Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime

doi:10.1364/OE.21.004093

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Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime

Cong Cao, Chuan Wang, Ling-yan He, and Ru Zhang »View Author Affiliations

Optics Express, Vol. 21, Issue 4, pp. 4093-4105 (2013)
http://dx.doi.org/10.1364/OE.21.004093

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Abstract

We investigate an atomic entanglement purification protocol based on the coherent state input-output process by working in low-Q cavity in the atom-cavity intermediate coupling region. The information of entangled states are encoded in three-level configured single atoms confined in separated one-side optical micro-cavities. Using the coherent state input-output process, we design a two-qubit parity check module (PCM), which allows the quantum nondemolition measurement for the atomic qubits, and show its use for remote parities to distill a high-fidelity atomic entangled ensemble from an initial mixed state ensemble nonlocally. The proposed scheme can further be used for unknown atomic states entanglement concentration. Also by exploiting the PCM, we describe a modified scheme for atomic entanglement concentration by introducing ancillary single atoms. As the coherent state input-output process is robust and scalable in realistic applications, and the detection in the PCM is based on the intensity of outgoing coherent state, the present protocols may be widely used in large-scaled and solid-based quantum repeater and quantum information processing.

1. Introduction

Quantum entanglement is a key resource in quantum information processing (QIP), especially in quantum communication, such as quantum teleportation [1

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef] [PubMed]

], quantum dense coding [2

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992). [CrossRef] [PubMed]

], quantum secret sharing [3

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–5

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004). [CrossRef]

], quantum key distribution (QKD) [6

A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef] [PubMed]

–8

X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008). [CrossRef]

], quantum secure direct communication (QSDC) [9

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002). [CrossRef]

–12

X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006). [CrossRef]

], and so on. In order to perform quantum communication safely and perfectly, remote parties usually require the maximally entangled states to set up quantum channel. However, the maximally entangled quantum systems will inevitably suffer from environment noise, which will decrease the quality of entanglement and make the quantum communication insecure. Therefore, in long-distance quantum communication and quantum communication network, we need the quantum repeater to link two remote quantum nodes. The concept of quantum repeater was proposed by Briegel et al.[13

H, J. Briegel, W. Dr, J. I. Cirac, and P. Zoller, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. Lett. 81, 5932–5935 (1998).

] in 1998, whose basic ideal is to divide the total transmission line into segments with a shorter length at the order of the attenuation length, then entanglement purification and entanglement swapping can be used to depress the effect of noise and extend the entanglement to longer distance. In 2001, Duan et al.[14

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef] [PubMed]

] proposed the famous DLCZ protocol and entanglement purification is an essential requirement in almost all the proposed quantum repeater protocols.

Entanglement purification is used to distill some high-fidelity maximally entangled quantum systems from a mixed state ensemble. In 1996, Bennett et al.[15

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996). [CrossRef] [PubMed]

] proposed the first entanglement purification protocol (EPP) to purify a Werner state, resorting to quantum controlled-NOT (CNOT) gates and bilateral rotations. Subsequently, Deutsch et al.[16

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996). [CrossRef] [PubMed]

] modified this protocol by adding two specific unitary transformations. As photons are often considered as the perfect flying qubits in entanglement distribution, EPP for entangled photonic systems have been widely discussed. In 2001, Pan et al.[17

J. W. Pan, C. Simon, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001). [CrossRef]

] proposed an EPP with only linear optical elements and single-photon detectors. They also accomplished the experimental demonstration later [18

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003). [CrossRef] [PubMed]

]. In 2002, Simon and Pan [19

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002). [CrossRef] [PubMed]

] introduced an EPP for currently available optical parametric down-conversion (PDC) source. In 2008, Sheng et al.[20

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008). [CrossRef]

] proposed an efficient EPP for a PDC source using cross-Kerr nonlinearity. The concept of deterministic entanglement purification protocol (DEPP) was proposed in 2010 based on the hyerentanglement [21

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010). [CrossRef]

–23

F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011). [CrossRef]

]. In 2011, Wang et al.[24

C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 0988–1002 (2011).

] proposed an interesting EPP using cross-Kerr nonlinearity by identifying the intensity of probe coherent beams. There are also some important EPPs for multipartite systems in Ref. [25

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998). [CrossRef]

, 26

F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011). [CrossRef]

Entanglement concentration, on the other hand, is used to distill a subset system in a maximally entangled state from less-entangled pure state systems. In 1996, Bennett et al.[27

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996). [CrossRef] [PubMed]

] proposed the first entanglement concentration protocol (ECP) based on the Schmidt projection method and some collective measurements. In 1999, Bose et al.[28

S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999). [CrossRef]

] presented an ECP via entanglement swapping. In 2000, Shi et al.[29

B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000). [CrossRef]

] developed this method with collective unitary evaluation. Focusing on entangled photonic systems, in 2001, Zhao et al.[30

Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001). [CrossRef]

] and Yamamoto et al.[31

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001). [CrossRef]

] proposed two similar ECPs with only linear optics, respectively. In 2008, Sheng et al.[32

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008). [CrossRef]

] proposed an ECP using cross-Kerr nonlinearity. Recently, two efficient single-photon-assisted ECPs for less-entangled photon pairs were proposed in Ref. [33

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012). [CrossRef]

, 34

F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012). [CrossRef]

], which are far different from the previous ECPs mostly based on the Schmidt projection method. These two protocols can obtain an optimal yield by iterating the concentration process several rounds with the assistance of ancillary single photons and cross-Kerr nonlinearities.

Recent years, entanglement purification and concentration for solid state systems have been widely discussed as the solid state qubits are easy to be stored and promising candidate for QIP which exhibits longer coherence time and more scalable. For example, in 2005, Yang proposed an EPP [35

M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005). [CrossRef]

] for arbitrary unknown ionic states via linear optics and an ECP [36

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005). [CrossRef]

] for unknown atomic states via entanglement swapping. Feng et al.[37

X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005). [CrossRef]

] proposed an EPP for conduction electrons with charge detection. In 2006, an ECP for unknown atomic states via cavity decay was proposed by Cao et al.[38

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006). [CrossRef]

]. Reichle et al.[39

R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443, 838–841 (2006). [CrossRef] [PubMed]

] also reported their experiment for two-atom entanglement purification. In 2007, Ogden et al.[40

C. D. Ogden, M. Paternostro, and M. S. Kim, “Concentration and purification of entanglement for qubit systems with ancillary cavity fields,” Phys. Rev. A 75, 042325 (2007). [CrossRef]

] introduced the entanglement concentration and purification for qubit systems encoded in flying atomic pairs. In 2011, Wang et al.[41

C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011). [CrossRef]

, 42

C. Wang, “Efficient entanglement concentration for partially entangled electrons using a quantum-dot and microcavity coupled system,” Phys. Rev. A 86, 012323 (2012). [CrossRef]

] proposed an EPP and an ECP for entangled electrons based on quantum-dot spins in optical micro-cavities. In 2012, Peng et al.[43

Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A 86, 034305 (2012). [CrossRef]

] proposed a concentration protocol for entangled atomic and photonic systems via photonic Faraday rotations.

