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Quantum repeaters based on atomic ensembles and linear optics

The distribution of quantum states over long distances is limited by photon loss. Straightforward amplification as in classical telecommunications is not an option in quantum communication because of the no-cloning theorem. This problem could be overcome by implementing quantum repeater protocols, which create long-distance entanglement from shorter-distance entanglement via entanglement swapping. Such protocols require the capacity to create entanglement in a heralded fashion, to store it in quantum memories, and to swap it. One attractive general strategy for realizing quantum repeaters is based on the use of atomic ensembles as quantum memories, in combination with linear optical techniques and photon counting to perform all required operations. Here we review the theoretical and experimental status quo of this very active field. We compare the potential of different approaches quantitatively, with a focus on the most immediate goal of outperforming the direct transmission of photons.


IV. Comparison and Discussion
The distribution of quantum states over long distances is essential for potential future applications of quantum technology such as long-distance quantum cryptography (Bennett and Brassard, Dec. 1984;Gisin et al., 2002) and quantum networks (Kimble, 2008;Nielsen and Chuang, 2000). In practice quantum channels such as optical fibers or free-space transmission are affected by loss and decoherence. This limits the distance over which quantum information can be transmitted directly by sending individual quantum systems (typically photons). In practice the most immediate problem is photon loss. For example, typical telecommunication optical fibers have losses of 0.2 dB/km in the optimal wavelength range around 1.5 µm. In a sense these losses are impressively low. For example, a piece of fiber that is 1 km long has a transmission of 95 percent. However, they become nevertheless very significant once one envisions distances of hundreds of kilometers or more. Even for very highrepetition rate sources (say 10 GHz, which is a very ambitious value for a source of quantum states), the rate of transmitted photons becomes exponentially low for such distances. For example, for 500 km one would have a rate of 1 Hz. The rate drops to 0.01 Hz for 600 km, and to 10 −10 Hz for 1000 km. The latter rate corresponds to 1 photon every 300 years.
In classical telecommunications this problem is overcome through the use of amplifiers ("repeaters"). Unfortunately straightforward amplification is not an option in quantum communication because of the no-cloning theorem (Dieks, 1982;Wootters and Zurek, 1982), which shows that noiseless amplification is impossible unless one restricts oneself to sets of orthogonal states, whereas the quantum nature (and thus the advantage) of protocols such as quantum key distribution arises precisely from the existence of non-orthogonal states. However, it turns out that the problem can be overcome using a more sophisticated method based on entanglement, which is known as the "quantum repeater" approach (Briegel et al., 1998).
Entanglement is one of the most counter-intuitive, nonclassical features of quantum physics. Bell's famous theorem (Bell, 1964;Mermin, 1993) states that entangled states cannot be simulated by local hidden variables, thus showing that entanglement lies at the heart of quantum non-locality. A very remarkable feature of entanglement is that it can be "swapped" (Zukowski et al., 1993). Given an entangled state between two systems A and B and another entangled state between systems C and D, it is possible to create an entangled state between systems A and D by performing a joint measurement of systems B and C in a basis of entangled states, followed by classical communication of the result to the location of system A and/or D. Entanglement between the two latter systems can be created in this way even though they may never have interacted.
Entanglement swapping can be seen as the generalization of quantum teleportation (Bennett et al., 1993) to entangled input states. Conversely, teleportation can be seen as a consequence of entanglement swapping. Performing a measurement on system A before the swapping (when A and B are entangled) projects system B into an un-entangled quantum state. Performing the same measurement after the swapping (when A and D are entangled) creates the same state in system D. But the described swapping procedure commutes with measurements on system A. This implies that the entanglement swapping procedure transfers the quantum state of system B to system D, without any quantum system physically moving from one location to the other.
Consider a great distance L, such that the overall transmission of the channel is forbiddingly small. As we have just seen, if one has an entangled state of two particles separated by the distance L at one's disposition, one can use this entangled state to teleport quantum states over this distance. One can also use it to perform entanglement-based quantum key distribution (Ekert, 1991) directly. The creation of entanglement over some great distance L can also be seen as a fundamental goal in itself, allowing one to extend tests of quantum non-locality to a new distance scale.
The key idea of the quantum repeater approach (Briegel et al., 1998) is that entanglement over the distance L can be created by entanglement swapping starting from two entangled pairs, each of which covers only half the distance, L 2 . Moreover, these entangled states can themselves be created starting from states covering a distance L 4 and so on. If one has a way of independently establishing entanglement for N = 2 n adjacent elementary links each covering a distance L 0 = L N , one can then efficiently create entanglement over a distance L with n levels of entanglement swapping operations, cf. Figure  1 (n is called the nesting level). For long distances the described protocol scales much better than direct transmission.
One essential requirement for the described approach is thus to be able to establish entanglement for the elementary links in a "heralded" way, i.e. one has to know when the entanglement has been successfully established. At first sight, the most direct approach would be to create entanglement between two systems locally and then send one of the two systems (e.g. a photon) to the distant location. However, the elementary links will still be quite long for realistic protocols, typically of order 100 km, corresponding to a transmission of order 10 −2 . Implementing heralding in such an approach would require being able to measure that the photon has arrived without destroying the entanglement, which is very difficult in In order to distribute entanglement over long distances, say between locations A and Z, one proceeds step by step : a) Entanglement is first created independently within short elementary links, say between the locations A and B, C and D, . . . , W and X, Y and Z. b) Entanglement is then swapped between neighboring links such that the locations A and D, . . . W and Z share entanglement. c) Entanglement swapping operations are performed successively in a hierarchical fashion until entanglement is distributed over the desired distance separating the locations A and Z. Yellow squares represents quantum memories. The dotted arrows connecting two remote memories indicate that they are entangled.
practice. A better approach is to create the entanglement "at a distance". For example, entanglement between one atom in A and another atom in B can be created via the detection of a photon that could have been emitted by either atom, provided that the measurement of the photon is performed in such a way that all "which-way" information is erased (Bose et al., 1999;Cabrillo et al., 1999).
(This can be seen as another application of the principle of entanglement swapping, cf. below.) The detection of the photon then serves as the heralding event for the creation of the entanglement between the two atoms. If the photon is lost in transmission, there is no detection and one knows that one has to try again. Another essential requirement for the quantum repeater protocol is that one has to be able to store the created elementary entanglement until entanglement has been established in the neighboring link as well, in order to then be able to perform the required entanglement swapping operation. The resulting higher-level entanglement again needs to be stored until the neighboring higher-level link has been established and so on. Thus quantum repeaters require the existence of "quantum memories" (Hammerer et al., 2008;Tittel et al., 2008). If such memories are not available, the only solution is to create entanglement in all links simultaneously. Such memory-less repeaters, also called "quantum relays", do not help to overcome the problem of photon loss, but can still be useful to alleviate other problems such as detector dark counts (Collins et al., 2005;Jacobs et al., 2002). Finally one has to be able to perform the required entanglement swapping operations between the quantum memories, i.e. to perform local joint measurements projecting onto entangled states between two memories. Such measurements are certainly possible if one has a way of performing general quantum gates (e.g. CNOT gates) between neighboring memories. However, this is generally a difficult task and it is thus of interest to consider dedicated, simpler solutions, e.g. entangling measurements that work only with a certain probability, cf. below.
The original quantum repeater protocol of (Briegel et al., 1998) furthermore contains "entanglement purification" (Bennett et al., 1996) steps that allow one in principle to purify the effects of any kind of decoherence. However, the implementation of such general entanglement purification requires the preparation of at least two initial pairs for every purified pair at any given nesting level for which purification is implemented, leading to significant overheads and thus to lower rates. This makes it advantageous to forgo full entanglement purification for simple architectures of just a few links, where it is not necessary for small, but realistic error probabilities per operation. In the present review our focus will be on such simple architectures, because they offer the most realistic chance in the short and medium term of achieving the most immediate goal of a quantum repeater, namely to outperform the quantum state distribution rate achievable by direct transmission.
A highly influential proposal for realizing quantum repeaters was made by (Duan et al., 2001). It is widely known as the DLCZ protocol (for Duan, Lukin, Cirac and Zoller). The authors showed how to meet all the above requirements using atomic ensembles as quantum memories, and linear optical techniques in combination with photon counting to perform all the required operations. The use of atomic ensembles as opposed to single quantum systems such as individual atoms as memories was motivated by the fact that collective effects related to the large number of atoms in the ensemble make it much easier to achieve a strong and controllable coupling between the memory and the photons that serve as long-distance quantum information carriers.
The basic process at the heart of the DLCZ protocol is the spontaneous Raman emission of a photon, which simultaneously creates a spin excitation in the atomic ensemble. This correlation between emitted photons and atomic excitations in each ensemble forms the basis for the generation of entanglement between distant ensembles (for each elementary link), which is done via a single photon detection that erases all "which-way" information, following the principle outlined above for the case of individual atoms. The spin excitations can be efficiently reconverted into photons thanks to a collective interference effect. This forms the basis for the entanglement swapping operations, which are again done by detecting single photons while erasing which-way information.
The DLCZ proposal inspired a large number of highly successful experiments, for example Kuzmich et al., 2003;Matsukevich and Kuzmich, 2004;van der Wal et al., 2003;Yuan et al., 2008), showing that the approach of using atomic ensembles, linear optics and photon counting is indeed very attractive from a practical point of view. Motivated both by the impressive experimental progress and by the growing realization that, while pioneering, the DLCZ protocol does not yet allow one to outperform the direct transmission of photons in practice, several papers have proposed significant improvements to the protocol, while using the same or very similar experimental ingredients. These proposals have in turn spurred new experimental investigations. Here we review this area of research.
We begin with a review of the theoretical proposals, starting with the DLCZ proposal in section II, including a discussion of its practical limitations, followed by a review of the most important proposed improvements in section III. In the DLCZ protocol, both entanglement generation and swapping are based on one single-photon detection each. Subsection III.A describes a protocol where entanglement is swapped based on two photon detections, leading to an improvement in the overall rate. Subsection III.B describes protocols where entanglement is generated based on two photon detections, leading to enhanced robustness with respect to phase fluctuations in the channel. Subsections III.C and III.D are devoted to multiplexing. Subsection III.C reviews the idea of using memories that can store multiple temporal modes. Their use in the present context is made possible by the realization that a DLCZ-type atomic ensemble can be emulated by combining a photon pair source and an "absorptive" quantum memory (i.e. a memory that can absorb and emit photons). This approach promises a great enhancement in the entanglement generation rate. Subsection III.D reviews work on spatial multiplexing, which would moreover significantly reduce the requirements on the memory time. Subsection III.E discusses a protocol based on single-photon sources, which can be effectively implemented with atomic ensembles, and which yields a significantly enhanced rate compared to the DLCZ protocol. Subsection III.F describes protocols that are based on effectively approximating ideal photon pair sources with atomic ensembles, leading both to enhanced rates and greatly enhanced robustness.
In section IV we compare the performance of different protocols quantitatively. Subsection IV.A is devoted to the entanglement distribution rates, whereas subsection IV.B discusses the robustness of the protocols with respect to storage time limitations, phase errors, and memory and detection inefficiencies. Subsection IV.C discusses complexity issues.
In section V we review the experimental status quo from the point of view of the different protocols described beforehand. In particular, subsection V.A is de-|e ∆ write pulse Stokes photon |g 1 |g 2 |e |g 1 |g 2

Write process Read process
Anti-Stokes photon read pulse FIG. 2 Basic level scheme for the creation of collective atomic excitations in atomic ensembles via spontaneous Raman emission (write process) and for their readout (read process), as proposed in the DLCZ protocol. Write process: All atoms start out in g1. A laser pulse off-resonantly excites the g1 − e transition, making it possible for a photon to be emitted on the e − g2 transition (with small probability). Read process: a resonant laser is applied on the g2 − e transition, promoting the single atomic excitation from g2 back to e, followed by collective emission on the e − g1 transition of a Stokes photon in a well-defined direction.
voted to experiments that realize elements of the DLCZ protocol, subsection V.B to experiments that are directed towards the creation and swapping of entanglement via two-photon detections. Subsection V.C discusses the implementation of quantum light sources compatible with ensemble-based quantum memories, while subsection V.D discusses the realization of the (absorptive) quantum memories themselves. Subsection V.E is devoted to photon detectors. Subsection V.F discusses different realizations of quantum channels. Finally subsection V.G discusses the issue of coupling losses.
In section VI we briefly review approaches to quantum repeaters using other ingredients besides or instead of atomic ensembles and linear optics, for example the use of single trapped ions or NV centers as quantum memories.
In section VII we give our conclusions and look towards the future.

II. THE DLCZ PROTOCOL
In this section we review the DLCZ protocol for quantum repeaters (Duan et al., 2001). We explain the basic physics underlying the protocol, followed by a description of its individual steps. We then evaluate the required time for long-distance entanglement distribution. Finally we discuss its limitations.

A. Basic Physics
The DLCZ protocol uses atomic ensembles that can emit single photons while creating a single atomic excitation which is stored in the ensemble. The photons can be used to entangle two distant ensembles. The atomic excitation can be efficiently converted into a photon thanks to collective interference, which is used for entanglement swapping and final use of the entanglement. Here we briefly describe the underlying physics, the next section explains the protocol.
The basic (idealized) scheme is as follows, cf. Fig. 2. In an ensemble of three-level systems with two ground states g 1 and g 2 and an excited state e all N A atoms are initially in the state g 1 . An off-resonant laser pulse on the g 1 −e transition (the write pulse) leads to the spontaneous emission of a Raman photon on the e − g 2 transition.
We will denote this photon as the Stokes photon, which corresponds to the usual Raman scattering terminology, provided that the energy of g 2 is slightly higher than that of g 1 . We will adopt this convention throughout this review. Detection of the Stokes photon in the far field, such that no information is revealed about which atom it came from, creates an atomic state that is a coherent superposition of all the possible terms with N A −1 atoms in g 1 and one atom in g 2 , namely where k w is the k vector of the write laser, k S is the k vector of the detected Stokes photon, and x k is the position of the k-th atom. In practice the amplitudes of the different terms may vary, depending on the laser profile and the shape of the atomic ensemble. A remarkable feature of such collective excitations that are of great interest for practical applications is that they can be read out very efficiently by converting them into single photons that propagate in a well-defined direction, thanks to collective interference (Duan et al., 2001;Laurat et al., 2006;Simon et al., 2007c). Resonant laser excitation of such a state on the g 2 − e transition (the read laser pulse) leads to an analogous state with N A − 1 atoms in g 1 and one delocalized excitation in e, but with supplementary phases e ikrx k , where k r is the k vector of the read laser and x k is the position of the k-th atom at the time of the readout (which may be different from its initial position x k if the atoms are moving).
All the terms in this state can decay to the initial state |g 1 ⊗N A while emitting a photon on the e − g 1 transition (the Anti-Stokes photon). The total amplitude for this process is then proportional to The conditions for constructive interference of the N A terms in this sum depend on whether the atoms are moving during the storage. If they are at rest (x k = x k for all k), then there is constructive interference whenever the phase matching condition k S + k AS = k w + k r is fulfilled, leading to a very large probability amplitude for emission of the Anti-Stokes photon in the direction given by k w + k r − k S . For atomic ensembles that contain sufficiently many atoms, emission in this one direction can completely dominate all other directions. This allows a very efficient collection of the Anti-Stokes photon Simon et al., 2007c). If the atoms are moving, there can still be constructive interference, provided that k S = k w and k AS = k r . We will come back to this point in section V.A. Note that there is no collective interference effect for the emission of the Stokes photon, since its emission by different atoms corresponds to orthogonal final states, e.g. the state |g 2 1 |g 1 2 ...|g 1 N A if the Stokes photon was emitted by the first atom etc. Full "which-way" information about the origin of the photon is thus stored in the atomic ensemble, making interference impossible (Scully and Zubairy, 2003). As a consequence the total emission probability for the Stokes photon is simply given by the sum of the emission probabilities for each atom, and there is no preferred direction of emission.
We have focused on the emission of a single Stokes photon into the mode of interest. However, since there is an ensemble of atoms, there are also amplitudes for the emission of two or more Stokes photons, accompanied by the creation of the same number of atomic excitations in g 2 . This dynamics can be described by the following Hamiltonian, where χ is a coupling constant that depends on the laser intensity, the number of atoms, the detuning and the transition strengths for the g 1 − e and e − g 2 transitions, a † is the creation operator for a Stokes photon and s † is the creation operator for an atomic excitation in g 2 . The vacuum state |0 for the mode s corresponds to the atomic state with all atoms in g 1 , the state s † |0 with one excitation in s corresponds to a state like in Eq.
(1) with one atom in g 2 etc. Here one focuses on one particular k vector for both the Stokes photon and the atomic excitation. This Hamiltonian, whose derivation is discussed in much more detail in (Hammerer et al., 2008), section II.A, thus describes the creation (and annihilation) of pairs of bosonic excitations. Note that it is formally equivalent to the Hamiltonian for the non-linear optical process of parametric down-conversion (Burnham and Weinberg, 1970;Hong and Mandel, 1985;Hong et al., 1987;Wu et al., 1986). Using operator ordering techniques developed by (Collett, 1988), one can show that, starting from an initial vacuum state for both modes a and s, it creates the following two-mode entangled state For small values of χt this can be expanded 6 Therefore, if the probability to emit one photon and create one atomic excitation is (χt) 2 , then there is a probability (χt) 4 to emit two photons and create two excitations etc. This possibility of creating multiple pairs of excitations, which becomes more significant as the probability of creating a single excitation is increased, is an important limiting factor in the quantum repeater protocols discussed in this review, cf. below.

B. Protocol
In this subsection we review the DLCZ protocol. We discuss the principle for the entanglement creation between two remote ensembles belonging to the same elementary link and the method for entanglement swapping between neighboring links. Finally we show how the created single-photon entanglement of the form |1 |0 +|0 |1 can be used to post-selectively obtain two-photon entanglement that is useful for applications.

Entanglement Creation for two Remote Atomic Ensembles
FIG. 3 (Color online) Entanglement creation between two remote ensembles located at A and B within the DLCZ protocol. The blue circles embedded in yellow squares represent DLCZ-type atomic ensembles which probabilistically emit Stokes photons (green dots). These photons are sent through long optical fibers (dotted line) to a central station.
The detection of a single Stokes photon at the central station in mode d ord, which could have come either from location A or B, heralds the storage of a single excitation (sa or s b ) in one of the two ensembles. Half circles represent photon detectors. The vertical bar represents a beam splitter (BS).
The procedure for entanglement creation between two remote locations A and B requires one ensemble at each location. The two ensembles are simultaneously excited such that a single Stokes photon can be emitted, corresponding to the state Here, we assigned bosonic operators a (b) and s a (s b ) to the Stokes photon and to the atomic excitation respec-tively associated to the ensemble A (B), φ a (φ b ) is the phase of the pump laser at the location A (B), and |0 is the vacuum state for all modes; O(p) represents the multi-photon terms discussed in the previous subsection. The Stokes photons are coupled into optical fibers and combined on a beam splitter at a central station between A and B. The modes after the beam splitter are d = where ξ a,b stand for the phases acquire by the photon on their way to the central station. The detection of a single photon in d for example, projects the state of the two atomic ensembles in A single atomic excitation is thus delocalized between A and B. This corresponds to an entangled state which can be rewritten as where |0 a(b) denotes an empty ensemble A (B), and |1 a(b) denotes the storage of a single atomic excitation. We have also defined θ ab = φ b − φ a + ξ b − ξ a . Taking into account detections both in d andd, the success probability of the entanglement creation is given by P 0 = pη d η t where η d is the photon detection efficiency and η t = exp(− L0 2Latt ) is the transmission efficiency corresponding to a distance of L0 2 , where L 0 is the distance between A and B (i.e. the length of the elementary link), and L att is the fiber attenuation length. (The losses of 0.2 dB/km mentioned previously, which are achievable in the telecom wavelength range around 1550 nm, correspond to L att = 22 km.) This way of creating entanglement by a single photon detection was inspired by similar proposals for entangling two individual quantum systems rather than two ensembles (Bose et al., 1999;Cabrillo et al., 1999). Note that it can be seen as an implementation of entanglement swapping. One starts with entangled states between the modes a and s a as well as b and s b as in Eqs. (4) and (5). The single photon detection at the central station projects onto an entangled state of the photonic modes a and b, creating entanglement between the stored modes s a and s b .

Entanglement Connection between the Elementary Links
Once entanglement has been heralded within each elementary link, one wants to connect the links in order to extend the distance of entanglement. This is done by successive entanglement swapping between adjacent links with the procedure shown in Fig. 4.
Consider two links AB and CD in which the ensembles A-B and C-D respectively are entangled by sharing a single excitation, as before. They are described by the 4 (Color online) Entanglement connection between two links A-B and C-D. The ensemble A (C) is initially entangled with B (D) as described by |ψ ab (|ψ cd ). The memories B and C are read-out and the resulting anti-Stokes photons are combined on a beam splitter. The detection of a single photon after the beam splitter, which could have come either from location B or C, heralds the storage of a single excitation (sa or s d ) in the ensembles A and D and projects them into an entangled state |ψ ad .
state |ψ ab ⊗|ψ cd , where |ψ kl are defined by the equation (7). The atomic excitations s b and s c that are probabilistically stored in the ensembles B and C are read-out with a strong, resonant light pulse to be converted back into "Anti-Stokes" photons associated to the mode b and c . These two modes are combined on a beam-splitter and the measurement of a single photon, e.g. in the mode 1 √ 2 (b + c ) will project the ensembles A and D into the entangled state By iterating successive entanglement swapping operations, it is possible to establish entanglement between more and more distant ensembles. We now analyze the effect of nonunit detector efficiency η d and memory efficiency η m on the entanglement swapping procedure. The detectors can give the expected click when two photons are stored in the memories B and C, but only one is detected. In this case, the created state contains an additional vacuum component where α 1 = 1 2−η and β 1 = 1−η 2−η . We have defined η as the product of the detector efficiency by the memory efficiency η = η d η m . The success probability for the first swapping is given by P 1 = η(1 − η 2 ). Similarly, one can show that the success probability for entanglement swapping at the (i+1)th level is given by P i+1 = α i η(1− αiη 2 ), where α i is the weight of the normalized entangled component in the state associated to the level i. It is connected to α i−1 by α i = αi−1 2−αi−1η . Using this last formula, one can easily show that after n nesting levels, the ratio βn αn = (1 − η)(2 n − 1). The relative weight of the vacuum component thus increases linearly with the number of elementary links N = 2 n composing the quantum repeater. We will see in what follows that in schemes where entanglement swapping is performed via two-photon detections, the vacuum components remain constant, cf. below.

Post-Selection of Two-Photon Entanglement
FIG. 5 (Color online) Post-selection of two-photon entanglement. Entanglement has been distributed independently within two chains (labeled by the subscript 1 or 2) such that the ensembles A1-Z1 and A2-Z2 share an entanglement.
The atomic excitations at the same location are read-out and the emitted anti-Stokes photons are combined into a beamsplitter and then counted. Measurements in arbitrary basis can be done by changing the beam-splitter transmission coefficients and phases.
Suppose that entanglement has been distributed over the desired distance, say between locations A and Z. The created entanglement, which consists of A and Z sharing a single delocalized excitation, is of limited use on its own, because it is difficult to perform measurements in any basis other than that of the Fock states |0 and |1 . This is why in the DLCZ protocol the created singleexcitation entanglement is now used as a building block for more directly useful two-photon entanglement.
One needs two ensembles at each location, labeled A 1 (Z 1 ) and A 2 (Z 2 ) for location A (Z). Entanglement between A 1 and Z 1 , and between A 2 and Z 2 have been established as presented before, such that we have the state 1 2 (a † 1 + e iθ1 z † 1 )(a † 2 + e iθ2 z † 2 )|0 . The projection of this state onto the subspace with one photon in each location is which is analogous to conventional polarization or timebin entangled states. The required projection can be performed post-selectively by converting the atomic excitations back into Anti-Stokes photons and counting the number of photons in each location. Measurements in arbitrary basis are possible by combining modes a 1 and a 2 (and also z 1 and z 2 ) on beam splitters with appropriate transmission coefficients and phases. The component |Ψ az of the mixed state ρ az distributed after n swapping operations is post-selected with the probability P ps = α 2 n η 2 /2. 8 C. Performance

Calculating the Entanglement Distribution Time
The general formula for calculating the time required for a successful distribution of an entangled state |Ψ az is The first factor is the waiting time at the elementary level, where L 0 = L/2 n is the length of the elementary link, L is the total distance, and n is the "nesting level" of the repeater, as introduced in section I. Entanglement creation attempts for elementary links only succeed with a probability P 0 . After every attempt, one has to wait to find out whether the attempt has succeeded (whether there was a photon detection in the central station). If not, the memory has to be emptied, and one tries again.
Assuming that the repetition rate is not limited by the speed of the write and read processes themselves, this leads to a basic period of L0 c , and also explains the factor 1/P 0 . Furthermore the total time is inversely proportional to the success probabilities at each level P i , and to the probability of successful post-selection at the end, P ps . The factors f 0 to f n , which all satisfy 1 ≤ f i ≤ 2, take into account the fact that for every i-th level swapping attempt one has to establish two neighboring links at level i − 1. This takes longer than establishing a single such link by a factor f i−1 . The precise values of these factors depend on the ensemble of success probabilities up to the given level. No analytic expression for them is known at this point for general i. However, it is easy to show that f 0 = 3 2 for P 0 1 (Brask and Sorensen, 2008;Collins et al., 2007). Intuitively, if the waiting time for a single link is T , one only has to wait a time T /2 for a success in one of two neighboring links. Then one still has to wait a time T for the second link. Moreover numerical evidence shows that setting f i = 3 2 for all i is a good approximation (Brask and Sorensen, 2008;Jiang et al., 2007b). We will use this approximation in the following to calculate and compare entanglement distribution times for different protocols. A more detailed justification for Eq. (12) and a more detailed discussion of the f i is given in Appendix A.
Plugging the expressions for the success probabilities P 0 , P i and P ps into Eq. (12), one finds This still contains the pair emission probability p. The value of p is constrained by the fact that there is a probability of order p 2 for the emission of two Stokes photons into the desired mode (associated with the creation of two atomic excitations), as discussed in section II.A. This leads to errors which reduce the fidelity F of the distributed state. For the present protocol these errors grow approximately quadratically with the number of links. To first order in p, F is always of the form 1−A n p(1−η), with, for example A 0 = 8, A 1 = 18, A 2 = 56, A 3 = 204, A 4 = 788. More details on the multi-photon error calculation are given in Appendix B. In the following we will assume that we can tolerate a fidelity reduction 1 − F = 0.1 due to multiphoton errors. The maximum allowed value of p for a given n can then be determined directly from the above values for A n .

