Towards experimental entanglement connection with atomic ensembles in the single excitation regime

We present a protocol for performing entanglement connection between pairs of atomic ensembles in the single excitation regime. Two pairs are prepared in an asynchronous fashion and then connected via a Bell measurement. The resulting state of the two remaining ensembles is mapped to photonic modes and a reduced density matrix is then reconstructed. Our observations confirm for the first time the creation of coherence between atomic systems that never interacted, a first step towards entanglement connection, a critical requirement for quantum networking and long distance quantum communications.


I. INTRODUCTION
The distribution of entanglement between different parties enables the realization of various quantum communication protocols, such as quantum cryptography, dense coding and teleportation [1,2]. Such distribution can rely on entanglement swapping, namely the teleportation of entanglement, which aims at entangling two distant systems which never interacted in the past. Important aspects of this striking feature have already been demonstrated with independent sources of entangled light. In the discrete variable regime, one can generate two independent pairs of polarization entangled beams and subject a superposition of two of the beams to a Bell-state measurement. The two remaining beams are then projected into an entangled state [3]. More recently, unconditional entanglement swapping has been achieved for continuous quantum variables of light [4,5].
However, to enable quantum communication over arbitrary long distances, entanglement needs to be stored in matter systems. In the quantum repeater architecture [6], entanglement is distributed by swapping through a chain of spatially separated entangled pairs of memories, leading to the possibility of scalable long-distance communication. Connecting entangled matter systems is thus a critical requirement for the practical realization of quantum networks. Along this line, generation of entanglement between atomic systems has been reported, including entanglement of the discrete internal states of two trapped ions [7], long-lived entanglement of macroscopic quantum spins [8] and, more recently, heralded entanglement between atomic ensembles in the single excitation regime [9]. However, no entanglement connection has been demonstrated so far with such matter systems. In this paper, we present our work towards entanglement connection of atomic ensembles and demonstrate for the first time the transfer of coherence between two atomic ensembles which never interacted.
The paper is organized as follows. Section II gives a brief overview of our matter building block, namely an atomic ensemble in the regime of single collective excitation. In section III, we discuss the principles of measurement-induced entanglement between excitation from two remote atomic ensembles, and connection of two pairs. The theoretical model developed in [9] to verify experimentally entanglement is summarized, and used to give insights into the connection process. The experimental setup is finally presented in section IV, together with the experimental results. We discuss in section V the experimental perspectives of realizing entanglement connection.

II. ATOMIC ENSEMBLE IN THE SINGLE EXCITATION REGIME
In 2001, Duan, Lukin, Cirac and Zoller (DLCZ) proposed an original approach to perform scalable longdistance quantum communications, involving atomic ensembles, linear optics and single photon detectors [10]. The building block is a large ensemble of N identical atoms with a Λ-type level configuration, as shown in figure 1. A weak light pulse, called write pulse, with frequency close to the |g → |e transition, illuminates the atoms and induces spontaneous Raman scattering into a photonic mode called field-1. For a low enough write power, such that two excitations are unlikely to occur, the detection of a field-1 photon heralds the storage of a single spin excitation distributed among the whole ensemble. The joint state of the atoms and field 1 can then be written as: where |n 1 stands for the state of the field 1 with n photons and p corresponds to the small probability of a single photon scattered into field 1 by the atoms illuminated by the write pulse. We define |0 a ≡ N i |g i and |1 a denotes a symmetric collective excitation, with A read pulse, on resonance with the |s → |e transition, can later, after a programmable delay, transfer this atomic excitation into another photonic mode, field 2.
The key element of the protocol is that this readout can be achieved with high efficiency thanks to a many-atom interference effect, called collective enhancement [11,12]. After readout, the ideal resultant state of the fields 1 and 2 is: The photon numbers are correlated, precisely as in the case of parametric down conversion. The lower is the excitation probability, the better is the approximation of the non-vacuum part by a photon pair, at the price of reduced count rates. In our group, the optically thick atomic ensemble is obtained from cold cesium atoms in a magneto-optical trap (MOT). At a frequency of 40 Hz, the magnetic field is switched off for 7 ms. After waiting about 3 ms for the magnetic field to decay, sequences of writing, reading and repumping processes are carried out for about 4 ms, with a period of 575 ns. The weak write pulses, with a 200 µm beam waist and linear polarization, are detuned 10 MHz below resonance. The read pulse is orthogonally polarized to the write pulse and mode-matched to it in a counter-propagating configuration. Both write and read pulses are 30 ns long. Fields 1 and 2 are collected into mode-matched fibers with a 3 • angle relative to the common direction defined by write and read beams [13], and with a waist of 50 µm defined by the backward projection of our imaging system into the sample. Before detection, field 1 passes through a filtering stage in order to filter out the photons that are spontaneously emitted when the atoms in the sample go back to |g , without creating the desired collective excitation.
Three parameters well characterize experimentally the system : how well the system is in the single excitation regime, how efficient is the retrieval of a single excitation and how long the excitation can be stored before retrieval while preserving its coherence. The first parameter can be determined by a measurement of the suppression of the two-photon component of the field 2 obtained from the retrieval of the excitation. Suppression below 1% of the value for a coherent state has been reported in our system [14]. The ability to efficiently retrieve the excitation is also critical. The probability to have a photon in field 2 in a single spatial mode at the output of the atomic ensemble once an event has been recorded for field 1 can be as high as 50%, leading to a probability around 12.5 % for having a detection event [14]. Last but not least, the writing and retrieval processes can be separated by a programmable delay. As this delay is increased, the above two quantities decay in a typical time scale around 10 to 20 µs. The principal cause for this finite coherence time is the residual magnetic field that inhomogeneously broadens the ground state levels of the atomic samples. Detailed theoretical and experimental studies of the decoherence have been reported in Refs. [12,15,16].