Cavity quantum electrodynamics (QED) systems are fundamental systems in quantum optics as well as excellent candidates for solid-based QIP [44

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565 (2001). [CrossRef]

]. By exploiting the atoms interacting with local cavities as quantum nodes and the photons transmitting between remote nodes as quantum-bus, we can set up a quantum network to realize large-scale QIP. In the past decades, many theoretical and experimental works have been reported in this field [45

S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87, 037902 (2001). [CrossRef] [PubMed]

–53

F. Mei, M. Feng, Y. F. Yu, and Z. M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009). [CrossRef]

]. However, most of them rely on the efficient single-photon input-output process in a high-Q cavity and the strongly coupling between the confined atoms and the high-Q cavity field. In 2009, An et al.[55

J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]

] reported their work to implement QIP using single-photon input-output process with respect to low-Q cavity based on the photonic Faraday rotation, which is considerably different from the high-Q cavity and strong coupling cases. Following this scheme, various works have been presented, such as quantum logic gate [56

Q. Chen and M. Feng, “Quantum gating on neutral atoms in low-Q cavities by a single-photon input-output process,” Phys. Rev. A 79 064304 (2009). [CrossRef]

], QIP in decoherence-free subspace [57

Q. Chen and M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A 82 052329 (2010). [CrossRef]

], quantum teleportation [58

J. J. Chen, J. H. An, M. Feng, and G. Liu, “Teleportation of an arbitrary multipartite state via photonic Faraday rotation,” J. Phys. B 43, 095505 (2010). [CrossRef]

], entanglement concentration [43

Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A 86, 034305 (2012). [CrossRef]

], and so on. We noticed that the coherent optical pulses are good candidate to replace the single photons to perform cavity input-output process in cavity QED [59

P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater using bright coherent light,” Phys. Rev. Lett. 96, 240501 (2006). [CrossRef] [PubMed]

, 60

T. D. Ladd, P. van Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006). [CrossRef]

]. Especially in 2010, Mei et al.[61

F. Mei, Y. F. Yu, X. L. Feng, Z. M. Zhang, and C. H. Oh, “Quantum entanglement distribution with hybrid parity gate,” Phys. Rev. A 82, 052315 (2010). [CrossRef]

] proposed an efficient entanglement distribution protocol among different single atoms confined in separated low-Q cavities by means of bringing cavity QED systems and coherent states together. Compared with other protocols also using cavity input-output process, this protocol does not need the confined atom strong coupling to a high-Q cavity, and extends the earlier protocols with single-photon to continuous variable regime, which could greatly relax the experimental requirement.

Motivated by recent progress on solid-based QIP with low-Q cavity, we investigate an atomic EPP based on the coherent state input-output process by working in low-Q cavity in the atom-cavity intermediate coupling region. The information of the entangled states are encoded in three-level configured single atoms confined in separated one-side optical micro-cavities. Through the coherent state input-output process, remote parties can construct the two-qubit parity check module (PCM) for the local atomic states, respectively, and then utilize the PCMs to distill a high-fidelity atomic entangled ensemble from an initial mixed state ensemble nonlocally. The scheme can further be used for atomic entanglement concentration, in which the two remote parties can concentrate the atom pairs in less-entangled pure states efficiently. Also by exploiting the PCM, we describe a modified ECP by introducing the ancillary single atoms in cavity QED. Compared with previous works, we use coherent optical pulse rather than single-photon to perform quantum channel, meanwhile, the coherent state input-output process is robust and scalable by working in low-Q cavity in the atom-cavity intermediate coupling region. Furthermore, the detection is based on the intensity of outgoing coherent states of the PCM, which is more efficient than single photon detection and can be easily achieved with current technologies. This paper is organized as follows: in Sec.II, we briefly introduced the coherent state input-output relation in low-Q cavity. In Sec.III and Sec.IV, we show the entanglement purification and concentration protocol between remote parties in detail. In Sec.V, we will discuss the experimental feasibility in realistic applications with current technologies. And the last section is the summary.

2. Coherent state input-output process in low-Q cavity QED regime

The idea to use the light-matter interaction in cavity QED based QIP was once presented in Ref. [54

C. Y. Hu, A. Young, J. L. OBrien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: Applications to entangling remote spins via a single photon,” Phys. Rev. B 78, 085307 (2008). [CrossRef]

, 55

J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]

] based on the state-dependent phase shift of an optical pulse reflected from a cavity coupled to a matter qubit. The theoretical analysis of the input-output relation in [54

, 55

J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]

] is based on the Jaynes-Cummings model. Here we replace the single-photon with coherent optical pulse to perform cavity input-output process. Suppose a three-level configured atom interacting with a one-side low-Q cavity driven by an input coherent optical pulse. The atom has two degenerate ground states |0〉 and |1〉, and an excited state |e〉. The qubit is encoded by different hyperfine levels |0〉 and |1〉. The transition |1〉 ↔ |e〉 for the atom is coupled to the cavity mode a and driven by the input field a_in, while the state |0〉 is decoupled from the cavity mode due to the large hyperfine splitting. The principle is shown in Fig. 1.

Fig. 1 Schematic diagram showing the principle of a coherent state |α〉 input-output process in low-Q cavity. |e〉, |0〉 and |1〉 are level structures of the atom. g is the coupling between the atom and the micro-cavity.

According to the discussion in Ref. [55

J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]

], under the assumption that the cavity decay rate κ is large enough to guarantee there is only a weak excitation by the input optical pulse on the atom initially prepared in the ground state. When

ω_{0} - ω_{p} ≫ \frac{γ}{2}

, the Heisenberg-Langevin equations for the internal cavity field and the atomic operation can be adiabatically solved, and the reflection coefficient of the input optical pulse are

r_{1} (ω_{p}) = \frac{[i (ω_{c} - ω_{p}) - \frac{κ}{2}] [i (ω_{0} - ω_{p}) + \frac{γ}{2}] + g^{2}}{[i (ω_{c} - ω_{p}) + \frac{κ}{2}] [i (ω_{0} - ω_{p}) + \frac{γ}{2}] + g^{2}},

(1)

and

r_{0} (ω_{p}) = \frac{i (ω_{c} - ω_{p}) - \frac{κ}{2}}{i (ω_{c} - ω_{p}) + \frac{κ}{2}},

(2)

when the atom is in state |1〉 and |0〉, respectively. Here ω₀ denotes the resonant frequency between excited state |e〉 and ground state |1〉. ω_c and ω_p are frequencies of the cavity and the input state, respectively. g is the atom-cavity coupling strength. γ is the atomic decay rate.