Comparison to Direct Transmission
We will now compare the entanglement distribution time for the DLCZ protocol to the time for quantum state distribution using direct transmission. We consider a fiber attenuation of 0.2 dB/km (η t = e −L0/(2Latt) , with L att = 22km) corresponding to typical telecom fibers and telecom wavelength photons. We furthermore take into account the reduced photon velocity within the fiber, c = 2 × 10 8 m/s. We assume equal memory and photonresolving detector efficiencies η m = η d = 0.9. This is certainly a demanding choice, however it is far from the performance levels typically required for fault-tolerant quantum computing, for example. We will describe and discuss the experimental status quo in some detail in section V. We optimize the nesting level n and thus the number of links 2 n for each distance L.
For direct transmission we assume a single-photon source with a repetition rate of 10 GHz, as we did in section I. This will be our reference all through this paper. This is certainly an ambitious value. It might one day be achieved for single-photon sources based on quantum dots in high-finesse semiconductor micro-cavities (Moreau et al., 2001;Santori et al., 2002), for example, where lifetimes can be of order 100 ps. However, for now these sources are not very efficient. Moreover, the source would have to operate at telecom wavelengths (i.e. around 1.5 µm) (Hostein et al., 2009;Ward et al., 2005). Of course, any such choice is somewhat arbitrary. However, it will become clear later on that our conclusions don't depend very strongly on the exact choice of rate for the reference source, essentially because the scaling with distance is very different for direct transmission and for quantum repeaters. As a consequence, the curve corresponding to direct transmission in our main comparison figure, Fig. 18 in section IV, has a much steeper slope than the curves corresponding to all the considered repeater protocols, so that it intersects them all in the distance range L = 500 to 650 km. Changing the reference rate would change the cross-over distances only slightly. The most important question in the short and medium term is what distribution rates a given repeater protocol can achieve in that distance range.
For the DLCZ protocol, we find a cross-over point of L = 630 km, with an entanglement distribution time T tot = 340 seconds at that distance, for n = 2 (4 links). The corresponding value of p = 0.01. Note that, thanks to the very different scaling, for longer distances, the repeater is much faster than direct transmission, for example for 1000 km T tot = 4100 seconds, compared to 10 10 seconds for direct transmission. Nevertheless, this result is somewhat disappointing. On the one hand, a single entangled pair every 340 seconds is of course a very low rate. Even more importantly, for the repeater to work, the memory storage time has to be comparable to the mentioned 340 seconds. In particular, it has to be long enough for the final post-selection to be possible, i.e. long enough to create two independent single-photon entangled states over the whole distance. This is extremely challenging. Briefly anticipating the detailed discussion in section V, the best current results for quantum memory times in DLCZ-type experiments with atomic gases are in the few ms range (Zhao et al., 2009a,b). A storage time of order 1 s was achieved in a solid state system for a memory protocol based on electromagnetically induced transparency , though not yet at the quantum level. Decoherence times as long as 30 seconds have been demonstrated for the same kind of solid state system (Pr:YSO) in . Going even further and really implementing the whole protocol at such timescales is likely to be extremely challenging. There is thus strong motivation to try to invent protocols that allow faster generation of long-distance entanglement. Section III is dedicated to various such proposals.

D. Discussion -Limitations
Motivated by the results in the last subsection, we now describe several limitations of the DLCZ protocol, which have become starting points for further developments that will be described in section V.
(i) There is a trade-off between high fidelity of the distributed state and high distribution rate. We have seen that the errors due to multiple emissions from individual ensembles grow quadratically with the number of elementary links N . In order to suppress these errors, one then has to work with very low emission probability p, which limits the achievable rate. This quadratic growth of the multi-photon errors is related to the fact that the vacuum component in the created single-photon entangled state grows linearly with N , cf. the discussion at the end of Appendix B. In sections III.A, III.B and III.F we will see schemes where the vacuum component remains constant, and, as a consequence, the multi-photon errors grow only linearly with N , thanks to the use of entanglement swapping operations based on two photon detections instead of a single photon detection. In section III.E we review a scheme where multi-photon errors are greatly reduced through the use of single-photon sources, which can be effectively realized with atomic ensembles.
(ii) The entanglement creation between two remote ensembles requires interferometric stability over long distances. To illustrate the challenge this represents, let us consider an elementary link with L 0 = 125km. The entanglement in equation (11) depends on the phase θ 2 − θ 1 , the contribution to which from the given elementary link can be rewritten as θ B2 (t 2 ) − θ B1 (t 1 ) − θ A2 (t 2 ) − θ A1 (t 1 ) . We have defined t 1 (t 2 ) as the moment where the first (second) single-photon entangled state Eq. (8) was created in the elementary link. The phase thus has to remain stable over the time scale given by the mean value of t 2 − t 1 which is L 0 /(cP 0 ). For the considered example, this gives t 2 −t 1 = 4.5s. Over such long time scales, both the phases of the pump lasers and the fiber lengths are expected to fluctuate significantly. This problem has to be addressed in any practical implementation of the protocol, either through active stabilization of the fiber lengths, or possibly through the use of self-compensating Sagnac-type configurations, cf. section V.F. The described problem stems from the fact that in the DLCZ protocol long-distance entanglement is generated via single photon detections. In sections III.B and III.F we review schemes where entanglement is instead generated via two-photon detections, greatly reducing the stability requirements for the channels.
(iii) We have argued before that in the DLCZ repeater protocol one is a priori limited to a single entanglement generation attempt per elementary link per time interval L 0 /c. In sections III.C and III.D we will describe how this limitation can be overcome using memories that can store a large number of distinguishable modes, cf. below.
(iv) For long communication distances to be realistic, the wavelength of the Stokes photons has to be in the optimal range for telecom fibers (around 1.5 µm). This either severely restricts the choice of atomic species or forces one to use wavelength conversion techniques (Tanzilli et al., 2005), which for now are not very efficient at the single photon level, mostly due to coupling losses. In section III.C we describe how this requirement can be overcome by separating entanglement generation and storage.

III. IMPROVEMENTS
In this section we review various improvements over the DLCZ protocol that have been proposed over the last few years. We only discuss architectures that use essentially the same ingredients, i.e. atomic ensembles, linear optics, and photon counting, but that use them in different ways in order to achieve improved performance. We have seen that in the DLCZ protocol both entanglement generation and swapping are based on a single photon detection. Subsection III.A describes a protocol where entanglement is swapped based on two photon detections, which leads to a constant (rather than growing) vacuum component in the created state, resulting in an improvement in the overall entanglement distribution rate. Subsection III.B describes protocols where entanglement is moreover generated based on two photon detections, leading to enhanced robustness with respect to phase fluctuations in the channel. Subsections III.C and III.D are devoted to multiplexing. Subsection III.C reviews the idea of using memories that can store multiple temporal modes. Such memories can be realized using inhomogeneously broadened atomic ensembles in certain solid-state systems. Their use in the present context is made possible by the realization that a DLCZ-type atomic ensemble can be emulated by combining a photon pair source with a memory that can absorb and emit photons. This approach promises a great enhancement in the entanglement generation rate. Subsection III.D reviews work on spatial multiplexing, which would be even more powerful than the temporal variety. Subsection III.E discusses a protocol based on single-photon sources, which can be effectively implemented with atomic ensembles, and which yields a significantly enhanced rate compared to the DLCZ protocol. Subsection III.F describes protocols that are based on effectively approximating ideal photon pair sources with atomic ensembles, leading both to enhanced rates and greatly enhanced robustness.

A. Entanglement Swapping via Two-Photon Detections
In the previous chapter, it has been pointed out that entanglement swapping based on single-photon detections leads to the growth (linear with the number of links) of vacuum components in the generated state, and to the rapid growth (quadratic with the number of links) of errors due to multiple emissions from individual ensembles. The vacuum components are detected at the post-selection level and thus reduce the achievable rate. The multi-photon errors reduce the fidelity of the distributed states. In order to suppress these errors, one has to work with very low emission probabilities, further reducing the distribution rate. One possible way of addressing this problem is the use of entanglement swapping operations that are based on two photon detections instead of a single detection (Chen et al., 2007b;Jiang et al., 2007b;Zhao et al., 2007). It turns out that in this case the vacuum component remains stationary under entanglement swapping, and the multi-photon related errors grow only linearly with the number of links. In the present subsection we review the proposal of (Jiang et al., 2007b), where the elementary entanglement is generated by single-photon detections as in the DLCZ protocol, but the entanglement swapping is based on two-photon detections. In the next subsection we discuss the proposals of (Chen et al., 2007b;Zhao et al., 2007) where the entanglement is generated by two photon detections as well.

First-Level Entanglement Swapping
The elementary links in the protocol of (Jiang et al., 2007b) have the same form as in the DLCZ protocol. Depending on the nesting level, two distinct swapping operations are performed. At the first swapping level,

Ah Av Bv
Ch Dh the principle of the entanglement connection is shown in Fig. 6. This requires two ensembles at each locations emitting photons with well defined polarization : the horizontally (vertically) polarized modes are produced from upper (lower) atomic ensembles A h and B h (A v and B v ). Suppose that the ensembles A h and B h (A v and B v ) are entangled as in the DLCZ protocol, i.e. based on the detection of a single Stokes photon at a central station which could have been emitted by either of the two ensembles. Further suppose that entanglement between C h and D h (C v and D v ) has also been heralded in the same way. The average time for the entanglement creation of these four links is T 0 = 25 12 1 P0 L0 c , where P 0 = pη d η t . The prefactor 25 12 can be obtained using the same methods as in Appendix A for four variables instead of two. In order to swap the entanglement toward the ensembles A and D, the spin-wave stored in the memories B h -B v and C h -C v are readout and the emitted anti-Stokes modes, labelled b h -b v and c h -c v , are combined at a central station where they are detected in modes Fig.  6. In the ideal case, a twofold coincident detection between d + andd + projects the state of the two remaining spin-wave modes nondestructively into This operation thus allows one to exchange single spinwave entanglement of the form (7) with more standard two-particle entanglement of the form (14). However, only four out of the 16 terms in the Schmidt decomposition of |ψ ahbh ⊗ |ψ avbv ⊗ |ψ chdh ⊗ |ψ cvdv have a contribution to the output state, the remainders being eliminated by projective measurement, reducing the success probability for entanglement swapping to 1/8. Taking into account nonunit detector efficiency and memory recall, one can get the expected coincident detection when more than two spin-waves are stored in the memories B and C but only two are detected. In this case, the created state contains additional terms including single spin-wave modes and a vacuum component where . We have introduced a superscript 1 to label the level of the entanglement swapping. The probability for the successful preparation of this mixed state is P 1 = 1 8 η 2 (2 − η) 2 . For further distribution over longer distances of the previous two spin-wave entangled state, one uses the setup shown in Fig. 7. To illustrate the higher-level swapping operations, suppose that two spin-wave entangled states of the form (14) are distributed between ensembles A-D and E-H leading to the state |Ψ ad ⊗ |Ψ eh . This entanglement can be swapped toward the ensembles A-H by combining two anti-Stokes photons at a central station, where one photon is released from the D ensembles, the other from the E ensembles, and performing a projective measurement onto the modes d h ± e v and e h ±d v . The twofold coincident detection between d h +e v and e h + d v for example, collapses the two remaining full memories into |Ψ ah = 1/ √ 2(s † ah s † hh + s † av s † hv )|0 . Using the same set of linear optics and detectors shown in Fig. 7, one can perform successive entanglement swap-ping operations, such that the state |Ψ az can be distributed over the full distance after n swapping steps, between the locations A and Z. Due to imperfect detection and memory efficiency, the distributed state ρ n az includes single spin-wave and vacuum modes. One can show that their weights c n 2 , c n 1 and c n 0 are unchanged compared to the weights c 1 2 , c 1 1 and c 1 0 . Indeed, the condition for having a stationary state is c 0 c 2 = 4(c 1 ) 2 , which is fulfilled by c 1 2 , c 1 1 and c 1 0 . This is in contrast to the DLCZ protocol, where the vacuum component is amplified (approximately doubled) through every entanglement swapping operation, cf. section II. The success probability for the ith entanglement connection is therefore given by P i = 2η 2 (c 1 2 /2 + c 1 1 ) 2 = η 2 2(2−η) 2 , for i > 1. The entangled component |Ψ az of the distributed state ρ n az can be post-selected subsequently. The probability for such a successful post-selection is P ps = η 2 c n 2 = η 2 (2−η) 2 .

Performance
Using the expressions of T 0 , P 1 , P i for n ≥ i > 1 and P ps , one can write T tot as As in the DLCZ protocol, one has to take into account the possible errors due to multiple-pair emissions within an elementary link. The fidelity of the distributed state that one wants, fixes the value for the success probability p of the Stokes emission that one can use to estimate the distribution rate based on eq. (16). It is shown in Ref. (Jiang et al., 2007b) that the errors grow only linearly with the number of elementary links, whereas they grow quadratically with the number of links when entanglement connection is based on single-photon detections, as seen before for the DLCZ protocol. This improved scaling is related to the fact that the vacuum component is stationary in the present protocol, since the errors in the final state arise from the interaction of the vacuum and multi-photon components, cf. Appendix B. The protocol of (Jiang et al., 2007b) begins to outperform direct transmission (with a 10 GHz single-photon source as before) for a distance of 610 km, achieving an entanglement distribution time of 190 seconds, cf. also Fig. 18 in section IV. Here we assume the same values of η m = 0.9 and η d = 0.9 as before, and the same desired final fidelity F = 0.9. The optimum number of links for this distance is 4. This is about a factor of 4 faster than the performance of the DLCZ protocol, however it is clearly still a very long time for creating a single entangled pair, which moreover is probably still out of reach for realistic quantum memory storage times. The advantage compared to the DLCZ protocol is larger for longer distances, but of course the overall entanglement distribution times are even longer.
The main reason why the improvement is so relatively modest is that errors in the elementary link due to multiple excitations still force one to work with low emission probability p. The optimum value for 4 links is p = 0.037 (Jiang et al., 2007b), compared to p = 0.010 for the DLCZ protocol. Multiple excitations are hard to detect in the entanglement generation process (which is the same as in the DLCZ protocol), because the corresponding Stokes photons have to propagate far and are lost with high probability. Moreover the entanglement generation based on a single photon detection also leads to the phase stability issues discussed at the end of the previous section. It is then natural to consider changing the elementary link. This is the topic of the next subsection.

B. Entanglement Generation via Two-Photon Detections
Simultaneously with the proposal by (Jiang et al., 2007b), several schemes were proposed where not only the entanglement swapping, but also the elementary entanglement generation step is done via a two-photon detection (Chen et al., 2007b;Zhao et al., 2007). This approach was inspired by earlier proposals for entanglement creation between distant atoms or ions based on twophoton detections Feng et al., 2003;Simon and Irvine, 2003). The main advantage of generating entanglement in this way is that long-distance phase stability is no longer required. In the protocols discussed so far the detection of a single photon which could have come from either of two distant locations creates a single delocalized atomic excitation whose entangled nature depends on the propagation phases of the photon for the two possible paths. In contrast, in the present case entanglement between two distant memories is generated by projecting two photons, one coming from each location, into an entangled state of their internal degrees of freedom. This operation, and thus the created longdistance entanglement, is insensitive to the propagation phases of the two photons, which only contribute an irrelevant global phase to the pair wave function.
In this subsection we will focus on the protocol presented in section III.B of (Chen et al., 2007b) because it is very similar to the protocol of (Jiang et al., 2007b) and achieves a better performance than the simpler protocol of , which we will also discuss briefly. The protocol of section III.C of (Chen et al., 2007b), which is based on the local preparation of entangled pairs, followed by two-photon entanglement generation and swapping, and its improved version by (Sangouard et al., 2008b), which achieve significantly better performance while being equally robust, are discussed separately in section III.F.

Principle
Interestingly, even though it was proposed simultaneously and independently, the protocol of Ref. (Chen et al., 2007b) section III.B can be presented as a simple variation of the protocol of (Jiang et al., 2007b) discussed in the previous subsection, in which the entanglement generation step of (Jiang et al., 2007b) is performed locally and the first entanglement swapping step of (Jiang et al., 2007b) is performed remotely.

Av Bv
Ch Dh FIG. 8 (Color online) Setup for entanglement creation based on two-photon detection as proposed in Ref. (Chen et al., 2007b) section III.B. The ensembles A h and B h , as well as Av and Bv, where all four ensembles are located at the same node AB, are entangled two by two as in the DLCZ protocol, cf. Fig. 3. The ensembles A h -B h (Av-Bv) store a single delocalized photon with horizontal (vertical) polarization. In a similar way, the ensembles C h and D h (Cv and Dv), located at a different node CD, have been entangled independently. The excitations stored in the ensembles B h , Bv, C h , Cv are read out and the resulting photonic modes are combined at a central station using the set-up shown. Ideally, the coincident detection of two photons in d+ andd+ projects nondestructively the atomic cells A-D into the entangled state |Ψ ad of Eq. (14).
At each node, one needs four ensembles, say A h , A v , B h and B v at location AB and C h , C v , D h and D v at location CD as in Fig. 7, except that the A and B ensembles are close to each other, but far from the ensembles C-D. At each node, the ensembles with identical subscripts are entangled by sharing a single spin-wave excitation leading to the state |ψ ahbh ⊗ |ψ avbv ⊗ |ψ chdh ⊗ |ψ cvdv .
The average waiting time for the creation of this product state is approximatively given by T prep = 25 12 1 rPs with r the repetition rate of the elementary sources and P s = pη d . The prefactor 25 12 is a very good approximation for P s 1. It can be obtained using the same methods as in Appendix A for four variables instead of two. Note that multiple emissions from the same ensemble are detected with high probability, in contrast to the two previous protocols, since the corresponding Stokes photon does not propagate far and thus does not undergo significant losses.
Entanglement over the distance between AB and CD is now generated using the setup shown in Fig. 8, i.e. by converting the atomic excitation stored in the B and C ensembles into anti-Stokes photons followed by the detection in modes d± andd ± . Taking into account imperfections of detectors and memories, the twofold coincident detection d + -d + projects the state of the ensembles A and D into ρ 0 ad = c 0 2 |Ψ ad Ψ ad | + c 0 1 (|s ah s ah | + |s av s av | + |s dh s dh | + |s dv s dv |) +c 0 0 |0 0|; where c 0 2 = 1 (2−ηηt) 2 , c 0 1 = (1−ηηt) 2(2−ηηt) 2 and c 0 0 = (1−ηηt) 2 (2−ηηt) 2 . The probability for the successful preparation of this mixed state is P 0 = 1 8 η 2 η 2 t (2 − ηη t ) 2 . Figure 7 shows how, using the same combination of linear optical elements as for the previous protocol, one can perform successive entanglement swapping operations in order to distribute the state ρ n az after n swapping operations. In analogy to above, the state ρ n az includes vacuum and single spin-wave components with unchanged weights with respect to the initial ones, i.e. c i 2 = c 0 2 , c i 1 = c 0 1 and c i 0 = c 0 0 where the superscript i refers to the ith swapping. The probability for the swapping operation to succeed is given by Finally, one can perform a post-selection of the entangled component |Ψ az of the state ρ n az . The probability for the successful post-selection is given by P ps = η 2 c n 2 = η 2 (2−ηηt) 2 .

Performance
Taking into account the expressions of P 0 , P i with n ≥ i ≥ 1 and P ps , one can write T tot as For this formula to be strictly valid, the time T prep required to prepare entanglement between local ensembles has to be negligible compared to the communication time, i.e. in our case T prep = 25 12rpη d L0 c . Otherwise, one has to replace L0 c by L0 c +T prep . For a realistic source repetition rate of 10 MHz, preparation time and communication time become comparable for p = 10 −3 . The authors did not quantify the multi-photon errors in the protocol in detail, making it difficult to say for which link number this value of p is attained. Unfortunately the results of (Jiang et al., 2007b) on the multi-photon errors cannot be taken over directly because the strong photon loss corresponding to long-distance propagation intervenes at different stages in the two protocols, even though they are otherwise formally equivalent. For the following estimate we take the simple formula Eq. (18), which gives a lower bound for the entanglement distribution time. Since two photons have to reach the central station, the square of the transmission η t intervenes in this formula, making the distribution time more sensitive to losses in the elementary link and thus favoring more and shorter links compared to the DLCZ protocol and the protocol of (Jiang et al., 2007b). We limit the total number of links to 16 in order to stay in a regime where it is reasonably plausible that entanglement purification may not be required. (It is worth noting that increasing the link number improves the rate by less than a factor of 2 in the distance range that we are focusing on.) We choose the same detection and memory efficiencies, η m = η d = 0.9, as before. With all the mentioned assumptions, the protocol starts to outperform direct transmission (with a 10 GHz source, as before) for a distance of 640 km, achieving an entanglement distribution time of 610 seconds, see also Fig. 18 in section IV. The performance in terms of rate is thus comparable to (but slightly worse than) for the DLCZ protocol.
One important reason for the long time required is that excess photon emissions (three or four photons) in the long-distance entanglement generation step typically remain undetected due to large fiber losses. As a consequence, the generated state has large vacuum and singlephoton components, which lead to small success probabilities for the subsequent swapping steps, and thus to a rather low overall entanglement distribution rate. This makes it very difficult to really profit in practice from the main advantage of the protocol, which is its increased robustness with respect to phase fluctuations in the fibers. We will see below that the protocols discussed in section III.F have the same advantage in robustness while achieving much faster entanglement distribution.

Other Protocol
As mentioned before, another protocol based exclusively on two-photon detections was proposed by  simultaneously with the work by (Chen et al., 2007b). In the scheme of Ref. , entanglement is directly generated over long distances, without a preceding local DLCZ-type step. Since only a small excitation probability can be used for each entanglement generation attempt in order to avoid multiphoton errors, and since after each attempt one has to communicate its success or failure over a long distance, the required entanglement generation time becomes significantly longer than for the DLCZ protocol. In fact, since the success probability for every entanglement generation attempt is proportional to p 2 (where p is the emission probability as before), the entanglement distribution time for this protocol is about a factor of 1 p 2 longer than for the protocol of section III.F. For typical link numbers p has to be smaller than 0.01 in order to avoid multi-photon errors, resulting in a factor of at least 10 4 between these two protocols, cf. also Fig. 18 in section IV.
So far we have seen that using improved protocols compared to the original DLCZ proposal it is possible to achieve moderately faster entanglement distribution or to eliminate the need for interferometric stability. However the achievable rates are still far too low. The next two sections are devoted to multiplexing, an approach which holds great promise for overcoming this key difficulty.

C. Photon Pair Sources and Multimode Memories
In this subsection we review an approach towards multiplexing  that starts from the realization that a DLCZ-type atomic ensemble can be emulated by the combination of a photon pair source and a quantum memory that can absorb and re-emit photons. By itself, this has the advantage of allowing greater wavelength flexibility for the memory compared to the DLCZ situation where the Stokes photon has to be emitted at a telecom wavelength. If the memory furthermore has the capacity of storing and re-emitting light in a (possibly large) number of different temporal modes, the approach described below promises greatly improved entanglement distribution rates. The implementation of such memories is within reach in certain solid-state atomic ensembles, as will be discussed in more detail in section V.D.

Separation of Entanglement Generation and Storage
The basic element of all the protocols discussed so far is an ensemble of three-level atoms that is coherently excited in order to generate a Stokes-photon by Raman scattering, heralding the storage of an atomic spin excitation which can later be reconverted into an anti-Stokes photon. Depending of the protocol, either the Stokes photon or the anti-Stokes photon is used to create entanglement between remote memories. One of these photon therefore has to propagate over long distances and we want its wavelength to match the telecom wavelengths where the fiber attenuation is small (around 1550 nm). This gives a significant constraint on the operating wavelength of the memory. None of the quantum memories that have been demonstrated so far work at this wavelength. Possible technological solutions include the use of wavelength conversion, which however so far is not very efficient at the single photon level (Tanzilli et al., 2005) (primarily due to coupling losses), or the use of Erbium-doped crystals as quantum memories, where first experimental (Staudt et al., 2007a(Staudt et al., , 2006(Staudt et al., , 2007b and theoretical (Ottaviani et al., 2009) investigations have been performed, but implementing the DLCZ protocol is still a distant and uncertain prospect.
A different approach for long-distance entanglement creation was proposed in . It combines pair sources and absorptive memories to emulate the DLCZ protocol, cf. Fig. 9. The basic procedure for entanglement creation between two remote locations A and B requires one photon-pair source and one mem- ory at each location. The sources are simultaneously and coherently excited such that each of them has a small probability p/2 to emit a pair, corresponding to the state here a and a (b and b ) are two modes, corresponding e.g. to two different directions of emission, cf. Fig. 9.
The O(p) term describes the possibility of multiple pair emissions. It introduces errors in the protocol, implying that p has to be kept small, in analogy with the DLCZ protocol. The modes a and b are stored in local memories whereas the modes a and b are combined on a beam splitter at a central station. The modes a and b should thus be at telecom wavelength, but there is no such requirement for the modes a and b . Similarly to the entanglement creation in the DLCZ protocol, the detection of a single photon after the beam splitter heralds the storage of a single photon in the memories A and B, leading to the state (8). Note that we have set the phases to zero for simplicity. The entanglement can be extended to longer distances by successive entanglement swapping as in the DLCZ protocol. The required photon pair sources could be realized with atomic ensembles. For example, (Chanelière et al., 2006) have proposed to use a specific atomic cascade in Rb for which the first photon has a wavelength of 1.53 µm. There are also convenient ways of implementing pair sources not based on atomic ensembles, notably parametric down-conversion (Burnham and Weinberg, 1970;Hong and Mandel, 1985;Hong et al., 1987;Wu et al., 1986) in non-linear optical crystals, which allows a lot of wavelength flexibility. Pair sources can also be realized based on the DLCZ protocol, by applying the write and read pulses with a small time interval or even simultaneously, cf. section V.
10 Entanglement creation with temporal multimode memories. The source can be triggered a large number of times in every communication time interval L 0 c . One mode from each pair is sent to the central station, the other one is stored in the multimode memory.
FIG. 11 Entanglement swapping with temporal multimode memories. One has to be capable of recombining at the beam splitter exactly those modes whose partners have participated in a successful entanglement generation at the lower level.

