# An integrated quantum repeater at telecom wavelength with single atoms in optical fiber cavities

**Part of the following topical collections:**

## Abstract

Quantum repeaters promise to enable quantum networks over global distances by circumventing the exponential decrease in success probability inherent in direct photon transmission. We propose a realistic, functionally integrated quantum-repeater implementation based on single atoms in optical cavities. Entanglement is directly generated between the single-atom quantum memory and a photon at telecom wavelength. The latter is collected with high efficiency and adjustable temporal and spectral properties into a spatially well-defined cavity mode. It is heralded by a near-infrared photon emitted from a second, orthogonal cavity. Entanglement between two remote quantum memories can be generated via an optical Bell-state measurement, while we propose entanglement swapping based on a highly efficient, cavity-assisted atom-atom gate. Our quantum-repeater scheme eliminates any requirement for wavelength conversion such that only a single system is needed at each node. We investigate a particular implementation with rubidium and realistic parameters for Fabry–Perot cavities based on \(\hbox {CO}_2\) laser-machined optical fibers. We show that the scheme enables the implementation of a rather simple quantum repeater that outperforms direct entanglement generation over large distances and does not require any improvements in technology beyond the state of the art.

## 1 Introduction

Heralded entanglement between particles separated by large distances is a valuable resource in quantum communication. The applications range from fundamental tests of quantum physics like loophole-free Bell tests [1, 2] to device-independent quantum key distribution (QKD) [3, 4]. The distribution of entanglement via the direct transmission of photons via fiber optics is practically impossible for large distances, because the inevitable losses in optical fibers cause an exponential decrease of the success rate with distance. This can be overcome by dividing the distance into smaller segments with quantum-repeater nodes in between [5].

The performance of quantum repeaters can be characterized by the rate at which pairs of quantum memories separated by a given distance can be entangled with high fidelity. The obvious benchmark for a quantum repeater to beat is the rate achievable via direct transmission. The critical parameter in the latter case is the attenuation length of the optical fiber, which is maximal at telecom wavelengths around 1.3 and 1.5 μm, where the absorption in optical fibers is low [6, 7]. In order to keep the technological overhead minimal, it is therefore essential to operate a quantum repeater at a telecom wavelength. Additional requirements for an efficient quantum repeater are quantum memories with coherence times by far exceeding the average time required for the protocol to succeed and high-efficiency and high-fidelity implementations of all subparts of the protocol. These include entanglement generation and entanglement distribution via photons as well as entanglement swapping using single-qubit operations, two-qubit operations and state readout. Against this backdrop, various systems have been proposed for the implementation of a quantum repeater, e.g., atomic ensembles [8], single neutral atoms and ions [9], nitrogen vacancy centers [10], quantum dots [9] and ion-doped solids [11].

Single atoms in optical cavities are especially promising [12], because they can be isolated from the environment to provide long coherence times [13] and have been shown to be an efficient light-matter interface [14]. Several subparts of a quantum-repeater protocol have been successfully implemented with these systems at near-infrared wavelengths, e.g., the generation of atom-photon entanglement [15], an atom-photon quantum gate [16] and the heralded storage of a photonic quantum bit [17]. Demonstrations of a quantum repeater that go beyond proof of concept will have to find a way to combine these operations with telecom-wavelength photons. This has recently sparked very active research in external devices that convert photonic qubits at wavelengths in the near-infrared to telecom wavelengths [18, 19]. An alternative route being pursued is the generation of an entangled photon pair via spontaneous parametric down-conversion, with one photon at telecom wavelength and the other in the near-infrared, followed by storage of the near-infrared photon in a quantum memory [20, 21]. While these strategies seem straightforward, they come at the price of a large technological overhead and reduced efficiency.

