1. Introduction

Quantum entanglement plays an important role in quantum communication, such as quantum teleportation [1], quantum dense coding (QDC) [2, 3], quantum key distribution [4–7], quantum secret sharing [8], quantum secure direct communication [9, 10], and so on. Quantum teleportation [1] requires a maximally entangled photon pair to set up the quantum channel for teleporting an unknown single-particle quantum state without moving the particle itself. The high capacity of QDC [2, 3] comes from the maximal entanglement of quantum systems. However, an entangled photon pair is usually produced locally and suffers inevitably from the environment noise (e.g., the thermal fluctuation, vibration, imperfection of an optical fiber, and birefringence effects [11]) in its distribution process between the two parties in quantum communication, which will degrade its entanglement or even make it in a mixed state. The process of quantum state storage will degrade the entanglement as well. This decoherence will decrease the security of quantum communication protocols and the fidelity of quantum teleportation. In a practical long-distance quantum communication network, a quantum repeater [12] is required to depress the decoherence by the destructive effect of the noise [13]. Entanglement distillation [14–34] is one of the key techniques in a quantum repeater protocol [12, 13] and it is used to distil highly entangled states from less entangled ones [14–16]. It includes entanglement purification and entanglement concentration.

Accurately, entanglement purification is used to obtain a subset of high-fidelity nonlocal entangled quantum systems from a set of those in a mixed state with less entanglement [14–23]. In 1996, Bennett et al. [14] proposed the first entanglement purification protocol (EPP) to purify a particular Werner state, resorting to quantum controlled-not (CNOT) gates. In 2001, Pan et al. [16] introduced an EPP with linear optical elements, by keeping the cases that the two photons owned by the same party have the identical polarization. In 2002, Simon and Pan [17] developed an EPP with linear optical elements and a currently available parametric down-conversion (PDC) source. Recently, Ren and Deng presented an EPP for spatial-polarization hyperentangled photon systems [20] and Sheng et al. proposed some deterministic EPPs [21–25]. Entanglement concentration is used to get some nonlocal maximally entangled systems from a set of systems in a partially entangled pure state. In 1996, Bennett et al. [26] presented the first entanglement concentration protocol (ECP) for two-particle systems, resorting to the Schmidt projection. In 2001, Zhao et al. [28] and Yamamoto et al. [29] proposed two similar ECPs independently with linear optical elements. In 2008, Sheng et al. [30] developed an ECP with cross-Kerr nonlinearity and its efficiency was improved largely by iteration of the entanglement concentration process. In 2013, Ren, Du, and Deng [32] presented an ECP for hyperentangled photon pairs with the parameter-splitting method based on linear optical elements. In 2014, Ren and Long [33] gave a general scheme for polarization-spatial hyperentanglement concentration of photon pairs. By far, some ECPs for atom systems [35–37] have been proposed.

The atomic ensemble system is one of the most promising candidates for quantum communication [38], due to the collective enhancement effect originating from the indistinguishability of the atoms interacting with radiations. Moreover, the time for the storage of a single photon in a cold atom ensemble is about several milliseconds [39] and atom ensembles can, in principle, be used as memory nodes for long-distance quantum communication network. For example, in [40], an atomic ensemble is used as a local memory node [39] for global quantum communication. Inspired by the repeater protocol in [40], a number of works for long-distance quantum communication based on atomic ensembles have been done [38]. In 2007, Chen et al. [41] introduced a fault-tolerant quantum repeater protocol with atomic ensembles and linear optics. In 2010, Zhao et al. [42] proposed a quantum repeater protocol for the atomic ensembles by utilizing the Rydberg blockade effect [43]. In 2011, Aghamalyan et al. [44] proposed a quantum repeater protocol based on the deterministic storage of a single photon in an atomic ensemble with the cavity-assisted interaction.

In the fault-tolerant quantum repeater [41], two atomic ensembles that associate with the photons in different polarizations are used as an effective memory node. The entanglement purification for the nonlocal atomic ensembles can be performed with the EPP for polarizing photons [16] when the quantum memories are involved [39]. In [42], two different single collective spin-wave excitation states of the atomic ensemble are used to encode a stationary qubit. The entanglement purification for the atomic ensembles in two different nodes is completed with the scheme proposed in [14], since the CNOT gate between the two atomic ensembles in each node is available, when they are placed within the blockade radius [43].