Suppose the initial input coherent optical pulse is prepared in the state |α〉 and the atom is prepared in a superposition state

\frac{1}{\sqrt{2}} (| 0 〉 + | 1 〉)

. The evolution of the whole state with the cavity-assisted transformation can be described as:

\frac{1}{\sqrt{2}} (| 0 〉 + | 1 〉) {| α 〉}_{in} \to \frac{1}{\sqrt{2}} (| 0 〉 {| α e^{i θ_{0}} 〉}_{out} + | 1 〉 {| α e^{i θ_{1}} 〉}_{out}),

(3)

here θ_i = arg(r_i) (i = 0, 1) are controlled by ω₀, ω_c, ω_p and g in the case of low-Q cavity (κ ≫ γ). Obviously, with the input-output process, a phase shift corresponds to the atomic state is generated on the output coherent state. By adjusting

ω_{0} - ω_{p} = ω_{c} - ω_{p} = \frac{κ}{2}

and

g = \frac{κ}{\sqrt{2}}

in the atom-cavity intermediate coupling region, we could get huge nonlinearity phase shifts

θ_{1} ≃ \frac{π}{2}

and

θ_{0} ≃ - \frac{π}{2}

, respectively [61

F. Mei, Y. F. Yu, X. L. Feng, Z. M. Zhang, and C. H. Oh, “Quantum entanglement distribution with hybrid parity gate,” Phys. Rev. A 82, 052315 (2010). [CrossRef]

3. Atomic entanglement purification using coherent input-output process in low-Q cavity

Now we start to explain our atomic EPP for bit-flip errors using the coherent state input-output process by working in low-Q cavity in the atom-cavity intermediate coupling region. The principle is shown in Fig. 2(a).

Fig. 2 (a) Schematic diagram showing the principle of the atomic EPP based on the coherent state input-output process in low-Q cavity QED system. BS denotes the 50:50 beam splitter, which transforms |α〉 |β〉 to

| \frac{α - β}{\sqrt{2}} 〉 | \frac{α + β}{\sqrt{2}} 〉

. D_i (i=1,2,3 and 4) is the detector on the ith output port. Delay is time delay setup. The two devices surrounded by the dashed line are two two-qubit parity check modules (PCMs) constructed by Alice and Bob, respectively. (b) The fidelity of the present EPP in each iteration round by iterating the entanglement purification process N (N = 1,2,3 and 4) rounds. F is the initial fidelity of the state |ϕ⁺〉. It is obvious to see that the new fidelity is large than F when

F > \frac{1}{2}

, altered with the increment of N, and approximately approaches to 1 when N ≥ 4.

Suppose the initial mixed state ensemble shared by two remote parties Alice and Bob can be described as

ρ_{a b} = F {| ϕ^{+} 〉}_{a b} 〈 ϕ^{+} | + (1 - F) {| ψ^{+} 〉}_{a b} 〈 ψ^{+} |,

(4)

here

{| ϕ^{+} 〉}_{a b} = \frac{1}{\sqrt{2}} ({| 0 〉}_{a} {| 0 〉}_{b} + {| 1 〉}_{a} {| 1 〉}_{b})

and

{| ψ^{+} 〉}_{a b} = \frac{1}{\sqrt{2}} ({| 0 〉}_{a} {| 1 〉}_{b} + {| 1 〉}_{a} {| 0 〉}_{b})

. The subscripts a and b represent the single atoms confined in separate cavities A and B owned by Alice and Bob, respectively. F is the initial fidelity of the state |ϕ⁺〉. By selecting two pairs of entangled atoms randomly, the four atoms are in the state |ϕ⁺〉_a1b1|ϕ⁺〉_a2b2 with a probability of F², in the state |ϕ⁺〉_a1b1|ψ⁺〉_a2b2 and |ψ⁺〉_a1b1|ϕ⁺〉_a2b2 with a probability of F(1 − F), and in the state |ψ⁺〉_a1b1|ψ⁺〉_a2b2 with a probability of (1 − F)². The two devices surrounded by dashed line in Fig. 2(a) are two two-qubit PCMs constructed by Alice and Bob, respectively. In the entanglement purification process, Alice first prepares a coherent optical pulse in the state

| \sqrt{2} α 〉

and lets it pass through a 50 : 50 beam splitter (BS) to generate |α〉₁ = |α〉₂ = |α〉. Then the coherent state |α〉₂ interacts with cavity A₁ and cavity A₂ sequentially as quantum channel. We have adjusted the parameters of the QED systems as

ω_{0} - ω_{p} = ω_{c} - ω_{p} = \frac{κ}{2}

and

g = \frac{κ}{\sqrt{2}}

in the case of low-Q cavity beforehand. After that, the coherent states in two arms of the PCM |α〉₁ and |α〉₂′ (output from cavity A₂) interfered with each other on another 50 : 50 BS and be detected by the intensity on output ports 1 and 2. Bob does the same operation as Alice simultaneously. By comparing the detected results with classical communication, Alice and Bob can distill a high-fidelity atomic entangled state in a deterministic probability.

For example, if the four atoms are in the state |ϕ⁺〉_a1b1|ϕ⁺〉_a2b2, the evolution of the whole system is

\begin{array}{l} {| ϕ^{+} 〉}_{a 1 b 1} {| ϕ^{+} 〉}_{a 2 b 2} {| \sqrt{2} α 〉}_{a} {| \sqrt{2} α 〉}_{b} \\ \to \frac{1}{2} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2}) {| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} {| - \sqrt{2} α 〉}_{3} {| 0 〉}_{4} \\ + \frac{1}{2} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2}) {| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4} . \end{array}

(5)

If Alice and Bob get the result

{| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} {| - \sqrt{2} α 〉}_{3} {| 0 〉}_{4}

by detecting the outgoing coherent states, the atomic entangled state will collapse to

\frac{1}{\sqrt{2}} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2})

. The two parties can make X basis measurement on both a2 and b2 atoms with the help of external classical field, respectively, then the original state |ϕ⁺〉 can be recovered. In detail, if both the measurement results on Alice’s and Bob’s sides are |+ X〉 or |− X〉, they will get the original state |ϕ⁺〉 on a1 and b1 atoms. If the results are not in correspondence with each other, they need a phase-flip operation to recover the original state. On the contrary, if Alice and Bob get the result

{| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4}

, they will obtain

\frac{1}{\sqrt{2}} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2})

, which can be used to distill |ϕ⁺〉 with X basis measurement and phase-flip operation in a similar way.

