## Abstract

Entanglement is the essential feature of quantum mechanics. Notably, observers of two or more entangled particles will find correlations in their measurement results that cannot be explained by classical statistics. To make it a useful resource, particularly for scalable long-distance quantum communication, the heralded generation of entanglement between distant massive quantum systems is necessary. We report on the creation and analysis of heralded entanglement between spins of two single rubidium-87 atoms trapped independently 20 meters apart. Our results illustrate the viability of an integral resource for quantum information science, as well as for fundamental tests of quantum mechanics.

Entanglement between distant stationary quantum systems will be a key resource for future applications in the field of long-distance quantum communication, such as quantum repeaters (*1*) and quantum networks (*2*). At the same time, it is an essential ingredient for new experiments on the foundations of physics, in particular for a first loophole-free test of Bell’s inequality (*3*–*5*). Central to all these applications is the heralded generation of entanglement, i.e., a signal is provided once an entangled pair is successfully prepared.

Until now, (unheralded) entanglement between separated massive quantum objects has been achieved for various systems (*6*, *7*), even over a distance of 21 m (*8*). Heralded entanglement has been demonstrated with cold atomic ensembles (*9*, *10*), single trapped ions (*11*, *12*), and diamond crystals (*13*), albeit over short distances in a single setup only. For the realization of heralded entanglement over long distances, single neutral atoms are promising candidates. In view of future applications, several important milestones have already been demonstrated for such systems: manipulation of atomic quantum registers (*14*), storage of quantum information (*8*, *15*–*17*), fast and highly efficient state analysis (*18*), deterministic quantum gates between nearby trapped atoms via Rydberg blockade (*19*, *20*), and distribution of light-matter entanglement over large distances (*21*).

We report on the preparation and analysis of heralded entanglement between two single ^{87}Rb atoms over a distance of 20 m via entanglement swapping (*22*). The scheme starts with entangling the spin of each of the two atoms with the polarization state of a spontaneously emitted photon (*23*). The photons are guided to a Bell state measurement (BSM) setup where the two-photon polarization state is projected onto an entangled state, thereby providing the heralding signal. In a final step, we evaluate the entanglement between the atomic spins.

Our experimental arrangement (Fig. 1A) consists of two independently operated experiments, here called trap 1 and trap 2 (*24*), which are situated in two laboratories and equipped with their own laser and control systems. In each experiment, we load a single ^{87}Rb atom into an optical dipole trap (*25*). The typical lifetime of a single trapped atom is 5 to 10 s, limited mainly by heating during the experimental process and collisions with background gas. Photons emitted by the atoms are coupled into single-mode optical fibers and guided to the BSM arrangement next to trap 1. The lengths of the optical fibers from trap 1 and trap 2 to the BSM are 5 and 30 m, respectively. To compensate for polarization drifts induced by temperature changes and mechanical stress in the 30-m fiber, an automatic polarization stabilization (*21*) is used. The interferometric BSM arrangement consists of a 50-50 single-mode fiber beam splitter (BS) with polarizing beam splitters (PBSs) in each of the output ports. Additional half- and quarterwave plates allow us to select the measurement basis for the BSM and the atom-photon entanglement measurements. Finally, photons are detected by four avalanche photodiodes (APDs).

First, we verify atom-photon entanglement in each experiment separately. The generation of an entangled atom-photon state starts by preparing the atom in the initial state 5^{2}S_{1/2}, |F = 1, *m*_{F} = 0〉 (Fig. 1B) via optical pumping. Then the atom is excited to the state 5^{2}P_{3/2}, |F′ = 0, *m*_{F}_{′} = 0〉 by a short optical pulse (full width at half-maximum pulse length = 21 ns). In the following spontaneous decay, the polarization of a single photon emitted into the collection optics (defining the quantization axis *z*) is entangled with the atomic spin (*23*), yielding the state

where |*L*〉, |*R*〉 denote the left- and right-circular and |*H*〉, |*V*〉 the horizontal and vertical linear polarization states of the photon. The atomic qubit is defined by the Zeeman states |*m*_{F} = +1〉 and |*m*_{F} = −1〉 of the ground level 5^{2}S_{1/2}, F = 1, which we associate with spin orientations |↑〉* _{z}* and |↓〉

*, respectively. Preparation and excitation of the atom are repeated until a photon is detected. Taking into account additional cooling periods required to counteract heating of the atom, the preparation and excitation of the atom can be performed 50 × 10*

_{z}^{3}times per second. The overall efficiency for detecting the photon after an excitation in trap 1 (trap 2) is η

_{1}= 0.9 × 10

^{−3}(η

_{2}= 1.25 × 10

^{−3}). These numbers include the excitation probability, the collection and coupling efficiencies as well as losses in the optics, and also the quantum efficiency of the photodetectors. Polarization analysis of the single photons is performed with the BSM arrangement, which also serves to monitor fluorescence of the atom inside the trap.