In this paper, we investigate the possibility of achieving the entanglement distillation for nonlocal atomic ensembles trapped in single-sided small cavities [44–48] by the input-output process of single photons. We give three efficient entanglement distillation protocols (EDPs) for nonlocal two-atomic-ensemble systems, including an optimal ECP for a partially entangled state with known parameters, an ECP for an unknown partially entangled pure state with a nondestructive parity-check detector (PCD) which works efficiently in a simple way, and an EPP for a mixed entangled state. Compared with the EDPs for atom systems by others [35–37], our EDPs for nonlocal atomic ensembles have the advantage of high fidelity and efficiency with current experimental techniques. Moreover, it is easier to implement our EDPs than the former and they are useful for quantum communication network with atomic ensembles acting as memories.

2. Nondestructive parity-check detector on two local atomic ensembles

2.1. An atomic-ensemble-cavity system

Let us consider an ensemble composed of N cold atoms trapped in a single-sided optical cavity [44–48], in which one of the mirrors is perfectly reflective and another one is of small transmission allowing for incoupling and outcoupling to light, shown in Fig. 1(a). The atom has a four-level internal structure, shown in Fig. 1(b). The two hyperfine ground states of a cold atom are denoted with |g〉 and |s〉. The excited state |e〉 and the Rydberg state |r〉 are two auxiliary states. The atomic transition between |s〉 and |e〉 is resonantly coupled to the polarized cavity mode a, which is nearly resonantly driven by the input photon in the polarization |h〉 with the frequency ω, while the transition between |g〉 and |e〉 is a dipole-forbidden one [49]. Initially, the atomic ensemble is prepared in the ground state |G〉 = |g₁,..., g_N 〉. An arbitrary unitary operation between the ground state |G〉 and the single collective spin-wave excitation state $|S〉=\frac{1}{\sqrt{N}}{\sum }_j|g_1,\dots ,s_j,\dots ,g_N〉$ can be performed efficiently with the Rydberg blockade effect of the state |r〉 by the collective laser manipulations on the ensemble [43, 50]. When the Rydberg blockade shift is Δ = 2π × 100MHz, the transition between |G〉 and |S〉 can be completed with an effective coupling strength Ω = 2π × 1MHz and the probability of nonexcited and doubly excited errors are about 10⁻³ − 10⁻⁴ [45, 51].

 

Fig. 1 (a) Schematic diagram for a single-sided cavity coupled to an atomic ensemble system composed of N cold atoms. (b) Schematic diagram for the level structure of a cold atom.

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In the frame rotating with the cavity frequency ω_c, the Hamiltonian of the whole system composed of an input photon and an atom ensemble inside a single-sided cavity can be expressed as (h̄ = 1) [49, 52]

Initially, when the ensemble is prepared in the state |S〉, the input photon is in the state |h〉, and the cavity mode is vacuum, with the Hamiltonian in Eq. (1), the evolution of the whole system can be confined in the first-order excitation subspace in a general state |Ψ(t)〉. Here

If the ensemble is initially in the state |G〉, it will be decoupled to the cavity mode and the input photon in the polarization |h〉 will be reflected by an empty cavity [52]. The reflection coefficient r₀(δ′) is

2.2. PCD on two local atomic ensembles with the input-output process of a single photon

The principle of our nondestructive PCD on two local atomic ensembles, say E_A1 and E_A2, is shown in Fig. 2. Suppose the ensembles E_A1E_A2 are in the state |φ〉_EA1EA2 = α₁|GG〉 + α₂|GS〉 + α₃|SG〉 + α₄|SS〉. A probe photon a in the polarization state $|\varphi 〉=\frac{1}{\sqrt{2}}\left(|h〉+|v〉\right)$ sent into the port a_in is split into two components |h〉 and |v〉 by the polarizing beam splitter PBS₁ and then is led into the two cavities which contain the ensemble E_A1 and the ensemble E_A2, respectively. The state of the system composed of the photon a and the ensembles E_A1E_A2 evolves as follows:

 

Fig. 2 Schematic diagram for a PCD on two ensembles E_A1 and E_A2. a_in is the input port of the photon. HWP_i (i = 1, 2) represents a half-wave plate whose optical axis is set to π/4 to perform the bit-flip operation σ_x = |h〉 〈v| + |v〉 〈h| on the photon. H represents a half-wave plate whose optical axis is set to π/8 and completes the Hadamard transformation. PBS is a polarizing beam splitter, which transmits the |h〉 polarization photon and reflects the |v〉 polarization photon, respectively.