Similarly, the evolution of the other three cases can be described by:

\begin{array}{l} {| ϕ^{+} 〉}_{a 1 b 1} {| ψ^{+} 〉}_{a 2 b 2} {| \sqrt{2} α 〉}_{a} {| \sqrt{2} α 〉}_{b} \\ \to \frac{1}{2} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2} {| 0 〉}_{b 2}) {| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4} \\ + \frac{1}{2} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 0 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 1 〉}_{b 2}) {| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| - \sqrt{2} α 〉}_{3} {| 0 〉}_{4}, \end{array}

(6)

\begin{array}{l} {| ψ^{+} 〉}_{a 1 b 1} {| ϕ^{+} 〉}_{a 2 b 2} {| \sqrt{2} α 〉}_{a} {| \sqrt{2} α 〉}_{b} \\ \to \frac{1}{2} ({| 0 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2} + {| 1 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2}) {| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4} \\ + \frac{1}{2} ({| 0 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2}) {| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| - \sqrt{2} α 〉}_{3} {| 0 〉}_{4}, \end{array}

(7)

and

\begin{array}{l} {| ψ^{+} 〉}_{a 1 b 1} {| ψ^{+} 〉}_{a 2 b 2} {| \sqrt{2} α 〉}_{a} {| \sqrt{2} α 〉}_{b} \\ \to \frac{1}{2} ({| 0 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 0 〉}_{b 2}) {| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} {| - \sqrt{2} α 〉}_{3} {| 0 〉}_{4} \\ + \frac{1}{2} ({| 0 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2} {| 0 〉}_{b 2} + {| 1 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} {| 1 〉}_{b 2}) {| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4} . \end{array}

(8)

Alice and Bob discard all the items only if both their detected results on outgoing coherent states are

| - \sqrt{2} α 〉 | 0 〉

(or

| 0 〉 | \sqrt{2} α 〉

). Finally, four atoms in the state |ψ⁺〉_a1b1|ψ⁺〉_a2b2 cannot be discarded as Alice and Bob can not distinguish the two cases that a₁b₁ and a₂b₂ both contain bit-flip errors. So the two cases |ϕ⁺〉_a1b1|ϕ⁺〉_a2b2 and |ψ⁺〉_a1b1|ψ⁺〉_a2b2 are preserved with probabilities of F² and (1 − F)², respectively. That is, based on the post-selection principle according to the detected results on outgoing coherent states, Alice and Bob can eventually preserved a new mixed state ensemble with a fidelity F′ = F²/[F² + (1 − F)²], which is larger than F when

F > \frac{1}{2}

Up to now, we have briefly introduced our atomic EPP for bit-flip errors. In this protocol, the PCM is designed to replace the CNOT gate in original EPP [15

], it can be used to distinguish |0〉|0〉 and |1〉|1〉 states from |0〉|1〉 and |1〉|0〉 states of the two atoms. The advantage of our PCM is the use of coherent state pulse from standard stabilized laser source as quantum channel, which do not need the atoms to interact with each other directly. Thus, this protocol is quantum nondestructive like the protocols with cross-Kerr nonlinearity [20

]. At the output ports, the detection is based on the intensity of outgoing coherent states, after two coherent states in two arms of the PCM interfered with each other on a 50:50 BS. Ideally, any response of the detector indicates that the coherent state is

| \pm \sqrt{2} α 〉

, which is different from |0〉, and the error probability of the coherent states comparison is P_error = exp(−2|α|²). Such detector could be simple photodiode.

Based on our PCM, the remaining entangled atoms can be expected to obtain a higher fidelity. That is, by iterating the purification process for the preserved atom pairs, the entanglement fidelity can be improved further. We calculate the fidelity in each iteration round by iterating the entanglement purification process N rounds and numerically simulated them in Fig. 2(b). We can conclude that it is altered with the increment of N and approximately approaches to 1 when N ≥ 4. Phase-flip errors may also occur during the interaction with environment noise. As discussed in Ref. [17

J. W. Pan, C. Simon, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001). [CrossRef]

, 18

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003). [CrossRef] [PubMed]

], phase-flip errors can be transformed into bit-flip errors with Hadamard operations. So we only discuss bit-flip errors entanglement purification here and the phase-flip errors can be purified in a next round. That is, this atomic EPP can purify an arbitrary mixed state ensemble.

4. Atomic entanglement concentration using coherent input-output process in low-Q cavity

The scheme in Fig. 2(a) can further be employed for atomic entanglement concentration, based on the discussion in Ref. [31

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001). [CrossRef]

]. Suppose the two pairs of unknown less-entangled pure state atoms shared by Alice and Bob can be described as:

\begin{array}{l} {| ϕ^{+} 〉}_{a 1 b 1} = m {| 0 〉}_{a 1} {| 0 〉}_{b 1} + n {| 1 〉}_{a 1} {| 1 〉}_{b 1}, \\ {| ϕ^{+} 〉}_{a 2 b 2} = m {| 0 〉}_{a 2} {| 0 〉}_{b 2} + n {| 1 〉}_{a 2} {| 1 〉}_{b 2}, \end{array}

(9)

here a and b represent the single atoms owned by Alice and Bob, respectively. |m|² + |n|² = 1. Our goal is to generate the original maximally entangled state |ϕ⁺〉 between Alice and Bob. In the entanglement concentration process, Alice and Bob prepare two coherent optical pulses in the state

| \sqrt{2} α 〉

to input the PCMs, respectively, and the evolution of the whole state is

\begin{array}{l} {| ϕ^{+} 〉}_{a 1 b 1} {| ϕ^{+} 〉}_{a 2 b 2} {| \sqrt{2} α 〉}_{a} {| \sqrt{2} α 〉}_{b} \\ \to (m^{2} {| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2} + n^{2} {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2}) {| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} {| - \sqrt{2} α 〉}_{3} {| 0 〉}_{4} \\ + m n ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2}) {| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4} . \end{array}

(10)

Alice and Bob can compare their detected results on outgoing coherent states with classical communication and only keep the instance corresponding to

{| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} {| 0 〉}_{3} {| \sqrt{2} α 〉}_{4}

. Then the four atoms are projected to the state

\frac{1}{\sqrt{2}} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} {| 1 〉}_{b 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2} {| 0 〉}_{b 2})

with a success probability of 2|mn|². After that, Alice and Bob can perform X basis measurement on a₂ and b₂ atoms, respectively. If both the measurement results on Alice’s and Bob’s sides are |+ X〉 or |− X〉, they will get the original state |ϕ⁺〉 on a1 and b1 atoms. If the results are not in correspondence with each other, they need a phase-flip operation to recover the desired state. In this method, the two parties do not need to know the coefficients of the less-entangled states beforehand [31

T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001). [CrossRef]

Moreover, if Alice know the coefficients m and n beforehand, the ECP can be further simplified with an ancillary particle, by exploiting a previously known protocol in Ref. [33

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012). [CrossRef]

]. The principle of the modified ECP is shown in Fig. 3(a). We can introduce an ancillary atom a2 confined in a low-Q cavity A2 on Alice’s side, whose parameters are same as the atom a1 confined in cavity A1 and the initial state is prepared in

{| ϕ 〉}_{a 2} = m {| 1 〉}_{a 2} + n {| 0 〉}_{a 2} .