In the following, we describe the implementation of the individual parts of the scheme. In Sect. 2 we propose a way to directly generate entanglement between single atoms and a telecom-wavelength photon using a cascaded scheme. We will estimate the performance of a particular implementation using \(^{87}\)Rb and realistic cavity parameters. In Sect. 3 we investigate the indistinguishability of the telecom photons created in this way and show that they are well suited for a photonic BSM that entangles the two remote single-atom quantum memories. Entanglement swapping by atomic BSM will be the topic of Sect. 4. In Sect. 5 we analyze the performance of the full repeater scheme and show that it can outperform entanglement generation based on direct transmission. We furthermore discuss the prospects of integrating entanglement purification in our scheme. While all required components can be implemented in our cavity-based approach, we show that under the assumption of realistic fidelities, entanglement purification is of little use for a quantum repeater over moderate distances with a small number of swap levels.

## 2 Heralded atom-photon entanglement at telecom wavelength

### 2.1 Cascaded entanglement scheme

We assume an atom with a level scheme compatible with the one depicted in Fig. 2b, which resembles the levels of bosonic alkali atoms. Essential for the protocol are two degenerate, long-lived states \(\vert {\pm 1}\rangle _f\), which are Zeeman sublevels with \(m_F=\pm 1\) and will be used to store the atomic qubit. These states are each coupled via a \(\pi\)-transition at wavelength \(\lambda _{\text{h}}\) to one of the two short-lived, intermediate states \(\vert {\pm 1}\rangle _i\), which also feature \(m_F=\pm 1\). The excited state \(\vert {e}\rangle\), with \(m_F=0\), can decay to \(\vert {\pm 1}\rangle _i\) at wavelength \(\lambda _{\text{t}}\). The atom is initialized in a third ground state \(\vert {g}\rangle\) with \(m_F=0\) and can be coupled to \(\vert {e}\rangle\) via a two-photon transition far detuned from an intermediate state \(\vert {k}\rangle\), which may be identical to \(\vert {i}\rangle\). The atom is placed at the intersection of two perpendicular cavity modes. The cavities are named heralding and entangling cavity and are resonant at \(\lambda _{\text{h}}\) and \(\lambda _{\text{t}}\), respectively. To describe the system, we choose the quantization axis to coincide with the axis of the entangling cavity. This way, the entangling cavity supports \(\sigma ^-\)- and \(\sigma ^+\)-polarization and the heralding cavity supports linear polarization modes, one of which needs to be aligned with \(\pi\)-polarization.

The proposed scheme is suitable for all bosonic isotopes of alkali atoms. Implementations with rubidium, cesium and francium are particularly interesting, because these elements have suitable transitions at wavelengths in telecom bands.

### 2.2 Implementation with \(^{87}\)Rb

To evaluate the performance of the proposed scheme with current technology, we investigate a particular implementation with \(^{87}\)Rb (Fig. 2c) more closely. We choose the hyperfine states \(\vert {F=1;m_F=0}\rangle\), \(\vert {F=2; m_F=-1}\rangle\) and \(\vert {F=2; m_F=+1}\rangle\) of the \(5^2S_{1/2}\) state as \(\vert {g}\rangle\), \(\vert {-1}\rangle _f\), and \(\vert {+1}\rangle _f\), respectively. As intermediate states \(\vert {\pm 1}\rangle _i\) the \(\vert {F'=1;m_F=\pm 1}\rangle\) substates of the \(5^2P_{1/2}\) manifold are used and the \(4^2D_{3/2}\vert {F''=1; m_F=0}\rangle\) state serves as the excited state \(\vert {e}\rangle\). The \(5^2P_{3/2}\) state can be employed as the intermediate state \(\vert {k}\rangle\) for the two-photon control. This requires a heralding cavity resonant at \(\lambda _{\text{h}}=795\,\hbox {nm}\) and will generate entangled photons at \(\lambda _{\text{t}}=1476\,\hbox {nm}\), which is in the S-band of optical fiber communication.