Protocol with Temporal Multimode Memories
In section II.D we pointed out that in the DLCZ protocol one is a priori limited to a single entanglement generation attempt per communication time interval L0 c , because the memories have to be emptied after every unsuccessful attempt. The same is true for the protocols of sections III.A and III.B. The architecture described in the present section is particularly well adapted for temporal multiplexing, which overcomes this limitation, cf. Fig. 10. If the memories can store not only one mode but a train of pulses, one can trigger the sources many times per communication time L 0 /c, potentially creating pairs into modes a i and a i (b i and b i ), where i (i = 1, ..., N m ) labels the respective "time-bin", and N m is the total number of temporal modes. All the modes a i and b i are stored in the respective memories at A and B. Any of the modes a i or b i can now give rise to a detection after the central beam splitter. This leads to an increase of the entanglement generation probability P 0 by a factor of N m (for N m P 0 1), which directly translates into an increase of the entanglement distribution rate by the same factor. The speed-up is thus achieved at the most elementary level, that of entanglement generation. As a consequence, the same principle could also be applied to other quantum repeater protocols, although the technological challenges vary depending on the protocol, cf. below.
In order to do entanglement swapping using multimode memories, one has to be able to recombine exactly those modes, whose partners have given rise to a detection, and thus a successful entanglement generation, in the respective links, cf. Fig. 11. If this is ensured, entanglement swapping can again proceed in analogy with the DLCZ protocol. Temporal multimode memories with the required characteristics can be realized for example based on the photon echo principle in inhomogeneously broadened solid-state atomic ensembles (notably in rare-earth ion doped crystals). A particularly promising approach towards the efficient realization of such memories is based on "atomic frequency combs" (Afzelius et al., 2009a). This is discussed in more detail in section V.D. In addition to the requirements on memory efficiency and storage time discussed previously, an important characteristic for such a multimode memory is its bandwidth, since this may limit the number of modes that can be stored in a given time interval L0 c , even if the memory is in principle capable of storing more modes. This is not a major limitation for the present protocol. In (Afzelius et al., 2009a) it is argued, for example, that a memory based on an Eu-doped crystal with a bandwidth of 12 MHz (limited by hyperfine transition spacings in Eu) would be capable of storing a train of N m = 100 pulses with a total length of 50 µs, which is still much shorter than the communication time for a typical link length L 0 = 100 km, which is of order 500 µs, taking the reduced speed of light in the fiber into account. Note that a first experimental demonstration of an interface with N m = 32 was recently performed in a Nd-doped crystal (Usmani et al., 2009). Assuming N m = 100, and η m = η d = F = 0.9 as before, the protocol of  starts to outperform direct transmission (assuming the usual 10 GHz single-photon source) for a distance of 510 km, achieving an entanglement distribution time of 1.4 seconds, using a repeater architecture with 4 elementary links, see also Fig. 18 in section IV. This is a significantly improved rate compared to the previous sections. Moreover this timescale is also much more compatible with realistically achievable quantum memory times Longdell et al., 2005), as mentioned before and discussed in more detail in section V.
The present protocol closely follows the original DLCZ protocol, in particular relying on entanglement generation via a single photon detection, leading to similar phase stability issues, even though they are somewhat reduced by the shorter timescale of entanglement distribution. It is natural to ask whether it is possible to implement multimode versions of more robust protocols, such as the ones of section III.B or the one (still to be discussed) of section III.F. This may well be possible. It is however more challenging than for the present protocol, mainly because the fastest robust protocols rely on the local preparation of stored single-photon (section III.B) or two-photon (section III.F) entanglement. Such preparation requires a lot of repetitions, which reduces the time available for temporal multiplexing per time interval L0 c . For NPRDs, the indistinguishability of one-and two-photon pulses requires a final projective measurement, which succeeds with probability 1=c 3 1 (see Ref. [3] for a detailed discussion). Consider a 1000 km communication link. Assume a fiber loss of 10= ln10 0:16 dB=km, 0 0:01, and 0:9. Taking N 3 (L 0 125 km) gives P 0 0:001. For concreteness, we treat the NPRD case, producing connection probabilities: P 1 0:698, P 2 0:496, P 3 0:311, and 0:206. Figure 3 demonstrates agreement with the exact predictions for n 1 and the approximate predictions for n > 1. The slight discrepancies for long memory times with larger n are uniform and understood from the simultaneous connection successes neglected in Eq. (6).
An N-level quantum repeater succeeds in entanglement distribution when it entangles the terminal nodes with each other. Figure 4 shows the entanglement distribution rate of a 1000 km N 3 quantum repeater as a function of the quantum memory lifetime. Remarkably, for multiplexing with n * 10, the rate is essentially constant for coherence times over 100 ms, while for the parallel systems, it de-

D. Spatially Multiplexed Memories
For temporal multimode memories as described in the previous section, the gain in entanglement distribution time is due to the lowest level of the repeater protocol, as P 0 is enhanced by a factor of N m . All higher levels of the protocol are unchanged with respect to the protocols without multiplexing. Ref. (Collins et al., 2007), which was published before Ref. , studies a more far-reaching form of multiplexing, which might be possible in particular in the spatial domain. They envision a situation where several different sub-ensembles of the DLCZ type can be addressed completely independently. In particular, photons can be retrieved independently from each sub-ensemble and combined at will. Most importantly, even when some of the sub-ensembles have been "filled", i.e. entangled atomic excitations that involve these sub-ensembles have been created, one can use others that are still empty to make new attempts at entanglement creation. There is no known way of implementing such a step-by-step "accumulation" of stored entanglement in the temporal multimode case. The temporal multimode memories discussed in the previous subsection, can be charged only once (although with a large number of modes), then they have to be read out before being useful again, cf. section V.
Ref. (Collins et al., 2007) compares such strongly multiplexed repeaters to the case where N r completely independent repeater architectures are used in parallel (for which the rate enhancement is exactly equal to N r , of course). They find a moderate advantage in terms of rate for the strongly multiplexed case, without explicitly quantifying the advantage. Note that (Jiang et al., 2007b) address the same question in section V of their paper and find a scaling of the entanglement distribution rate with N 1.12 r instead of N r , which is consistent with the modest improvement found by (Collins et al., 2007). However, (Collins et al., 2007) also show that there is a very significant advantage for the multiplexed approach in terms of the necessary memory time. Whereas for strictly parallel repeaters the necessary memory times are determined by the waiting times for each repeater individually (and thus are extremely long), the multiplexed architecture leads to greatly reduced requirements on the storage times, cf. Figure 12. First experimental efforts towards spatial multiplexing are described in section V.B.2. Further theoretical work includes Vasilyev et al., 2008). The ideal in the long run would clearly be to combine temporal and spatial multiplexing in the same system, in order to maximize potential quantum repeater rates. Multiplexing of different completely independent frequency channels is another attractive possibility, in particular for the inhomogeneously broadened solid-state ensembles that are being investigated in the context of temporal multimode storage.

E. Single-Photon Source based Protocol
In section II and Appendix B it was shown that multiphoton emission events impose significant limitations on the performance of the DLCZ protocol. Motivated by this fact,  suggested a protocol based on single-photon sources, which makes it possible to eliminate such errors. The protocol was conceived for ideal single-photon sources. However, a good approximation of such a source can be implemented with atomic ensembles. The resulting scheme leads to a significantly improved entanglement distribution rate compared to the DLCZ protocol. In the following we first describe the ideal protocol and its performance, then we discuss how to implement a single-photon source to good approximation with atomic ensembles.

Principle
The architecture of the scheme proposed in Ref.  is represented in Fig. 13. The locations A and B contain each one single-photon source and one memory. When they are excited, each of the two sources ideally creates one photon which is sent through a beam splitter with reflection and transmission coefficients α and β satisfying |α| 2 + |β| 2 = 1, such that after the beam-splitters, the state of the two photons is (αa † + βa † )(αb † + βb † )|0 , which can be developed as The modes a , b are stored in local memories. The modes a, b are coupled into optical fibers and combined on a beam splitter at a central station, with the modes after the beam-splitter denoted by d = 1 √ 2 (a + b) and d = 1 √ 2 (a − b), as before. We are interested in the detection of one photon, for example in mode d.
13 (Color online) Entanglement creation between two remote ensembles located at A and B based on single-photon sources. Blue circles represent single-photon sources. A single-photon is generated at each location and sent through an asymmetric beam splitter with small transmission and high reflectivity, leading to a superposition of modes a and a (b and b ) at location A (B). The modes a and b are stored in local memories whereas a and b are sent to a central station where they are combined on a 50/50 beam splitter. The detection of a single photon at the central station heralds the storage of the second one within the memories with high probability, due to the asymmetry of the local beam splitters. This creates the entanglement of the two remote memories which share a single excitation.
separately the contributions from the three terms in Eq.
(20). The term a † b † |0 which corresponds to two full memories, cannot generate the expected detection and thus does not contribute to the entanglement creation. The term (a † b † + a † b † )|0 may induce the detection of a single photon in mode d with probability α 2 β 2 η t η d . Such detection creates the desired state 1 √ 2 (a † + b † )|0 associated to entangled memories. Finally the term a † b † |0 may also produce a single photon in mode d if one of the two photons is lost. The probability this term produces the expected detection is approximately β 4 η t η d , since for long distances η t 1. This detection heralds the vacuum state |0 for the remaining modes a and b . Taking into account all these contributions, the state created by the detection of a single photon in mode d is thus given by The state |ψ ab describes the entanglement of the two memories located at A and B, while the vacuum state |0 corresponds to empty memories. We emphasize that none of the three terms in Eq. (20) leads to an error of the form a † b † |0 . This is a crucial difference compared to the DLCZ protocol, cf. Appendix B. By considering both detections in mode d andd, one can show that the success probability for entanglement creation in an elementary link is P 0 = 2p 1 β 2 η t η d with p 1 the probability that the source emits one photon (p 1 = 1 in the ideal case).
The further steps are as for the DLCZ protocol: Neighboring links are connected via entanglement swapping, creating the entanglement between two distant locations A and Z. One shows that the success probability for entanglement swapping at the i-th level is given by . Moreover each location contains two memories, denoted A 1 and A 2 for location A etc. Entangled states of the given type are established between A 1 and Z 1 , and between A 2 and Z 2 . By post-selecting the case where there is one excitation in each location, one generates an effective state of the form The probability for a successful projection onto the state Eq. (22) is given by The vacuum component in Eq. (21) does not contribute to this final state, since if one of the two pairs of memories contains no excitation, it is impossible to detect one excitation in each location. The vacuum components thus have no impact on the fidelity of the final state. This is not the case for components involving two full memories as in Refs. (Duan et al., 2001) and , which may induce one excitation in each location and thus decrease the fidelity. Note that vacuum components, which exist for the single-photon source protocol already at the level of the elementary links, occur for the DLCZ protocols as well, starting after the first entanglement swapping procedure.

Performance
As indicated before, the absence of fundamental errors proportional to the entanglement creation probability leads to significantly improved entanglement distribution rates for the single-photon source protocol with respect to the DLCZ protocol. We now discuss this improvement quantitatively. The weight of the vacuum component at each nesting level is larger in the singlephoton source protocol, and thus the success probabilities P i (with i ≥ 1) for entanglement swapping are somewhat lower. However, the probability P 0 can be made much larger than in the photon-pair source protocols. Overall, this leads to higher entanglement distribution rates, as we detail now. Taking into account the expression of P 0 , P i and P ps , one can show that the total time required for entanglement distribution with the single-photon protocol is Assuming η m = η d = p 1 = 0.9 as before, the present protocol starts to outperform direct transmission (with a 10 GHz single-photon source) for a distance of 580 km, achieving an entanglement distribution time of 44 seconds, with a repeater composed of 4 links and a beam splitter transmission β 2 = 0.16, see also Fig. 18 in section IV. This is about an order of magnitude faster than the DLCZ protocol. The significant improvement compared to the DLCZ protocol can be understood as due to higher value of P 0 (9.2 × 10 −3 as opposed to 3.4 × 10 −4 for DLCZ). On the other hand the vacuum component is larger in the present protocol, reducing the success probability for the swapping operations, which is why the improvement is not as large as the difference in P 0 .

Implementing the Single Photon Source Protocol with Atomic Ensembles
In section II we explained how the emission of a Stokes photon in a DLCZ-type atomic ensemble creates a single stored atomic excitation, and how this atomic excitation can subsequently be reconverted into a photon. This implies that an ensemble charged with a single excitation can serve as a single-photon source. The possibility of multi-photon emissions, which can go undetected even for photon-number resolving detectors since they do not have perfect efficiency in practice, means that this source is not ideal, but that there is a two-photon contribution with an amplitude equal to p 2 = 2p(1 − η d )η m , where p is the emission probability for the Stokes photon, η m is the memory efficiency, and η d is the efficiency of the detector that detects the Stokes photon and thus announces that the ensemble is charged. If one chooses p sufficiently small, one can therefore realize a very good approximation to an ideal single-photon source. However, this means that the excitation triggering the potential Stokes photon emission has to be repeated many times before the ensemble is successfully charged. In an implementation one has to check whether this imposes significant limits on the distribution rate. Fortunately, this is not the case for realistic repetition rates for the Stokes emission, say 10 MHz. For a repeater with 4 links one can show that the maximum value of p 2 compatible with F = 0.9 is p 2 = 0.0011, giving p = 0.006 for the emission probability. With a 10 MHz repetition rate, the ensemble will thus be charged on average every 18 µs. Comparing to a typical communication time L0 c of order 750 µs for a 150 km link, this even leaves considerable scope for temporal multiplexing, provided one has appropriate multimode memories. It should be emphasized in particular that in the present protocol both the source and the memory have to be at telecom wavelength. Note that a single-photon source can also be realized by combining a photon pair source (which can be ensemble-based, but also e.g. based on parametric downconversion) and an absorptive memory in analogy with the approach described in section III.C, see also section V.
FIG. 14 (Color online) A single-photon source whose output is partially stored in a memory, as required in the protocol of  described in section III.E, can be emulated with a DLCZ-type atomic ensemble, in which an atomic excitation is first created, accompanied by a Stokes photon emission, and then partially read out through the application of a read pluse with an area that is smaller than π.
The same principle is also used in the protcol of (Sangouard et al., 2008b) described in section III.F.

Alternative Implementation via Partial Readout
We presented the protocol as consisting of the creation of single photons, followed by their partial storage, i.e. the storage of one of the two output modes of a beam splitter. Alternatively, once the ensemble that serves as the single-photon source has been charged by the emission of a Stokes photon as described before, it can be partially read out, i.e. the atomic spin waves can be partially converted back into propagating photons, see Fig. 14. In principle this could be done using read pulses whose area is smaller than the standard π, chosen to give the same values of α and β as above. There is a subtlety concerning this idea in the usual DLCZ-type experiments because the Anti-Stokes photon is typically emitted during the duration of the read pulse, such that it seems difficult to assign a fixed pulse area to the read. However, a Rabi oscillation regime was nevertheless observed by (Balic et al., 2005) and theoretically described by (Kolchin, 2007) even for ensembles with large optical depth. It may thus be possible nevertheless to pick a pulse area that corresponds to the desired values of α and β. The described idea certainly works for other kinds of memories, where readout pulse and emission are separated in time, such as the memories based on the photon echo principle discussed in section V.D.2, which is particularly relevant if the single-photon source is realized by combining a photon pair source and an absorptive memory as discussed above. The partial readout approach discussed here is important for the protocol described in the following section III.F as well, see below.

F. Protocols based on Local Generation of Entangled Pairs and Two-Photon Entanglement Swapping
In the previous section we saw that for quantum repeaters where the entanglement creation is based on a single photon detection, it is advantageous to have an ideal single-photon source, or even a good approximation of such a source realized with atomic ensembles. Analogously, it is fairly natural to ask whether it may be possible to achieve an efficient repeater protocol with entanglement creation based on two-photon detections if one had an ideal photon pair source, or a good approximation of such a source implemented with atomic ensembles. This is indeed a fruitful approach.
The first such protocol was proposed in section III.C of (Chen et al., 2007b), without evaluating its performance. Ref. (Sangouard et al., 2008b) proposed an improved version of the same approach and showed that this leads to a powerful quantum repeater protocol, which is both robust under phase fluctuations, and achieves the best entanglement creation time of all known non-multiplexed protocols with ensembles and linear optics, cf. below.
In order to effectively realize a single-pair source, Reference (Chen et al., 2007b) proposed to generate entangled pairs of atomic excitations locally by using four single-photon sources (which, as we saw in the previous section, can be realized with DLCZ-type ensembles), linear optical elements, and two quantum memories based on Electromagnetically Induced Transparency (EIT), cf. Fig. 11 of Ref. (Chen et al., 2007b). Four photons are emitted by the ensembles serving as sources, two of them are detected, two are absorbed again by the EIT memories. This double use of ensembles (emission followed by storage) leads to relatively large errors (vacuum and single-photon contributions) in the created state if the memory efficiencies are smaller than one. These errors then have a negative impact on the success probabilities of the entanglement generation and swapping operations, and thus on the overall time needed for long-distance entanglement distribution.
The proposal of Ref. (Sangouard et al., 2008b), which is based on partial readout of the memories, allows one to produce entangled pairs of atomic excitations with higher-fidelity. Indeed, this scheme does not use any emission followed by storage. For the same memory and detection efficiency, it thus leads to higher quality entangled pairs compared to the method of Ref. (Chen et al., 2007b), and as a consequence to a significantly improved rate for the overall quantum repeater protocol. In the following we describe the proposal of Ref. (Sangouard et al., 2008b) in more detail and evaluate its performance.

Local Generation of Entangled Pairs of Atomic Excitations
The proposed setup for the generation of high-fidelity entangled pairs requires four atomic ensembles. As before, the four ensembles are repeatedly excited independently with a repetition rate r until four Stokes photons are detected, heralding the storage of an atomic spinwave in each ensemble. The Stokes photons have a well defined polarization : the horizontally (vertically) polarized modes are labeled by a † h and b † h , (a † v and b † v ) and are produced from upper (lower) atomic ensembles A h and 15 (Color online) Setup for generating high-fidelity entangled pairs of atomic excitations. Yellow squares containing a blue circle represent atomic ensembles which probabilistically emit Stokes photons (green dots). The conditional detection of a single Stokes photon heralds the storage of one atomic spin-wave excitation. In this way an atomic excitation is created and stored independently in each ensemble. Then all four ensembles are simultaneously read out partially, creating a probability amplitude to emit an Anti-Stokes photon (red dots). The coincident detection of two Anti-Stokes photons in d+ andd+ projects non-destructively the atomic cells into the entangled state |Ψ ab of Eq. (24). Fig. 15. The associated single atomic spin excitation are labeled by s † ah , s † av , s † bh or s † bv depending on the location. The average waiting time for successful charging of all four ensembles is approximately given by T = 25 12rp , as can be shown by the same methods that are used in Appendix A. Thanks to the independent creation and storage, it scales only like 1/p, with p the probability for a Stokes photon to be emitted.
Once all ensembles are charged, the four stored spinwave modes are then partially converted back into photonic excitations, leading to the state (αa refer to the emitted anti-Stokes photons from memories located at A h and A v (B h and B v ) respectively. The discussion concerning the implementation of partial readout at the end of the preceding section also applies in the present case. The released anti-Stokes photons are combined at a central station where they are detected in using the setup shown in Fig. 15. In the ideal case, a twofold coincident detection between d + andd + projects the state of the two remaining spin-wave modes nondestructively onto The stored atomic excitations can be reconverted into photons as desired. In the proposed quantum repeater protocol, one excitation (e.g. the one in the B ensembles) is reconverted into a photon right away and used for entanglement generation. The other excitation is reconverted later for entanglement swapping or for the final 20 use of the entanglement. Note that the setup can also be used as a source of single photon pairs, if both excitations are converted into photons simultaneously. Given an initial state where all four memories are charged, the probability for a coincidence between d + andd + is given by 1 2 α 4 β 4 . Since the twofold coincidences with the appropriate one-qubit transformation also collapse the state of the atomic ensembles into |Ψ ab , the overall success probability for the entangled pair preparation is given by P s = 2α 4 β 4 .
We now analyze the effect of nonunit detector efficiency and memory recall efficiency. The waiting time for the memories to be charged is now T η = T /η d = 25 12rpη d . Furthermore, the detectors can now give the expected coincidences when three or four anti-Stokes photons are released by the memories, but only two are detected. In this case, the created state contains additional terms including single spin-wave modes and a vacuum component, We have introduced a superscript s for"source". The probability for the successful preparation of this mixed state is P η s = 2η 2 α 4 (1 − α 2 η) 2 . The fidelity of the conditionally prepared state is equal to the two-photon component c s 2 = β 4 /(1 − α 2 η) 2 . As can be seen from the two previous equations, there is a tradeoff on the readout coefficients α, β. The creation of an entangled state with a high fidelity favors α ≈ 0, whereas a high success probability favors α ≈ β ≈ 1/ √ 2.

Repeater Protocol using Two-Photon Detections
We now describe how this source of heralded pairs can be inserted within a quantum repeater protocol. The setup for entanglement creation between two remote sources involving the ensembles AB and CD is shown in Fig. 16. The central station is identical to the one used for the higher-level entanglement swapping operations in the protocol of (Jiang et al., 2007b) see also section 6, Fig. 7, and for all swapping operations in the protocol of section III.B of (Chen et al., 2007b), see also section III.B of the present review. Two anti-Stokes photons are combined at a central station, where one photon is released from the B ensembles and the other from the C ensembles and a projective measurement is performed into the modes D bc 16 (Color online) Entanglement creation based on twophoton detections using two locally prepared entangled pairs. The excitations stored in the ensembles B h , Bv, C h , Cv are read out and the resulting photonic modes are combined at a central station using the set-up shown. Ideally, the coincident detection of two photons in d+ andd+ projects nondestructively the atomic cells A-D into the entangled state |Ψ ad . |Ψ ad . Due to imperfections, the distributed state ρ 0 ad includes vacuum and single spin-wave modes. The weights c s 2 , c s 1 , c s 0 of the source state ρ s ab satisfy c s 0 c s 2 = 4(c s 1 ) 2 such that they are unchanged after the entanglement creation, as before. We thus have c 0 2 = c s 2 , c 0 1 = c s 1 and c 0 1 = c s 0 . The success probability for the entanglement creation is given by P 0 = 2η 2 η 2 t (c s 2 /2 + c s 1 ) 2 .

Ah
Using the same set of linear optical elements and detectors (see Fig. 17), one can perform n successive entanglement swapping such that the state ρ n az is distributed between the distant locations A and Z. In analogy to above, the distributed state ρ n az includes vacuum and single spin-wave components with unchanged weights with respect to the initial ones, i.e. c n 2 = c s 2 , c n 1 = c s 1 and c n 0 = c s 0 . From the expression of P 0 and keeping in mind that the entanglement swapping operations are performed locally such that there are no transmission losses, one deduces the success probability for the i-th swapping, P i = 2η 2 (c s 2 /2 + c s 1 ) 2 . The two-spin-wave component of the distributed mixed state |Ψ az is finally post-selected with the probability P ps = c s 2 η 2 .

Performance
From the expressions of P 0 , P i (with i ≥ 1) and P ps , one can rewrite T tot as For realistic values of the repetition rate (say r = 10 MHz) the source preparation time T s = 3T η 2P η s will be comparable to the communication time L0 c . With the usual assumptions η m = η d = F = 0.9, the protocol starts to outperform direct transmission with a 10 GHz singlephoton source for a distance of 560 km, with an entanglement distribution time of 15 seconds, for a repeater with 8 links, p = 0.013 and α 2 = 0.26, see also Fig. 18 in section IV. Note that for these values T s = 380 µs and L0 c = 350 µs, with c = 2 × 10 8 m/s in the fiber as usual.
Since the repetition rate is already limited by the source preparation time, multiplexing is difficult in the present case. However, the protocol achieves the best performance of all known non-multiplexed protocols using only atomic ensembles and linear optics, and is moreover robust with respect to channel phase fluctuations thanks to the use of two-photon detections for entanglement generation. Multiplexing would become possible if the source preparation could be accelerated, in particular if an ideal photon pair source (such as a single-atom cascade) in combination with an appropriate memory was available.

IV. COMPARISON AND DISCUSSION
We now compare the performance of the various proposed protocols in more detail. The first subsection is devoted to the time needed for entanglement distribution. Then we review the robustness of the protocols with respect to several important technological imperfections. Finally we briefly discuss the complexity of implementing the proposed protocols. Fig. 18 shows the time required for distributing a single entangled pair as a function of distance for the protocols discussed in detail in the previous sections. We have again chosen a final target fidelity F = 0.9. It should be noted that this takes into account only errors due to multi-photon emission, which occur in all the discussed protocols. In practice there are other sources of errors in addition, such as imperfect mode overlap and phase fluctuations, that affect different protocols differently, as we have previously discussed, requiring e.g. different degrees The quantity shown is the average time needed to distribute a single entangled pair with a final fidelity F = 0.9 for the given distance. We assume losses of 0.2 dB/km, corresponding to telecom fibers at a wavelength of 1.5 µm. A: as a reference, the time required using direct transmission of photons through optical fibers with a single-photon generation rate of 10 GHz. B: original DLCZ protocol that uses single-photon detections for both entanglement generation and swapping (Duan et al., 2001). C: protocol of (Jiang et al., 2007b) which uses entanglement swapping based on two-photon detections. D: the protocol of section III.B of (Chen et al., 2007b) which first creates single-photon entanglement locally using singlephoton detections, and then generates long-distance entanglement using two-photon detections. E: protocol of ) that uses photon pair sources (which can be realized with ensembles) and multi-mode memories to implement a temporally multiplexed version of the DLCZ protocol.