(11)

Based on the parity check operation on a1 and a2 atoms, the evolution of the whole state is

\begin{array}{l} {| ϕ^{+} 〉}_{a 1 b 1} {| ϕ 〉}_{a 2} | \sqrt{2} α 〉 & \to (m^{2} {| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 1 〉}_{a 2} + n^{2} {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 0 〉}_{a 2}) {| 0 〉}_{1} {| \sqrt{2} α 〉}_{2} \\ + m n ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2}) {| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2} . \end{array}

(12)

By choosing the detected result is

{| - \sqrt{2} α 〉}_{1} {| 0 〉}_{2}

, the initial state is projected into a three-atom maximally entangled state

\frac{1}{\sqrt{2}} ({| 0 〉}_{a 1} {| 0 〉}_{b 1} {| 0 〉}_{a 2} + {| 1 〉}_{a 1} {| 1 〉}_{b 1} {| 1 〉}_{a 2})

with a success probability of 2|mn|². Then Alice can perform X basis measurement on the ancillary atom a2. If the measurement result is |+ X〉, Alice and Bob will get the original state |ϕ⁺〉 on a1 and b1 atoms. If the measurement result is |− X〉, they need a phase-flip operation to recover the original state. On the contrary, if the detected result is

{| 0 〉}_{1} {| \sqrt{2} α 〉}_{2}

, Alice and Bob will obtain a less-entangled pure state

\frac{1}{\sqrt{m^{4} + n^{4}}} (m^{2} {| 0 〉}_{a 1} {| 0 〉}_{b 1} + n^{2} {| 1 〉}_{a 1} {| 1 〉}_{b 1})

after performing X basis measurement and phase-flip operation, which can be seen as another initial state and be concentrated in the next round by preparing a new ancillary atomic state. That is, Alice can iterate the concentration process several rounds to improve the success probability further. We have calculated the success probability in each iteration round and shown them in Fig. 3(b), which is essentially the same as figure 4 in Ref. [33

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012). [CrossRef]

]. In this modified ECP, Alice need to know the information about the initial state beforehand. The PCM and X basis measurement are only needed on Alice’s side, there is no need to exchange any measurement results with Bob. This greatly simplifies the complication of classical communication. The parity check operation does not consume any entanglement resource, if we define the yield of maximally entangled states Y as the ratio of the number of maximally entangled atom pairs and the number of initial less-entangled atom pairs, this modified ECP can obtain an optimal one.

Fig. 3 (a) Schematic diagram showing the principle of the modified ECP based on coherent state input-output process in low-Q cavity. a2 is an ancillary atom confined in a low-Q cavity A2, whose parameters are same as the atom a1 confined in cavity A1. BS denotes a 50:50 beam splitter, which transforms |α〉 |β〉 to

| \frac{α - β}{\sqrt{2}} 〉 | \frac{α + β}{\sqrt{2}} 〉

D_i (i = 1, 2) is detector on the ith output port. Delay is time delay setup. (b) The success probability of the modified ECP in each iteration round by iterating the entanglement concentration process N (N=1,2,3 and 4) rounds. m is the coefficient of the initial state. It is obvious to see that the success probability is relatively high when N ≥ 4.

Fig. 4 The success probability of the present EPP and ECP with respect to

\frac{γ}{g} = 0.05

, 0.1 0.15 and 0.2 when η_D = 0.7,

η = \frac{1}{3}

, α = 3, the transmission rate through other optical components is 0.9. Here we only perform the entanglement purification and concentration process one round. The success probability of entanglement purification in an ideal condition is F² +(1 − F)². After considering the error probabilities P_e1 and P_e2, the success probability will decrease as shown in Fig. 4(a). The success probability of entanglement concentration is 2|αβ|², which will decrease such as in Fig. 4(b). As discuss above, the error probabilities P_e1 and P_e2 do not affect the fidelity of the mixed state based on the post-selection principle in the EPP, and the success probabilities in the ECP can be improved further by iterating the modified protocol several rounds.

5. Experimental feasibility

The key element of the EPP and ECP is the PCM constructed by low-Q cavities with coherent optical pulse. According to the discussions above, the detuning of the input coherent state with respect to the atomic resonance frequency and the cavity mode are required to be the same as

ω_{0} - ω_{p} = ω_{c} - ω_{p} = \frac{κ}{2}

. Thus, the frequency of the input coherent state ω_p should be set to satisfy this condition when the atom and cavity mode have been adjusted in resonant interaction. Suppose the cavities in our protocols are Fabry-Perot (F-P) cavities, then the atom-cavity coupling strength depends on the atomic position, which can be described as

g (\vec{r}) = g_{0} cos (k_{z} c) exp [- r_{⊥}^{2} / ω_{c}^{2}],

(13)

here g₀ is the peak coupling strength, r_⊥ is the radial distance of the atoms with respect to the cavity axis, ω_c and k_c are the width and the wave vector of the Gaussian cavity mode, respectively. In 2005, Nuβmann et al.[63

S. Nuβmann, M. Hijlkema, B. Weber, F. Rohde, G. Rempe, and A. Kuhn, “Submicron positioning of single atoms in a microcavity,” Phys. Rev. Lett. 95, 173602 (2005). [CrossRef]

] reported their experiment to precisely control and adjust individual ultracold ⁸⁵Rb atoms coupled to a high-finesse optical cavity. In 2007, Fortier et al.[64

K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, “Deterministic loading of individual atoms to a high-finesse optical cavity,” Phys. Rev. Lett. 98, 233601 (2007). [CrossRef] [PubMed]

] realized deterministic loading methods of single ⁸⁷Rb atoms into the cavity by incorporating a deterministic loaded atom conveyor. Colombe et al.[65

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef] [PubMed]

] reported their experiment which could realize strong atom-field coupling for Bose-Einstein condensates (BEC) in a fiber-based F-P cavity on a chip. The ⁸⁷Rb BEC can be positioned deterministically anywhere within the cavity and localized entirely within a single antinode of the standing-wave cavity field. These excellent experiments have proved that we can manipulate the position of a single atom and tune the atom-cavity coupling strength, and then control the reflectivity of the input coherent state to obtain the desired phase shifts. In our proposed protocols, if r_⊥ = 0, the atom-cavity coupling strength g = g₀ cos(k_zc) should be matched with the cavity decay rate κ as

g = \frac{κ}{\sqrt{2}}

. Suppose the ⁸⁷Rb atom resonant frequency

ω_{0} = \frac{2 π c}{λ}

at λ= 780nm. Parameters of the cavity are same as in Ref. [65

] with a cavity length L = 38.6μm, cavity decay rate κ = 2π × 53MHz, peak coupling strength g₀ = 2π × 215MHz, finesse f = 37000 which correspond to a longitudinal mode number n = 99. We can estimate the appropriate atomic longitudinal coordinate

z = n \frac{λ}{2} + 173 n m

. As the input coherent optical pulse in the state

| \sqrt{2} α 〉

is split by a BS into two arms in the PCM, one of them has to interact with the atom-cavity coupled system twice to obtain the joint-state-dependent phase shift. The success of our protocols relies on the two-path interference of the coherent optical pulses on another BS. Thus, we should introduce the time-delay setup (optical fiber in a suitable length, for example) to keep the lengths of the two paths to be stable at subwavelength.