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3. Entanglement concentration for partially entangled atomic ensembles

3.1. Optimal ECP for two nonlocal atomic ensembles in a known partially entangled state

The principle of our optimal ECP for a pair of nonlocal partially entangled two-atomic-ensemble system E_AE_B is shown in Fig. 3. E_A and E_B belong to two parties in two nonlocal memory nodes in a quantum communication network, say Alice and Bob, respectively. This ECP is used to distill probabilistically a nonlocal two-ensemble system in a maximally entangled Bell state from that in a partially entangled pure state with known parameters.

 

Fig. 3 Schematic diagram of our optimal ECP for a nonlocal two-atomic-ensemble system in a partially entangled state with known parameters. Alice and Bob are two parties in two nonlocal memory nodes in a quantum communication network. E_A and E_B are the two nonlocal atomic ensembles which belong to Alice and Bob, respectively. The UBS is an unbalanced beam splitter with the reflection coefficient R = α/β.

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Suppose the system E_AE_B is initially in the following partially entangled state [27, 31]

After PBS₁, the photon b will be reflected by the cavity or the mirror M, and a reflection operator R̂ = |h〉 〈h|(−|G〉 〈G| + |S〉 〈S|) is introduced. Subsequently, Bob performs a Hadamard operation on the photon b with the half-wave plate H₁. The state of the composite system composed of the photon b and the ensembles E_AE_B evolves into |ϕ〉_ABb,

To complete the nonlocal optimal ECP for the entangled ensembles E_AE_B, Bob performs another Hadamard operation on the photon b with H₂ and the state of the system becomes

Our optimal ECP can be generalized to distill maximally entangled N-ensemble GHZ states from a partially entangled GHZ-class pure state. Let us, for example, consider the case that Alice (E_A), Bob (E_B), ···, and Charlie (E_C) share a partially entangled N-ensemble state

3.2. ECP for atomic ensembles in a partially entangled state with unknown parameters

The principle of our ECP for nonlocal atomic ensembles in a partially entangled pure state with unknown parameters is shown in Fig. 4. The two atomic ensembles E_A1 and E_A2 belong to Alice, and the two atomic ensembles E_B1 and E_B2 belong to Bob. Suppose these two identical pairs of partially entangled ensembles E_A1E_B1 and E_A2E_B2 are respectively in the states [26, 28–30]

 

Fig. 4 Schematic diagram of our ECP for a nonlocal two-atomic-ensemble system in a partially entangled state with unknown parameters, achieved by the input-output process of a single photon. Bob completes the parity-check measurement on the ensembles E_B1 and E_B2 with a PCD, assisted by a single photon.

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After Bob performs a parity-check measurement on his two atomic ensembles E_B1 and E_B2, if the photon detector D_v clicks (an odd-parity outcome is obtained), the state of the four-atomic-ensemble system E_A1 E_B1 E_A2 E_B2 is projected into a maximally entangled four-qubit GHZ state ${|\mathrm{\Phi }〉}_{E_{A_1}{E}_{B_1}{E}_{A_2}{E}_{B_2}}=\frac{1}{\sqrt{2}}{\left(|GSSG〉-|SGGS〉\right)}_{A_1{B}_1{A}_2{B}_2}$, which takes place with the probability 2|αβ|². By taking a Hadamard operation H_E $\left[|G〉\leftrightarrow \frac{1}{\sqrt{2}}\left(|G〉+|S〉\right),|S〉\leftrightarrow \frac{1}{\sqrt{2}}\left(|G〉-|S〉\right)\right]$ on both the ensembles E_A2 and E_B2, Alice and Bob evolve the four-atomic-ensemble system into