The proposed scheme requires two cavities with small mode volumes and intersecting modes. We propose to use a Fabry–Perot cavity based on \(\hbox {CO}_2\) laser-machined optical fibers [23] as the entangling cavity, because this type of cavity combines low mode volumes, high finesse and excellent optical access due to small external dimensions. Using telecom fibers as substrates also provides direct fiber integration, which facilitates long-distance communication. Fiber cavities can support degenerate polarization eigenmodes [24]. This is assumed for the entangling cavity, such that the atomic state gets entangled with the polarization of the emitted telecom photon, but not with its frequency. Starting from realistic cavity parameters (see “Appendix 1” for details), we calculate an atom-light coupling rate of \(g_{\text{t}}=2\pi \times 70\,{\text{MHz}}\), a cavity field decay rate through the outcoupling mirror of \(\kappa _{\text{t}}^{\text{oc}}=2\pi \times 95\,{\text{MHz}}\), and an additional cavity decay caused by other losses of \(\kappa _{\text{t}}^{\text{l}}=2\pi \times 8\,{\text{MHz}}\) for the entangling cavity. Photons leaving this cavity through the output coupler couple to the mode of a standard single-mode telecom fiber with an efficiency of 0.96.

For the heralding cavity, fiber integration is not necessary, and we therefore propose to use \(\hbox {CO}_2\) laser-machined glass plates as mirror substrates [25]. Degeneracy of its polarization eigenmodes is undesirable, because the mode with polarization orthogonal to \(\pi\) can enhance wrong decay paths (shown as dashed arrows in Fig. 2). This is detectable, because it does not result in a \(\pi\)-polarized herald photon. Consequently, the failed entangling attempt is discarded, thereby reducing the efficiency. To avoid this problem, we suggest to employ a heralding cavity with a large frequency splitting of the polarization eigenmodes [26], such that the mode orthogonal to \(\pi\)-polarization is far detuned from the atomic transition. We assume the heralding-cavity parameters to be \(\{ g_{{\text{h}}} ,\kappa _{{\text{h}}} ^{{{\text{oc}}}} ,\kappa _{{\text{h}}}^{{\text{l}}} \} = 2\pi \times \{ 16.3,11.9,1.5\} {\mkern 1mu} {\text{MHz}}\). The total cavity decay rates are given by \(\kappa _{\text{t}}=\kappa _{\text{t}}^{\text{oc}}+\kappa _{\text{t}}^{\text{l}}\) and \(\kappa _{\text{h}}=\kappa _{\text{h}}^{\text{oc}}+\kappa _{\text{h}}^{\text{l}}\). The cooperativity of the atom-cavity system is \(C_{\text{t(h)}} = g_{\text{t(h)}}^2 / \left( \kappa _{\text{t(h)}} \varGamma _{\text{t(h)}}\right) = 25~(3.4)\) for the entangling (heralding) atom-cavity system. Here, \(\varGamma _{\text{t}} = 2\pi \times 1.92\,{\text{MHz}}\,(\varGamma _{\text{h}}=2\pi \times 5.75\,{\text{MHz}})\) is the decay rate of the \(4^2D_{3/2}\)\((5^2P_{1/2})\) state of \(^{87}\)Rb.

The second method is a Monte Carlo wave-function approach [27, 28, 29], which yields information about correlations between the cavity outputs. From the latter, we calculate the overall success probability \(p_{\text{ht}}\), i.e., the probability to obtain a telecom photon in the optical fiber and a herald photon leaving the heralding cavity through the output coupler. For the parameters chosen here, we find \(p_{\text{ht}}=0.57\), basically independent of the length of the control pulse (see Fig. 4). The exception are very short control pulses, for which we calculate lower success probabilities, because the control pulse gets spectrally broad enough to excite the \(4^2D_{3/2}\vert {F''=3; m_F=0}\rangle\) state. The main loss channels are spontaneous decay of the \(5^2P_{1/2}\) state (probability 0.24) and parasitic losses in the entangling and heralding cavity (probability 0.08 and 0.07, respectively). By increasing the coupling between atom and heralding cavity and decreasing the parasitic losses in both cavities, these loss channels could be minimized. Thus, with improvements in technology, the scheme could be performed almost deterministically.