To evaluate atom-photon entanglement, conditioned on the detection of the emitted photon, the internal spin state of the atom is read out (*23*). The detection process consists of a Zeeman state-selective stimulated Raman adiabatic passage (STIRAP) (*26*) with subsequent hyperfine state detection. This process can be considered as a projection of the atom onto the state cos(γ)|↑〉_{x}_{}+ sin(γ)|↓〉* _{x}*, where γ is the angle of linear polarization of the STIRAP pulse defining the measurement basis (the angle of the corresponding direction of the atomic spin is 2γ). In atom-photon correlation measurements, we register event numbers

*S*,

*S*′ ∈ {|↑〉, |↓〉} are the eigenstates of the spin of the atom and the photon along their respective measurement directions defined by γ and δ. Figure 2 shows the resulting correlations between atomic spin and photon polarization measurements for both traps separately. In these measurements, the photon was detected in

*H*/

*V*basis (δ = 0°) and in ±45° basis (δ = 45°), while the atomic measurement angles α (trap 1) and β (trap 2) were varied between 0° and 180°. The visibilities

*V*

^{(δ)}of the correlation curves obtained by least-squares fits are

*V*

_{1}

^{(0°)}= 0.869 ± 0.006,

*V*

_{1}

^{(45°)}= 0.900 ± 0.006 for trap 1, and

*V*

_{2}

^{(0°)}= 0.895 ± 0.004,

*V*

_{2}

^{(45°)}= 0.901 ± 0.005 for trap 2, where the given errors are the expected statistical 1σ deviations. These high visibilities, limited mainly by the quality of the atomic state read-out, demonstrate that atom-photon entanglement is reliably generated and detected with high fidelity in both traps.

The second crucial condition for preparing a highly entangled state of two trapped atoms is a high-fidelity Bell state measurement of the photons, i.e., projecting them onto maximally entangled states. We use interferometric Bell state analysis based on the Hong-Ou-Mandel effect (*27*). This two-photon detection scheme does not require interferometric stability on a wavelength scale, thereby relaxing the experimental requirements for long-distance quantum communication. In general, at a beam splitter, bunching (antibunching) of two photons in a symmetric (antisymmetric) state enables one to identify Bell states. In our case, a coincidence in detectors *H*_{1}*V*_{1} or *H*_{2}*V*_{2} (Fig. 1A) signals projection of the photons onto the state *H*_{1}*V*_{2} or *H*_{2}*V*_{1} indicate projection onto the state *28*–*30*). Thus, by detecting one of the four coincidences mentioned above, we project the incoming photons unambiguously on a Bell state, thereby heralding the generation of entanglement between the separated atoms.

The visibility of the two-photon interference, which determines the fidelity of the Bell state measurement, depends crucially on temporal, spatial, and spectral indistinguishability of the arriving photons. Experimentally, the temporal overlap is achieved by synchronizing the two excitation procedures in trap 1 and trap 2 to better than 500 ps, which is far below the lifetime of the excited atomic state of 26.2 ns, and by exactly matching the shapes of the interfering wave packets (Fig. 1C). The single-mode fiber beam splitter guarantees spatial mode overlap of unity. Frequency differences of the emitted photons are minimized by zeroing all relevant fields (*31*). Further reduction of the fidelity of the Bell state measurement arises from two-photon emission by a single atom due to off-resonant excitation of the atom to the 5^{2}P_{3/2}, F′ = 1 level (see Fig. 1B and fig. S1) if the first photon is emitted already within the duration of the excitation pulse. However, owing to the structure of the involved atomic levels, with a probability of 78.1% a polarization-entangled state *32*). These events are registered as coincidences *H*_{1}*H*_{2} and *V*_{1}*V*_{2} and do not herald projection onto a Bell state. Reduction of the fidelity is therefore due to the remaining two-photon emissions and to dark counts of the detectors. On the basis of additional calibration measurements, we estimate a fidelity of the Bell state projection of at least 92% (*32*).