If the outcome of the parity-check measurement on E_B1 E_B2 is an even-parity one (the D_h detector clicks), the four-atomic-ensemble system is projected into the partially entangled state ${|\xi 〉}_{E_{A_1}{E}_{B_1}{E}_{A_2}{E}_{B_2}}=\frac{1}{{\left|\alpha \right|}^4+{\left|\beta \right|}^4}{\left({\alpha }^2|GSGS〉+{\beta }^2|SGSG〉\right)}_{A_1{B}_1{A}_2{B}_2}$. Alice and Bob can measure the ensembles E_A2 and E_B2 with the basis {|G〉,|S〉} after a Hadamard operation H_E on both the two ensembles E_A2 and E_B2, and they will obtain the two-atomic-ensemble systems E_A1 E_B1 in the state ${|{\phi }^{\prime }〉}_{E_{A_1}{E}_{B_1}}=\frac{1}{{\left|\alpha \right|}^4+{\left|\beta \right|}^4}{\left({\alpha }^2|GS〉+{\beta }^2|SG〉\right)}_{A_1{B}_1}$ with or without a phase-flip operation σ_z on E_A1. They can perform the ECP in the second round by replacing α and β with ${\alpha }^{\prime }\equiv \frac{{\alpha }^2}{{\left|\alpha \right|}^4+{\left|\beta \right|}^4}$ and ${\beta }^{\prime }\equiv \frac{{\beta }^2}{{\left|\alpha \right|}^4+{\left|\beta \right|}^4}$, respectively, when two copies of two-atomic-ensemble systems in the state |φ′〉_EA1EB1 are available. It is not difficult to calculate the success probability of the ECP in the second round ${\eta }_{c^{\prime }}^{ii}=\frac{2{\left|\alpha \beta \right|}^4}{{\left({\left|\alpha \right|}^4+{\left|\beta \right|}^4\right)}^2}$. After two rounds of entanglement concentration, the total success probability is ${\eta }_{c_t^{\prime }}^i={\eta }_{c^{\prime }}^i+\left(\frac{1-{\eta }_{c^{\prime }}^i}{2}\right){\eta }_{c^{\prime }}^{ii}$. Certainly, the method described above can be cascaded to the n-th round of concentration in the ideal case that the photon loss of the input-output process and the decoherence of the ensembles are negligible, and the total success probability of the ECP can be further improved.

4. Entanglement purification for atomic ensemble systems with PCDs

In general, a quantum system in a maximally entangled Bell state degrads into a mixed entangled state in its distribution between two memory nodes and its storage. In this time, the parties should use an EPP to improve the entanglement of their nonlocal quantum systems for creating a high-fidelity quantum channel in a quantum communication network.

Suppose that the two-ensemble systems shared by Alice and Bob are in a mixed entangled state ρ_EAEB [16–19],

The principle of our EPP for two-atomic-ensemble systems with PCDs is shown in Fig. 5. Alice and Bob choose two pairs of two-ensemble systems ρ_EA1EB1 and ρ_EA2EB2 in each time for purification. The four-ensemble system E_A1E_B1E_A2E_B2 can be viewed as the mixture of four pure states |Ψ₄〉₁ = |ψ⁺〉_EA1EB1 ⊗ |ψ⁺〉_EA2EB2, |Ψ₄〉₂ = |ϕ⁺〉_EA1EB1 ⊗ |ψ⁺〉_EA2EB2, |Ψ₄〉₃ = |ψ⁺〉_EA1EB1 ⊗ |ϕ⁺〉_EA2EB2, and |Ψ₄〉₄ = |ϕ⁺〉_EA1EB1 ⊗ |ϕ⁺〉_EA2EB2 with the probabilities $f_0^2$, f₀(1 − f₀), f₀(1 − f₀), and (1 − f₀)², respectively. Alice prepares a single photon a in the state ${|\phi 〉}_a=\frac{1}{\sqrt{2}}\left(|h〉+|v〉\right)$ and Bob prepares a single photon b in the state ${|\phi 〉}_b=\frac{1}{\sqrt{2}}\left(|h〉+|v〉\right)$. They send their photons a and b into the cavities through the ports a_in and b_in, respectively. By choosing the outcomes with the same parity, they can improve the fidelity of the entanglement of the atomic ensembles shared. Let us detail the principle of our EPP as follows.

 

Fig. 5 Schematic diagram of our EPP for atomic ensembles with PCDs.