III. MEASUREMENT-INDUCED ENTANGLEMENT AND CONNECTION OF ATOMIC ENSEMBLES
Starting from this building block, DLCZ proposed in their seminal paper to generate and store entanglement for excitation in two remote ensembles and then to connect two pairs. This section presents these measurementinduced schemes, which rely on quantum interference in the detection of a photon emitted by one of the ensembles. After establishing entanglement, directly or via connection, a difficult experimental task is to prove the entanglement [17]. A robust model developed in [9] is then presented.

BS
FIG. 2: Setup for (a) entanglement generation between two atomic ensembles and (b) entanglement connection between two pairs. In both cases, the entanglement results from the interference of photonic modes at a 50/50 beamsplitter, with outputs directed towards single-photon detectors. The photonic modes consist in (a) field 1 modes from both ensembles or (b) field 2 modes after reading of one ensemble of each pair.

A. Entanglement between two ensembles
Let us consider now two atomic ensembles, for which fields 1 are superposed on a 50/50 beamsplitter, in an indistinguishable way, with the outputs directed towards two single-photon detectors ( Figure 2). The detection of a field 1 photon from either of the two ensembles results in an entangled state with one excitation shared coherently between the two ensembles. Such measurementinduced entanglement has been demonstrated recently in our group for excitation stored in two atomic ensembles separated by 3 meters [9].
In more details, after two write pulses are sent into the two ensembles simultaneously, the scattered fields 1 and ensembles are in the product state : Here β R and β L correspond to overall propagation phases determined by the write pulses. Detection of a photon in either detector then projects the state of the ensembles as follows, in the ideal case where higher-order terms are neglected: ρ(|Ψ ) ≡ |Ψ Ψ|, T r 1L1R stands for tracing over the states of fields 1 L and 1 R , a 1L and a 1R are the annihilation operators associated with fields 1 L and 1 R , θ = θ R − θ L the difference phase shift between the two field 1 paths from the ensemble to the beamsplitter, and finally the overall phase η = (β L − β R ) + (θ L − θ R ). This phase η is the sum of the phase difference of the write beams at the L and R ensembles and the phase difference acquired by fields 1 in propagation from the ensembles to the beamsplitter. To achieve entanglement, this phase has to be kept constant. In order to meet this stringent and challenging requirement in the initial demonstration reported in [9], the different phases have been independently controlled and actively stabilized by using auxiliary fields. Finally, the ± sign in Eq. 4 comes from the π phase difference between the two outputs of a beam splitter: depending on which detector records the heralding event, two different entangled states are generated, and stored for subsequent utilization.

