Robust and efficient quantum repeaters with atomic ensembles and linear optics

In the last few years there has been a lot of interest in quantum repeater protocols using only atomic ensembles and linear optics. Here we show that the local generation of high-fidelity entangled pairs of atomic excitations, in combination with the use of two-photon detections for long-distance entanglement generation, permits the implementation of a very attractive quantum repeater protocol. Such a repeater is robust with respect to phase fluctuations in the transmission channels, and at the same time achieves higher entanglement generation rates than other protocols using the same ingredients. We propose an efficient method of generating high-fidelity entangled pairs locally, based on the partial readout of the ensemble-based memories. We also discuss the experimental implementation of the proposed protocol.

Figure 1

(Color online) Comparison of different quantum repeater protocols that all use only atomic ensembles and linear optics. The quantity shown is the average time needed to distribute a single entangled pair for the given distance. (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. (B) The original DLCZ protocol that uses single-photon detections for both entanglement generation and swapping [2]. (C) The protocol of Ref. [10], Sec. III B, which first creates entanglement locally using single-photon detections, and then generates long-distance entanglement using two-photon detections. (D) Protocol of Ref. [11] that uses quasi-ideal single photon sources (which can be implemented with atomic ensembles, cf. text) plus single-photon detections for generation and swapping. (E) Protocol of Ref. [10], Sec. III C, that locally generates high-fidelity entangled pairs and uses two-photon detections for entanglement generation and swapping. (F) The proposed new protocol which follows the approach of Ref. [10], Sec. III C, but uses an improved method of generating the local entanglement. The performance of the protocol of Ref. [9] is close to curve B for this distance range (the authors announce a factor of 2 improvement over DLCZ at 640 km and a factor of 5 at 1280 km). For all the curves we have assumed memory and detector efficiencies of 90%. The numbers of links in the repeater chain are optimized for curves B and D, e.g., giving four links for 600 km and eight links for 1000 km for both protocols. For curves C, E, and F, we imposed a maximum number of 16 links (cf. text), which is used for all distances for curve C and for distances greater than 400 km for curves E and F.

Figure 2

(Color online) Setup for generating high-fidelity entangled pairs of atomic excitations. Yellow squares represent atomic ensembles which probabilistically emit Stokes photons (green dots). The conditional detection of a single Stokes photon heralds the storage of one atomic spin-wave excitation. In this way an atomic excitation is created and stored independently in each ensemble. Then all four ensembles are simultaneously read out partially, creating a probability amplitude to emit an anti-Stokes photon (red dots). The coincident detection of two photons in $d_{+}$ and ${\tilde{d}}_{+}$ projects nondestructively the atomic cells into the entangled state $|{\Phi }_{\text{ab}}⟩$ of Eq. (1); $d_{+}\text{−}{\tilde{d}}_{-}$, $d_{-}\text{−}{\tilde{d}}_{+}$, and $d_{-}\text{−}{\tilde{d}}_{-}$ coincidences, combined with the appropriate one-qubit transformations, also collapse the state of the atomic cells into $|{\Phi }_{\text{ab}}⟩$. Half circles represent photon detectors. Vertical bars within squares label polarizing beam splitters (PBSs) that transmit (reflect) $H$ $\left(V\right)$-polarized photons. The central PBS with a circle performs the same action in the $\pm 45°$ basis.

Figure 3

(Color online) (A) Long-distance entanglement creation using two four-ensemble sources as shown in Fig. 2. The A and D ensembles are entangled by the detection of two photons emitted from the B and C ensembles, using the same setup as in Refs. [9, 10, 12]. Note that the AB source is separated from the CD source by a long distance. (B) Entanglement swapping. The same set of linear optical elements allows one to entangle the A and H ensembles belonging to two adjacent elementary links. Note that the D and E ensembles are at the same location.