A. Entanglement Distribution Time
We have assumed a memory that can store N = 100 temporal modes. F: protocol of  that uses quasiideal single photon sources (which can be implemented with atomic ensembles) plus single-photon detections for generation and swapping. G: protocol of (Sangouard et al., 2008b), which creates high-fidelity entangled pairs locally and uses two-photon detections for entanglement generation and swapping, thus following the approach of section III.C of (Chen et al., 2007b), but using an improved method of generating the local entanglement. We have assumed a basic source repetition rate of 10 MHz, which is a limiting factor for this protocol. For all the curves we have assumed memory and detector efficiencies of 90%. We imposed a maximum number of 16 links, which is larger than or equal to the optimal link number for all protocols apart from curve D, for which the effect is also less than a factor of two.
of fiber length stabilization depending on whether entanglement is generated via single-photon or two-photon detections.
Schemes that use two-photon detections for longdistance entanglement generation are more sensitive to photon losses than schemes that use single-photon detections for the same purpose. As a consequence, twophoton protocols favor larger numbers of elementary links for the same distance compared to the single-photon schemes. In Fig. 18 we have limited the maximum number of links to 16, to make the protocols more compa-rable, and to have link numbers for which it is plausible that entanglement purification may not be necessary. In the shown distance range, this has no effect on the performance of single-photon entanglement generation based protocols (curves B, C, E and F), which favor fewer links, nor on the two-photon detection based protocol of (Sangouard et al., 2008b) (curve G). It has a slight effect (of order a factor of 2) for the protocol of (Chen et al., 2007b) (curve D). Fig. 18 shows that all protocols start to outperform direct transmission somewhere in the range 500 to 650 km. It also shows the significant differences in the required entanglement distribution time that we have already seen in section III. Focusing first on protocols that create entanglement by single photon detection, one can see the improvement in going from the original DLCZ protocol (curve B) to the protocol of Ref. (Jiang et al., 2007b) (curve C) and then to the protocol of Ref  (curve F). On the other hand, for protocols where entanglement is created by two-photon detection, one sees that the protocol of Ref. (Chen et al., 2007b) section III.B (curve D) already achieves a performance that is fairly similar to the DLCZ protocol (while significantly improving its robustness), whereas the protocol of Ref. (Sangouard et al., 2008b) is significantly faster (curve G). However, even the fastest protocol (curve G) still requires very long times for entanglement distribution, which not only leads to very low rates of quantum communication, but also is extremely taxing in terms of quantum memory requirements. Note that the twophoton detection based protocol of  is slower than the protocols shown, cf. section III.B.
Curve E, which corresponds to the multi-mode memory based protocol of Ref. , emphasizes the advantage of multiplexing. In this protocol, which is essentially a multiplexed version of the DLCZ protocol, entanglement is created by single-photon detections, requiring long-distance phase stability. Multiplexing the other protocols is more challenging in terms of source repetition rate and memory bandwidth, as we discussed in the previous section, but certainly worth investigating in detail. We have focused on temporal multiplexing because this seems particularly promising in practice, however other forms of multiplexing (spatial, frequency) may be possible as well, and promise additional benefits in addition to improved rate, such as greater robustness with respect to storage time limitations (Collins et al., 2007), cf. section III.D.

B. Robustness
We have mentioned the importance of long memory times and of long-distance phase stability (for most of the discussed protocols) repeatedly in the previous sections. Here we will briefly review what is known about the effects of imperfections in these respects on the performance of the various quantum repeater protocols. We will also discuss imperfections of other important parameters, in particular memory and detection efficiencies. Let us note that multi-photon emission errors are not an imperfection in the same sense, but inherent to the ensemble-based protocols. For many protocols they directly determine the achievable rates by forcing one to work with a certain value of the emission probability p, which is why we studied them already in the previous sections. Eliminating them would be possible with different resources, such as ideal single-photon sources or single-pair sources.

Storage Time
We have emphasized that it is essential for the storage time to be long enough to allow the highest-level entanglement swapping (or the final post-selection in most protocols) to be performed. This means that the memory time has to be comparable to the total entanglement creation time. (Razavi et al., 2008) have recently studied quantitatively how the performance of quantum repeaters declines if this is not the case. They find, for example, that the repeater rate declines as a power of exp(− L/cτ ), where L is the total distance, c the speed of light and τ the memory time, in the regime where τ L/c. As discussed in section III.D, (Collins et al., 2007) pointed out that certain kinds of multiplexing can greatly reduce memory time requirements, whereas simply running several repeaters in parallel does not. Developing quantum memories with long storage times is thus essential for the implementation of quantum repeaters. In certain solid-state atomic ensembles storage times over 1 second  and coherence times up to 30 seconds  have already been demonstrated. This, as well as the experimental status quo for atomic gases, will be discussed in more detail in section V.

Phase Stability and Entanglement Purification
As we have already discussed, phase stability is particularly important for protocols based on single-photon detections. Not only the fiber links have to be interferometrically stable over long time scales, but also the laser phases. The relevant timescale is given by the creation of entanglement in an elementary link, not over the whole distance, but this will typically also be at least in the ms range. This requires stabilization of the channel, e.g. through active feedback or through the implementation of self-compensating Sagnac-type setups, and distribution of a phase reference. This is an active field of investigation, which will be reviewed in more detail in section V.F. Any phase error not eliminated by stabilization will typically be amplified by a factor of 2 in every entanglement swapping operation, similarly to the vacuum and multi-photon components discussed in previous sections.
The difficulty is less severe for two-photon detection based protocols, where propagation and laser phases only contribute to an irrelevant global phase. However, if fiber length fluctuations become too large, they reduce the overlap between the two photons, which also leads to phase errors. Active stabilization is thus likely to still be required, but less precision is sufficient. The level of stabilization required depends on the coherence length of photons used in a given implementation, which will typically be on the scale of meters. Again any remaining errors are amplified in every entanglement swapping step. This is why we have limited the number of links in our comparison above. The experimental status of single-photon and two-photon visibilities is discussed in more detail in section V, in particular sections V.B.1 and V.F.
There is of course the possibility of using entanglement purification, as originally discussed by (Briegel et al., 1998). This introduces a supplementary layer of complexity, causing a further slowdown, which is why we haven't included it explicitly in our discussions and comparisons. We believe that for the most immediate goal of beating direct transmission it will probably be a better strategy to minimize all errors and do without purification. However, purification procedures have now been developed that can be used in all the discussed protocols. On the one hand, a protocol for the entanglement purification of photon pairs with linear optics was proposed by (Pan et al., 2001). The protocol was adapted to parametric down-conversion sources by (Simon and Pan, 2002), leading to an experimental realization (Pan et al., 2003). This topic is well reviewed in . On the other hand, single-photon entanglement plays an essential role in several of the protocols discussed in this review (Duan et al., 2001;Jiang et al., 2007b;Sangouard et al., 2007;Simon et al., 2007b). A protocol for the purification of single-photon entanglement with linear optics has recently also been proposed (Sangouard et al., 2008a). The effects of phase errors and the inclusion of entanglement purification in repeater protocols were studied in (Chen et al., 2007b;Jiang et al., 2007b;Zhao et al., 2007) with some quantitative results.

Memory Efficiency
Up to now, we have characterized the performance of protocols by considering a memory efficiency of η m = 0.9. It is important to know how the entanglement distribution rates vary with the memory efficiency. In Fig. 19, the average time for the distribution of an entangled state over 600 km is plotted as a function of the memory efficiency, with all other parameters as before, in particular η d = 0.9. It clearly appears that because single-photon detection based protocols require less memories, they are less sensitive to non-unit memory efficiency than protocols based on two-photon detections. The main conclusion from Fig. 19 however is the enormous importance of The letters refer to the same protocols as in Fig. 18. highly efficient memories in order to achieve reasonable entanglement distribution rates. For example, a reduction in the memory efficiency from 90 % to 89 % leads to an increase in the entanglement distribution time by 10-14 %, depending on the protocol. This is understandable because the memory efficiency intervenes in every entanglement swapping operation. Intrinsic memory efficiencies above 80 % have already been achieved (Simon et al., 2007c), however overall efficiencies are typically much lower due to coupling losses. The experimental status quo will be reviewed in more detail in section V, in particular sections V.A.1, V.D and V.G.

Photon Detection Efficiency
It is also interesting to know the influence of the photon detector efficiency on the protocol performance. Hith-erto, we have considered photon detectors capable of resolving the photon number, and with an efficiency of η d = 0.9. Fig. 20 shows the average time for the distribution of one entangled pair for various photon detector efficiencies, with all other parameters as before, in particular η m = 0.9. Since they require less detectors, the protocols based on single-photon detections are more robust with respect to photon detector inefficiency than protocols based on two-photon detections. Again the main conclusion is that the detection efficiency is clearly very important, as it too intervenes in every swapping operation. For example, a reduction in detection efficiency from 0.90 to 0.89 leads to an increase in the entanglement distribution time that ranges from 7 % for the single-photon source based protocol of  (curve F) to 19 % for the local entangled pair based protocol of (Sangouard et al., 2008b).
Photon-number resolving detectors with efficiencies as high as 95 % have been demonstrated (Lita et al., 2008). We review the experimental status quo concerning photon detectors in more detail in section V.

Dark Counts
Realistic detectors do not only have imperfect efficiency, but also a certain level of dark counts, which however depends strongly on the type of detector used. The effect of dark counts on quantum repeater protocols has been analyzed for specific protocols, in particular for the single-photon source based protocol by  and for the DLCZ protocol by (Brask and Sorensen, 2008), who studied the impact of a number of imperfections. It is interesting to note, as was already remarked in section I, that "memory-less" repeaters, also known as quantum relays, can help alleviate the effects of dark counts on the transmission of quantum states (Collins et al., 2005;Jacobs et al., 2002).

C. Complexity
Quantifying and comparing the complexity of different quantum repeater protocols is not a straightforward task. One simple way of assessing complexity is counting elements. For example, outperforming direct transmission with the protocol of Ref.  will require at least 2 links, each of which has 4 sources and 4 multi-mode memories. The cross-over occurs for a distance of 510 km, for an entanglement distribution time of 2.8 seconds; a repeater with 4 links is a bit faster, achieving a time of 1.4. seconds. Similar resource counts for the protocols discussed in the present review are given in (Sangouard et al., 2008b).
However in practice the number of elements is not the only (nor necessarily the most important) consideration. We have already emphasized the importance of phase stabilization, for example, where the required level of preci-sion is different for single-photon or two-photon detection based protocols. There are other distinctions where it is less clear which side is favored. For example, temporal multi-mode memories would typically be realized in solid-state atomic ensembles at cryogenic temperatures. On the other hand, DLCZ-type experiments so far were performed with atomic gases, requiring optical cooling and trapping.
Our overall conclusion is that outperforming direct transmission appears possible with repeater architectures of quite moderate complexity. However, the individual components have to be excellent. For example, successful quantum repeaters will probably require storage times of several seconds, memory and detection efficiencies of 90 percent or more, length-stabilized long-distance fiber links, and minimal coupling losses between the various local components, cf. section V.

V. IMPLEMENTATIONS
We will now review experiments that are relevant to the different quantum repeater architectures described above. We first review in section V.A experiments that are directly relevant to the DLCZ protocol itself. Section V.B is devoted to the experiments relevant to the protocols based on two-photon entanglement generation and swapping. In section V.C, we review the various quantum light sources at the single photon level that are compatible with ensembles based quantum memories before describing in section V.D light storage experiments in atomic ensembles, in particular the storage of single photons. Section V.E is devoted to the various single photon detectors that may be used in a quantum repeater architecture. Section V.F briefly describes the quantum channels that may be used in a quantum repeater, in particular optical fibers (including phase stabilization) but also free space links. Finally, we mention in section V.G an important practical and technological aspect, the coupling losses. In all the section V, we are mainly concerned with experiments performed with photon counting in the single excitation regime. In particular the various experiments demonstrating the storage and teleportation of quantum continuous variables of light in atomic ensembles performed with homodyne detection, e.g. (Appel et al., 2008;Cviklinski et al., 2008;Honda et al., 2008;Julsgaard et al., 2001Julsgaard et al., , 2004Sherson et al., 2006) will not be discussed here. This area of research has been reviewed recently in (Hammerer et al., 2008).

A. DLCZ Protocol
The publication of the article of DLCZ in 2001 triggered an intense experimental effort to realize the basic elements of this protocol. Over the last few years there have been a large number of DLCZ type experiments in atomic gases. In this subsection, we review these exper-iments. We start in section V.A.1 by the experimental realization of the fundamental building block: the generation of strong quantum correlations between emitted Stokes photons and stored collective spin excitations, followed by the efficient mapping of the stored excitation into an anti-Stokes photon. We then describe in section V.A.2 experiments demonstrating heralded entanglement between remote atomic ensembles. Section V.A.3 describes the experimental realization of an elementary segment of DLCZ quantum repeater. Finally, section V.A.4 is devoted to an experiment attempting to demonstrate entanglement swapping between DLCZ ensembles.

Creation of Correlated Photon Pairs with a Programmable Delay
(i) Quantifying quantum correlations As we have seen in section II.A, the number of Stokes photons |n S emitted during the spontaneous Raman process is in the ideal case strongly correlated with the number of collective spin excitations |n a stored in the level |g 2 . The joint atom-photon state can be written as (cf Eq. (5)): (27) This state is sometimes referred as a two-mode squeezed state. The probability p of creating a pair Stokes photoncollective excitation is directly proportional to the write laser intensity. In practice, various sources of noise can degrade the quantum correlations and it is important to experimentally quantify these correlations. The first step is to convert the atomic collective excitations into anti-Stokes photons, with a read laser. The correlations between Stokes photons and stored excitations will now be mapped into correlations between Stokes and anti-Stokes fields, which can be measured by photon counting techniques. In particular, the various probabilities p S ,p AS to detect a Stokes and an anti-Stokes photon, respectively and the joint probabilities p S,AS , p S,S , p AS,AS to detect a pair Stokes-anti-Stokes, two Stokes and two anti-Stokes photons in a given trial can be easily accessed.
With these measured probabilities, it has been shown that there exists a well defined border between classical and quantum fields (Clauser, 1974;Kuzmich et al., 2003). Specifically, for classical fields, we have the following Cauchy Schwartz inequality: is the normalized cross-correlation function between Stokes and anti-Stokes fields, and g(S, S) = p S,S /p 2 S and g(AS, AS) = p AS,AS /p 2 AS ) are the normalized autocorrelation function for the Stokes and anti-Stokes fields, respectively.
For a perfect two mode squeezed state as in Eq. (27), the non-conditional Stokes and anti-Stokes fields exhibit thermal statistics and hence bunching, with g S,S =g AS,AS =2. In that case, a measured value of g S,AS > 2 is a signature of non classical correlations between Stokes and anti-Stokes fields.
While a formal proof of non classical correlations requires the measurement of the Cauchy-Schwartz inequality, the measurement of a cross correlation function g S,AS > 2 already gives strong evidence of non classical behavior. This is because in practice, the presence of background noise such as leakage of excitation lasers and dark counts decreases the bunching of the non conditional fields, such as g S,S =g AS,AS <2. In addition, the cross correlation function is very important, since many parameters crucial for applications, such as the autocorrelation function of the heralded anti-Stokes photon (Chen et al., 2006;Chou et al., 2004;Laurat et al., 2006;Matsukevich et al., 2006a), the visibility of two photon interference (Chen et al., 2007a;de Riedmatten et al., 2006), the visibility of the Hong-Ou-Mandel interference Yuan et al., 2007) and the visibility in measurement induced entanglement experiments  are directly related to g S,AS . For a perfect two-mode squeezed state, the normalized crosscorrelation function is linked to the probability p of creating a Stokes-anti-Stokes pair as : for p 1. In that case the finite value of g S,AS is due to multiple pair creation. As mentioned in Section II.B, multiple pair creation is a major source of errors in the DLCZ architecture that should be minimized by working in the regime of high quantum correlations. For an ideal state, this regime can be in principle accessed by using very low pump power. This is true of course as long as the background noise is negligible. A major experimental challenge of this type of experiment is to preserve the quantum character of the emitted single photons, which requires excellent filtering of the various sources of noise, such as leakage of the excitation lasers, fluorescence and stray light. The strong correlations between Stokes and anti-Stokes photons enable the generation of heralded single photon states. For example, the detection of a Stokes photon by detector D S projects the anti-Stokes field on a single photon state. The quality of this single photon state can be measured with a Hanbury-Brown-Twiss setup, i.e. by splitting the anti-Stokes field at a beam splitter and recording the detection events in the two output modes with detectors D 1 and D 2 (see Fig. 21). As shown by (Grangier et al., 1986), the single photon character of the conditional anti-Stokes field can be characterized with the auto-correlation function: where the various p's correspond to the probability of a joint detection event in the corresponding combination of detectors D S , D 1 and D 2 . For classical fields, a Cauchy-Schwartz inequality leads to α ≥ 1. For coherent fields, we have α=1 while α=2 for thermal fields. In contrast, for a perfect conditional single photon field α=0. The measurement of the "anticorrelation parameter" α thus provides a way to quantify the two photon suppression of the conditional field with respect to a coherent field.
(ii) Initial experiments The first enabling step towards a practical realization of the DLCZ quantum repeater is the observation of non classical correlations between the Stokes and anti-Stokes fields emitted with a controllable delay by one atomic ensemble. The first experiments have been performed simultaneously in 2003 in Caltech (Kuzmich et al., 2003) and Harvard (van der Wal et al., 2003). The Caltech experiment used ensembles of cold Cesium atoms in a magneto-optical trap (MOT) and observed quantum correlations in the single excitation regime. The write and read pulses were separated by 400 ns and sent in a collinear co-propagating configuration through the ensemble. A challenging aspect of the experiment was to separate the classical pulses from the weak nonclassical fields, since they were temporally and spatially overlapped, and their frequencies were only 9 GHz apart. In the first experiment the filtering had three stages. First, the Stokes (anti-Stokes) fields was separated from the write (read) pulse in a polarizing beam splitter (PBS) right after the MOT chamber. Later, the leakage of the excitation pulses that still escapes the PBS in the wrong direction was spectrally filtered by optically pumped vapor cells. Finally, the Stokes (anti-Stokes) field was distinguished from the read (write) pulse by temporal gating of the detection.
The non classical character of the fields was demonstrated using the Cauchy-Schwartz inequality of (Eq.28).
A value of R = 1.84 ± 0.06 > 1 was measured, thereby demonstrating that the Stokes and anti-Stokes fields in the single photon regime were non classically correlated. The size of the violation of the inequality was limited mostly by uncorrelated fluorescence from individual atoms in the atomic sample and by the residual leakage of excitation pulses.
The Harvard experiment used a hot vapor of Rubidium atoms. The first experiment was carried out in the regime of high excitation number (10 3 − 10 4 ) (van der Wal et al., 2003). Strong intensity correlation between the Stokes and anti-Stokes fields were observed and their quantum nature was demonstrated by an analysis of the fluctuation spectral density with respect to the shot-noise (or vacuum-state) limit. In more recent experiments, non classical correlations were also observed in the single excitation regime with hot vapors (Eisaman et al., 2005. Since the initial experiments, tremendous progress has been made on several fronts in various experiments. We shall now review relevant experiments, based on three important properties of the DLCZ source: the quality of non-classical correlations between Stokes and anti-Stokes photons, the efficiency of atomic to photonic conversion for the Anti-Stokes field (called retrieval efficiency) and the storage time of the stored excitation.

(iii) Quantum Correlations
In the initial experiment (Kuzmich et al., 2003) the measured g S,AS was only slightly above 2. Various improvements in the experiments have allowed the observation of substantially higher quantum correlations between Stokes and anti-Stokes field. A first important step was to set the write laser slightly off resonance to avoid uncorrelated fluorescence. A further improvement was to use a four-level scheme of excitation, in which write and read pulses are 42 nm apart. This allows a fourth filtering stage by narrow-bandwidth optical filters, and the study of correlations with temporally overlapped write and read pulses. Experiments in that regime yielded values of g S,AS of order 10 Felinto et al., 2005;Polyakov et al., 2004). More recently, (Chen et al., 2006) have achieved a value of g S,AS =100 with an improved version of the setup of (Kuzmich et al., 2003). Another important step was achieved by using an off axis geometry in which the Stokes (anti-Stokes) photon is collected with a small angle with respect to the write (read) beam direction. This configuration was first used in the classical regime in (Braje et al., 2004), and in the quantum regime in (Matsukevich et al., 2005). This geometry allows for a very efficient spatial filtering of the write and read lasers, which decreases the background light by a substantial amount. Much higher values of crosscorrelation function have been obtained in this case, e.g. g S,AS 300 in (Matsukevich et al., 2006a) and g S,AS 600 in .
Note that the detection of Stokes and anti-Stokes photons with an angle cannot be used with hot atomic vapors, due to the motion of atoms (see below). In addition, the detuning of the write pulse in order to be off resonance must be much larger than in cold gases, because of the doppler inhomogeneous broadening (which is approximately 500 MHz). Consequently the write pulse intensity is also much larger than in cold gases. For these reasons, the filtering of the write and read pulses is more challenging in hot vapors than in cold gases. Strong non classical correlations have nevertheless been observed by using a counterpropagating configuration for the write and read beams (Eisaman et al., 2005;Walther et al., 2007).
(iv) Retrieval efficiency The single spin excitation stored in the ensemble can in principle be retrieved with unit efficiency in a well defined spatio-temporal mode, due to the collective enhancement effect, as discussed in section II.A. The retrieval efficiency is defined as the probability to have an anti-Stokes photon in a well defined spatio-temporal mode at the output of the atomic ensemble conditioned on the successful detection of a Stokes field. In practice however, several factors can limit the retrieval efficiency. For example, it depends on the available optical depth and read beam power. The collective interference can also be decreased by various dephasing effects due e.g. to spatial intensity profile mismatch between read beam and stored excitation, inhomogeneous broadening (Ottaviani et al., 2009) or atomic motion (for atomic motion, see below and section II.A). The effect of atomic motion has been discussed in the the While early experiments suffered from low retrieval efficiencies, progress has been made on several fronts, leading to retrieval efficiencies of 50 % in free space  and of more than 80% in cavities (Simon et al., 2007c). Fig. 22 shows the measured retrieval efficiency as a function of the optical depth with the atomic ensemble inserted inside an optical cavity (Simon et al., 2007c). Note that the conditional probability to detect an anti-Stokes photon is usually much lower than these values, due to the various passive losses and to detection inefficiency (see section V.G).
(v) Storage time A crucial parameter for using DLCZ quantum memories in a repeater scheme is the storage time of the collective spin excitation, which is limited by the coherence time of the ground state transition |g 1 -|g 2 . There are several factors that affect the storage time at different timescales. The main factors can be divided in two classes: the inhomogeneous broadening of the spin transition and the atomic motion. We are now going to describe these effects in more detail.
(a) Inhomogeneous broadening of the spin transition. proaches its fundamental limit g wr & 1=n w . The large value of g wr corresponds to strongly nonclassical writeread correlations -a necessary condition for sub-Poissonian noise of the read photons. To verify the single-photon character of the read field conditioned on having detected a write photon, we measure the conditional second-order read autocorrelation function g rrjw with two detectors. At n w 0:153, we find g rrjw 0:2721 < 1, clearly demonstrating that the source produces single photons. While the result agrees with the expected value g rrjw g ww n w 0:3 for this value of n w , the error bar for this time-consuming three-photon measurement remains relatively large due to the low detection efficiency stemming from cavity losses. After completing the experiments described below, we cleaned the deposited cesium off of the mirrors, which reduced the cavity losses (and the effect of detector dark counts) by a factor of 7 and extended the high-recovery region of Fig. 2 down to n w 0:005. We then measure g rrjw 0:158 at n w 0:007.
The analysis of Fig. 2 shows that the quantity of fundamental interest, the single-magnon conversion , in the region of negligible write backgrounds (0:04 n w 0:4), is well approximated by R c =1 n w . (Here we make use of g ww g th ww 2 and n w m w .) In the following, we evaluate this expression measured at fixed write photon number n w to examine the magnon-photon interface.
increase in read photon emission time in the dark-state rotation regime (by a factor of N) for atoms with multiple excited states increases the dephasing ls and reduces the conversion. The predicted conversion 0 ls including all  3 (color online). Magnon-photon conversion efficiency versus read optical depth N at a write-read delay of 120 ns. The optical depth is extracted from the write scattering rate and known intensities and detunings. The dashed line shows the predicted conversion 0 for a three-level system; the solid line is the prediction from a model including dephasing from additional excited states.