B. Entanglement connection
When two pairs of atomic ensembles are prepared in such an entangled state (figure 2), one can connect the pairs by sending strong read pulses into one ensemble of each pair. The fields 2 resulting from this readout are then brought to interference at a 50/50 beamsplitter. Again, a single click on either detector prepares the remaining ensembles in an entangled state [10].
After independent preparation of entanglement for the pairs {L, I1} and {R, I2} and perfect reading of the states of the ensembles I1 and I2, the joint state of the fields 2 and the ensembles can be written, neglecting higher order terms: where the phases resulting from the entanglement generation and the readout process are given by with δ Ij the phase of the read beam at the Ij ensemble. Fields 2 I1 and 2 I2 are then mixed on a 50/50 beamsplitter, and detection of a photon in either detector projects the remaining two ensembles L and R into: which can be written as where ξ = ζ R,I2 − ζ L,I1 + γ. This overall phase is the sum of the phase difference for entanglement generation for each pair, the phase difference between the two read beams up to the two ensembles and the phase difference of the generated fields 2 from the ensembles to the beamsplitter. The vacuum part comes from the probability of reading the two excitations at the same time, leaving no remaining excitation in the system. In the ideal case, the connection succeeds 50% of the time. Let us underline also that, significantly, the absolute phases do not necessarily need to be stabilized to succeed in the connection. Only the overall phase ξ must be kept constant. This feature is exploited in the proposed experimental setup, where passive stability is found to be enough to meet this requirement.
The generated state given by Eq. 7 is what DLCZ called an "effective maximally-entangled state"(EME) as any state of this form would be purified to a maximally entangled state in the proposed entanglement-based communication scheme [10]. The vacuum coefficient only influences the success probability, but not the overall fidelity of the long-distance communication. This important feature is known as "built-in purification".

C. Experimental verification of entanglement
To experimentally verify the entanglement between the two atomic ensembles, L and R as sketched in figure 3, a solution is to map the delocalized atomic excitation into a field state by applying simultaneous strong read pulses. For perfect state transfer, the entangled state of the atoms would be mapped to an ideal entangled state of the two photonic modes.
However, the presence of various noises, the vacuum contribution (coming from a finite retrieval efficiency or also a finite success probability in the case of the swapping), as well as higher order terms, have to be taken into account. In order to prove experimentally the generation of entanglement at the atomic level, our group has developed in Ref. [9] a robust, model-independent determination of entanglement based upon quantum tomography of the fields 2. As entanglement cannot be increased by local operations on either of the two ensembles, the entanglement for the state of the ensembles will always be greater than or equal to that measured for the light fields. The model consists of reconstructing a reduced density matrix, ρ, obtained from the full density matrix by restricting it to the subspace where no more than one photon populates each mode. It can be shown that this reduced density matrix exhibits less or equal entanglement than the full one. The model will thus lead to a lower bound of the entanglement, enabling an unambiguous determination of the presence of entanglement, at the price of eventually underestimating its actual magnitude.
The reduced density matrix can be written as: in the photon-number basis |n |m , with {n, m} = {0, 1}. p ij is the probability to find i photons in mode 2 L and j in mode 2 R , d is the coherence term between the |1 |0 and |0 |1 states, and P = p 00 + p 01 + p 10 + p 11 . From this density matrix, one can calculate the concurrence C, which is a monotone measurement of entanglement [18]: Let us underline, as d 2 ≤ p 10 p 01 , a necessary requirement for C > 0 is that there is a suppression of two-photon events relative to the square of the probability of single photon events for the fields 2 : h ≡ p 11 /(p 10 p 01 ) < 1. Experimentally, the density matrix can be reconstructed by using two different configurations, as sketched in figure 3. The diagonal elements are determined by measuring individual statistics, i.e. by detecting independently each field. The coherence term can be measured by combining the fields 2 on a 50/50 beamsplitter and recording the count rate as a function of the phase difference between them. This results in an interference fringe with a visibility V . It can been shown that d ≃ V (p 10 + p 01 )/2. Together, this two-stage measurement gives access to the concurrence C.