### 2.3 Fidelity

At the start of the protocol the atom needs to be initialized in \(\vert {g}\rangle\). This can be achieved by optical pumping, which is susceptible to experimental imperfections. Because there is no fundamental limit and various strategies are possible, we assume that the atom is initialized in \(\vert {g}\rangle\) with unit efficiency and the control pulse addresses the state \(\vert {e}\rangle\) without exciting any other state. Under these conditions, the fidelity of the intermediate atom-photon state with the ideal state \(\vert {\varPsi _1}\rangle\) is determined solely by the geometry of the system and we assume any photon leaving the entangling cavity to be in the correct entangled state. This is a very good approximation if the \(5^2P_{1/2}\) state of \(^{87}\)Rb is used as the intermediate state, because undesired processes are far off-resonant. The transition from the intermediate to the final state is unambiguously heralded by a photon with the correct polarization at wavelength \(\lambda _{\text{h}}\). Therefore, conditioned on the detection of a herald photon, high-fidelity entanglement generation should be possible. In the following, we study two potentially detrimental effects and demonstrate that their influence on the fidelity of the entangled state is marginal for the particular implementation and parameters chosen here.

#### 2.3.1 Second polarization mode of the heralding cavity

Degenerate polarization eigenmodes of the heralding cavity reduce the efficiency of the protocol. Besides this obvious effect, the unwanted resonant mode of the herald cavity also has a subtle but detrimental impact on the fidelity, because it leads to a state-dependent probability for herald generation (cf. “Appendix 2”).

In the implementation with \(^{87}\)Rb (Sect. 2.2), the effect is small, because the unwanted mode of the heralding cavity couples only weakly to the state with \(m_F=0\) compared to the coupling to the states with \(m_F\ne 0\). This is due to the specific branching ratios of the states involved. Even if the orthogonal polarization mode is degenerate with the \(\pi\)-polarized heralding mode, the reduction in fidelity is only 0.15 % (see “Appendix 2”). The effect is minimized if there is a large frequency splitting between the polarization eigenmodes, as also required to maximize the efficiency.

#### 2.3.2 Free-space decay

If the atom decays via free-space emission, a state different from the desired one might be created. If no herald photon is emitted, the resulting state can be discarded. This results in a reduced efficiency but leaves the process fidelity unaffected. But, if the intermediate states \(\vert {\pm 1}\rangle _i\) can decay back to the initial state \(\vert {g}\rangle\) or any other state that can be excited by the control, there is a chance for multi-photon events. Reexcitation of \(\vert {e}\rangle\) can result in a second telecom photon, which leaves the system in an undesired entangled state, but can still result in the generation of a herald. There are two ways to minimize the generation of multiple telecom photons. The first one is to choose the intermediate states in such a way that spontaneous decay preferably puts the atom in states that are not excited by the control laser. The second one is to execute the scheme quickly on the time scale of atomic decay. In the limit of fast excitation [30], for example with a picosecond laser, the detrimental effect would be eliminated. However, this would remove the ability to influence the shape of the photonic wave packet with the control lasers and a broadband pulse might excite other states, which do not couple to the entangling cavity and thus reduce the efficiency.

We employ the Monte Carlo wave-function approach to calculate an upper bound for multi-photon events for the parameters of Sect. 2.2 and a short Gaussian control pulse of 5.9 ns FWHM. We assume the worst case, namely that all decays to states outside the simulated system will result in a decay to the initial state, from which the atom can be efficiently excited again by the control laser. We compare the number of calculated quantum trajectories which result in multiple photons being generated to the number of desired trajectories and find that less than 0.4 % of the herald photons are accompanied by more than one telecom photon. More realistically, the atom will also decay to other states, which might not be excited as efficiently as the initial state, such that the number of multi-photon events should be lower than this upper bound.

