Heralded Mapping of Photonic Entanglement into Single Atoms in Free Space: Proposal for a Loophole-Free Bell Test

An obvious way to entangle two atoms located at remote locations is to produce a pair of entangled photons half-way between the two atoms, to sent one photon to each location and to subsequently map the photonic entanglement into the atoms. The efficiency of this process is, however, fundamentally limited due to overall transmission losses. We propose a method to herald the success of the mapping operation in free space without destroying nor revealing the stored quantum state. Interestingly for a Bell test, the heralding signal does not open the detection loophole provided the measurement choice is performed once the heralding is obtained only. We show through a detailed feasibility study that this approach could provide an attractive alternative to Bell tests where the atom--atom entanglement is created from atom--photon entanglement using an entanglement swapping operation.


1
x,y=0 (−1) x y p (a = b|x y) − p (a = b|x y) > 2. (1) In this case, Alice and Bob are forced to conclude that the observed correlations are non-local, i.e. they got correlated results that are locally random, but cannot be reproduced by a shared classical randomness. All the Bell experiments realized so far point to the conclusion that this non-locality is, indeed, an element of the physical reality, but they were all subjected to loopholes. There are basically two loopholes, the detection loophole and the locality loophole. The latter is closed if the measurement choice on Alice's side and the measurement result on Bob's side, and vice versa, are spacelike separated. If this condition is not fulfilled, the particles could simply communicate the measurement settings they experience to choose the results accordingly. The former is related to the inefficiency of detections. The particles could take advantage of the undetected events to answer when the measurement settings are in agreement with a predetermined strategy only. The locality loophole was addressed in experiments with entangled photons [2] and the detection loophole with ions, atoms and photons [3][4][5]. Closing both in a single experiment would not only be the end of a long history of disputes, but like many fundamental findings, it would open the way to fascinating applications. For example, closing the detection loophole over tens of km would provide unique opportunities to quantum-key-distribution protocols where the security does not rely on the device that is used to generate the key [6] 5 .

State of the art
Photons are naturally suited for closing the locality loophole. They are fast, easy to guide and can be produced at high repetition rates. However, the overall detection efficiency has to be higher than 82.8% if one wants to close the detection loophole from the inequality (1), the socalled Clauser-Horne-Shimony-Holt (CHSH) inequality [1], with maximally entangled states. Considering realistic noise, achievable coupling into optical fibres and detection efficiencies, one rapidly becomes aware that closing the detection loophole with photons between spacelike separated locations is a very challenging task. Some looked for specific states or peculiar Bell inequalities offering a better resistance to inefficiencies [7]. Other studied the possibility of photonic Bell tests with homodyne measurements to overcome the problem of the single-photon detection inefficiency [8]. But the most promising approach for a loophole-free Bell test uses atom-photon entanglement [9], the photon allowing for the distribution of entanglement over long distances and the state of an atom being detected with an efficiency close to one. A lot of experimental effort has been devoted to the creation of a single photon from a single atom where the photon polarization is entangled with internal states of the atom [10][11][12]. Such entanglement has further been used to entangle remote atoms through an entanglement swapping operation [4,13,14] 6 . Hopefully, these impressive experimental results could lead to the first loopholefree Bell test in a near future.
Experimental activities investigating the resonant interaction in free-space of a single atom with single photons produced through the spontaneous parametric down conversion (SPDC) have emerged in parallel [15][16][17]. In [18], the interaction of single heralded SPDC photons with a single atom has been demonstrated and in [19], the possibility of obtaining atom-photon entanglement from the absorption by a single atom of a photon belonging to a polarizationentangled SPDC pair has been shown. One of the next great challenges could naturally be the creation of entanglement between remote atoms, by producing a pair of entangled photons halfway between two atoms and subsequently mapping the photonic entanglement into the atoms. The efficiency of this process is, however, fundamentally limited due to transmission losses. The mapping efficiency further decreases the entanglement creation rate. Lloyd et al [20] proposed a way to herald the success of the entanglement creation by exciting a cycling transition and by detecting the resulting fluorescence detection. This heralding method has been implemented in [21]. By further embedding the atoms in high-finesse cavities, the authors of [20] end up with an efficient yet technologically demanding architecture for quantum networking. We here focus on free-space interaction and propose a fast and simple alternative method to herald the success of the mapping process without revealing the stored quantum state. Although the proposed heralding process is probabilistic, we show that it does not open the detection loophole provided that the heralding signal is obtained before the measurement choice. We believe that the proposed scenario is a potential candidate for the first loophole-free Bell test. Mapping polarization entanglement into the internal states of two atoms. The success of the absorption process is heralded locally by the detection of a single photon emitted spontaneously from an excited state. By choosing properly the detection basis, the heralding signal does not reveal the polarization of the absorbed photon and leads to the heralded creation of two entangled remote atoms (see the main text for details).

