanglement distribution rate with atomic ensembles and linear optics. Curve C and D: protocol based on single ions with 8 and 16 links, respectively. We have assumed a success probability for the ion to emit a photon of $p=90\%$ requiring high-finesse cavity (cf. text).

Figure 3

(Color online) Robustness of a repeater based on single trapped ions with respect to the success probability for an ion to emit a photon $p$ (photon source efficiency) which includes the probability to prepare the ion into the cavity mode and coupling into the fiber, as well as the frequency conversion to match the telecom wavelength. The quantity shown is the average time for the distribution of an entangled pair over 1000 km for a repeater with 16 elementary links [Eq. (5)].

Figure 4

Relevant levels for ${{}^{40}\text{C}\text{a}}^{+}$.

Figure 5

(Color online) Setup for temporal multiplexing. Chains of ions are transported through cavities such that the ions are excited one by one when they interact with the cavity mode. The ions of the chain B are alternatively excited in the upper cavity for entanglement creation between A and B or in the lower cavity for entanglement creation between B and C. If entanglement has been established between the $m^{\text{th}}$ ions for the link A-B and between the $n^{\text{th}}$ ions for B-C, entanglement swapping is done by performing a Bell state analysis on the $m^{\text{th}}$ and $n^{\text{th}}$ ions of the B chain.