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FIG. 22 Retrieval efficiency versus read optical depth at a write-read delay of 120 ns with the atomic ensemble inserted into an optical cavity (Simon et al., 2007c). The dashed line shows the predicted retrieval efficiency for a three-level system; the solid line is the prediction from a model including dephasing from additional excited states.
In most experiments to date, the spin excitation was stored in a hyperfine state containing a Zeeman state manifold. In absence of external magnetic fields, all the Zeeman states with different m F are degenerate. In practice however, it is difficult to suppress external magnetic fields completely. For experiments with cold atoms for example, a strong magnetic field gradient is needed in order to trap the atoms. This leads to an inhomogeneous broadening of the spin transition. In that case, the collective spin excitation undergoes a strong inhomogeneous dephasing, which effectively suppresses the collective enhancement necessary for efficient retrieval of the photons. This decoherence mechanism was studied in detail in . The inhomogeneous broadening due to the the trapping magnetic field gradient leads to storage times of the order of a few hundreds nanoseconds (Kuzmich et al., 2003;Matsukevich and Kuzmich, 2004;Polyakov et al., 2004). A direct solution to this problem is to switch off the magnetic field gradient during the write/read sequence. This solution was tested experimentally and led to an increase of storage time by two orders of magnitude (of order of 10µs) (Black et al., 2005;Chen et al., 2006;Felinto et al., 2005;Matsukevich et al., 2006aMatsukevich et al., , 2005. The storage time was in that case mostly limited by the residual magnetic field. It is experimentally difficult to further decrease residual magnetic fields. Another solution to avoid the inhomogeneous broadening of the spin transition is to use first order magnetically insensitive hyperfine transitions, known as 'clock transitions', connecting two specific Zeeman states. Such transitions exists in Cs and Rb atoms. This requires to prepare all the atoms in a specific Zeeman state (typically with m F =0). This preparation can be implemented with optical pumping techniques. A storage time of 1 ms has been recently demonstrated using this technique in a collinear configuration (Zhao et al., 2009a).
While turning off the trapping magnetic field allows . c, (|1,1 ,|2,−1 ) at θ = 0.2 • . In this case we take into account the effect of loss of atoms and fit the data by using g S,AS (δt) = 1 + Cexp(−δt 2 /τ 2 D )/(1 + Aδt 2 ), with A the fitting parameter obtained from the collinear configuration. The fitted lifetime for each case is (a) τ D = (61 ± 2) µs, (b) τ D = (144 ± 9) µs, (c) τ D = (283 ± 18) µs. By reducing the angle, the lifetime is increased from 25 to 283 µs, which implies that the decoherence is mainly caused by the dephasing induced by atomic random motion. d, The measured lifetime τ D as a function of detection angle θ , where the horizontal error bars indicate measurement errors in the angles. The solid line is the theoretical curve with T = 100 µK. The experimental results are in good agreement with the theoretical predications. The error bars indicate ±1 standard deviation.
collisions between atoms may also affect the phase of the SW. However, the collision rate can be estimated by Γ ∼ nv s σ ∼ 1 Hz, where the atomic density n = 10 10 cm −3 and the s-wave scattering cross-section σ = 8πa 2 with the scattering length a = 6 nm. Thus, on the timescale of milliseconds, the collisions can be safely ignored.
To further confirm that the decoherence is mainly caused by the dephasing induced by atomic motion, we increase the wavelength of the SW by decreasing the detection angle (see Fig. 1d). In this way, according to the above model, the dephasing is suppressed and the storage time will be extended. In our experiment, we select angles of θ = 1.5 • ,0.6 • and 0.2 • and measure the lifetime of the quantum memory for each configuration. Note that for θ = 0.2 • the write (read) and Stokes (anti-Stokes) lights with the same polarization cannot be spatially separated, and therefore we use another pair of 'clock states' (|g = |1,1 ,|s = |2,−1 ) by preparing the atoms in |1,1 . In this case, the Stokes (anti-Stokes) photon is σ + (σ − ) polarized and thus can be separated by a Glan-laser prism from the write (read) pulse.
The experimental results are shown in Fig. 3a-c. As expected, the dephasing of the SW dominates. The lifetime increases from 25 to 283 µs when reducing the angle owing to the extended wavelength of the SW. Our results clearly show that the dephasing of the SW is extremely sensitive to the small angle between the write beam and Stokes modes, and that the long-wavelength SW is robust against the dephasing induced by atomic random motion. Note that, for θ = 0.2 • , the data are fitted taking into account the effect of loss of atoms. The measured lifetime τ D is shown in Fig. 3d as a function of angle θ. The solid line is the theoretical curve τ D = (1/ kv s ), with v s = 0.1 m s −1 corresponding to a temperature of T 100 µK. The good agreement between theory and experiment implies that our work provides an alternative approach to measure the temperature of an atomic ensemble. Moreover, as the lifetime is sensitive only to the velocity of the atoms in the interaction region, which is determined by the waist of the detection mode and is controllable, our method can also be used to measure the velocity distribution of the atomic ensemble by making measurements in different regions.
To further suppress the dephasing and achieve a longer storage time, we use the collinear configuration (θ = 0 • ), where we have the maximum wavelength of the SW of λ 4.4 cm, which yields a computed lifetime of τ D = 72 ms. In this case, the decoherence due to the loss of atoms, which causes a typical lifetime of a few hundred microseconds, is isolated as the principal source of decoherence. This mechanism can be estimated by calculating the average time for the atoms leaving the pencil shaped interaction region, where the thermal motion in the radial direction dominates. At temperature T , an atomic cloud with a cross-section radius r 0 expands according to r 2 (δt ) = r 2 0 + v 2 r δt 2 , with the average speed in the radial direction v r = √ 2k B T /m. The retrieval efficiency can be given by γ (δt ) = r 2 0 /r 2 (δt ) = 1/(1 + v 2 r δt 2 /r 2 0 ). Thereby, when γ (τ L ) = 1/e, only a fraction 1/e of the atoms remain in the interaction region, giving a lifetime of τ L (1.31r 0 /v r ). For r 0 = 100 µm as the waist of the detection mode and T = 100 µK, a direct calculation yields τ L = 950 µs, which is much smaller than the dephasing time due to atomic motion and decoherence induced by the magnetic field. The experimental result is shown in Fig. 4, where the 'clock states' (|1,1 ,|2,−1 ) are used as well. Our data show a lifetime of τ L = (1.0 ± 0.1) ms, when the retrieval efficiency has dropped to 1/e. The experimental result is in good agreement with the theoretical estimate.
In our experiment, we have isolated and identified different decoherence mechanisms of the quantum memory for single excitations and thoroughly investigated the dephasing of the stored SW by varying its wavelength. Moreover, we have successfully realized a long-lived quantum memory for single collective excitation with the storage time of 1 ms, which is 30 times longer than the best result reported so far 10 , and is long enough for photon transmission over 100 kilometres. In our experiment, the coherence time of the quantum memory is limited by the decoherence due to loss of atoms, which can be suppressed by lowering the temperature through optical molasses. A storage time of 3 ms is achievable by reducing the temperature to 10 µK. This will be the upper limit for an atomic memory with a magneto-optical trap, because a NATURE PHYSICS | VOL 5 | FEBRUARY 2009 | www.nature.com/naturephysics 97 FIG. 23 (a-c), The cross-correlation gA,AS versus the storage time δt for different angles θ between the write beam and the Stokes field. (Zhao et al., 2009a). By reducing the angle, the lifetime is increased from 25 to 283 µs, which implies that the decoherence is mainly caused by the dephasing induced by atomic random motion. (d) The measured lifetime τD as a function of detection angle theta, where the horizontal error bars indicate measurement errors in the angles. The solid line is the theoretical curve with T=100 µK. The experimental results are in good agreement with the theoretical predictions. a strong reduction in the inhomogeneous broadening of the spin transition, it has a major drawback: in that case, the atoms are no more trapped and are free to fly away, which severely decrease the available optical depth in the time scale of a few ms. To overcome this problem, it was suggested to use an optical dipole trap to maintain a sufficient atom density. This solution was tested experimentally in . The storage time was however limited to a few tens of µs by atomic motion since the experiment was performed in the configuration with an angle between the write (read) and the Stokes (anti-Stokes) fields (see below)

(b) Atomic motion
Another important cause of decoherence is the motion of atoms. This is obviously a bigger problem for experiment with hot gases, but is also a strong limitation for cold ensembles, as we shall see. The motion of atoms can cause two different problems. The first one is the diffusion of the atoms out of the excitation region during the storage of the spin excitation. This is the prime cause of decoherence for the experiments with hot gases realized to date. For hot gases, this leads to coherence time of a few µs (Eisaman et al., 2005, while for cold ensembles the diffusion time is of order of 1 ms . This diffusion can be mitigated by using bigger beams and/or colder atoms. There is however a much more severe effect of the atomic motion: the disturbance of the phase of the collective spin excitation. In section II.A we have seen that the motion of the atoms is not a problem for the phases of the collective state, as long as a collinear configuration with k w = k s , and k r = k as is used. For all other configurations, the motion of the atoms will induce a dephasing that depends on the angle between k w and k s , as was nicely demonstrated experimentally in (Zhao et al., 2009a). The wavelength of the stored spin wave can be written as : where θ is the angle between the write beam and the Stokes field. The time scale of the dephasing can be estimated by calculating the average time to cross 1/2π of the wavelength of the spin wave, leading to storage lifetime of τ ∼ (Λ/2πv) with v = k B T /m the one dimensional speed of the atoms, where k B is the Boltzmann constant, T the temperature and m the mass of the atoms. The reduced wavelength of the spin wave due to the angle θ severely limits the achievable storage time. For example, for a typical θ = 3 • and for T=100 µK, we find Λ = 15µm and τ = 25µs. (Zhao et al., 2009a) confirmed this prediction experimentally by measuring the storage time as a function of θ using a clock transition in a cold Rb ensemble, as shown in Fig. 23. For θ=0 • (collinear configuration) they achieved a storage life time of order 1 ms.
In practice however, there is a great advantage of using a non-collinear configuration since it enables a very efficient spatial filtering to suppress the excitation beams in the quantum channel. In that case, the only way to avoid motion induced dephasing is to suppress the atomic motion. With atomic gases, one possibility is to load the atoms into an optical lattice. (Zhao et al., 2009b) have demonstrated a DLCZ quantum memory using a clock transition in rubidium atoms confined in a one dimensional optical lattice. They achieved a storage lifetime exceeding 6 ms, which is currently the longest storage lifetime observed in the single photon regime (see Fig. 24). A light storage experiment with bright coherent pulses based on Electro-magnetically induced transparency (EIT) has also been recently demonstrated with Rb atoms confined in a 3 dimensional optical lattice, leading to a storage lifetime of 240 ms (Schnorrberger et al., 2009). Another possibility may be to use colder atoms, for example a Bose Einstein condensate where collective coherences in the high excitation number (> 10 4 ) regime have been created and stored recently (Yoshikawa et al., 2007(Yoshikawa et al., , 2009). These two techniques are however complex and technically demanding. Another potential solution, which may be more practical, is the use of atomic ensembles in the solid state, implemented with rare-earth ion doped solids. In such a medium, the atoms behave as a 'frozen gas' and a storage lifetime exceeding 1 s has been demonstrated, though not yet in the quantum regime . This is a linear combination of the read spin wave associated with the clock transition and the idler field propagating along the quantization axis z, and linearly polarized in the x direction; these are described by creation operatorsŝ † andφ † , respectively. The form of the polariton operator shows that adiabatic variation of the y-polarized read field Rabi frequency Ω causes reversible conversion between the propagating idler field and the read spin wave. The collective Rabi frequency associated with the idler transition c → b is given by κ √ n, where κ is the dipole coupling strength and n is the atomic number density.
Our goal is to convert the write spin wave, heralded by signal photodetection, into the idler field, with high efficiency. It is therefore essential to have a large overlap between the write and read spin waves. To maximize this overlap, the signal and idler spatial mode functions should be matched and the condition k i = k w − k s + k r satisfied 29 , where k i and k r are wave vectors for the idler and read fields, respectively. The overlap is also influenced by the atomic state preparation, angular momentum quantum numbers and transition strengths of the atomic levels a, b and c. For F a = 1,F b = F c = 2 with atoms prepared in the |+ (upper clock) state, the write and read spin wave operators are equal, implying maximum efficiency.
As noted earlier, ballistic expansion limits the quantum memory time, and to increase it into the millisecond regime we load the atoms into a one-dimensional optical lattice as shown in Fig. 1. The period of the lattice 25 µm is shorter than the spin grating wavelength Λ λ/θ ≈ 50 µm, determined by the angle θ ≈ 0.9 • between the write and signal fields of wavelength λ = 795 nm.
We outline in the Methods section the experimental procedure used to determine the retrieval efficiency and to demonstrate the quantum character of the memory through the α-parameter 30 . In Fig. 2, we show the retrieval efficiency as a function of storage time for an unpolarized atomic sample loaded into the lattice. We observe fast oscillations in the first two hundred microseconds followed by a slow decay on the scale of several milliseconds. factors g a ≈ −0.5018, g b ≈ 0.4998 and δg ≡ g a + g b = −0.002; B 0 is the magnetic field. We note that because ω 1,−1 ∝ δg , it is about 500 times smaller than ω 1,1 ≈ ω 2,0 . The inset to Fig. 2 shows that the fast oscillations, associated with the 2 ↔ 0 and 1 ↔ 1 coherences, have a period of 2.8 µs corresponding to B 0 ≈ 0.25 G. The expected 1.4 ms period oscillation of the slow 1 ↔ −1 magnetic coherence, which has a small transition weight for this configuration, is not visible for this field. We have, however, observed this modulation by increasing the magnetic field to 2 G. The 0 ↔ 0 coherence of the clock transition survives the decay of the magnetic coherences to a storage time of 6 ms (with a small contribution from the 1 ↔ −1 coherence). The solid lines are fits to the data from the theory discussed in the Methods section, where the frequencies and decay times are treated as adjustable parameters. The ratio of maximum efficiency at short times to the efficiency at times longer than 200 µs is predicted, in the absence of atomic motion, to be (32/13) 2 ≈ 6.1, whereas the observed value is approximately 13. As we discuss further below, the effects of atomic motion result in an extra dephasing of the spin wave, which may account for this observation. The asymptotic retrieval efficiency is clearly limited by the initial population p 0 = 1/5 of the clock state |+ in the unpolarized sample.
To further increase the efficiency at long times, we optically pump the atoms into the |+ state (see the Methods section). In this situation, much higher retrieval efficiencies at long storage times are observed. We note, however, that optical pumping results in a reduction of overall atomic number by a factor of 2-3, which must be accounted for when comparing the relative efficiencies. In Fig. 3a, two sets of data are shown, corresponding to maximum trap depths U 0 = 40 µK and U 0 = 80 µK, respectively, with a longer coherence time in the former case. All magnetic coherences are strongly suppressed by optical pumping, and, on the timescale shown, the fast magnetic coherences are completely dissipated, leaving primarily the clock hyperfine coherence. Atoms in the |+ and |− clock states experience different, spatially varying light shifts in the lattice. The observed millisecond-scale decoherence of the clock spin wave may be attributed to the atomic motion in the lattice potential, accompanied by the phase broadening resulting from the differential light shifts 31 . Assuming a single harmonic trap, a formula for the decay of the clock transition coherence of the form (1 + (t /T c ) 2 ) −3/2 was derived 31 .
Our lattice instead has a distribution of trap depths and the atoms do not necessarily perform simple harmonic motion. Within each well of the lattice the atoms are expected to be thermally distributed; however, they are not necessarily thermalized with respect to the global potential. Nevertheless, our data fit the formula well, and from this we extract the decay times T c (7, 5) ms for U 0 (40, 80) µK, respectively. According to the model of ref. 31, the decay time T c corresponds to the (homogeneous lattice) equilibrium temperature T = 2 /(ξ k B T c ), where ξ is the ratio of the ground-state hyperfine splitting to the effective detuning of the lattice light-here ξ = 6.8 × 10 −5 -resulting in values of (30,42) µK. In Fig. 3b, we show short-time dynamics for the optically pumped sample and, as expected, observe much lower visibility oscillations than those in Fig. 2. The data suggest the clock state |+ population p 0 ≈ 0.85, and p ±1 ≈ 0.07, assuming that p ±2 ≈ 0. In Fig. 3c, optical pumping enables us to observe motional dephasing on the scale of a few hundred microseconds, followed by a damped evolution towards the 1 ms timescale, the efficiency dropping by about 50% (this typical behaviour is also to be expected in the data of Fig. 3a, but is not shown here). In common with Fig. 2, we attribute this fast dephasing to atomic oscillations along the lattice axis x . A simple one-dimensional model of harmonic motion yields the characteristic behaviour shown in Fig. 3c if we average over a distribution of oscillator frequencies corresponding to a distribution of trap depths. We have also observed this qualitative behaviour in (3 + 1)-dimensional Monte Carlo simulations that include the full atomic orbits.  FIG. 24 (a) Experiment demonstrating a DLCZ quantum memory with an atomic ensembles loaded into an 1 D optical lattice, from (Zhao et al., 2009b). Between 10 5 and 10 6 sub-Doppler-cooled 87Rb atoms are loaded into an optical lattice (see the Methods section), and detection of the signal field, generated by Raman scattering of the write laser pulse (reddetuned by 20 MHz), heralds the presence of a write spin wave excitation. A resonant read/control field converts the surviving atomic excitation into an idler field after a storage period Ts. The inset shows the atomic level scheme of 87Rb with levels a and b being the hyperfine components of the ground 5 S 1/2 level, and level c being a hyperfine component of the excited 5 P 1/2 level. (b) Retrieval efficiency (including detection) as a function of storage time for atoms optically pumped in clock states in the optical lattice for two different lattice depths U0 ( Diamonds,U0 80 µK; circles U0 40 µK circles.) The intrinsic retrieval efficiency at the output of the ensemble is roughly four times larger.

Heralded Entanglement between two Atomic Ensembles
A crucial step towards the implementation of a quantum repeater is the demonstration of heralded entanglement between two spatially separated atomic ensembles in the single excitation regime. This was first demonstrated in  with two cold Cs atomic ensembles in two vacuum chambers separated by 3 meters. Entanglement for excitation stored in remote ensembles was created by a quantum interference in the detection of light emitted by the quantum memories. Fol- lowing the DLCZ protocol described in section II.B, the two ensembles are simultaneously and coherently excited by a weak write beam and the two Stokes fields created by spontaneous Raman scattering are collected into single mode optical fibers and mixed at a beam splitter, forming a long Mach-Zehnder interferometer. If the two Stokes fields are indistinguishable, the information about the origin of the photon is erased and a detection after the beam splitter projects the ensembles in the ideal case onto an entangled state with one de-localized excitation, of the form: the phase of the laser at ensemble A and B respectively, and ξ A,B the phase acquired by the Stokes photons from the ensembles to the beam spitter. In order to generate a measurable entangled state, it is important that the phase θ AB is stable during the duration of the experiment. In , it was actively stabilized using an auxiliary laser. The state of Eq. (33) is an idealized state. In practice, various sources of noise can turn the heralded state into a mixed state. For example, due to the probabilistic nature of the spontaneous Raman process, there is an unavoidable finite probability to create higher order terms with two or more excitations (see section II.A). Non perfect filtering of the excitation light will also alter the heralded state. In order to prove entanglement experimentally, it is therefore crucial to demonstrate the single excitation character of the atomic state as well as the coherent superposition of the de-localized excitation (van Enk et al., 2007).  have devised a way to prove unambiguously the entanglement of the heralded atomic state, based on quantum tomography. They reconstructed the density matrix of the stored state in a Hilbert space spanned by the state is the state with n excitations in ensemble A and n excitation in ensemble B. In order to measure the density matrix, the atomic state is first transferred into a photonic state and the state of the atoms is inferred from the state of the electromagnetic fields. The diagonal terms of the density matrix are measured by direct photon counting, while the coherences are inferred from an interference measurement with the conditional anti-Stokes photons (see Fig. 25). The density matrix can then be used to calculate the amount of entanglement using an entanglement measure, for example the concurrence C , where C = 0 for unentangled states, and C = 1 for maximally entangled states (Wootters, 1998). Using this technique,  were able to demonstrate measurement induced entanglement between the two spatially separated atomic ensembles, albeit with a low concurrence (of order C = 2 · 10 −2 at the output of the ensembles). The main reason for the low concurrence in the first experiment was the limited retrieval efficiency (10%). This was considerably improved in a more recent experiment with two ensembles in the same MOT . The concurrence was measured as a function of the cross correlation function (see Fig. 26), with a maximum value of g S,AS = 60. At this value, a concurrence of C = 0.35 ± 0.1 has been measured at the output of the ensemble, leading to an inferred C = 0.9 ± 0.3 for the atomic state. The decoherence of the stored entangled state was also analyzed in this experiment, with entanglement persisting for at least 20µs.

Elementary Segment of DLCZ Quantum Repeater
Number state entanglement of the form of Eq. (33) is not practical for performing quantum communication tasks, such as quantum key distribution. It is indeed difficult to implement single qubit rotations in the excitation number basis. The solution proposed by DLCZ to this problem is the implementation of two chains of entangled number state ensembles in parallel (see section II.B). In the resulting interference are recorded as a function of the relative phase between the 2 U;D fields. It can be shown that d ' Vp 10 p 01 =2 Vp c =2 [7], where V is the visibility of the interference fringe.
To investigate the scaling of entanglement with excitation probability , we determine C for various values of for fixed memory time 200 ns. As increases, higher order terms in the expansion of Eq. (2) cannot be neglected, precisely as in parametric down conversion. A convenient parameter to assess the excitation regime of each ensemble is the normalized intensity cross correlation function g 12 between field 1 and field 2 [16], defined as g 12 p 12 =p 1 p 2 with p 12 the joint probability for detection events from field 1 and 2 in a given trial and p i the probability for unconditional detections in field i. In the ideal case, this function is related to the excitation probability by g 12 1 1=, where g 12 > 2 defines the nonclassical border in the ideal case [14] and g 12 2 being the single-excitation regime for the ensembles.
Expressing the two-photon component for the two ensembles as p 11 p 2 c p 2 c =g 12 , we rewrite the concurrence as where g 12 is for either ensemble alone, with g U 12 g D 12 g 12 assumed. The visibility V can be expressed in terms of g 12 as the higher order terms act as a background noise. With 1=2p 1 p 2 a good estimation for the background, the visibility can be written as [25] V ' p 12 ÿ p 1 p 2 p 12 p 1 p 2 g 12 ÿ 1 g 12 1 ; ( 6) where is the overlap between fields 2 U;D [28]. In the limit of near zero excitation, as g 12 goes to infinity, the concurrence reaches its asymptotic value given by the retrieval efficiency p c [29]. Figure 2 presents our measurements of the concurrence C as a function of g 12 . As the excitation probability is decreased, g 12 increases as does the entanglement. The threshold to achieve C > 0 is found to be g 0 12 ' 7, corresponding to a probability p ' 1:2 10 ÿ2 per trial for the creation of the heralded entangled state and to a preparation rate 2 kHz. Note that C 0 (or C not greater than zero) does not imply that there is no entanglement, only that any possible entanglement is not detected by our protocol, which provides a lower bound for the entanglement. More importantly, in an infinite dimensional Hilbert space, entangled states are dense in the set of all states [30], so that zero entanglement is not provable for an actual experiment by way of the concurrence.
To confirm the model leading to Eq. (6), the inset gives the measured visibility V as a function of g 12 . The solid line is a fit according to Eq. (6) with free parameter , leading to an overlap 0:95 0:01, in agreement with the value 0:98 0:03 obtained from an independent two-photon interference measurement. With the fitted value of and with the independently determined value of the conditional probability p c 0:135 0:005 from measurements performed on each ensemble separately, we compare our measurements of C with the prediction of Eq. (5) (solid line in Fig. 2) and find good agreement. Table I provides the diagonal elements of the density matrix 2 U ;2 D and the concurrence for the case g 12 60 4 corresponding to a probability to create entanglement p 9 10 ÿ4 per trial (160 Hz). A value C 0:092 0:002 is directly measured at detectors D 2a , D 2b without correction. By way of the independently determined propagation and detection efficiencies, we infer the density matrix output 2 U ;2 D for fields 2 U , 2 D at the output of the ensembles, from which we obtain a concurrence C output 2 U ;2 D 0:35 0:10. This value exceeds the published state of the art by 2 orders of magnitude [7]. This leap underlines the progress obtained in terms of suppression of the two- FIG. 2 (color online). Concurrence C as a function of the normalized cross correlation function g 12 , for the two possible heralding events (detection at D 1a or D 1b ). The solid line is obtained from Eq. (5) with the fitted overlap (see inset) and assuming an independently-measured retrieval efficiency at 13.5%. The dotted line corresponds to C 0 . Inset: Average visibility of the interference fringe between the two field-2 modes. The solid line is a fit using the expression given by Eq. (6), with the overlap fitted to 0:95 0:01. TABLE I. Diagonal elements and concurrence of the density matrices for fields 2 U;D , without and with correction for propagation losses and detection efficiencies. The last column provides the estimated elements and concurrence for the atomic state by considering the readout efficiency at g 12 60 4.  . Concurrence C (without correcting for propagation and detection losses) as a function of the normalized cross correlation function gS,AS (denoted here g12, for the two possible heralding events (detection at D1a or D1b after the beam splitter). Inset: Average visibility of the interference fringe between the two field-2 modes.) this way, it is possible to create effective two-excitation entangled state by post selection when the two chains are combined at the remote locations. This architecture also considerably relaxes the constraints for phase stability. If the light fields for the two chains are combined together and multiplexed in the same quantum channel, the phase of the quantum channel must be constant only during the time interval ∆t between the successful entanglement generation in the two chains. An advantage of this scheme compared to two photon schemes (see below) is that the entanglement by single photon detection can be generated independently in the two chains, which leads to higher generation rates for the elementary link. The realization of this scheme  was the first experimental demonstration of an elementary segment of a quantum repeater. The experiment was realized with 4 cold atomic ensembles, as shown in Fig. 27. The two quantum nodes were in different apparatuses, about 3 m apart. Each node was composed of two ensembles about 1mm apart, implemented by addressing two different parts of a cold atomic cloud with orthogonal polarizations. The Stokes light emitted by the ensembles was recombined and coupled in the same single mode fiber, with two orthogonal polarization (polarization multiplexing). The light from the two nodes was then mixed at a beam splitter, followed by polarization optics to separate the two polarizations corresponding to the two chains of ensembles. The entanglement was generated in a heralded fashion, using a conditional control  that stopped sending excitation pulses once a successful detection was obtained for the corresponding chain. In this way, the entanglement Here, |n〉 x is the n-photon state for mode x, where x ∈ {2 LU ,2 RU ,2 LD ,2 RD }, and h U and h D are the matrix equivalent to the one for a maximally entangled state of the form of the last term in Eq. 2. Notably, the absolute phases h U and h D do not need to be independently stabilized. Only the relative phase h = h U -h D must be kept constant, which no logic is implemented, this process resulted in a 19-fold enhancement in the probability of generating this overall state from the four ensembles (SOM text).
The second requirement-stability of the relative phase h-could be accomplished by active stabilization of each individual phase h U ,h D , as in (20). Instead of implementing this challenging technical task (which ultimately would have to be extended across longer chains of ensembles), our setup exploits the passive stability between two independent polarizations propagating in a single interferometer to prepare the two ensemble pairs (27). No active phase stabilization is thus required. In practice, we found that the passive stability of our system was sufficient for operation overnight without adjustment. Additionally, we implemented a procedure that deterministically sets the relative phase h to zero.
We also extended the original DLCZ protocol (Fig. 1) by combining fields (2 LU , 2 LD ) and (2 RU , 2 RD ) with orthogonal polarizations on polarizing beamsplitters PBS L and PBS R to yield fields 2 L and 2 R , respectively. The polarization encoding opens the possibility of performing additional entanglement purification and thus superior scalability (28,29). In the ideal case, the resulting state would now be effectively equivalent to a maximally entangled state for the polarization of two photons where |H 〉 and |V 〉 stand for the state of a single photon with horizontal and vertical polarization, respectively. The sign of the superposition in Eq. 3 is inherited from Eq. 1 and is determined by the particular pair of heralding signals recorded by (D 1a ,D 1b ) and (D 1c ,D 1d ). The entanglement in the polarization basis is well suited for entanglement-based quantum cryptography (11, 25), including security verifica- Node "Left" Node "Right" . The atomic state is thus stored while waiting for the second ensemble pair to be prepared. After both pairs of ensembles U,D are entangled, the control logic releases strong read pulses to map the states of the atoms to Stokes Field 2 fields through |s〉→ |e〉→ |g〉. Fields 2 LU and 2 LD are combined with orthogonal polarizations on the polarizing beamsplitter PBS L to yield field 2 L ; fields 2 RU and 2 RD are combined with orthogonal polarizations on the polarizing beam splitter PBS R to yield field 2 R . If only coincidences between fields 2 L and 2 R are registered, the state is effectively equivalent to a polarization maximally entangled state.  . The inset at the bottom left shows the relevant atomic levels for the 6 S 1/2 → 6 P 3/2 transition in atomic cesium, as well as the associated light fields. With this setup, a photo-detection event at either detector D1a or D1b indicates entanglement between the collective excitation in LU and RU, and a photo-detection event at either detector D1c or D1d indicates entanglement between the collective excitation in LD and RD (20). Two orthogonal polarizations in one fiber beamsplitter implement BSU and BSD, yielding excellent relative path stability. A heralding detection event triggers the control logic to gate off the light pulses going to the corresponding ensemble pair (U or D) by controlling the intensity modulators (I.M.). The atomic state is thus stored while waiting for the second ensemble pair to be prepared. After both pairs of ensembles U,D are entangled, the control logic releases strong read pulses to map the states of the atoms to photons that are combined with orthogonal polarizations on the polarizing beamsplitter PBSL and PBSR. If only coincidences between the fields at both nodes are registered, the state is effectively equivalent to a polarization maximally entangled state.) could be obtained independently in the two chains (up to the limited memory time of about 10µs in the present experiment).
Once the two chains of ensembles have been successfully entangled, the stored excitations are retrieved simultaneously in all ensembles, and the retrieved light is combined at each node. The desired effective two-photon state entangled is finally post-selected by conserving only the events where one detection is present at each node. The effective entanglement was verified by violating a Bell inequality with the two fields.
Note that the optical phases were not actively stabilized in this experiment. The passive phase stability of the quantum channels during the time interval corresponding to the memory time (10 µs) was good enough to ensure a proper phase compensation.