## 3 Remote entanglement of two systems

The creation of remote entanglement between two of the quantum-repeater nodes described in the previous section can be achieved by an optical BSM [31, 32] on the polarization of the two telecom-wavelength photons that are entangled with the atom at their respective repeater node. The BSM is based on two-photon interference and therefore requires the telecom photons to be indistinguishable in their temporal, spectral and spatial properties, despite their remote origins. Single atoms in optical cavities can be controlled to exactly defined conditions, enabling indistinguishable photon-generation processes at remote locations [33]. However, possible temporal correlations of the telecom photon with the herald photon have to be considered. Emission via a two-photon cascade in free space shows a clear order of the photons. Emission of a first photon from the upper part of the cascade is followed by an exponential decay of the photon belonging to the lower part. Conditioned on the arrival time of the second photon, the first photon will have an exponentially rising envelope, with a sharp drop to zero at the detection time of the second photon [34]. Telecom photons corresponding to herald photons detected at different times will thus have different arrival-time distributions, which renders the photons distinguishable. Postselecting on events where the herald photons were detected with the same delay relative to the control pulse would reestablish indistinguishability, but also severely limit the efficiency.

The cavity for the herald photon offers a way out: If the lifetime of the heralding cavity is very long compared to the wave-packet envelope of the generated telecom photon, the former will determine the wave packet and therefore the detection-time distribution of the herald photon. Consequently, the correlations between the detection time of the herald photon and the wave-packet shape of the telecom photon will be erased by the heralding cavity, thereby restoring indistinguishability between telecom photons of different origin. The ideal implementation requires an entangling cavity with large coupling between atom and telecom photon and large cavity decay rate. The heralding cavity should have a smaller decay rate, and a smaller coupling between atom and herald photon can be tolerated.

### 3.1 Numerical calculation of the photon indistinguishability

In order to quantify the indistinguishability of the telecom photons, we use the Monte Carlo wave-function approach to generate arrival-time pairs of telecom and herald photons. These pairs are used as input to a kernel density estimator in order to estimate the probability distribution of telecom photons conditioned on a photon leaving the heralding cavity at a specific time. As the kernel, we use a two-dimensional Gaussian function with the bandwidth along the two axes set to \(6\kappa _{\text{t}}\) and \(6\kappa _{\text{h}}\), respectively. This bandwidth should be larger than the bandwidths of the processes occurring in the cavities, so the resulting probability distribution is likely undersmoothed. The resulting telecom-photon probability, conditioned on the detection time of the herald photon, is shown in the inset of Fig. 4 for the same parameters used in the simulation depicted in Fig. 3. Under the assumption that the processes at remote locations are identical in all other aspects, we calculate the interference contrasts expected in a Hong–Ou–Mandel experiment [33, 35] for the telecom photons conditioned on photons leaving the heralding cavity at different times. Weighting the resulting contrasts with the probability distribution for the herald photons yields the average contrast *C*, which is 0.97 in this case. This two-photon interference contrast can be converted into a remote-entanglement fidelity \(F= {\frac{1}{2}}(1 + C)\) under the assumption that all other processes are perfect [36]. Thus, a fidelity of close to 0.99 should be achievable with our model parameters.

We repeat the calculation for Gaussian control pulses of different width (Fig. 4). For long control pulses, the emission time of the herald photon is correlated with the emission time of the telecom photon, which reduces the indistinguishability and thus the interference contrast of the telecom photons. For very short control pulses, there is little correlation and the interference contrast is therefore high; the success probability, however, is not maximal. Both the success probability and the interference contrast are near their respective maximum for control pulses with a FWHM between 5 ns and 10 ns. This is therefore an ideal point of operation, with minimal sacrifices in the tradeoff between efficiency and fidelity. Because of the use of undersmoothed probability distributions, the values for the contrast are limited by the variance of the Monte Carlo method and therefore present a lower limit to the theoretically achievable contrast.

The width of the envelope of telecom and herald photons is well above the timing resolution of commercially available single-photon counters. It is therefore possible to postselect events using the arrival times of telecom and herald photons. This enhances the interference contrast and thereby the entanglement fidelity, however at the cost of a reduced total success probability. In the limit of identical detection time of the herald photons and in the absence of any other effect that makes the telecom photons distinguishable, an interference contrast of unity is reached.