D. Entanglement connection revisited
The principle of entanglement connection has been explained previously in the ideal case where higher order terms and vacuum contributions are neglected. Let us consider now the more general case, which can be described by the previous approach. We consider two pairs of entangled ensembles and consider that the fields 2 after reading can be described by the same density matrix ρ ′ with diagonal elements p ′ ij . The relevant question is now what will be the expression of ρ, the reduced density matrix for the fields 2 of the two remaining ensembles after the connection.
Let us assume that p ′ 10 = p ′ 01 . To later normalize the events conditioned on swapping, one needs to first determine the probability to have a click heralding the connection at one output of the beamsplitter. To the first order, this quantity can be written as: The factor 1/2 corresponds to the 50% chance that the photon be reflected or transmitted at the beamsplitter, while the factor 2 results from the symmetry of the scheme where the photon can come from either ensemble. One can then evaluate, after the reading of the two remaining ensembles, the probability to have one photon for one mode and zero for the other, when a swap event has been detected: The terms inside the parenthesis correspond to one photon in mode 2 L and zero in 2 R (or the other way round), and all the other combinations for 2 I1 and 2 I2 which can give a swapping event. The final factor 1/2 comes from the fact, already established before in the ideal case, that the swapping succeeds, to the first order, 50% of the time. Finally, in a similar way, the probability to have one photon in each mode is given by: The main feature which appears here is that the weight of the two photon component stays the same, while the single-photon component is divided by two. As a result, if one calculates for the connected pairs the new suppression h of two-photon events relative to the square of the probability for single photon events as a function of the initial h ′ for each entangled pair: h ∼ 4h ′ . This result points out the difficulty which could arise in the experimental demonstration of entanglement connection: one needs to start with atomic ensembles entangled with a very low two-photon component, at the price of low count rates and statistics.

IV. EXPERIMENTAL SETUP AND MEASUREMENT RESULTS
In this section, we present a scheme that permits us to investigate entanglement connection between two pairs of atomic ensembles, without the requirement of any active phase stabilization. Experimental results are finally given. heralds entanglement between the collective excitation in I2 and R. The Down pair is prepared in a similar fashion via events at D1c, D 1d . A heralding detection event triggers the control logic to gate off the light pulses going to the corresponding ensemble pair by controlling intensity modulators (IM ). The atomic state is thus stored while waiting for the second ensemble pair to be prepared. After both pairs of ensembles have been prepared, the control logic releases strong read pulses. Fields 2I2 and 2I1 (2R and 2L) are combined with orthogonal polarizations on polarizing beam splitters. Fields 2I2 and 2I1 are detected with the half-wave plate (λ/2)c at 22.5 • , which is equivalent to a 50/50 beamsplitter configuration. The fields 2 from the remaining ensembles are characterized conditionally on a detection event heralding the connection. The two configurations of figure 3 correspond to two different angles, 0 • and 22.5 • , of the half-wave plate (λ/2).

A. Experimental setup
The experimental setup is depicted in Figure 4. Two parallel pairs of atomic ensembles are first prepared independently, following the measurement-induced method detailed in section III. This preparation stage is sped up by real-time conditional control [16,19]: a detection event at either pair triggers intensity modulators that gate off all lasers pulses going to the corresponding pair of ensembles, thereby storing the associated state. After successfully preparing both pairs, strong read pulses are sent into the ensembles. The fields 2 I1 and 2 I2 are brought to interfere and a detection event on either detector heralds the connection process. Thanks to the conditional control, a 20-fold enhancement is obtained in the probability to establish the connection, leading to a rate of connection around 4Hz. Depending on the combinations of field 1 and field 2 detector clicks, two different entangled states are generated for the two remaining ensembles, denoted by + and −, with a π phase-shift between them.
As pointed out before, the process of connection be-tween the two remaining ensembles, which never interacted in the past, only requires the stability of the relative phase ξ over trials. This overall phase is defined as the phase difference between the absolute phase of all the paths (write beams, field 1, read pulses, and field 2 on the connection side) for the upper pair and the ones for the lower pair. Instead of actively stabilizing all individual phases as it was performed in [9] where two ensembles were involved, this requirement is fulfilled in our setup by exploiting the passive stability between two independent polarizations propagating in a single interferometer [19]. All the paths for the upper and lower pairs are common, except inside a small interferometer where orthogonal polarizations are separated to define the two ensembles on each side [20]. Operation over more than 24 hours is possible without any adjustment as the phase does not change by more than a few degrees. As a result, no active phase stabilization is required, simplifying significantly the experimental investigation of the connection process. Note that although the present configuration is sufficient to demonstrate the principle of the connection, an experiment where the final pair of ensembles L and R are  [9], since in that case all the paths would be distinct. Our configuration for passive stability is better suited to the case of parallel chains of ensembles, as in the original proposal of DLCZ.