Entanglement Connection
Entanglement connection is obviously a crucial step in order to extend the entanglement distance in quantum repeater architectures. While many experimental demonstrations of entanglement swapping have been realized with entangled photons (Halder et al., 2007;Kaltenbaek et al., 2009;Pan et al., 1998;de Riedmatten et al., 2005), only one attempt has been made so far to demonstrate the swapping of entanglement with ensemble quantum memories . The experiment was realized in a setup similar to the one of ref . Heralded number state entanglement was again generated independently in two chains of ensembles. One ensemble of each chain is then read-out simultaneously and the retrieved light is combined at a beam splitter. A detection after the beam splitter transfers the entanglement to the remaining ensembles, which have never interacted. While the authors have been able to demonstrate the transfer of a substantial amount of quantum coherence, the demonstration of entanglement was not possible using the method developed in . Actually, this experiment highlights one of the main limitation of the DLCZ protocol: the quadratic propagation of two photon errors with the number of links. In order to keep the two-photon error low enough to demonstrate entanglement, the excitation probability has to be kept to a very low level. The resulting count rate was not high enough to be able to determine the two excitation probability, as required for the quantum tomography, in a reasonable time.

B. Entanglement Creation and Swapping based on Two-Photon Detections
We will now review experiments that are particularly relevant to the schemes where the entanglement creation and/or connection are based on two-photon detections.

Two Photon Quantum Interference from Separate Ensembles
Two-photon quantum interference plays an essential role in all of the protocols discussed in this review. Depending on the protocols, it is used for entanglement swapping, see sections III.A, III.B and III.F, and for entanglement generation in the elementary links, see sections III.B and III.F. It also intervenes in the protocols that are primarily based on single-photon detections (see sections II, III.C, III.E) in the final step, where twophoton entanglement is post-selected. At the heart of two-photon interference lies the Hong-Ou-Mandel effect (Hong et al., 1987). Due to the bosonic nature of photons, two indistinguishable photons mixed at a beam splitter stick together (photon bunching) and always exit in the same spatial mode. This is due to a destructive interference between the probability amplitude of both photons being reflected and both transmitted. This effect manifests itself by the absence of coincidence detection between the two output modes of the beam splitter when the two photons are made indistinguishable, a property known as Hong-Ou-Mandel (HOM) dip. The observation of a HOM dip is an efficient way to quantify the degree of indistinguishability of photons generated by spatially separated atomic ensembles.
In 2006, Felinto and coworkers  reported the first observation of a two photon quantum interference with photons emitted by separate atomic memories. The authors used heralded single photons generated independently in two cold Cs ensembles. Write pulses were sent in the two ensembles simultaneously and Stokes light was collected in an optical fiber and sent to a single photon detector. In order to be able to address independently the two clouds, the authors used a conditional control that stopped sending write pulses in the corresponding ensemble when a Stokes photon was detected. In this way, they could generate single spin excitations independently in the two ensembles. After a time corresponding to the memory time of the device, the two spin excitations were converted into single photons, coupled in a single-mode optical fiber and combined at a beam splitter in order to measure the two photon quantum interference. A visibility of 70% was measured for the Hong-Ou-Mandel dip. From this, value, taking into account the loss of visibility due to the remaining two-photon contribution, the authors inferred an indistinguishability of 90% between the two photons.
Beyond the two photon interference, this experiment was also the first one to show that the use of a quantum memory could increase the generation rate of quantum state of light in separated sources. The conditional control resulted in a 28-fold increase in the probability of obtaining a pair of single photons, relative to the case without memory.
A similar setup with Rb atoms was used in a more recent experiment by (Yuan et al., 2007). The Hong-Ou-Mandel dip was measured by varying the relative delay between the two read pulses. In this way, the authors could infer the coherence time of the photons, 25 ns. A HOM visibility of 80 %was obtained, also limited by twophoton contributions. The authors also measured the HOM dip in the frequency domain, by changing the relative detuning between the two read beams. They found a similar visibility and a dip width of 35 MHz, in accordance with the time measurement.
In the two experiments mentioned above, the quantum interference is realized with conditional anti-Stokes single photons retrieved from the stored excitations. However, if the creation of entanglement is realized with a two-photon detection (as in the protocol of ), the interference will take place between the Stokes photons. This configuration was tested experimentally by Chanelière and coworkers (Chanelière et al., 2007) with two cold Rb ensembles separated by 5.5m. The two photon interference was measured by recording the coincidence rate after the beam splitter for photons combined with the same and with orthogonal polarization. When the two unconditional Stokes photons are combined at the beam splitter, a HOM visibility of 33 % is observed. This low visibility reflects the fact that the non-conditional Stokes fields are thermal fields. The probability of creating two Stokes photons in the same ensemble is equal to the probability of creating one Stokes photon in each ensemble. This is similar to what has been observed with two separate parametric down conversion sources (de Riedmatten et al., 2003). However, if only those cases are taken into account where the stored excitations are converted into anti-Stokes photons and detected (using a four photon delayed coincidence procedure), then the conditional Stokes fields are single photon fields, and a high visibility HOM dip can be achieved, provided that the two fields are indistinguishable. The authors observed a visibility of 86 ±3%.
In order to keep a high fidelity in a quantum repeater architecture, it is essential that the visibility of the Hong-Ou-mandel interference is very high. The dip visibility indeed determines the fidelity of the swapping operations. The errors acquired during each swapping will then grow linearly with the number of links. The visibility of the Hong-Ou-Mandel interference for the experiments described here is still too low in that context. However, it is mainly limited by the two photon components, which is already taken into account in the theoretical description of the protocols in section III. By working in a lower excitation regime, it should thus be possible to significantly improve the visibility. Besides the two photon components, other factors can also decrease the visibility, such as waveform or polarization distinguishability. In that context, it is informative to look at the experiments demonstrating Hong-Ou-Mandel interferences with independent parametric down conversion sources. The visibility has steadily improved over the last few years with the best result so far being 0.96 (Kaltenbaek et al., 2009). Note that the two photon error is suppressed only in the very low excitation regime. That regime can be experimentally accessed if the measurement is done with two photons from the same source. In that case, visibilities approaching unity (0.994) have been measured (Pittman and Franson, 2003).

Entanglement between a Photon an a Stored Excitation
Entanglement between photons and atomic excitations plays an important role in a number of repeater protocols, cf. sections III.B and III.F. This entanglement can be realized in different ways: by encoding one logical qubit in two ensembles and by using internal spin states. We now describe these techniques in more detail.

(i) Collective excitations in different spatial modes
This first technique uses two collective excitations in different ensembles/spatial modes to encode one logical atomic qubit. It was first proposed and experimentally realized in (Matsukevich and Kuzmich, 2004). In this experiment two nearby ensembles A and B within the same atomic cloud are simultaneously excited with orthogonally polarized write beams. Similarly to Eq. (6), the joint state of the atom-photon system after the Raman excitation can be written: The resulting orthogonally polarized Stokes fields in different spatial modes are then combined into a single spatial mode at a polarizing beam splitter. Neglecting vacuum and higher order terms, the state can then be written: where 1 a,b represents the terms with one collective spin excitation in ensemble A and B, respectively, and |H (|V ) is a photon with horizontal (vertical) polarization. By measuring the Stokes photon in the polarization basis, Matsukevich and Kuzmich were able to project the atomic ensembles into a superposition state. Then they showed that the atomic qubit could be mapped into a photonic qubit with a fidelity exceeding classical thresholds, by simultaneously reading out the two ensembles.
In 2007, this technique was used to demonstrate the quantum teleportation of a polarization qubit carried by a weak photonic coherent state onto a matter qubit implemented with two cold Rb ensembles  An interesting development has been proposed in 2007 (Chen et al., 2007b). Instead of using two separate ensembles, the authors used a single ensemble, but collected the Stokes photon in two different spatial modes separated by a small angle, as shown in Fig (28). After rotating the polarization of one of the mode by 90 • , the two modes are combined at a PBS. The read beam is then sent in a counter-propagating way. Thanks to the phase matching, the anti-Stokes photon are emitted in the opposite direction of their respective Stokes photon. The entanglement between the two photons was verified by a violation of a Bell inequality, persisting for a storage time of 20 µs.
The advantage of the entanglement with collective excitation in different spatial modes is that the relative probability of excitation of the two modes can be tuned, contrary to the entanglement with internal spin states (see below). The drawback is that it requires interferometric stability between the two spatial modes.
The experiments mentioned here can also be seen as an elementary realization of spatial multiplexing, as described in section III.D.

(ii) Internal spin states
The second possibility to generate entanglement between light and stored excitation uses entanglement be-Stokes fields (jei ! jbi, Laguerre-Gauss LG 00 mode, 70 m waist) AS L and AS R induced by the write beam via spontaneous Raman scattering are collected at 3 relative to the propagating direction of the write beam. This also defines the spatial mode of the atomic ensemble L and R. With small excitation probability, the atom-light field can be expressed as [2] where m 1 is the excitation probability of one collective spin in ensemble m (m L, R), and m p ji AS i b i m denote the i-fold excitation of the anti-Stokes light field and the collective spin in atomic ensemble.
Because of the momentum conservation, the overall k vector of the collective excitation after the spontaneous Raman scattering isk atom k W ÿk AS , wherek AS andk W are the wave v respectively. I during the sto pulse is applie correlated Sto field. The wav k R k atom , wh beam [22]. Th Under the cou beams (shown To characte correlation g 2 A correlation, be As two anti-S different spati and S R can be mode-matched correlation g A quantum corre matched field correlation is o these two diffe new approach We adjust t ( L R on PBS 1 and s in Fig. 1. Ne excitations, th and the atomi j where jHi=jV of the single a collective spi propagating p fields before t photon entang polarization e parametric do To verify th and the atomi with 75 ns c counter-propa a controllable excitation bac After comb Fig. 1), the su fields is the f state FIG. 1 (color online). Illustration of the scheme of the experiment setup and the relevant energy levels of the 87 Rb atoms. Cold 87 Rb atoms captured by MOT are initially prepared in state jai. A weak horizontal polarized write pulse W illuminates the atom cloud. The spontaneous Raman scattered anti-Stokes field AS L and AS R with vertical polarization are collected at 3 to the propagating direction of the write beam, defining the spatial mode of the atomic ensembles L and R, respectively. The AS R mode is rotated to be horizontal polarized, combined with AS L mode on a polarizing beam splitter PBS 1 and sent to the polarization analyzer. This creates the entanglement between the polarization of the anti-Stokes field and the spatial modes of spin excitation in the atomic ensemble. To verify the entanglement after a storage time , a vertical polarized read pulse counter-propagating with write pulse is applied to retrieve the spin excitation to the Stokes fields S L and S R . The polarization of S L is rotated by 90 , combined with S R on PBS 2 and sent to the polarization analyzer. to generate entanglement between a photon a stored atomic excitation using the two modes approach (Chen et al., 2007b). A weak horizontal polarized write pulse illuminates the cold 87 Rb atom cloud. The spontaneous Raman scattered anti-Stokes field ASL and ASR with vertical polarization are collected at ±3 • to the propagating direction of the write beam, defining the spatial mode of the atomic ensembles L and R, respectively (Note that in this experiment the storage state has a lower energy than the initial state. Hence, the field generated by the write beam is called anti-Stokes field and the collective field generated by the read beam is called Stokes field, contrary to the convention used in this review). The ASR mode is rotated to be horizontal polarized, combined with ASL mode on a polarizing beam splitter PBS1 and sent to the polarization analyzer. This creates the entanglement between the polarization of the anti-Stokes field and the spatial modes of spin excitation in the atomic ensemble.
tween the polarization of the Stokes photon and the internal spin state of the ensemble (Matsukevich et al., 2005;de Riedmatten et al., 2006). This technique is similar to the one used to entangle single atoms or ions and emitted photons Feng et al., 2003;Simon and Irvine, 2003). Suppose an ensemble of three-level atoms with two ground state |g 1 and |g 2 , and one excited state |e . The atoms are initially in |g 1 . A weak write pulse induces spontaneous Raman scattering via the excited state, and the excitation is transferred to |g 2 while emitting a Stokes photon. The emission of this photon can follow two different decay path, leaving the atoms in different spin states. For example, for a circularly polarized write beam and for atoms initially in |g 1 , m F , the spontaneous Raman scattering can lead to spin excitation in |g 2 , m F with emission of a Stokes photon with polarization σ − , and to a spin excitation in |g 2 , m F + 2 with a σ + polarized Stokes photon. As long as the final states are indistinguishable in all other degrees of freedom, the non vacuum part of the joint state of the light is given by: The coefficient η m F is given by the Clebsch-Gordan coefficient for the relevant transition. In the more general case where the initial state is an incoherent mixture of that various |g, m F , the collective atomic states are mixed states. This scheme was first demonstrated in (Matsukevich et al., 2005), with a cold Rb ensemble. The entanglement was verified by the violation of a Bell-CHSH inequality between the Stokes and anti-Stokes photon, after a storage time of 200 ns. More recent results with cold Cs atoms  have led to a violation of Bell inequality close to the quantum limit, and to the measurement of decoherence of entanglement, with violation of Bell inequality up to 20 µs.
Besides these two techniques, two other experiments have demonstrated entanglement between collective matter qubits and photonic qubits. The first one is based on frequency encoded photonic qubits and dual species atomic ensembles . The matter qubit basis consists in single collective excitations in each of the co-trapped atomic species (Rb 85 and Rb 87 ). The second experiment demonstrated entanglement between the orbital angular momentum of the Stokes photon a the stored excitation (Inoue et al., 2006).  reported the experimental realization of an elementary segment of quantum repeater, following the protocol of . In that protocol, the entanglement creation is based on two photon detection. This requires the combination of light matter entanglement and of two photon quantum interference. Probabilistic entanglement between stored excitations and emitted photons is first generated simultaneously in remote atomic ensembles. The two Stokes photons are then combined at a beam splitter (or a Polarizing beam splitter) in the middle station for the Bell State measurement (BSM). A successful BSM projects the atomic ensembles in an entangled state. This stored atomic entangled state can be retrieved on demand by simultaneously reading out the memories. , used two cold Rb ensembles connected by 6 m and 300 m of optical fibers. In each ensemble, they created entanglement between the polarization of the Stokes photon and the spatial mode of the stored excitations, as in Eq. 35, using the technique introduced in (Chen et al., 2007b). The two Stokes photons were then combined at a polarizing beam splitter, thus analyzing the projection on the Bell state:

Elementary Segment of Quantum Repeater
A successful BSM projects the atomic ensembles in the state: where |L i and |R i corresponds to an excitation in spatial mode L and R with i = 1, 2 denoting the remote quantum nodes. In order to verify the entanglement, atomic qubits are converted into photonic qubits. In this scheme, the double excitations in each ensemble induce spurious events in the BSM that do not result in successful entanglement swapping. Hence, the probability to successfully project the ensembles in the state of Eq. (38) conditioned on a BSM is 1/2 .
The events that lead to a state with two excitations in one ensemble and none in the other can be eliminated by post-selection during the entanglement verification stage. More importantly they can be in principle discarded after the first entanglement connection, with a properly designed BSM .
In the present experiment, the quality of the postselected atomic state was good enough to violate a Bell inequality with the retrieved photons when the two ensembles were connected by 6 m of fiber for a storage time of up to ∼ 4µs. When the ensembles were connected by 300 m of fiber, a post-selected fidelity F = T r (ρ exp |Φ + 1,2 Φ + |) = 0.83 ± 0.02 was measured.

Deterministic Local Generation of Entanglement
As mentioned in section III.B, the local generation of high fidelity pairs of entangled ensembles is an important capability for the implementation of robust quantum repeaters architectures, in particular for the realization of the protocol of (Chen et al., 2007b) section III.B. We briefly mention here two deterministic schemes to create number state entanglement between two ensembles that may be useful in this context. The first experiment is based on the adiabatic transfer of one excitation between two ensembles using a quantum bus in a optical cavity . The second experiment is based on the absorption of a delocalized one photon state by two atomic ensembles using Electromagnetically-induced Transparency (EIT) (Choi et al., 2008).
Note that these techniques, while very effective for deterministically entangling nearby ensembles, are very sensitive to loss, and are thus not suitable to generate a significant amount of entanglement in remote ensembles.

C. Quantum Light Sources compatible with Ensembles based Quantum Memories
An alternative to DLCZ-like quantum memories, where the photon source and the memory are implemented within the same atomic ensembles is to use different systems for the generation and for the storage of quantum light, cf. section III.C. This configuration has the distinct advantage that the photon to be stored and the photon to be transmitted over long distances can have different wavelengths. The separation of entanglement creation and storage also allows the use of a class of quantum memories that are well adapted to the storage of multiple temporal modes (see section V.D.2).
In order to be compatible with ensemble based quantum memories, the single photons must be at the resonance frequency of the atoms and have a narrow spectrum that match the quantum memory bandwidth (typically between 10 MHz and 100 MHz). Several ways have been proposed to create quantum light with such specific properties. The first technique, described in section V.C.1, is based on the generation of photons pairs and heralded single photons with atomic ensembles. We also briefly mention two other promising approaches based on parametric down conversion (section V.C.2) and on single quantum emitters (section V.C.3).

Photon Pair and Single Photon Sources based on Atomic Ensembles
A natural way to create single photons or photon pairs compatible with ensemble based quantum memories is to use the same atomic ensembles as photon source.

(i) Photon pair sources
In principle, any DLCZ-like memory such as those described in section V.A.1, can be used as a source of nonclassical photon pairs, since the Stokes and anti-Stokes fields are strongly correlated. While the previously described experiments were performed in the pulsed regime with write and read beams separated in time, photon pair creation has also been demonstrated in the continuous regime. In 2005, (Balic et al., 2005) reported a four wave mixing experiment which generated counterpropagating paired photons with coherence time of about 50 ns and linewidth of about 9MHz using cw write and read lasers in a cold Rb ensemble. They used a four level system with two hyperfine ground state. The wavelength of the Stokes and anti-Stokes photons were 780 nm and 795 nm, respectively. The waveforms of the photon pairs were shown to be controllable at a rudimentary level by changing the read beam Rabi frequency. The photons were coupled into opposing single mode fibers at a rate of 12000 pairs/sec and violated the Cauchy-Schwartz inequality by R = 400. Using a similar setup, (Kolchin et al., 2006) generated photon pairs with 5 ns coherence time with the use of a single driving laser. This is possible by using a far of resonance (6.8GHz) Raman transition for the Stokes photon generation. Simultaneously, (Thompson et al., 2006) reported the generation of nearly identical photon pairs of 1.1 MHz bandwidth at a rate of 5 ×10 4 pairs per second from a cold Cs atomic ensemble inside a low finesse (F=250) optical cavity. In order to generate degenerate photons, the authors also used a single driving laser, but two Zeeman ground state level, instead of two hyperfine ground state level. The two photons were shown to be non classically correlated (R = 760). The were also shown to be indistinguishable to a large degree by performing a Hong-Ou-Mandel type experiment. The results of (Balic et al., 2005) were recently improved by the use of a two dimensional magneto optical trap with optical depth as high as 62 (Du et al., 2008). Photon pairs with coherence time up 900 ns with a subnatural linewidth of 0.75 MHz were generated. The authors observed that 74 % of the Stokes photons were paired, and a strong violation of the Cauchy-Schwartz inequality (R = 11600).
The previously described DLCZ-type photon sources do not offer much flexibility with the photon wavelength (see section II.A). In particular, it is not possible to directly create a photon at 1.5 µm, as required for quantum repeater applications. As mentioned in section III.C, one possibility is to use wavelength conversion techniques. Another interesting possibility for creating non-degenerate photon pairs in ensembles is based on atomic cascade transitions. (Chanelière et al., 2006) have demonstrated an entangled pair of 1530 nm and 780 nm photons generated from an atomic cascade transition in a cold Rb ensemble (see Fig. 29). While the 1530 nm photon can be transmitted with low loss in optical fibers, the 780 nm is naturally suited for mapping to an ensemble based Rb memory. They authors observed polarization entanglement between the two photons and superradiant temporal profiles for the photon at 780 nm. A quantum repeater at telecommunications wavelengths with long-lived atomic memory is proposed, and its critical elements are experimentally demonstrated using a cold atomic ensemble. Via atomic cascade emission, an entangled pair of 1:53 m and 780 nm photons is generated. The former is ideal for long-distance quantum communication, and the latter is naturally suited for mapping to a long-lived atomic memory. Together with our previous demonstration of photonic-to-atomic qubit conversion, both of the essential elements for the proposed telecommunications quantum repeater have now been realized. DOI: 10.1103/PhysRevLett.96.093604 PACS numbers: 42.50.Dv, 03.65.Ud, 03.67.Hk A quantum network would use the resources of distributed quantum-mechanical entanglement, thus far largely untapped, for the communication and processing of information via qubits [1,2]. Significant advances in the generation, distribution, and storage of qubit entanglement have been made using laser manipulation of atomic ensembles, including atom-photon entanglement and matterlight qubit conversion [3], Bell inequality violation between a collective atomic qubit and a photon [4], and light-matter qubit conversion and entanglement of remote atomic qubits [5]. In each of these works, photonic qubits were generated in the near-infrared spectral region. In related developments, entanglement of an ultraviolet photon with a trapped ion [6] and of a near-infrared photon with a single trapped atom [7] have been demonstrated. Heterogeneous quantum network schemes that combine single-atom and collective atomic qubits are being actively pursued [8]. However, photons in the ultraviolet to the near-infrared range are not suited for long-distance transmission over optical fibers due to high losses.