## 4 Entanglement swapping

To connect separate repeater links, entanglement swapping has to be performed. This requires a BSM on two quantum memories, each of which is entangled with a remote node. The small size of the fiber-based entangling cavities enables the placement of two of them in one and the same heralding cavity (Fig. 1). The heralding cavity can then be used for the collection of herald photons from both entangling cavities. To this end, the creation of atom-photon entanglement has to be alternated between the two atoms, which is possible by addressing the atoms with individual control beams and selective detuning, e.g., via a local light shift. The remote-entanglement procedure described in the previous section can thus be repeated individually for each atom until it has succeeded for both.

A quantum repeater requires quantum memories with long coherence times. The coherence between the Zeeman states \(\vert {-1}\rangle _f\) and \(\vert {+1}\rangle _f\) is limited by fluctuations of the effective magnetic field [37]. States with spin-orbit angular momentum \(J=1/2\), different hyperfine quantum number and the same magnitude but opposite sign of the magnetic quantum number feature a reduced differential Zeeman shift. Consequently, coherent superpositions of these states have a strongly reduced sensitivity to magnetic field fluctuations [38]. We therefore propose to use a microwave pulse to state-selectively transfer one of the Zeeman states (e.g., \(\vert {-1}\rangle _f\)) to the other hyperfine ground state with the same \(m_F\) with high fidelity. The corresponding final qubit states of \(^{87}\)Rb are \(\vert {F=1;m_F=-1}\rangle\) and \(\vert {F=2;m_F=+1}\rangle\). At a moderate magnetic field of about 3.23 G, the two atomic states experience the same first-order Zeeman shift and a coherence time of several seconds has been observed [13, 39].

As the heralding cavity can couple to both atoms, it is a natural choice for the implementation of an interaction mechanism for entanglement swapping [40, 41, 42]. We propose to use a quantum gate between the two atoms based on the reflection of a resonant single photon from the cavity [16, 41]. As only one of the two hyperfine ground states couples to the heralding cavity, this results in a state-dependent phase shift of \(\pi\) on the two-atom state, equivalent to a controlled-Z quantum gate [43].

To perform entanglement swapping, we start with a Hadamard single-qubit rotation on one of the atoms and then apply the controlled-Z gate and subsequently a Hadamard gate on each of the two atoms. This maps the four atomic Bell states unambiguously onto four separable atomic states. These can be detected with unity efficiency and high fidelity by performing cavity-assisted hyperfine state detection [44, 45] on each of the atoms. This entanglement swapping projects the remote nodes into one of the four Bell states. The measurement result identifies the created Bell state. Therefore, conditional single-qubit operations at the remote nodes might then be used to rotate the entangled state into a specific target state. Although realistic implementations of reflection-based gates have a failure probability, the success of the gate is heralded by the detection of the reflected photon. The method requires a single-sided cavity and equal reflectivities of the empty cavity and the coupled atom-cavity system. These requirements are compatible with the design criteria of the heralding cavity posed by the entanglement scheme presented in Sect. 2.

## 5 Quantum-repeater performance

The techniques described in the previous sections can be used to implement a quantum repeater protocol with entanglement generation, entanglement distribution and entanglement swapping. To assess the performance of such a repeater, we consider two atoms trapped in remote cavities separated by the total distance *L*, which are to be entangled. We calculate the rate at which heralded entangled pairs are produced at these end points with the help of repeater nodes separated by the distance \(L_0\) and compare it to the rate achievable without repeater nodes. If all processes succeed with high fidelity, fidelity differences between the protocols are marginal and the entangled pair rates can be compared directly. In the case of experimental imperfections, the reduced fidelity has to be compensated by a higher entangled-pair rate, e.g., to generate a secret key of the same length (see Sect. 5.3).