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If the four-ensemble system E_A1E_B1E_A2E_B2 is in the state |Ψ₄〉₁, Alice and Bob will obtain the outcomes with the same parity when they measure the parity of their atomic ensembles with our PCDs. If both Alice and Bob obtain an even-parity outcome, the state of the four-ensemble system becomes ${|{\mathrm{\Psi }}_4^{\prime }〉}_1=\frac{1}{\sqrt{2}}{\left(|GSGS〉+|SGSG〉\right)}_{A_1{B}_1{A}_2{B}_2}$. If both Alice and Bob obtain an odd-parity outcome, the state of the four-ensemble system becomes ${|{\mathrm{\Psi }}_4^{″}〉}_1=\frac{1}{\sqrt{2}}{\left(|GSSG〉+|SGGS〉\right)}_{A_1{B}_1{A}_2{B}_2}$. These two instances take place with the same probability $f_0^2/2$. After Alice and Bob measure the states of their ensembles E_A2 and E_B2 with the basis $\left\{\frac{1}{\sqrt{2}}\left(|G〉\pm |S〉\right)\right\}$, the entangled two-ensemble pair E_A1E_B1 in the state |ψ⁺〉_EA1EB1 is obtained with or without a single-qubit operation on either of the ensemble E_A1 or E_B1.

If the four-ensemble system E_A1E_B1E_A2E_B2 is in the state |Ψ₄〉₂ or |Ψ₄〉₃, Alice and Bob cannot obtain the outcomes with the same parity. That is, when one obtains an even-parity outcome of the PCD on his/her two atomic ensembles, the other obtains an odd-parity outcome. Alice and Bob will obtain the states |ψ⁺〉_EA1EB1 and |ϕ⁺〉_EA1EB1 with the same probability f₀(1 − f₀)/2 after they perform the measurement on their ensembles E_A2 and E_B2 and operate the ensembles E_A1 and E_B1 with or without a single-qubit operation.

If the system E_A1E_B1E_A2E_B2 is initially in state |Ψ₄〉₄, Alice and Bob will also obtain the outcomes with the same parity when the PCDs are applied. If the outcomes are all even, the state of the four-ensemble system becomes ${|{\mathrm{\Psi }}_4^{\prime }〉}_4=\frac{1}{\sqrt{2}}{\left(|GGGG〉+|SSSS〉\right)}_{A_1{B}_1{A}_2{B}_2}$. If the outcomes are all odd, the state of the four-ensemble system becomes ${|{\mathrm{\Psi }}_4^{″}〉}_4=\frac{1}{\sqrt{2}}{\left(|GGSS〉+|SSGG〉\right)}_{A_1{B}_1{A}_2{B}_2}$. These two instances take place with the same probability (1 − f₀)²/2. After Alice and Bob measure the states of the ensembles E_A2 and E_B2 with the basis $\left\{\frac{1}{\sqrt{2}}\left(|G〉\pm |S〉\right)\right\}$, they project E_A1E_B1 into the state |ϕ⁺〉_EA1EB1 with or without a single-qubit operation on either of the ensemble E_A1 or E_B1.

By keeping the instances in which Alice and Bob obtain the outcomes with the same parity, the state of the remained two-atomic-ensemble systems E_A1E_B1 will be project into ρ′,

5. Efficiencies and fidelities of our entanglement distillation protocols

In the previous sections, our three efficient EDPs for two-atomic-ensemble systems are proposed by using the ideal input-output process of a single photon as a result of cavity QED. With the optimal ECP and the efficient ECP, the parties in the quantum communication network can get the target systems in the maximally entangled state. The fidelity of the systems after they are purified with our EPP is just the same as that by the original EPP with the CNOT gates when only the bit-flip is involved [14]. Then, we would like to discuss the performance of our protocols when the practical input-output process of a photon is considered, in which the incident photon of finite bandwidth is used and the practical coupling strength g and the decay rate γ of the single collective excitation state |E〉 are taken into account.

Recently, Colombe et al. [53] demonstrated the strong atom-field coupling in an experiment in which each ⁸⁷Rb atom in Bose-Einstein condensates is identically and strongly coupled to the cavity mode with a fibre-based cavity [54]. In this experiment, all the atoms are initialized to be the hyperfine Zeeman state |5S_1/2, F = 2, m_f = 2〉. The dipole transition of ⁸⁷Rb |5S_1/2, F = 2〉 ↦ |5P_3/2, F′ = 3〉 is resonantly coupled to the cavity mode with the maximal single-atom coupling strength g₀ = 2π × 215MHz. Meanwhile, the cavity photon decay rate is κ = 2π × 53MHz and the atomic spontaneous emission rate of |5P_3/2, F′ = 3〉 is γ_e = 2π × 3MHz. The parameters are available for our protocols, when we initialize all the ⁸⁷Rb atoms to be the state |g〉 ≡ |5S_1/2, F = 1, m_f = 1〉 and encode |s〉 ≡ |5S_1/2, F = 2, m_f = 2〉 and |e〉 ≡ |5p_3/2, F′ = 3, m_f = 3〉. The transition between the collective states |S〉 and |E〉 is resonantly coupled to the cavity mode, while the transition between |G〉 and |E〉 is dipole-forbidden. When the input-output process of a single photon is involved, we should use the practical reflection operator R̂(δ′), instead of the ideal reflection operator R̂ shown in Eq. (10), to describe the reflection process of the |h〉 polarized photon by the cavity,