By combining all methods described above, we can generate and characterize entanglement between two distant atoms. In each of the two experiments, a single atom is captured and the atom-photon entangling sequences are repeated until two photons are detected within a time window of 120 ns in the BSM arrangement. With a coincidence probability of 0.54 × 10^{−6} and a repetition rate of 50 kHz, and by taking into account the fraction of time when an atom was present in each of the traps of 0.35, we arrive at an atom-atom entanglement rate of about 1/106 s^{−1}. A valid twofold detection, i.e., registration of |Ψ^{±}〉_{Ph}, heralds projection of the atoms onto the state _{c} = 75 μs (*17*) and the coherence time of the entangled atom-atom state, which we expect to be at least τ_{c}/2, and thus does not limit the quality of our experiment (agreeably, the atom-atom entanglement rate and τ_{c} need substantial improvement for future quantum repeater scenarios).

To evaluate the atom-atom entanglement, we perform measurements of the atomic spins in two bases. We have chosen analysis angles α = 90° and α = 135°, while β is varied in steps of 22.5° between 90° and 180°, or between 45° and 135°, respectively. The obtained correlations are shown for the detection of the photonic |Ψ^{−}〉_{Ph} state (Fig. 3, A and B) and for the |Ψ^{+}〉_{Ph} state (Fig. 3, C and D). By fitting sinusoidal functions to the data points, we obtain visibilities *V*^{(α)} of ^{−}〉_{AA} state and^{+}〉* _{AA}* state, respectively. For estimation of the fidelity, we assume that the visibility in the third (unmeasured) conjugate basis is equal to the lower of the two measured ones, arriving at

*F*

_{Ψ−}= 0.811 ± 0.028 and

*F*

_{Ψ+}= 0.815 ± 0.028. These numbers prove that in both cases, an entangled state of the two atoms is generated. Moreover, the average visibilities

One of our main goals is to enable a future loophole-free test of Bells inequality (*3*). Inserting the data from the above measurements into*S* = |〈σ_{α}σ_{β}〉 + 〈σ_{α′}σ_{β}〉| + |〈σ_{α}σ_{β′}〉 − 〈σ_{α′}σ_{β′}〉| from the Clauser-Horne-Shimony-Holt-inequality *S* ≤ 2, which holds for local-realistic theories (*33*). For the data from Fig. 3, using the settings α = 135°, β = 67.5°; α = 135°, β′ = 112.5°; α′ = 90°, β′ = 112.5° together with α′ = 90°, β′′ = 157.5° (replacing β = 67.5°), for all four heralding signals we obtain an *S* value exceeding the limit of 2. Because a measurement result is obtained for each and every heralding signal, the average value of *S* = 2.19 ± 0.09 for the first time yields definite violation without relying on the fair sampling assumption for a macroscopic distance.

In this experiment, we have demonstrated heralded entanglement between two atoms 20 m apart. It was high enough to violate a Bell inequality, showing its suitability for quantum information applications such as device-independent quantum cryptography (*34*). The design of trap 2 allows rather straightforward extension of the distance between the two traps to at least several hundred meters, limited only by transmission of photons in the optical fiber connection. Two distant entangled atoms form the elementary link of the quantum repeater, enabling efficient long-distance quantum communication. Together with efficient and fast atomic state detection (*18*), this experiment forms the basis for the first loophole-free Bell experiment, answering the long-standing question on whether a local realistic extension of quantum mechanics can be a valid description of nature.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/337/6090/72/DC1

Materials and Methods

Figs. S1 and S2

Table S1

Reference (*35*)

## References and Notes

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Materials and methods are available as supplementary materials on
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**Acknowledgments:**We thank A. Deeg, C. Jakob, and C. Kurtsiefer for help during the early stages of the experiment. This work was supported by the European Union Project Q-Essence and the Bundesministerium für Bildung und Forschung Project QuORep. J. H. acknowledges support by Elite Network of Bavaria through the excellence program Quantum Computing, Control and Communication.