### 5.1 Entangled-pair rate

*N*parallel processes with individual success probability

*p*to have each succeeded once is [46]

*N*entangled pairs. The atomic BSM is not completely deterministic, but succeeds with probability \(p_{\text{es}} = R p_{\text{p}} \eta _{\text{h}}\), where

*R*is the reflectivity of the herald cavity and \(p_{\text{p}}\) is the efficiency of the SPS that generates the photon to be reflected. Therefore, we consider two different strategies for entanglement swapping. The first one is to swap all entangled pairs at once and restart the whole protocol when one entanglement swapping attempt fails (with probability \(1-p_{\text{es}}\)). In that case, the average number of attempts \(\langle n(N) \rangle\) until the protocol has succeeded over the distance \(L=N L_0\) is

The second strategy is to swap entangled pairs as soon as possible. Upon failure, all entangled pairs that were not part of the failed entanglement-swapping attempt are kept and entanglement between the others is reestablished. Obviously, for \(N>2\) this results in a higher rate of entangled pairs, but the memories also need to store the entangled states for a longer time, leading to increased decoherence. To calculate \(\langle n\left( N\right) \rangle\) for this strategy, we perform a Monte Carlo simulation with \(10^6\) runs per calculated point.

To be quantitative, we take \(p_{\text{ht}}\) as calculated in Sect. 2.2 for a control pulse with 5.9 ns FWHM, which combines high efficiency with high indistinguishability as estimated in Sect. 3.1. Correcting this value for the slightly reduced coupling of atoms not in the center of the herald cavity yields \(p_{\text{ht}} = 0.53\) (see “Appendix 1”), which is the value we use throughout this section. \(L_{\text{a}} = 22\,{\text{km}}\) and \(c_{\text{f}} = 2 \times 10^5\,{\text{km\slash s}}\) are typical parameters for commercial telecom optical fibers. We assume detectors with an efficiency of \(\eta _{\text{t}} = \eta _{\text{h}} = 0.8\) which is within range of current technology [47]. The reflectivity of the cavity is \(R=0.61\) for the considered heralding cavity (“Appendix 1”). Single atoms trapped in optical cavities have been shown to be highly efficient SPSs [48], and with state-of-the-art cavity parameters they could certainly achieve an efficiency of \(p_{\text{p}}=0.8\), which is the value we assume for the SPS used in the atomic BSM. We further consider a conservative cycle time of \(\tau = 100\,\upmu\)s. This would result in a repetition rate of 10 kHz over very short distances. For larger separations, the finite \(c_{\text{f}}\) and the resulting communication time have to be accounted for, which reduces the repetition rate. For example, at \(L_0=80\) km the repetition rate drops to 2 kHz, dominated by the communication time.

### 5.2 Required storage time

### 5.3 Secret-key rate

An especially important application of a quantum repeater is QKD. Device-independent QKD is enabled by heralded entanglement between remote nodes. The raw key extracted from an imperfect entangled state can be converted into an unconditionally secret key by classical postprocessing. In the limit of infinitely long keys, the secret-key rate can be obtained by multiplying the entangled pair rate with the secret fraction of the final state [49]. The latter can be calculated from the indistinguishability of the telecom photons characterized by their interference contrast and the fidelity of an entangled state produced by the atomic BSM with perfect input states (see “Appendix 4”). We assume that the coherence time of the memories is so long that we can neglect degradation due to decoherence. For an interference contrast of \(C=0.97\), the BSM fidelity needs to be 95 % (89 %) to yield a secret fraction of 0.5 (0.25). As long as the fidelity is above 83 %, a secret key can be extracted. The secret fraction decreases if additional elementary links are inserted. With four elementary links and \(C=0.97\) interference contrast, the BSM fidelity needs to be 99 % (97 %) to retain a secret fraction of 0.5 (0.25). In this case, the fidelity needs to be above 95 % for a nonzero secret fraction.