Furthermore, though our protocols can be realized in the context of low-Q cavities (κ ≫ γ), the value of the Q should not be too low for avoiding reducing the strength of the atom-field coupling. In order to guarantee a higher efficiency, the ultralow-Q cavity may not be suitable for our protocols, though the condition

g = \frac{κ}{\sqrt{2}}

may also be satisfied in this case. In the experiment, the Q value and the decay rate κ are closely related to the cavity finesse f. For the F-P cavity such as in Ref. [65

], we can set the coupling strength to peak coupling strength g₀, then decrease the cavity finesse to 4×10³ magnitude to satisfy

g = \frac{κ}{\sqrt{2}}

, and the relevant Q factor

Q = \frac{ω_{c}}{2 κ}

will decreases to 4 × 10⁵ magnitude. In Ref. [66

B. Dayan, A. S. Parkins, E. Takao Aoki, P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008). [CrossRef] [PubMed]

], Dayan et al. realized intermediate atom-cavity coupling in the micro-toroidal resonator (MTR) “bad cavity”. The Q value in this work is more than 10⁴ in the low-Q cavity limit (

κ ≫ \frac{g^{2}}{κ} ≫ γ

), which is also very suitable in our protocols. The requirement of the Q value in our protocols should be larger than 10². Moreover, we would like to point out that although the Q can be low, a high Q/V is usually required in cavity QED systems, in which V is the mode volume. So semiconductor systems may be better than atomic systems in some cavity QED based QIP for the mode volume is usually much smaller in semiconductor systems [59

, 60

There are still some imperfections in realistic applications. For example, small perturbations of the parameters of the cavity may lead to detuning of the cavity mode with respect to the atomic resonance frequency (ω_c ∼ ω₀). The variation of the atomic position in the cavity may lead to the mismatch of the coupling strength (

g ~ \frac{κ}{\sqrt{2}}

). These will slightly change the phase shifts θ₀ and θ₁ in the coherent state input-output process. We detect the intensity of outgoing coherent state instead of the direct homodyne measurement to overcome this imperfection. Experimental schemes, such as optical lattice [67

J. A. Sauer, K. M. Fortier, M. S. Chang, C. D. Hamley, and M. S. Chapman, “Cavity QED with optically transported atoms,” Phys. Rev. A 69, 051804(R) (2004). [CrossRef]

] or electromagnetic field [68

A. B. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Coupling a single atomic quantum bit to a high finesse optical cavity,” Phys. Rev. Lett. 89, 103001(2002). [CrossRef] [PubMed]

] can be used to fix the atoms stably in the cavities. The photon loss occurs due to the absorption and scattering of the cavity mirror and the fiber. After considering the photon loss rate η and the efficiency of the detectors η_D, the error probability coming from the detections can be defined to P_e1 = exp[−2η_D(1 − η)|α|²]. Moreover, the atomic decay rate γ may also effect the success probability of the protocols. The error probability caused by atomic decay rate in the intermediate coupling region in low-Q cavity is

P_{e 2} = \frac{γ}{g}

[55

J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]

,56

Q. Chen and M. Feng, “Quantum gating on neutral atoms in low-Q cavities by a single-photon input-output process,” Phys. Rev. A 79 064304 (2009). [CrossRef]

,62

F. Mei, Y. F. Yu, X. L. Feng, S. L. Zhu, and Z. M. Zhang, “Optical quantum computation with cavities in the intermediate coupling region,” Europhys. Lett. 91, 10001 (2010). [CrossRef]

]. Suppose η_D = 0.7,

η = \frac{1}{3}

, α = 3, the transmission rate through other optical components is 0.9, then we can numerically simulated the success probability in the present EPP and ECP with respect to

\frac{γ}{g}

in Fig. 4. Here we only perform the entanglement purification and concentration process one round. In our EPP, the success probability in an ideal condition is F² + (1 − F)². After considering the error probabilities P_e1 and P_e2, the success probability will decrease as shown in Fig. 4(a). However, is does not affect the fidelity of the mixed state based on the post-selection principle. In our ECP, the success probability 2|mn|² will decrease such as shown in Fig. 4(b), while we can utilize our modified protocol by introducing ancillary atoms in cavity QED to improve the success probability further.

Our protocols are based on the PCM by working in low-Q cavity in the atom-cavity intermediate coupling region, which is considerably different from the high-Q cavity and strong coupling cases [36

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005). [CrossRef]

, 38

Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006). [CrossRef]

, 40

C. D. Ogden, M. Paternostro, and M. S. Kim, “Concentration and purification of entanglement for qubit systems with ancillary cavity fields,” Phys. Rev. A 75, 042325 (2007). [CrossRef]

]. In the previous works [36

M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005). [CrossRef]

, 40

C. D. Ogden, M. Paternostro, and M. S. Kim, “Concentration and purification of entanglement for qubit systems with ancillary cavity fields,” Phys. Rev. A 75, 042325 (2007). [CrossRef]

], the atom is used as the flying qubit, where in our protocol, the atom is used as stationary qubit. Meanwhile, the sophisticated single-photon source and single-photon detector have been successfully avoided. The present protocols can further be extended to multi-atom Greenberger-Horne-Zeilinger (GHZ) state or W state purification or concentration, following some ideas in multipartite EPP and ECP for entangled photonic systems.

6. Summary

In summary, we have proposed an atomic entanglement purification protocol based on the coherent state input-output process in low-Q cavity QED system. The information of the entangled states are encoded in three-level configured single atoms confined inside separate one-side optical micro-cavities. Through the coherent state input-output process, remote parties can construct the two-qubit PCM for the local atomic states, respectively. And then they can utilize the PCMs to distill a high-fidelity atomic entangled ensemble from a initial mixed state ensemble nonlocally. The scheme can also be used for atomic entanglement concentration, in which the two remote parties can concentrate the atom pairs in less-entangled pure states efficiently. Also by exploiting the PCM, we describe a modified ECP by introducing the ancillary single atoms in cavity QED. The proposed protocols only require the coherent states as quantum channel, and the detection is based on the intensity of outgoing coherent states, which can be easily achieved with current technologies. Meanwhile, the coherent state input-output process is robust and scalable by working in low-Q cavity in the atom-cavity intermediate coupling region. Thus, the atomic EPP and ECP may be widely used in large-scaled and solid-based quantum repeater and QIP protocols. Moreover, the PCM is exploited instead of the CNOT gates, which could be widely used in entanglement generation, Bell-state analysis, quantum cloning, and so on.