Quantum Telecommunication Based on Atomic Cascade Transitions
In this Letter, we propose a telecommunications wavelength quantum repeater based on cascade atomic transitions in either (1) a single atom or (2) an atomic ensemble. We will first discuss the latter case, with particular reference to alkali metals. Such ensembles, with long-lived ground level coherences can be prepared in either the solid [9] or gas [4] phase. For concreteness, we consider a cold atomic vapor confined in high vacuum. The cascade transitions may be chosen so that the photon (signal) emitted on the upper arm has telecommunication range wavelength, while the second photon (idler), emitted to the atomic ground state, is naturally suited for mapping into atomic memory. Experimentally, we demonstrate phase-matched cascade emission in an ensemble of cold rubidium atoms using two different cascades: (a) at the signal wavelength s 776 nm, via the 5s 1=2 ! 5d 5=2 two-photon excitation; (b) at s 1:53 m, via the 5s 1=2 ! 4d 5=2 two-photon excitation. We observe polarization entanglement of the emitted photon pairs and superradiant temporal profiles of the idler field in both cases.
We now describe our approach in detail and at the end we will summarize an alternative protocol for single atoms.
Step (A).-As illustrated in Fig. 1(a), the atomic sample is prepared in level jai, e.g., by means of optical pumping. For an atomic ensemble qubit, an incoherent mixture of Zeeman states can be sufficient [4]. The upper level jdi, which may be of either s or d type, can be excited either by one-or two-photon transitions, the latter through an intermediate level jci. The advantage of two-photon excitation is that it allows for noncollinear phase matching of signal and idler photons; single-photon excitation is forbidden in electric dipole approximation and phase-matched emission is restricted to a collinear geometry (this argument implicitly assumes that the refractive index of the vapor is approximately unity). Ideally the excitation is two-photon detuned from the upper level jdi, creating a virtual excitation.
FIG. 1 (color online). (a) The atomic structure for the proposed cascade emission scheme involving excitation by pumps I and II. Pump II and the signal photons lie in the telecommunication wavelength range when a suitable level of orbital angular momentum L 0 or L 2 is used as level jdi. For atomic rubidium, the signal wavelength is 1:32 m (6s 1=2 ! 5p 1=2 transition), 1:37 m (6s 1=2 ! 5p 3=2 transition), 1:48 m (4d 3=25=2 ! 5p 1=2 transition), 1:53 m (4d 3=25=2 ! 5p 3=2 transition). For atomic cesium, the signal wavelength is 1:36 m (7s 1=2 ! 6p 1=2 transition), 1:47 m (7s 1=2 ! 6p 3=2 transition). For Na and K the corresponding wavelengths are in the 1:1-1:4 m range. (b) Schematic of experimental setup based on ultracold 85 Rb atomic gas. For s 776 nm, phase matching results in the angles " 0 " 1 , while for s 1:53 m, " 0 2" 2 . P 1 and P 2 are polarizers; D1 and D2 are detectors. (ii) Single photon sources The strong correlation between Stokes and anti-Stokes fields enables the generation of heralded single photon fields with programmable delay. The detection of the Stokes field heralds the presence of a stored excitation that can be converted into a single photon field at a pro-grammable time, as already described previously. This was first reported by Chou et al (Chou et al., 2004). The single photon character of the emitted Anti-Stokes field, conditioned on the detection of a Stokes photon was demonstrated with a Hanbury-Brown-Twiss setup, leading to an anticorrelation parameter (Grangier et al., 1986) ( see Eq. 31) α = 0.24 ± 0.05. Improvements in the filtering of the Stokes and anti-Stokes photons with cold ensembles (see section V.A.1) led to improved suppression of the two-photon component (Chanelière et al., 2005;Chen et al., 2006), with the best values of α close (Matsukevich et al., 2006a) or below 0.01 ) (see Fig. 30). Another interesting possibility is to use the heralded source of photons combined with a measurement feedback protocol for implementing a deterministic single photon source, as was first proposed and demonstrated in (Matsukevich et al., 2006a) and later in (Chen et al., 2006;Zhao et al., 2009b).
For hot gases, the filtering is more challenging and the two photon components of the heralded anti-Stokes field reported so far are sensibly higher that with cold ensembles, with the lowest value being α = 0.1 ± 0.1 (Eisaman et al., 2005;Walther et al., 2007).
FIG. 30 Suppression of the two-photon component of the anti-Stokes field conditioned on the detection events in the Stokes field, measured by the autocorrelation parameter α (denoted here w) as a function of gS,AS (denoted here g12), taken from . The lowest value obtained is 0.007 ± 0.003.

Narrow Band Photon Pair Sources based on Parametric Down Conversion
A well known technique to generate correlated or entangled photon pairs is based on spontaneous down conversion in non linear crystals (Burnham and Weinberg, 1970;Hong and Mandel, 1985;Hong et al., 1987;Wu et al., 1986). The bandwidth of the photon generated with this method are however usually of the order of 10 THz, very far from atomic memory bandwidth, of order 10 to 100 MHz. Two techniques have been proposed to achieve the extreme reduction in photon bandwidth required to efficiently map the light produced by spontaneous down conversion in atomic memories. The first one is based on cavity enhanced down conversion (Ou and Lu, 1999). In that case, the non linear crystal is placed inside an optical cavity and the light is emitted only in the cavity modes. The system is operated as an optical parametric oscillator far below threshold. The reduction in spectral bandwidth is compensated by the enhancement in the efficiency of the photon pair generation due to the cavity resonance. Without further filtering, the output of the cavity is spectrally multimode (Kuklewicz et al., 2006;Scholz et al., 2009;Wang et al., 2004;Wolfgramm et al., 2008). Ideally, a single cavity mode should be selected, such that the bandwidth of the parametric light is given by the width of the cavity mode. This can be done by using etalons or filter cavities with different free spectral range. (Neergaard-Nielsen et al., 2007) generated single high purity heralded single photons of 8 MHz bandwidth at 860 nm using such a system. More recently, the generation of single mode polarization entangled photons at 780 nm with 10 MHz bandwidth was reported (Bao et al., 2008). In order to profit from the enhanced generation efficiency, the cavity must be resonant with the two down conversion modes. Hence, all the experiments demonstrated so far have worked in the regime of degenerate or near-degenerate photon pairs.
The second technique is based on passive filtering. In that case, the massive reduction of bandwidth is accompanied by a corresponding reduction in conversion efficiency. It is thus extremely challenging to implement such a filtering with traditional non linear crystals. A possible solution is the use of highly efficient waveguide sources (Tanzilli et al., 2001) based on periodically poled (PP) crystals, such as Lithium Niobate (LN) or KTiOPO 4 (KTP). These sources feature a conversion efficiency about four orders of magnitude larger than conventional bulk crystals. Using a PPLN waveguide together with fiber Bragg grating filters, (Halder et al., 2008) have generated photon pairs at telecommunication wavelength with a bandwidth of 1.2 GHz. With 7 mW of pump power, the source achieved a spectral radiance of 0.08 photon pairs per coherence time. A photon pair source adapted for a rubidium atomic memory was also recently realized with a PPLN waveguide. The 10 THz bandwidth of the parametric fluorescence was first reduced to 23 GHz using a holographic grating, and further decreased to 600 MHz using an etalon before detection (Akiba et al., 2007). In a more recent experiment (Akiba et al., 2009), a third filtering stage was introduced with a moderate finesse optical cavity and the bandwidth of the photon pairs was reduced to 9 MHz. 37

Single Photon Sources based on Single Quantum Emitters
We now briefly mention some single quantum systems that generate single photons with the required spectral properties to be compatible with atomic memories. The advantage of using single quantum emitters is that the photons can be emitted on demand. The first example is single trapped atoms. Single photons have been emitted deterministically from single atoms trapped in high finesse optical cavities (Hijlkema et al., 2007;McKeever et al., 2004) and in free space (Darquie et al., 2005). More recently, entangled photons have been generated from a single atom trapped in an optical cavity (Weber et al., 2009). All these experiments have been performed with Cs or Rb atoms emitting at 852 nm and 780 nm, respectively. The emitted photons are therefore naturally suited for mapping in a corresponding atomic ensemble, but frequency conversion is required for long distance transmission in optical fiber. Single photons have also been generated by trapped single ions (Dubin et al., 2007;Keller et al., 2004;Maunz et al., 2007). Some solid state emitters at cryogenic temperature may also be interesting. A first example is single quantum dots embedded in microcavities. Quantum dots offer more flexibility for the wavelength of the emitted photon, in particular, single photons sources at telecommunication wavelength based on quantum dots have been demonstrated (Hostein et al., 2009;Ward et al., 2005;Zinoni et al., 2006). The demonstrated photon bandwidths are however still far to large to match the memory bandwidths. A promising solid state alternative is based on single dye molecules embedded in a solid state matrix. Fourier transformed single photons at 590 nm with a bandwidth of 17 MHz have recently been generated (Lettow et al., 2007). The use of a different dye molecule may offer some wavelength flexibility. Finally a last possible candidate is based on N-V center in diamond, where close to Fourier limited single photons have recently been observed, with lifetimes of about 10 ns. (Batalov et al., 2008)

D. Storage of Single Photons in Atomic Ensembles
We now review the protocols that have been proposed and demonstrated experimentally for the storage of single photons in ensembles of atoms. Section V.D.1 is devoted to quantum memories based on Electro-Magnetically-Induced Transparency (EIT) , and section V.D.2 describes another more recent class of quantum memories based on photon echoes . A more general review about lightmatter quantum interfaces can be found in (Hammerer et al., 2008).

Quantum Memories based on Electromagnetically-Induced Transparency
A well known technique for mapping quantum state of light into atomic states is based on Electromagnetically Induced Transparency (EIT) Lukin, 2000, 2002;. It has been the subject of several reviews  so we only give here the basic feature of the technique, and review the relevant experiments. EIT is a quantum interference effect that renders an opaque atomic medium transparent thanks to a control laser field. This effect is associated with a strong dispersion in the atomic medium that leads to strong slowing and compression of the light. It has been shown that by turning off the control laser when the optical pulse is completely compressed in the atomic memory, it is possible stop the light ,i.e. to map the state of the light onto collective spin excitations of the atoms Lukin, 2000, 2002). The atomic state can then be converted into light again by turning on the control laser. This scheme has been demonstrated with bright coherent light both in ultra-cold atoms (Liu et al., 2001) with storage time of about 1 ms and in hot atomic vapor (Phillips et al., 2001) with storage time up to 0.5 ms.
More recent experiments in hot atomic gases have led to storage and retrieval efficiencies close to 50% (Novikova et al., 2007;Phillips et al., 2008) for storage times of order of 100µs. These experiments implemented the optimal control strategy introduced in (Gorshkov et al., 2007a,b,c,d). Very recently, a stopped light experiment has been reported with ultra-cold atoms loaded in a 3D optical lattice. A storage time of 240 ms was reported (Schnorrberger et al., 2009), albeit with a low storage and retrieval efficiency (0.3%). Stopped light has also been observed in rare earth doped solids ( a Pr:Y 2 SiO 5 crystal) (Turukhin et al., 2001). The coherence time of the hyperfine ground state of Pr:Y 2 SiO 5 without external magnetic fields is 500µs. It can be however dramatically increased by using an appropriate small magnetic field (Fraval et al., 2004) and by using dynamical control of decoherence . Using these techniques,  have demonstrated a stopped light experiment (with efficiency of about 1%) in a Pr:Y 2 SiO 5 crystal with storage times exceeding 1 s, which is the longest light storage time reported do date.
All the above mentioned experiments were realized with bright coherent pulses. The first experiments of storage and retrieval of single photon fields, which represent an important milestone, were published in 2005 simultaneously by two groups. (Eisaman et al., 2005) used a hot atomic vapor to generate conditional single photons fields with the required wavelength and bandwidth properties. The single anti-Stokes photon pulses were then sent in another distant hot atomic ensemble where both slow and stopped light was observed. A storage and retrieval efficiency of 10% for short delays and a storage time of 1 µs were demonstrated, with the non classical character of the stored field persisting for 500 ns. (Chanelière et al., 2005) used two cold Rb atomic ensembles as single photon source and quantum memory respectively. In this experiment, the single photon field is implemented with the Stokes field, conditioned on a subsequent anti-Stokes detection. The single photons were then directed to another ensemble through 100 m of optical fiber to be stored. In order to minimize the noise from the excitation and control beams, the two beams are applied with a small angle. After a programmable time, the atomic excitation is converted back into a single photon. The storage and retrieval efficiency for a storage a time of 500 ns is 6%. The single photon character of the stored and retrieved field is verified explicitly by a Hanbury-Brown-Twiss experiment after a storage of 500 ns (with a minimum anticorrelation parameter α = 0.36 ± 0.11), while non-classical correlations between the retrieved Stokes photon and the anti-Stokes photon are observed for storage time exceeding 10 µs. The storage and retrieval efficiency of single photons in cold atomic ensembles has been recently increased to 17% for a storage time of 1 µs (Choi et al., 2008), see Fig. 31. (Matsukevich et al., 2006b) have shown that the EIT technique also allows the mapping of a polarization qubit carried by a single photon. They generated probabilistic entanglement between the polarization of the Stokes photons and the internal spin state of the stored excitation, as in (Matsukevich et al., 2005). The polarization of the Stokes photon was then mapped onto a distant atomic ensemble, resulting in the probabilistic entanglement of two remote matter qubits. The entanglement was demonstrated in a post-selected fashion by converting the two stored qubits in photonic qubits and by measuring polarization correlations between the two photons, resulting in a violation of a Bell inequality. L in ,R in . To avoid dissipative absorption for the fields in modes L in ,R in for our choice of polarization 26 , we spin-polarize the atomic ensemble into one Zeeman sublevel of a hyperfine ground state jF 5 4, m F 5 0ae. Initially, the strong control fields V c (L,R) (t) (resonant with 6S 1/2 , F 5 3 « 6P 3/2 , F9 5 4 transition) open transparency windows V c (L,R) (0) in L a ,R a for the signal modes. As the wave packet of the signal field propagates through each ensemble, the control fields V c (L,R) (t) are turned off in 20 ns by an electro-optical intensity modulator, thus coherently transforming the fields of the respective signal modes to collective atomic excitations within L a ,R a . This mapping leads to heralded entanglement between L a ,R a . After a user-defined delay, chosen here to be 1.1 ms, the atomic entanglement is converted back into entangled photonic modes by switching on the control fields V c (L,R) (t) (see Methods). Alternate algorithms for quantum control of the conditional readout allow extensions for entanglement connection and distribution by way of asynchronous preparation 6,18 .
For a given optical depth c, there is an optimal Rabi frequency V c (t) for the control field. In our experiment, c and V c (0) are 15 and 24 MHz, respectively. An example of our measurements of the EIT process for a single ensemble is presented in Fig. 2. Because of finite c, the small length (,3 mm) of the ensemble and the turn-off time of the intensity modulator, there is considerable loss in the storage process, as (t) are transformed to circular polarizations with the same helicity along each path L, R, and copropagate with an angle of 3u. c, Quantum interface for reversible mapping. Photonic entanglement between L in ,R in is coherently mapped into the memory ensembles L a ,R a by switching V c (L,R) (t) off adiabatically. After a programmable storage time, the atomic entanglement is reversibly mapped back into optical modes L out ,R out by switching V c (L,R) (t) on. Relevant energy diagrams for the storage and retrieval processes are shown in the insets. States | aae, | bae are the hyperfine ground states F 5 4, F 5 3 of 6S 1/2 in atomic caesium; state | cae is the hyperfine level a, Input. The data points are the measured probabilities p c for the signal field, a single photon generated from a separate atomic ensemble 13 . The red solid line represents a gaussian fit of 1/e width 28 ns. b, Storage and retrieval. The points around t 5 0 ms represent 'leakage' of the signal field due to the finite optical depth and length of the ensemble. The points beyond t 5 1 ms show the retrieved signal field. The overall storage and retrieval efficiency is (17 6 1)%. The blue solid line is the estimated Rabi frequency V c (t) of the control pulse. The red solid curve is from a numerical calculation solving the equation of motion of the signal field in a dressed medium 23 . Error bars give the statistical error of 1s for each point. FIG. 31 Storage and retrieval of single photon fields using EIT in a cold Cs ensemble (Choi et al., 2008). The points around τ = 0µs represent 'leakage' of the signal field due to the finite optical depth and length of the ensemble. The points beyond τ = 1µs show the retrieved signal field. The overall storage and retrieval efficiency is (17 ± 1)%. The blue solid line is the estimated Rabi frequency of the control pulse.
EIT storage of degenerate non classical light generated by parametric down conversion has also been recently demonstrated (Akiba et al., 2009(Akiba et al., , 2007. Correlated photon pairs and heralded single photons were generated in the PPLN waveguide source described in section V.C.2, and stored in a cold Rb ensemble. In the first experiment (Akiba et al., 2007), the parametric fluorescence light injected into the ensemble was spectrally much broader (600 MHz) than the EIT transparency window (8.3 MHz) and the authors demonstrated frequency filtered storage. The small part of the light that spectrally matched the EIT transparency window was stopped in the ensemble and thus delayed relative to the rest of the light, which propagated through the atoms. The retrieved photons (with 8% efficiency) could thus be detected using temporal filtering. The authors demonstrated that the coherence time of the light retrieved after a storage of 400 ns was increased to 35 ns, compared to 0.2 ns for the incident broadband light. Moreover, the conservation of the non-classical character of the stored and retrieved light was confirmed by measuring superbunching. In the second experiment (Akiba et al., 2009), the bandwidth of the incident photons was reduced to 9 MHz with an moderate finesse optical cavity. The filtered degenerate parametric fluorescence was stored for 300 ns in a cold Rb ensemble, before being retrieved with a storage and retrieval efficiency of ∼14 %. The non classical character of the retrieved light was verified by the experimental violation a classical inequality for photon counts introduced by the authors. This confirms that the non classical properties of the degenerate parametric fluorescence are conserved during the storage in the ensemble. The storage and retrieval of conditional single photons states was also demonstrated. The retrieved field was shown to exhibit anti-bunching (with an autocorrelation parameter α = 0.52 ± 0.30).
Protocols for storing light using off resonant Raman coupling to long lived schemes have also been proposed (Ham, 2008;Nunn et al., 2007), but not yet demonstrated experimentally. A detailed study of the theory of photon storage in optically thick ensembles of three levels atoms in a Λ configuration has been recently published in a series of papers (Gorshkov et al., 2007b(Gorshkov et al., ,c,d, 2008. To end this section, we discuss the multimode properties of EIT based storage. It has been shown that the number of modes N m that can be efficiently stored using EIT scales as N m ∼ √ d, where d is the optical depth of the sample (Nunn et al., 2008). This means that extremely high value of d are needed in order to store multiple temporal modes. The reason of this poor scaling is that high efficiency storage of many temporal modes requires at the same time a very slow group velocity in order to compress all the pulses in the sample before turning off the control fields and large transparency windows in order to store short photons.

Photon Echo based Quantum Memories
We are now going to review other quantum storage protocols based on photon echo techniques. Contrary to EIT based protocols which rely on transparency, these protocols rely on the reversible absorption of a single photon pulse in an inhomogeneously broadened media. After absorption, the single photon state is mapped onto a single collective atomic excitation at the optical transition, where z i is the position of atom i and δ i is the detuning of atom i with respect to the central frequency of the photon. This collective state rapidly dephases, since each term acquires a phase e iδit . The goal of the quantum memory protocols described here is to engineer the atomic system such that this inhomogeneous dephasing can be reversed. If this rephasing can be implemented, the light is re-emitted in a well defined spatio-temporal mode when the atoms are all in phase again, as a result of a collective interference between all the emitters. The rephasing of the dipoles can be triggered by optical pulses, as it is the case in traditional photon echo techniques. These techniques, while very successful to store classical light (Lin et al., 1995) and as a tool for high resolution spectroscopy (Macfarlane, 2002), suffer from strong limitations for the storage of single photons. In particular, it has been shown that the unavoidable fluorescence due to the atoms excited by the strong optical rephasing pulse blurs the single photon state and reduce the fidelity of the storage to an unacceptable level (Ruggiero et al., 2009).
We here describe two modified photon echo approaches that allow in principle the storage and retrieval of single photon fields with unit efficiency and fidelity. The first one is based on Controlled Reversible Inhomogeneous Broadening (CRIB), and the second one on Atomic Frequency Combs (AFC).

(i) Controlled Reversible Inhomogeneous Broadening
The theory of CRIB based quantum memories has been already reviewed elsewhere , and we only give here short explanation of the principle, before discussing experimental progress.
The idea of CRIB is to trigger the collective re-emission of light absorbed by an ensemble of atoms by reversing the detuning of each emitter at a given time τ after the absorption, such that δ i → −δ i . In this way, the state of the atoms evolves as : and all the atomic dipoles are in phase again when t = τ , leading to a collective emission a the time 2τ after absorption.
The initial proposal exploited the fact that the natural doppler broadening in a hot gas of atoms can be automatically reversed by using control pulses with opposite direction (Moiseev and Kröll, 2001). Achievable storage times are however limited in hot gases, due to the dephasing induced by atomic motion. Three groups then described how this protocol could be extended to store single photons in the optical regime in solid state materials (Alexander et al., 2006;Kraus et al., 2006;Nilsson and Kröll, 2005), typically in rare-earth doped solids. The implementation of CRIB in solids first requires to isolate a narrow (ideally homogeneously broadened) absorption peak within a large transparency window. This can be achieved by spectral hole burning techniques (Alexander et al., 2006;Rippe et al., 2008;Sellars et al., 2000;de Sèze et al., 2003). This line must then be artificially broadened in a controlled way, in order to spectrally match the photon to be stored. To this end, one can exploit the fact that some solid state materials have a permanent dipole moment which gives rise to a linear Stark effect. The resonance frequency of the atoms can then be controlled with moderate external electric fields, and a controlled broadening can be induced by applying an electric field gradient. The photon can then be absorbed by the broadened peak and stored in the excited state. After absorption, inhomogeneous dephasing takes place. The re-emission can then be triggered by changing the polarity of the electric field, which reverses the detunings and leads to the rephasing of the dipoles. In that case, the photon is emitted in the forward direction. It has been shown that in this configuration and for a broadening applied transversally with respect to the propagation direction, the storage and retrieval efficiency is given by : where d is the optical depth of the atoms after broadening and f (t) is coherence profile in the excited state, given by the Fourier transform of the initial absorption peak. The maximal storage and retrieval efficiency in that configuration is 54%, limited by the reabsorption of the echo by the optically thick transition. In addition, the storage time in the excited state is limited by the finite achievable width of the initial absorption peak. In order to overcome these limitations, it has been proposed to transfer the excitation to an empty longlived ground state level, using optical control pulses. The storage time is now given by the coherence time of the ground state level, which may be much longer than the excited state coherence time. Moreover, if the excitation is brought back to the excited state after storage using a counterpropagating control pulse, phase matching will enable the CRIB echo to be emitted backward. (Kraus et al., 2006) have shown that this backward read-out is equivalent to a time reversal of the Maxwell Bloch equations, which effectively suppresses re-absorption. In that case, the storage and retrieval efficiency is given by (San-gouard et al., 2007): with g(t) = f (t − T S ), where T S is the time spent in the long lived ground state. We see that η B can reach 100% for sufficiently high d.
Recently it has been shown theoretically that the efficiency of the storage and retrieval can reach 100 % even without the transfer to the ground state, using only a two level system (Hétet et al., 2008b). This requires the application of a longitudinal broadening, i.e. a broadening where the frequency of the atoms varies along the propagation direction. Such a broadening can be obtained by the use of a longitudinal electric field gradient. This configuration is sometimes referred as 'longitudinal CRIB' or 'Gradient Echo Memory'.
The multimode properties of CRIB have been studied in (Nunn et al., 2008;Simon et al., 2007b). It has been shown that, contrary to EIT, the number of modes N m that can be stored with high efficiency is proportional to the initial optical depth, N m ∼ d.
The first proof of principle demonstration of the CRIB scheme with bright coherent states was realized in a Europium doped solid (Alexander et al., 2006). The Europium ions doped in the solid state matrix have an optical transition at 580 nm, and a level structure with three hyperfine states in the ground and excited states. The authors used optical pumping techniques to create a narrow absorption peak with a width of 25 kHz, within a 3 MHz wide transparency window. The absorption of the peak was approximately 40 %. This peak was then broadened with a gradient of electric field implemented with four electrodes in a quadrupole configuration, thanks to the linear Stark effect. The broadened spectral feature was excited using 1 µs optical pulses and the polarity of the electric field was reversed after a programmable time τ . After a further delay τ , two-level Stark echoes were observed, with a decay time of about 20 µs. In an another experiment, the same authors stored and recalled a train of 4 pulses (Alexander et al., 2007). They also showed that the phase information of the input pulses was preserved during the storage. In these experiments, only a very small part of the incident pulses were re-emitted in the Stark echo (between 10 −5 and 10 −6 ). This low efficiency can be partly explained by the small absorption of the broadened peak (about 1% ).
In a more recent experiment, the same group demonstrated an improved storage and retrieval efficiency of 15 % using a Praseodymium doped crystal, which features an optical transition with larger oscillator strength and consequently larger absorption (Hétet et al., 2008b). Using a longer crystal they could achieve an efficiency as high as 45%. Very recently, a CRIB experiment has been demonstrated at the single photon level, using an Erbium doped crystal absorbing at the telecommunication wavelength of 1536 nm ).
Finally, an interesting variation has been proposed, where the reversible inhomogeneous broadening is not on the optical transition, but on the Raman transition between two ground state levels (Hétet et al., 2008a;Moiseev and Tittel, 2008). The light is mapped on the atoms by detuned Raman coupling to long lived ground states. A proof of principle demonstration with bright pulses has been reported in a rubidium vapor (Hétet et al., 2008a), where the Raman resonance line was broadened by a magnetic field gradient.