### 5.4 Entanglement purification

Imperfections in the substeps required for the operation of a quantum repeater lead to an error in the final state that scales exponentially with the number of nodes. This can be overcome via entanglement purification [5], which uses multiple states that are not maximally entangled to generate a single entangled state with higher fidelity. The purification protocol described by Deutsch et al. [50] could be implemented in the repeater architecture proposed here by employing the single-qubit rotations, the atom-atom gate mechanism, and atomic state detection procedures proposed for entanglement swapping (see Sect. 4). The quantum gates between two atoms in the same herald cavity would be complemented by a remote gate between two atoms in spatially separated herald cavities, also mediated by the reflection of a single photon [43], thereby enabling multiple rounds of entanglement purification using several instances of the proposed repeater node.

Entanglement purification can only increase the fidelity of the final state if the errors introduced by the required operations are significantly smaller than the errors present in the initial states. If entanglement purification and entanglement swapping build on the same gate mechanism, they will suffer from similar imperfections. For the very few entanglement swapping gates necessary for two and four elementary links, entanglement purification can therefore only lead to an increase in fidelity if considerable errors are introduced by other parts of the repeater protocol. An important source of error is loss of interference contrast *C* in the entanglement distribution process, caused by either the generation of partially distinguishable photons or errors introduced during fiber transmission. Taking an interference contrast of \(C=0.97\) (see Sect. 3.1) as an example and applying the error models described in “Appendix 4,” we find that the entanglement purification protocol described by Deutsch et al. [50] increases the fidelity of the final state only if the atom-atom gate error is less than 4 % for two elementary links (\(N=2\)) and less than 7 % for four elementary links (\(N=4\)).

In the case of perfect gates, any target fidelity below unity can in principle be reached. This comes, however, at the cost of a significant additional overhead, because more quantum memories are required, and the final entangled pairs are produced at a lower rate. For applications in QKD, the secret-key rate is a convenient measure that allows comparing repeater strategies with and without entanglement purification. Employing the methods described in “Appendix 4,” we calculate that even with perfect gate operations, i.e., efficiency and fidelity of unity, entanglement purification is beneficial only if the interference contrast is very low, namely \(C\le 0.55\) in case of \(N=2\) and \(C\le 0.83\) in case of \(N=4\). This has to be compared to our expected interference contrast of \(C=0.97\) (see Sect. 3.1) and the fact that high-fidelity transport of polarization qubits in optical fibers over the distances considered here is possible [6].

We therefore conclude that for a realistic repeater implementation over a few hundred kilometers, entanglement purification is unlikely to be beneficial and the best strategy is to focus on high-fidelity implementations without entanglement purification [51, 52].

## 6 Conclusion

Past experiments with single atoms in optical cavities have underlined their excellent prospects for applications in quantum communication [14, 16, 33]. The distances that can be bridged by direct photon transmission at near-infrared or visible wavelengths is, however, limited. We have shown how this limit could be overcome by employing, first, operation at telecom wavelength and, second, a realistic, efficient, and highly-integrated quantum-repeater concept.

Operation at telecom wavelength is essential for long-distance fiber-optic communication without the need for wavelength conversion. Our simulations show that creation of entanglement between single atoms and telecom-wavelength photons with an efficiency of 0.57 is possible with \(^{87}\)Rb atoms and current cavity technology. The designated S-band transition enables integration into existing fiber-optic networks. Remote entanglement can be established by a photonic BSM with a state fidelity of 99 %, because the generated photons are highly indistinguishable. Very efficient entanglement swapping can be achieved by a two-atom quantum gate employing the heralding cavity. Thus entanglement generation, quantum memories and entanglement swapping can be realized with single atoms in optical cavities, and all additional components necessary to complete a simple quantum repeater are commercially available. The highly efficient, yet heralded operations enable the generation of remote entanglement four times faster than direct transmission at a distance of 100 km. This simple quantum repeater can be extended by inserting two more nodes, providing a speedup of 82 over 200 km distance, compared to a system without repeater. The proposed scheme does not rely on assumptions about future technology and thus provides a clear and realistic path toward the experimental demonstration of such a quantum repeater.

## Notes

### Acknowledgments

Open access funding provided by Max Planck Society. This work was supported by the Bundesministerium für Bildung und Forschung via IKT 2020 (Q.com-Q) and the European Union (Collaborative Project SIQS).

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