5.1. The practical efficiency and the fidelity of our optimal ECP

For our optimal ECP, only a pair of atomic ensembles in a partially entangled pure state with known coefficients are used and the success of this protocol is heralded by the instance in which either the detector D_h or D_v clicks. Its practical efficiency η_c, shown in Fig. 6 (a), is

 

Fig. 6 The efficiencies η_c, η_c′, and η_p of our optimal ECP, efficient ECP, and EPP for two-atomic-ensemble systems, respectively. Here |α|² + |β|² = 1.

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Fig. 7 The fidelities of our EDPs for two-atomic-ensemble systems as the functions of the scaled coupling strength g/κ and the coefficient of the initial state α for our optimal ECP or the initial fidelity f₀ for our EPP. Here the scaled detuning δ′/κ = 0.0566. (a) The fidelity of our optimal ECP F_c. (b) The fidelity of our EPP F_p. Here |α|² + |β|² = 1.

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For the ideal input-output process, r₀ = −1 and r = 1, the efficiency ${\eta }_c^i$ and the fidelity $F_c^i$ of our optimal ECP are

5.2. The practical efficiency and the fidelity of our efficient ECP

With the practical reflection operator, the fidelity of the efficient ECP with unknown parameters is still maximal and F_c′ = 1 when the two pairs of the partially entangled pure states are identical, because the parties keep the instances in which a |v〉 polarized photon is detected by Bob, and the detector D_v does not click if both of the two ensembles E_B1E_B2 are in the same state, e.g., |GG〉 (|SS〉). The efficiency (success probability) of the ECP, shown in Fig. 6 (b), depends on the reflection coefficients and it can be expressed as

In the ideal case, r₀ = −1 and r = 1, the efficiency ${\eta }_{c^{\prime }}^i$ of our optimal ECP is

5.3. The practical efficiency and the fidelity of our EPP

In our EPP for the nonlocal atomic ensembles E_A1 E_B1 and E_A2 E_B2, the coincidental clicks of the detectors D_h (D_v) and D′_h (D′_v), are used to denote the instances which are kept by Alice and Bob, followed by some local operations and measurements on the two ensembles E_A2E_B2. That is, the success probability with the practical reflection operator becomes

When r₀ = −1 and r = 1, the efficiency ${\eta }_p^i$ and the fidelity $F_p^i$ of our EPP are

5.4. The performance of our EDPs with current experimental parameters

The efficiencies and the fidelities of our optimal ECP, efficient ECP, and EPP with the experimental parameters (κ, γ)/2π = (53, 3.0)MHz [53] are shown in Figs. 6 and 7, respectively, as the functions of the scaled coupling strength g/κ and the coefficient of the initial state α (f₀ for EPP), where the scaled detuning δ′/κ = γ/κ = 0.0566 is used. One can see that the performances (both the efficiencies and the fidelities) of our EDPs become better with a larger scaled coupling strength g/κ. Both the fidelities F_c and F_p of the optimal ECP and the EPP increase with the coupling strength g for g/κ < 0.8 and are almost robust to the variation of g for g/κ > 0.8 which leads to g²/κγ > 11.3. For example, when g/κ > 0.4 and α > 0.2, the efficiency η_c′ of our efficient ECP η_c′ > 0.065 which is 84.3% of the ideal efficiency of this ECP ${\eta }_{c^{\prime }}^i=2{\left|\alpha \beta \right|}^2$ when α = 0.2. When g/κ > 0.8 and α > 0.2, ${\eta }_{c^{\prime }}/{\eta }_{c^{\prime }}^i>95.6\%$. When g/κ > 0.8 and f₀ > 0.7, the fidelity and the efficiency of our EPP become F_p > 0.843 and η_p > 0.531 which are 99.7% and 91.6% of those with the ideal input-output process, respectively. Colombe et al. [53] demonstrated the strong atom-field coupling with g/κ > 4, which means our EDPs are feasible with current techniques.