(ii) Atomic Frequency Combs
In order to fully exploit temporal multiplexing in quantum repeater architectures, the memory should be able to store many temporal modes with high efficiency. For EIT based quantum memories, this requires extremely high and currently unrealistic value of optical depth. The scaling is better for CRIB based quantum memories but the required optical depth are still very high (e.g. 3000 for 100 modes with 90% efficiency) . Recently, (Afzelius et al., 2009a) proposed a new scheme, where the number of stored modes does not depend on the initial optical depth . The scheme is based on "atomic frequency combs" (AFC).
2 homogeneous linewidth γ h and a large inhomogeneous broadening Γ in (Γ in /γ h 1). This is the case for REdoped solids that we will consider for physical implementation in Sec. V. The |g − |e transition is spectrally shaped such that the atomic density function consists of a series of narrow peaks spanning a large frequency range (atomic frequency comb, or AFC). This can be done by frequency-selectively transferring atoms from |g to a metastable state |aux , for instance a third hyperfine state, through optical pumping techniques (see Sec. V). Note that the maximum spectral width of the AFC is limited by the level spacings between the ground state levels.
We then consider a single-photon input field, having a spectral distribution γ p larger than the AFC peak separation ∆, but narrower than the total width of the AFC Γ, which is in resonance with the |g − |e transition. If the atomic density integrated over the photon bandwidth is high enough the photon can be totally absorbed by the AFC, although the spectral density of atoms is concentrated to narrow peaks. This can be understood in terms of the Heisenberg energy-time uncertainty relation. For the time scale of the absorption, which is of the order of the input pulse duration τ = 1/γ p , the optical transition will have an uncertainty of the order of γ p ∆. This causes a spectral averaging of the sharp AFC structure into a smooth distribution, allowing for uniform absorption over the photons bandwidth (see also Sec. III).
After absorption (at t = 0) the photon is stored as a single excitation de-localized over all the atoms in the ensemble that are in resonance with the photon. The state is described by a collective Dicke state [24] |ψ = N j=1 c j e iδj t e −ikzj |g 1 · · · e j · · · g N , where z j is the position of atom j, k is the wave-number of the light field (for simplicity, we only consider a single spatial mode defined by the direction of propagation of the input field), δ j the detuning of the atom with respect to the laser frequency and the amplitudes c j depend on the frequency and on the spatial position of the particular atom j.
The collective state can be understood as a coherent excitation of a large number of AFC modes by a single photon. These modes are initially (at t = 0) in phase The input mode is completely absorbed and coherently excites the AFC modes, which will dephase and then rephase after a time 2π/∆, resulting in a photon-echo type coherent emission. A pair of control fields on |e − |s allow for long-time storage as a collective spin wave in |s , and on-demand read-out after a storage time Ts. details) can reach 54% in the forward direction (limited only by re-absorption). But if the re-emission is forced to propagate in the backward direction, by a proper phase matching operation (see below), the process can reach 100% efficiency.
The process described so far only implements a QM with a fixed storage time. In order to allow for ondemand read-out of the stored field (which is a necessary requirement for use in quantum repeaters) and long-term storage, the single collective excitation in |e is transferred to a ground state spin level |s . This can be done by applying an optical control field on |e − |s , for instance a short π pulse. The excitation is thereafter stored   FIG. 32 The principles of the atomic frequency comb (AFC) quantum memory (Afzelius et al., 2009a). (a) An inhomogeneously broadened optical transition |g − |e is shaped into an AFC by frequency-selective optical pumping to the |aux level. The peaks in the AFC have width γ (FWHM) and are separated by ∆, where we define the comb finesse as F = ∆/γ . (b) The input mode is completely absorbed and coherently excites the AFC modes, which will dephase and then rephase after a time 2π/∆, resulting in a photon-echo type coherent emission. A pair of control fields on |e − |s allow for longtime storage as a collective spin wave in |s , and on-demand read-out after a storage time Ts.
The idea of AFC is to tailor the absorption profile of an 41 inhomogeneously broadened solid state atomic medium with a series of periodic and narrow absorbing peaks of width γ separated by ∆ (see Fig. 32). The single photon to be stored is then collectively absorbed by all the atoms in the comb, and the state of the light is transferred to collective atomic excitations at the optical transition. After absorption, the atoms at different frequencies will dephase, but thanks to the periodic structure of the absorption profile, a rephasing occurs after a time 2π/∆ which depends on the comb spacing. When the atoms are all in phase again, the light is re-emitted in the forward direction as a result of a collective interference between all the emitters. In order to achieve longer storage times and on-demand retrieval of the stored photons, the optical collective excitation can be transferred to a long lived ground state before the re-emission of the light. This transfer freezes the evolution of the atomic dipoles, and the excitation is stored as a collective spin wave for a programmable time. The read out is achieved by transferring back the excitation to the excited state where the rephasing of the atomic dipoles takes place. If the two control fields are applied in a counterpropagating way, the photon is re-emitted backward. In that case, it has been shown that the re-absorption of the light can be suppressed thanks to a collective quantum interference. In that configuration, the theoretical storage and retrieval efficiency, assuming that the decoherence in the long lived ground state is negligible and a perfect transfer, is given by: where d is the peak optical depth, F = ∆/γ is the finesse of the AFC. We see that η AF C tends towards unity for sufficiently large d and F .
The number of temporal modes N m that can be stored in an AFC quantum memory is proportional to the ratio between the storage time in the excited state 2π/∆ and the duration of the stored photons, which is inversely proportional to the total AFC bandwidth Γ = N p ∆, where N p is the total number of peaks in the AFC. Hence, we see that N m is proportional to N p and is independent of the optical depth. The total bandwidth of the AFC is however limited by the ground and excited state level spacings.
The AFC protocol has been used to realize the first demonstration of a solid light matter interface at the single photon level . The authors demonstrated the coherent and reversible mapping of weak light fields with less than one photon per pulse on average onto an ensemble of 10 millions Neodymium atoms naturally trapped in a solid (a Nd:YVO4 crystal cooled to 3 K). They also showed that the quantum coherence of the incident weak light field was almost perfectly conserved during the storage, as demonstrated by performing an interference experiment with a stored time-bin qubit (see Fig. 33). Finally, they also demonstrated experimentally that the interface makes it possi-ble to store light in multiple temporal modes (4 modes). The storage and retrieval efficiency was low (about 0.5 %) in this experiment, mainly limited by the unperfect preparation of the atomic frequency comb and by unperfect optical pumping. In a more recent experiment with Zehnder interferometer 21 . If the time, t, between the weak pulses matches the time between the two readouts, the re-emission from the sample can be suppressed or enhanced depending on the phase, W. The visibility of this interference is a measure of the coherence of various values of n n between 0.4 and 1.7, which demonstrates the high coherence of the storage process, even at the single-photon level. This excellent phase preservation results from the collective enhancement 22 and the almost complete suppression of background noise.
We now analyse in more detail the efficiency performance of our interface using the model developed for the AFC quantum memory 11 . The efficiency of the storage and retrieval in the forward mode, for an ideal AFC, is given by (Supplementary Information) The first factor represents the collective interaction, which is proportional to the square of the number of atoms in the spectral grating. 7. For small storage times (,400 ns), an oscillation in the efficiency is clearly visible. This is a quantum beat due to the interaction of the electronic spin of the Nd 31 ion with the nuclear spin of the surrounding vanadium ions (called a super-hyperfine interaction). For longer storage times, the decay is exponential with decay constant of 220 ns. This interaction also limits the minimal width of the absorption peaks. Error bars represent statistical uncertainties in photon count (61 s.d.). are stored and analysed using the interface. The analysis is performed by projecting the time-bin qubit on a fixed superposition basis, which here is achieved by two partial readouts (see text for details bright pulses, (Chanelière et al., 2009) demonstrated improved performance in terms of efficiency (9 %) using a Thulium doped YAG crystal. The multimode capacity has also been recently increased by almost one order of magnitude compared to  in a Nd doped Y 2 SiO 5 crystal. A train of 32 pulses has been stored in the crystal for 1.5 µs (Usmani et al., 2009). In the previous experiments, only the first part of the protocol was demonstrated (i.e. the coherent mapping onto collective excitation at the optical transition and collective re-emission at a predetermined time). Hence these experiments did not allow for on demand-read out. Very recently, a proof of principle demonstration of the full AFC protocol including the transfer to a long lived ground state has been demonstrated in a Praseodymium doped Y 2 SiO 5 crystal (Afzelius et al., 2009b).

E. Detectors
Highly efficient single photon detectors with photon number resolution are important for all the quantum repeater protocols presented in this review. The most common and most practical single photons detectors are based on semiconductor avalanche photodiodes (APD). They offer the advantage of being operated without cryo-genic cooling (using Peltier elements) but their present performance is insufficient for use in a practical quantum repeater architecture. Silicon APDs feature detection efficiencies above 60 % in the wavelength range from 600 nm to 800 nm and low dark count rates (< 50 Hz). InGaAs APDs, which can detect photons at 1550 nm, feature a much worse efficiency to noise ratio. Moreover, standard APD operation usually does not allow photon number resolution. It was recently demonstrated however that using a technique to measure very weak avalanches at the early stage of their development, it is possible to obtain photon number resolution (Kardynal et al., 2008). ) demonstrated a visible light photon counter with an efficiency of 88% with an associated dark count rate of 20 kHz using avalanches across a shallow impurity conduction band in silicon at cryogenic temperature. This type of detectors has also been shown to allow for photon number resolution .
A new type of detector based on superconducting devices has recently shown promising performances. Detectors based on superconducting NbN nanowires (Gol'tsman et al., 2001), usually called Superconducting Single Photon Detectors (SSPD) feature a very low dark count rate and an excellent temporal resolution for high speed counting at cryogenic temperature (< 4K). An efficiency of 57 % at 1550 nm and of 67% at 1064 nm has recently been demonstrated (Rosfjord et al., 2006), by inserting the SSPD into an optical cavity and by using an anti-reflection coating. Photon number resolving capability has also been demonstrated recently (Divochiy et al., 2008) in these devices.
The most advanced single photon detector in term of efficiency and photon number resolution is based on superconducting transition edge sensors. For example, Ref. (Lita et al., 2008) has recently reported photon-number resolving detectors with 95% efficiency at the optimal wavelength for telecom fibers (1556 nm), with a temporal response time of about 1 µs. The drawbacks of these detectors are the slow response time and the fact that they have to be operated at very low temperature (100 mK), which necessitates sophisticated and expensive cooling techniques. Note that the slow response time is not necessarily a problem in single mode quantum repeater architectures, since the repetition rate is given by the communication time. For architectures based on temporal multiplexing however, it is important that the detector response time is fast enough to discriminate between the successive temporal modes.
Finally, we mention that high efficiency photon number resolving single photon detection based on atomic ensembles has also been proposed. These schemes have however not been demonstrated experimentally (Imamoglu, 2002;James and Kwiat, 2002).

F. Quantum Channels
In this section, we describe the quantum channels that can be used to transmit single photons to remote locations for the initial entanglement generation. The main focus in this review is on optical fibers. A detailed review about the use of optical fibers as quantum channel in quantum cryptography experiments can be found in (Gisin et al., 2002). In particular, the effect of birefringence leading to polarization mode dispersion and of chromatic dispersion are discussed. The importance of these effects decreases with the transmitted photon bandwidth and can thus be considered negligible for the narrow bandwidth photons required in quantum repeater architectures. Hence, we concentrate here on two other aspects that are crucial for quantum repeaters: loss and phase stability.
In a single mode optical fiber, light is guided thanks to the refractive index profile across the section of the fiber. To ensure single mode operation, the core of the fiber is small (diameter of order of a few wavelength). Over the last 30 years, a considerable effort has been made in order to reduce the transmission losses (initially several dB per km). Today, installed commercial fibers feature an attenuation of 0.35 dB/km at 1310 nm, and of 0.2 dB/km at 1550 nm. The loss around 800 nm is 2 dB/km. Recent developments have led to the fabrication of ultra-low loss optical fibers, with attenuation as low as 0.16 dB/km (Stucki et al., 2009). Note that the loss remains exponential and that quantum repeater architectures will be useful to increase the transmission rates even if optical fibers with lower losses are developed (unless the attenuation can be reduced dramatically, which seems unlikely in the foreseeable future).
In quantum repeater architectures where the entanglement generation is based on singe photon detection, such as the DLCZ scheme, the phase acquired in long fiber links must remain constant for times that are typically of order of seconds. (Minar et al., 2008) have studied the phase stability of installed fiber links for quantum repeater applications. They have found that the phase of 36 km long Mach Zehnder interferometer in urban environment remains stable at an acceptable level (0.1 rad) for a duration of around 100 µs, which provides information about the time scale available for active phase stabilization. Note that a phase noise of 0.1 rad at 1550 nm corresponds to a fiber length fluctuation of 25 nm, and thus to a timing jitter of 0.12 fs.
The stabilization of phase noise in optical fibers is also relevant for other applications. There is currently an active area of research aiming at the transmission of frequency references over large distances in optical fibers, in order to synchronize or compare remote opticalfrequency atomic clocks (Coddington et al., 2007;Foreman et al., 2007b;Musha et al., 2008;Newbury et al., 2007). In this case, the phase noise in the fiber link directly translates into a spectral broadening of the frequency reference. To preserve the precision of optical clocks, the light should be transmitted with subfemtosecond jitter over long distances through the fiber. Noise cancellation schemes have been developed, which work well as long as the phase noise is negligible during the round trip time of the fiber (see (Foreman et al., 2007a) for a recent review).
Another interesting possibility to achieve a phase stable operation is to excite the two remote memories in a Sagnac interferometer configuration (Childress et al., 2005;Minar et al., 2008). In this way, the excitation lasers for the two memories and the emitted photons travel the same path in a counterpropagating fashion. Hence as long as the phase fluctuations are slower than the travel time, the phase difference is cancelled automatically. (Minar et al., 2008) have shown that high visibility first order interference (V> 98%) could be achieved without any active stabilization for fiber loops as long as 70 km in an urban environment.
Optical fibers are not the only way of implementing a quantum channel. The long-distance distribution of photons through free space is also an active field of investigation. Single photons and entangled photon pairs have been transmitted over distances as great as 144 km (Fedrizzi et al., 2009;Ursin et al., 2007). Free-space channels are also subject to significant loss for long distances. For example, the total channel loss in (Fedrizzi et al., 2009) was 64 dB, where the attenuation was dominated by turbulent atmospheric effects. An interesting extension of this approach to long-distance quantum communication is the use of satellites, in which case only a small part of the photon path is in the atmosphere and the dominant losses are due to beam divergence. Realistic links would involve fast-moving low-orbit satellites. There are several feasibility studies, both theoretical (Bonato et al., 2009;Villoresi et al., 2004) and experimental (Peng et al., 2005;Villoresi et al., 2008). We would like to emphasize that the quantum repeater principle can be applied to any kind of lossy channel, including satellite-based transmission.

G. Coupling Losses
In section IV.B, we have studied the performances of various quantum repeater architectures with respect to the quantum memory and detector efficiencies. In practice, other kinds of loss need to be taken into account: the passive losses in the optical elements and the fiber coupling losses. These losses are of crucial importance for the performance of quantum repeaters. Passive loss between the memory and the detector (together with the loss between the photon source and the memory for experiments with absorptive memories) affects the repeater performances in the same way as the memory efficiency. One may thus introduce an effective memory efficiency that take the passive loss into account.
To illustrate the importance of passive losses, consider the experiment of (Simon et al., 2007c) which demon-strated a DLCZ like memory with a cold Cs ensemble in an optical cavity. This experiment has demonstrated the highest intrinsic retrieval efficiency to date (η R = 84%). However, if one takes into account the probability to escape the cavity (T = 0.17), the transmission efficiency of the interference filter used in the experiment (q 1 =0.61) and the fiber coupling efficiency (q 2 = 0.65), the effective retrieval efficiency, given by the conditional probability to have an anti-Stokes photon in front of the detector is η ef f R = η R T q 1 q 2 0.06. In free space experiments the highest η ef f R measured in a DLCZ source so far is of order of 25% .
We emphasize that in order to build a practical quantum repeater that beats direct transmission, all these passive losses must be considerably reduced.

VI. OTHER APPROACHES TOWARDS QUANTUM REPEATERS
There is a significant number of proposals for realizing quantum repeaters using ingredients other than atomic ensembles and linear optics. An exhaustive treatment of these proposals would be a task for another review paper comparable to this one. Here we will restrict ourselves to a very brief overview. One class of proposals keeps atomic ensembles and linear optical processing as important ingredients, but supplements them with additional non-linear elements such as the Kerr effect (He et al., 2008) or light shift induced blockade (Shahriar et al., 2007).
However, most alternative proposals involve individual quantum systems. One natural system to consider is trapped ions, since quantum information processing in general is extremely well developed in this system. There is a number of proposals for entangling distant ions or atoms, both via single-photon (Bose et al., 1999;Cabrillo et al., 1999) and two-photon detections Feng et al., 2003;Simon and Irvine, 2003), some of which have already been realized experimentally . Recently quantum repeaters with trapped ions have been analyzed more quantitatively, including a discussion of possible multiplexing ). This analysis shows that ionbased repeaters have the potential to significantly outperform atomic-ensemble based approaches, notably because all entanglement swapping operations can be performed with unit probability.
Another active field of research on quantum repeaters investigates the potential of single quantum systems in the solid state. In particular, (Childress et al., 2005(Childress et al., , 2006) developed a detailed proposal for a quantum repeater architecture adapted to the use of NV centers in diamond or of quantum dots as quantum memories. The basic entanglement generation step in this proposal is based on single-photon detections. A protocol based on two-photon detections that uses spins in quantum dots as quantum memories was proposed by . The "hybrid quantum repeater" approach of (van Loock et al., 2006(van Loock et al., , 2008 that combines the transmission of coherent states with the use of individual quantum systems is also primarily motivated by solidstate systems. Note that the present review is very focused on the most immediate goal of outperforming direct transmission, thus emphasizing simple protocols that are close to current experimental capabilities. There are also numerous contributions to the field of quantum repeaters that take a longer-term and/or more abstract view. Examples include work on the use of dynamic programming (Jiang et al., 2007a), error correcting codes (Jiang et al., 2008), decoherence-free subspaces (Dorner et al., 2008), and an analysis of the role of memory errors (Hartmann et al., 2007).

VII. CONCLUSIONS AND OUTLOOK
Since the seminal DLCZ paper, there has been significant progress towards the realization of quantum repeaters with atomic ensembles and linear optics both on the theoretical and on the experimental front. On the theoretical side, various improved protocols have been proposed, improving both the achievable entanglement distribution rate and the robustness. Quantifying the expected performance of the various protocols one finds that the prospect of efficient multiplexing in particular seems to make it realistic to implement a simple quantum repeater that outperforms the direct transmission of quantum states in the not too distant future. Spurred in part by the original DLCZ proposal, in part by more recent ideas, experiments are progressing rapidly. On the one hand, elementary links for certain quantum repeater protocols have already been demonstrated, though far from the performance required to be practically useful. On the other hand, very impressive values have been achieved for key parameters such as storage time, memory efficiency or multi-mode capacity, though not yet simultaneously in a single system. Besides requiring the capacity to generate, store and swap entanglement, constructing a viable quantum repeater will also require technological elements such as stabilized long-distance fiber links, and the virtual elimination of coupling losses between the various components.
We suspect that the first quantum repeater that beats direct transmission will probably be realized using atomic ensembles, linear optics and photon counting. In the longer run, this approach may well be overtaken by other systems with increased capabilities. New capabilities may come from modifying ensemble-based approaches, for example by including techniques based on continuous variables, where good quantum memories have already been demonstrated (Julsgaard et al., 2004). In a recent experiment, coherent-state type entanglement was created using photon counting (Ourjoumtsev et al., 2009), suggesting that hybrid approaches combining pho-ton counting and homodyne detection may be promising. Another promising modification may be the use of ensembles in optical lattices, where long light storage times have recently been demonstrated for the first time (Schnorrberger et al., 2009), and where more advanced information processing may be possible compared to conventional ensembles. However, it is also quite conceivable that single quantum systems, as mentioned briefly in section VI, will eventually be more powerful than ensembles. Whichever approach may turn out to be the most adapted, based on recent progress we believe that in the long run intercontinental entanglement will not be out of reach. interesting discussions. We gratefully acknowledge support by the EU Integrated Project Qubit Applications, the Swiss NCCR Quantum Photonics, and the ERC Advanced Grant QORE. (ii) Waiting for a success in two neighboring elementary links.
In order to be able to attempt the first swapping, entanglement creation has to succeed in two neighboring elementary links. The corresponding waiting time is the maximum of two waiting times, each of which follows the distribution given for the elementary link. We will denote the distribution for this combined waiting time byp(n) and its expectation value by ñ . One has p(n) = p(n) 2 + 2p(n) It is easy to see that the expression inside the parenthesis is bounded by 1. As a consequence, one must have ñ ≤ 2 n .
Obviously one also has n ≤ ñ , since waiting for two independent successes has to take at least as long as waiting for a single success, so defining f ≡ ñ n one certainly has 1 ≤ f ≤ 2.
(iii) First entanglement swapping The next step is the first entanglement swapping, which succeeds with a probability P 1 . Neglecting the time step required for the swapping itself, the mean waiting time for a success will be equal to ñ with probability P 1 . This is for the case where the swapping works right away. It will be 2 ñ with probability (1 − P 1 )P 1 (the second swapping attempt is successful) etc. Taking into account all possibilities, the waiting time for a successful swapping, and thus for the establishment of entanglement over the length of 2 elementary links is (iv) Higher levels in the repeater protocol.
We have seen that at the lowest level of the repeater the average waiting time for having a success in two neighboring links is essentially exactly 3 2 times longer than the average waiting time for one link. The situation is more complicated for higher levels of the repeater, because the corresponding distribution of waiting times p(n) for each individual link is no longer a simple exponential distribution. In fact, its form at each level of iteration depends not only on P 0 , but also on the success probabilities for entanglement swapping at all lower levels (P i ). To our knowledge, nobody has so far succeeded in obtaining useful analytical results for the general case. However, numerical evidence (Brask and Sorensen, 2008;Jiang et al., 2007b) suggests that 3 2 is still a good approximation for the factor f . One certainly always has 1 ≤ f ≤ 2. Let us also note that the exact value of f for each level has a relatively small impact on log-scale comparison plots such as those in this paper.
(v) Second entanglement swapping and general formula.
Whatever the exact value of ñ 1 , the same argument as for the first swapping shows that the expectation value for the waiting time for a successful second-level swapping, and thus for the establishment of entanglement spanning four elementary links is where P 2 is the success probability for the second-level swapping. The general formula for the waiting time after l levels (corresponding to 2 l elementary links) is thus (in the limit of small P 0 ) n l = f 0 f 1 ...f l−1 P 0 P 1 P 2 ...P l , with f 0 = 3 2 , and 1 ≤ f i ≤ 2 for all f i . The final post-selection step in protocols based on single-photon detections can be treated in full analogy to entanglement swapping. Again one needs two lowerlevel copies to create one higher-level one, so there is an f factor and a success probability P ps .

APPENDIX B: Multiphoton Errors in the DLCZ Protocol
Here we explain in more detail how to calculate the multi-photon errors in the DLCZ protocol (Jiang et al., 2007b;Minar et al., 2007). The starting point is Eq. (6), where we now explicitly write the O(p) terms, but set the phases φ a = φ b = 0, The O( √ p) terms give the results that are desired in the protocol. The higher orders in p give rise to errors.

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The probability for a detector that is photon number resolving, but that has non-unit efficiency η, to detect a single photon in modeb, given that there are n photons present in that mode, is p n = nη(1 − η) n−1 . (B2) In the entanglement generation step η = η d η t , with η t 1, such that one can approximate p 1 = η and p 2 = 2η. As a consequence, the state of the two modes s a and s b conditional on one click in either d = 1 √ 2 (a + b) or d = 1 √ 2 (a − b) (for simplicity we also set the phases ξ a = ξ b = 0) is ρ AB = |ψ + ψ + | + p 2 (|20 20| + |11 11| + |02 02|) where with |01 = |0 A |1 B etc. Here we assume that a corrective phase shift of π between s a and s b has been applied in the case of a detection ind. When these states are used to create entanglement between non-neighboring stations using entanglement swapping, the errors are amplified. Suppose we have established states ρ AB = k,l,k ,l ρ AB kl,k l |kl k l | (B5) and ρ CD . Entanglement swapping proceeds by reconverting the atomic modes s b and s c into photonic modes b and c , and then combining the modes b and c on a beam splitter such that b = 1 √ 2 (b +c), c = 1 √ 2 (b −c). Entanglement swapping also works with a single click.
The new state created in the entanglement swapping step is: The number of iterations depends on the nesting level of the repeater (which depends on the distance to be covered). We denote the state after the highest-level swapping operations by ρ AZ . The final step of the protocol is the projection onto one photon on each side (cf. before), which in the ideal case could be written as where P 1 A 1 Z is the projector onto the states that have exactly one photon in location A (taking A 1 and A 2 together) and one photon in location Z. In practice, the detectors are again not perfect (cf. above). We therefore have to consider the probabilities for single detections following Eq. (B2). The correct formula for the final density matrix is then σ AZ klmn,k l m n = ρ A1Z1 km,k m ρ A2Z2 ln,l n p k+l p m+n p k +l p m +n .
Here k, l, m, n refer to the modes a 1 , a 2 , z 1 , z 2 etc. The fidelity is given by the overlap of this state, properly renormalized, with the ideal final state.
To second order in p, the results are the following. Given as function of the nesting level n, the fidelity F (n) has the form where the coefficients for the lowest values of n are A 0 = 8, A 1 = 18, A 2 = 56, A 3 = 204, A 4 = 788, B 0 = 37, B 1 = 250, B 2 = 2966, B 3 = 43206, B 4 = 669702.
The dependence on (1−η) indicates that ideal photonnumber resolving detectors in combination with perfect memories would allow one to identify all undesirable multi-photon events, and thus to eliminate the considered errors. From n = 3 onwards, A n scales approximately like 2 2n or equivalently like N 2 , where N = 2 n is the number of links. This scaling becomes virtually exact for large values of n, see Fig. 34. Similarly, B n scales approximately like N 4 .
The scaling with N 2 of the multi-photon errors is related to the fact that the size of the vacuum component in the state scales linearly with N . In fact, the errors in the final post-selected two-photon state that lead to the given fidelity reduction arise from the combination of the multi-photon (i.e. two-photon) component for one pair of ensembles with the vacuum component for the other pair of ensembles, with both permutations contributing. The coefficients An and Bn as a function of n. One sees the scaling with N 2 = 2 2n and N 4 = 2 4n respectively.