Heralded mapping of photonic entanglement into single atoms: principle
Consider the scenario presented in figure 1 where two atoms, located at remote locations A and B, contain each a double -system of levels. Further consider that they are initially prepared in a coherent superposition of two Zeeman levels A photon pair source at a central station located half-way between two atoms is excited such that with a small probability p, an entangled pair is created, corresponding to a state Here, a + and a − (b + and b − ) are bosonic operators associated to two orthogonal polarizations propagating towards Alice's (Bob's) location, e. g. in optical fibres, and |0 is the vacuum state.
The O( p) term introduces errors in the protocol, leading to the requirement that p has to be kept small enough, cf below. If a pair is created, the corresponding two photons can both be absorbed when they reach their destinations. Two successful absorptions transfer Alice's and Bob's atoms in excited states and map the photonic entanglement into an atomic entanglement ψ at abs,− = 1 In principle, the maximum achievable probability p abs for a twofold absorption is equal to 1 4 pη 2 t , with η t being the transmission efficiency from the source to one of the atoms. The pre-factor 1 4 comes from the fact that out of the initial state (2), on average only every second photon can be absorbed. The coupling into the fibre η c and the absorption efficiency η abs further limit p abs to 1 4 pη 2 c η 2 t η 2 abs .
Once they are excited, the atoms can spontaneously decay into ground states g A + or g A − (g B + or g B − ) by emitting a photon with the corresponding polarization a + or a Hence, the detection of one spontaneous photon at each location serves as a heralding signal for the success of the mapping process and moreover, the entanglement is preserved if the reemitted photons are detected in the appropriate basis. For example, the detection of two photons, one with the polarization a H = 1 The probability to obtain the heralding signal after absorption is η 2 d where η d is the efficiency with which a spontaneous photon is detected from a single atom. Since twofold detection in also projects the two atoms into ψ at herald,− (up to a unitary), the overall efficiency for the twofold heralding is given by p herald = 1 4 pη 2 c η 2 t η 2 abs η 2 d . After the detection of the heralds, Alice and Bob can choose their measurement setting, i.e. they perform a rotation on the two level system {g + , g − } before measuring the state of their atom through state-selective fluorescence (electron shelving) or ionization. To close the locality loophole, it is important that the distance L separating Alice and Bob is such that L/c (where c is the velocity of light in vacuum) is larger than the time it takes to know the atomic state once the measurement setting is chosen. Independent of this is the other characteristic time scale, i.e. the time it takes to receive a twofold herald, which can be very long.
We emphasize that the heralding process does not open the detection loophole if Alice and Bob choose the measurement settings x and y only after the spontaneously emitted photon is detected. This simply reduces to a pre-selection and none of the inefficiencies mentioned so far enters in the detection efficiency required to close the detection loophole. In particular, contrarily to the situation without pre-selection, there is no limitation (other than technical ones) on the efficiency with which the photon states are mapped to the atoms, if one is willing to lower the atomic entanglement-creation rate [22]. Note also that Alice does not need to know whether Bob got the heralding signal when she chooses her measurement setting. The detection of one spontaneous photon at each location decides that a given run is going to contribute to the data of the CHSH inequality test, and the measurement settings can be determined locally from the polarization of the detected photon.

Heralded mapping of photonic entanglement into single atoms: practical implementation
We now discuss a practical implementation of the heralded entanglement distribution and Bell test in more detail. For concreteness, we consider implementing the scheme using two distant single trapped and laser-cooled 40 Ca + ions. The 40 Ca + ion is the only single atomic system so far which has been coupled to entangled SPDC photons [17][18][19]. A scheme of its relevant atomic levels and transitions is shown in figure 2. Based on the experimental work reported in [15][16][17][18][19] we assume that the entangled photons are created at 854 nm, resonant with the transition from the metastable D 5/2 level to P 3/2 . Photon loss in optical fibres at this wavelength is of the order of 1 dB km −1 . This means that for L = 3 km, both photons will reach Alice's and Bob's locations with a probability η 2 t ≈ 0.5. This also translates into an upper bound for the time delay between the measurement choice and the measurement result of 10 µs. The heralding photons are assumed to be emitted on the P 3/2 to S 1/2 transition at 393 nm. The other levels and transitions presented in figure 2 are employed for state preparation and detection [21].
A weak magnetic field B defines a quantization axis and thereby, together with the direction of propagation of the incoming photons k, possible polarization bases [19]: when k B, absorption of circularly polarized photons leads to m = ±1 transitions; when k ⊥ B, photons with polarization B induce m = 0 transitions, and photons with ⊥ B drive a superposition of the two | m| = 1 transitions.
The initial state (2) is implemented by preparing each atom in a coherent superposition of sublevels |D, ± 5 2 in D 5/2 . This is achievable, for example, by coherent excitation on the S 1/2 to D 5/2 quadrupole transition used as optical qubit in quantum logic experiments.
Then, the ions are exposed to the photons at 854 nm. The final state (4) after emission of the heralding photon involves the magnetic sublevels of the S 1/2 state. Faithful mapping of the absorbed photon to the atomic state requires that the two emission pathways be indistinguishable. Since they happen on transitions with different m, the corresponding photons have to be projected on the same emitted polarization ε 393 , as mentioned in the previous section. Detection in this case is optimal along the quantization axis in a linear polarization basis; by using two detectors for two orthogonal polarizations, the highest possible efficiency η d will be obtained. Current experiments [21,23] achieved the values η abs = 2.5 × 10 −4 and η d = 1.55% (using a photon collection of 5.5% and photon detectors with an efficiency of 28%). By using parabolic mirrors [24] to enhance the coupling to near unity, a maximal value of η abs ≈ 6% can be achieved (limited by the oscillator strength of the transition, cf figure 2) while η d can be of the order of 30%. Furthermore we assume that η c = 70%. All these numbers together yield an efficiency for the twofold heralding of the order of p herald ≈ 2 × 10 −5 p.