Acknowledgments

This work is supported by the National Fundamental Research Program Grant No. 2010CB923202, Specialized Research Fund for the Doctoral Program of Education Ministry of China No. 20090005120008, the Fundamental Research Funds for the Central Universities, China National Natural Science Foundation Grant Nos. 61177085 and 61205117.

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8.	X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008). [CrossRef]
9.	G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002). [CrossRef]
10.	F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A 68, 042317 (2003). [CrossRef]
11.	C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005). [CrossRef]
12.	X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A 74, 054302 (2006). [CrossRef]
13.	H, J. Briegel, W. Dr, J. I. Cirac, and P. Zoller, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. Lett. 81, 5932–5935 (1998).
14.	L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef] [PubMed]
15.	C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996). [CrossRef] [PubMed]
16.	D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996). [CrossRef] [PubMed]
17.	J. W. Pan, C. Simon, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London) 410, 1067–1070 (2001). [CrossRef]
18.	J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003). [CrossRef] [PubMed]
19.	C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89, 257901 (2002). [CrossRef] [PubMed]
20.	Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008). [CrossRef]
21.	Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044305 (2010). [CrossRef]
22.	X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010). [CrossRef]
23.	F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A 83, 062316 (2011). [CrossRef]
24.	C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 0988–1002 (2011).
25.	M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075–R4078 (1998). [CrossRef]
26.	F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011). [CrossRef]
27.	C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996). [CrossRef] [PubMed]
28.	S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999). [CrossRef]
29.	B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000). [CrossRef]
30.	Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001). [CrossRef]
31.	T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001). [CrossRef]
32.	Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008). [CrossRef]
33.	Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012). [CrossRef]
34.	F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012). [CrossRef]
35.	M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A 71, 012308 (2005). [CrossRef]
36.	M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A 71, 044302 (2005). [CrossRef]
37.	X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A 71, 064301 (2005). [CrossRef]
38.	Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A 73, 014303 (2006). [CrossRef]
39.	R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature 443, 838–841 (2006). [CrossRef] [PubMed]
40.	C. D. Ogden, M. Paternostro, and M. S. Kim, “Concentration and purification of entanglement for qubit systems with ancillary cavity fields,” Phys. Rev. A 75, 042325 (2007). [CrossRef]
41.	C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011). [CrossRef]
42.	C. Wang, “Efficient entanglement concentration for partially entangled electrons using a quantum-dot and microcavity coupled system,” Phys. Rev. A 86, 012323 (2012). [CrossRef]
43.	Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A 86, 034305 (2012). [CrossRef]
44.	J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565 (2001). [CrossRef]
45.	S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87, 037902 (2001). [CrossRef] [PubMed]
46.	L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004). [CrossRef] [PubMed]
47.	Y. F. Xiao, X. M. Lin, J. Gao, Y. Yang, Z. F. Han, and G. C. Guo, “Realizing quantum controlled phase flip through cavity QED,” Phys. Rev. A 70, 042314 (2004). [CrossRef]
48.	K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87 (2005). [CrossRef]
49.	J. Cho and H. W. Lee, “Generation of atomic cluster states through the cavity input-output process,” Phys. Rev. Lett. 95, 160501 (2005). [CrossRef] [PubMed]
50.	J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darqui, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature 440, 779–782 (2006). [CrossRef] [PubMed]
51.	H. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and R. Gerhard, “A single-photon server with just one atom,” Nat. Phys. 3, 253–255 (2007). [CrossRef]
52.	T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488–490 (2007). [CrossRef] [PubMed]
53.	F. Mei, M. Feng, Y. F. Yu, and Z. M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009). [CrossRef]
54.	C. Y. Hu, A. Young, J. L. OBrien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: Applications to entangling remote spins via a single photon,” Phys. Rev. B 78, 085307 (2008). [CrossRef]
55.	J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]
56.	Q. Chen and M. Feng, “Quantum gating on neutral atoms in low-Q cavities by a single-photon input-output process,” Phys. Rev. A 79 064304 (2009). [CrossRef]
57.	Q. Chen and M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A 82 052329 (2010). [CrossRef]
58.	J. J. Chen, J. H. An, M. Feng, and G. Liu, “Teleportation of an arbitrary multipartite state via photonic Faraday rotation,” J. Phys. B 43, 095505 (2010). [CrossRef]
59.	P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater using bright coherent light,” Phys. Rev. Lett. 96, 240501 (2006). [CrossRef] [PubMed]
60.	T. D. Ladd, P. van Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys. 8, 184 (2006). [CrossRef]
61.	F. Mei, Y. F. Yu, X. L. Feng, Z. M. Zhang, and C. H. Oh, “Quantum entanglement distribution with hybrid parity gate,” Phys. Rev. A 82, 052315 (2010). [CrossRef]
62.	F. Mei, Y. F. Yu, X. L. Feng, S. L. Zhu, and Z. M. Zhang, “Optical quantum computation with cavities in the intermediate coupling region,” Europhys. Lett. 91, 10001 (2010). [CrossRef]
63.	S. Nuβmann, M. Hijlkema, B. Weber, F. Rohde, G. Rempe, and A. Kuhn, “Submicron positioning of single atoms in a microcavity,” Phys. Rev. Lett. 95, 173602 (2005). [CrossRef]
64.	K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, “Deterministic loading of individual atoms to a high-finesse optical cavity,” Phys. Rev. Lett. 98, 233601 (2007). [CrossRef] [PubMed]
65.	Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef] [PubMed]
66.	B. Dayan, A. S. Parkins, E. Takao Aoki, P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008). [CrossRef] [PubMed]
67.	J. A. Sauer, K. M. Fortier, M. S. Chang, C. D. Hamley, and M. S. Chapman, “Cavity QED with optically transported atoms,” Phys. Rev. A 69, 051804(R) (2004). [CrossRef]
68.	A. B. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Coupling a single atomic quantum bit to a high finesse optical cavity,” Phys. Rev. Lett. 89, 103001(2002). [CrossRef] [PubMed]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.5568) Quantum optics : Quantum cryptography
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: December 3, 2012
Revised Manuscript: January 17, 2013
Manuscript Accepted: January 18, 2013
Published: February 11, 2013

Citation
Cong Cao, Chuan Wang, Ling-yan He, and Ru Zhang, "Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime," Opt. Express 21, 4093-4105 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4093