B. Characterization of the states generated upon connection
The generated state is analyzed by using the tomography technique explained in section III C. Conditioned upon a connection event, the density matrix ρ of the fields 2 is reconstructed following the two required steps: the measurement of the diagonal elements and the determination of the coherence term. Table I gives the measured diagonal elements deduced from the records of photo-electric counts, for both generated states, after a connection event. Unit detection efficiency is assumed, which can only lead to a smaller value for the concurrence than the actual field concurrence for finite detection efficiency. From theses values, one can deduce the suppression h of the two-photon events relative to the square of the probability for single-photon events. We find h + = 0.7 ± 0.1 < 1 and h − = 0.8 ± 0.1 < 1. From independent measurements, we inferred the h ′ parameter for each pair before connection to be h ′ = 0.20±0.05. The experimentally determined values of h are thus consistent with the expression h = 4h ′ established previously. As pointed out before, this relation arises from the intrinsic feature that the connection succeeds only 50% of the time. This can be seen in the quantities 2p 01 ∼ 4% and 2p 10 ∼ 6%, which should be equal to half the retrieval efficiency. The retrieval efficiencies (including detection) independently measured for each ensemble were both around 10%.
In order to access the coherence term, figure 5 shows the probability to have a detection event on either output of the beam splitter, normalized to the sum of these events, as a function of the phase-shift between the fields 2 L and 2 R . Practically, the relative phase is scanned by adjusting the phases of the two classical read beams via birefringent waveplates. The visibilities are found to be V + = 64 ± 3% and V − = 59 ± 3%. A simple model [19] predicts for our excitation probability a visibility equal to 65%±10%. By taking into account the measured overlap of the photon wavepacket for fields 2 deduced from a two-photon interference [19], 0.90 ± 0.05, the expected visibility can be roughly estimated to be 55 ± 10% if all the reduction is attributed to a non-perfect overlap. In the absence of conditioning, the visibility drops to near zero, the residual visibility (below 3%) being explained by finite polarization extinction ratio in our setup. This result demonstrates for the first time the creation of coherence between two atomic ensembles which never interacted in the past. The reconstructed density matrices are shown in figure 6.
With these data in hand, the concurrences C can be estimated for both states: These values show finally the absence of entanglement, or at least, that our entanglement measurement, which provides a lower bound of the atomic entanglement, cannot detect entanglement in this particular case. One can correct from detection efficiencies and propagation losses [9], but any zero concurrence will stay zero by this correction.
The h values confirm anyway that the connected systems are barely in the regime where the two-photon events are suppressed relative to single photon events. One needs to start with smaller h ′ for the initial pairs. h ′ as low as 0.05 can be obtained routinely for each pair in our lab but the count rate to characterize the connection would be prohibitively low.

V. DISCUSSION AND PERSPECTIVES
In summary, we have presented a possible scheme to demonstrate entanglement connection between atomic ensembles which never interacted in the past. Such striking capability is a critical requirement for the future development of elaborate quantum networks. Our investigation has shown for the first time the creation of coherence upon the connection process. This result validates our proposed setup, in particular its passive phase stability, and constitutes a significant step towards the entanglement connection of matter systems.
To finally generate and prove entanglement connection between the remaining ensembles, very stringent condition on the suppression of the two-photon component needs to be satisfied, at the sacrifice of the count rate in our current setup. Overall, the figure of merit of any elaborate experiment is the product of the probability to prepare the entangled state at each write pulse and the coherence time. Improvements in these two directions have to be explored. The first one can be addressed by, for instance, multiplexing the atomic ensembles. One can imagine to use spatially-resolving detectors, namely array of single-photon detectors, and adaptive optical systems to reconfigure the optical interconnects. Improving  the coherence time is a second critical direction as more elaborate protocols are involved. It would require better nulling of the residuals magnetic fields and also the use of improved trapping techniques [12] like a large dipoletrap, as a magneto-optical trap will be rapidly limited by the diffusion of the atoms outside the excitation region. An increase by two orders of magnitude, from tens of µs to ms, would enable for instance to demonstrate the entanglement connection in our current setup in a few hours of data taking. All together, these improvements would enable deeper investigation of experimental quantum networking, and will definitely lead to fruitful insights into the distribution and processing of quantum information.