Expected atom-atom entanglement visibility
Taking the main experimental limitations into account, we now estimate the fidelity of the heralded atom-atom entangled state with respect to the singlet state (5). Our approach starts with the emission of photon pairs. Due to possible multiple pair emission inherent in SPDC processes, the photon-photon entangled state can be written ρ ph where 1 stands for the identity and V ph is the visibility of the interference that would be obtained if Alice chooses the measurements σ x for example and Bob rotates his measurement basis in the {x z} plane. Additional errors (with corresponding probability e) occurring at the further stage of the experiments degrade the visibility according to V → (1 − e)V. Once a photon pair is created, it propagates to Alice's and Bob's locations. We assume that the polarization is actively controlled such that the error on the polarization is of the order of e pol = 1%. The photonic state is then mapped to the atoms. The heralded mapping operation (one photon to one atom) is estimated to be inaccurate at the same order, e map = 1%. Dark counts of the heralding detectors add negligible noise. The dominant error happens when one detector clicks because of a photon detection but the other one produces a dark count. The probability of this erroneous event is given by p dark = pη c η t η abs η d η dark where η dark = R dark t, R dark being the dark count rate (assumed to be 30 counts per second) and t is the coincidence window (up to 20 ns with respect to the photon coherence time of 7 ns). The corresponding error is given by e dark = p dark p dark + p herald 10 −3 . The resulting atom-atom entangled state is thus expected to be of the form where the visibility of the atomic entanglement is given by V at

Performing the Bell test
Once the heralding signal is obtained, Alice (or Bob) needs to be able to detect the state of her (his) ion in various bases. The necessary rotations between |S, + 1 2 and |S, − 1 2 are performed in up to 10 µs by a magnetic field at radio frequency. The detection then proceeds by electron shelving of one S 1/2 sublevel into D 5/2 and measuring resonance fluorescence from the S 1/2 ↔ P 1/2 transition. Following [26], such a measurement takes in average 145 µs with a photon collection of 0.22% and the mean accuracy of this procedure was experimentally determined to be 99.99%. However, assuming a global detection efficiency of η d = 30% as before, the measurement time reduces to the time it takes to perform to the local rotation (10 µs). This is fast enough to close the locality loophole with a distance of 3 km. The measurement accuracy could realistically be of 1 − e det = 99.95%.

Expected violation of the Clauser-Horne-Shimony-Holt inequality
The resulting CHSH value is expected to be given by with the statistical uncertainty where V stands for V at (1 − 2e det ) 2 [27]. In principle, the value of p needs to be optimized since a high CHSH value favours p ≈ 0 whereas a high heralding probability favours p ≈ 1. However, practical considerations limit the pair production to approximately 5 × 10 5 s −1 , i. e. p = 4 × 10 −3 in a coincidence window of t = 7 ns. This leads to V at ≈ 96% and S exp ≈ 2.73 so that N = 65 events are necessary to conclude about the violation of the CHSH inequality with a confidence level above 99.7% (three standard deviations). The duration of the state preparation (T ≈ 25 µs) mainly defines the repetition rate and this translates into an overall acquisition time of N T / p herald ≈ 6 h and 30 min.
In order to rule out local models as plausible explanations of an observed Bell inequality violation, it is convenient to estimate the probability that the observed statistics be produced by such a model [28,29]. Since the deviation of the Bell values that can be obtained by local models in presence of finite statistics need not be described appropriately by the experimental uncertainty S exp , we rely on the following bound [29]: P(S exp |local model) exp −N (S exp − 2) 2 /32 .
This quantity is bounded by 0.05 for the above value of S exp , whenever N > 181. A clear demonstration that the observed statistics are not the result of a local model is thus possible in a bit more than 18 h.

Conclusion
Our proposal opens a way to test Bell's inequalities in a loophole-free realization. Specifically, it offers an interesting alternative to Bell tests where atom-atom entanglement is created by means of an entanglement swapping operation. The latter is being pursued by several groups and recent advances are very promising [4]. Time will tell which one will allow one to answer a question lively debated about non-locality. From a more applied perspective, our proposal may find applications in quantum key distribution, either for implementing more secure protocols [6] (seefootnote5)orforextendingquantumkeydistributionoverthousandsofkilometersusing quantum repeaters [20,30,31]. This supposes, however, significant efficiency improvement in order to reach interesting key distribution rates.
Note added in proof. Related work has been carried out independently by Brunner et al [32].