6. Discussion and summary

In our EDPs, the perfect single-sided cavity, consisting of an ideal mirror with 100% reflection and a partially reflective mirror, might remain challenging. One can take the methods developed in [47,48,54] to implement an approximately single-sided cavity. The non-zero transmission of the ideal mirror along with the mirror scattering and absorption will lead to the photon loss of a probability p_loss. An additional photon filtering mechanism of the probability p_loss on the |v〉 polarized photon is needed to make the input-output process faithful [55], although this slightly decreases the overall success probability. When the photon pulse duration T satisfies Tκ ≫ 1, the temporal mode of the output pulse is basically the same as that of the input one leading to a faithful input-output process [49]. In fact, with the maximal detuning δ′_max = γ = 0.0566κ, the fidelities and the efficiencies of our three EDPs can achieve the values higher than 90% of those obtained with the ideal PCDs. Certainly, we assume that the local operation on the single photon with linear-optical elements is ideal, similar to all the existing EPPs and ECPs [14–34].

Compared with the previous ECPs for atom systems [35–37], our optimal ECP and efficient ECP for nonlocal atomic ensembles have some advantages. Our optimal ECP for atomic ensembles in a partially entangled pure state with known parameters has the optimal success probability as the same as that of the ECP for two-photon systems [32] and it is achieved by the detection of a single photon which interacts with a single-sided cavity one time. It can be used to distill the entanglement of each pair of atomic ensembles and does not require additional atomic ensembles, which relaxes the difficulty of its implementation in experiment largely. Our efficient ECP can be used to distill a subset of atomic ensembles in a maximally entangled state from those in a partially entangled pure state with unknown parameters, resorting to a PCD which is constructed in a simple way and involves only one effective input-output process of a single photon, not two or more. Moreover, the success of our efficient ECP is heralded by the individual detection of one |v〉 photon in each node, independent of the scaled coupling strength g/κ, which is far different from the existing cavity-involved ECPs [35–37]. By iteration of the concentration process, our efficient ECP has the maximal success probability, compared with other ECPs for the quantum systems in an entangled pure state with unknown parameters.

Our EPP for nonlocal atomic ensembles in a mixed entangled state is more efficient than that in [41], where the entanglement purification for atomic systems is completed with the EPP for two-photon systems in [16] conditioned on the effective quantum memory [39]. In [42], a CNOT gate for the two ensembles in each node is used to perform the EPP and it doubles the efficiency, compared with that in [41], while the two ensembles are required to be placed so close to each other that the CNOT gate can be performed faithfully. This requirement is not needed when the PCDs are used to perform our EPP and the efficiency of our EPP equals to that in [42] in the ideal case.

In actual implementations, errors can always take place. The efficiencies and the fidelities of our three EDPs are influenced by some experimental factors, such as the detector’s efficiency, the decay of the radiation to non-cavity modes, the impurities of the single-photon sources, and so on. For heralded single-photon sources based on the probabilistic correlated photon generation with a PDC source, we should make the average pair production levels much less than one to avoid producing multiple pairs that lead to the impurity in the heralded channel [56]. Besides, with the development of the deterministic single-photon source [56], the impact of the impurities of the single-photon sources can be further reduced. The photon loss due to the cavity mirror scattering and absorption and the nonunit efficiency of the detectors will decrease the efficiency of the input-output process and thus will reduce the efficiency (success probability) of our EDPs. Fortunately, all our EDPs succeed conditioned on the detection of a single photon and the instances with photon loss can be picked out according to the response of the detectors.

In summary, we have investigated the entanglement distillation for two-atomic-ensemble systems in single-sided cavities for the first time and proposed three efficient entanglement distillation protocols, including an optimal ECP for a partially entangled pure state with known parameters, an efficient ECP for an unknown partially entangled pure state with a PCD which is constructed in a simple way, and an EPP for a mixed entangled state. These EDPs have higher fidelity and efficiency with current experimental techniques, compared with the existing EDPs for atomic systems and atomic-ensemble systems, and they are useful for quantum communication network and quantum repeaters.