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References

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993). [CrossRef] [PubMed]
C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett.69, 2881–2884 (1992). [CrossRef] [PubMed]
M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A59, 1829–1834 (1999). [CrossRef]
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L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A69, 052307 (2004). [CrossRef]
A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett.67, 661–663 (1991). [CrossRef] [PubMed]
C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys.Rev. Lett.68, 557–559 (1992). [CrossRef] [PubMed]
X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A78, 022321 (2008). [CrossRef]
G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A65, 032302 (2002). [CrossRef]
F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A68, 042317 (2003). [CrossRef]
C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A71, 044305 (2005). [CrossRef]
X. H. Li, F. G. Deng, and H. Y. Zhou, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. A74, 054302 (2006). [CrossRef]
H, J. Briegel, W. Dr, J. I. Cirac, and P. Zoller, “Improving the security of secure direct communication based on the secret transmitting order of particles,” Phys. Rev. Lett.81, 5932–5935 (1998).
L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414, 413–418 (2001). [CrossRef] [PubMed]
C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett.76, 722–725 (1996). [CrossRef] [PubMed]
D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett.77, 2818–2821 (1996). [CrossRef] [PubMed]
J. W. Pan, C. Simon, and A. Zellinger, “Entanglement purification for quantum communication,” Nature (London)410, 1067–1070 (2001). [CrossRef]
J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003). [CrossRef] [PubMed]
C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett.89, 257901 (2002). [CrossRef] [PubMed]
Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A77, 042308 (2008). [CrossRef]
Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A82, 044305 (2010). [CrossRef]
X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A82, 044304 (2010). [CrossRef]
F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A83, 062316 (2011). [CrossRef]
C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput.11, 0988–1002 (2011).
M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A57, R4075–R4078 (1998). [CrossRef]
F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A84, 052312 (2011). [CrossRef]
C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A53, 2046 (1996). [CrossRef] [PubMed]
S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A60, 194–197 (1999). [CrossRef]
B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A62, 054301 (2000). [CrossRef]
Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A64, 014301 (2001). [CrossRef]
T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A64, 012304 (2001). [CrossRef]
Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A77, 062325 (2008). [CrossRef]
Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A85, 012307 (2012). [CrossRef]
F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A85, 022311 (2012). [CrossRef]
M. Yang, W. Song, and Z. L. Cao, “Entanglement purification for arbitrary unknown ionic states via linear optics,” Phys. Rev. A71, 012308 (2005). [CrossRef]
M. Yang, Y. Zhao, W. Song, and Z. L. Cao, “Entanglement concentration for unknown atomic entangled states via entanglement swapping,” Phys. Rev. A71, 044302 (2005). [CrossRef]
X. L. Feng, L. C. Kwek, and C. H. Oh, “Electronic entanglement purification scheme enhanced by charge detections,” Phys. Rev. A71, 064301 (2005). [CrossRef]
Z. L. Cao, L. H. Zhang, and M. Yang, “Concentration for unknown atomic entangled states via cavity decay,” Phys. Rev. A73, 014303 (2006). [CrossRef]
R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, “Experimental purification of two-atom entanglement,” Nature443, 838–841 (2006). [CrossRef] [PubMed]
C. D. Ogden, M. Paternostro, and M. S. Kim, “Concentration and purification of entanglement for qubit systems with ancillary cavity fields,” Phys. Rev. A75, 042325 (2007). [CrossRef]
C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A84, 032307 (2011). [CrossRef]
C. Wang, “Efficient entanglement concentration for partially entangled electrons using a quantum-dot and microcavity coupled system,” Phys. Rev. A86, 012323 (2012). [CrossRef]
Z. H. Peng, J. Zou, X. J. Liu, Y. J. Xiao, and L. M. Kuang, “Atomic and photonic entanglement concentration via photonic Faraday rotation,” Phys. Rev. A86, 034305 (2012). [CrossRef]
J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys.73, 565 (2001). [CrossRef]
S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett.87, 037902 (2001). [CrossRef] [PubMed]
L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett.92, 127902 (2004). [CrossRef] [PubMed]
Y. F. Xiao, X. M. Lin, J. Gao, Y. Yang, Z. F. Han, and G. C. Guo, “Realizing quantum controlled phase flip through cavity QED,” Phys. Rev. A70, 042314 (2004). [CrossRef]
K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London)436, 87 (2005). [CrossRef]
J. Cho and H. W. Lee, “Generation of atomic cluster states through the cavity input-output process,” Phys. Rev. Lett.95, 160501 (2005). [CrossRef] [PubMed]
J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darqui, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature440, 779–782 (2006). [CrossRef] [PubMed]
H. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and R. Gerhard, “A single-photon server with just one atom,” Nat. Phys.3, 253–255 (2007). [CrossRef]
T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science317, 488–490 (2007). [CrossRef] [PubMed]
F. Mei, M. Feng, Y. F. Yu, and Z. M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A80, 042319 (2009). [CrossRef]
C. Y. Hu, A. Young, J. L. OBrien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: Applications to entangling remote spins via a single photon,” Phys. Rev. B78, 085307 (2008). [CrossRef]
J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A79, 032303 (2009). [CrossRef]
Q. Chen and M. Feng, “Quantum gating on neutral atoms in low-Q cavities by a single-photon input-output process,” Phys. Rev. A79064304 (2009). [CrossRef]
Q. Chen and M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A82052329 (2010). [CrossRef]
J. J. Chen, J. H. An, M. Feng, and G. Liu, “Teleportation of an arbitrary multipartite state via photonic Faraday rotation,” J. Phys. B43, 095505 (2010). [CrossRef]
P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater using bright coherent light,” Phys. Rev. Lett.96, 240501 (2006). [CrossRef] [PubMed]
T. D. Ladd, P. van Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New J. Phys.8, 184 (2006). [CrossRef]
F. Mei, Y. F. Yu, X. L. Feng, Z. M. Zhang, and C. H. Oh, “Quantum entanglement distribution with hybrid parity gate,” Phys. Rev. A82, 052315 (2010). [CrossRef]
F. Mei, Y. F. Yu, X. L. Feng, S. L. Zhu, and Z. M. Zhang, “Optical quantum computation with cavities in the intermediate coupling region,” Europhys. Lett.91, 10001 (2010). [CrossRef]
S. Nuβmann, M. Hijlkema, B. Weber, F. Rohde, G. Rempe, and A. Kuhn, “Submicron positioning of single atoms in a microcavity,” Phys. Rev. Lett.95, 173602 (2005). [CrossRef]
K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, “Deterministic loading of individual atoms to a high-finesse optical cavity,” Phys. Rev. Lett.98, 233601 (2007). [CrossRef] [PubMed]
Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature450, 272–276 (2007). [CrossRef] [PubMed]
B. Dayan, A. S. Parkins, E. Takao Aoki, P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science319, 1062–1065 (2008). [CrossRef] [PubMed]
J. A. Sauer, K. M. Fortier, M. S. Chang, C. D. Hamley, and M. S. Chapman, “Cavity QED with optically transported atoms,” Phys. Rev. A69, 051804(R) (2004). [CrossRef]
A. B. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Coupling a single atomic quantum bit to a high finesse optical cavity,” Phys. Rev. Lett.89, 103001(2002). [CrossRef] [PubMed]

Ahmadi, P.
An, J. H.
Auffeves, A.
Becher, C.
Bennett, C. H.
Bernstein, H. J.
Bertet, P.
Berthiaume, A.
Beugnon, J.
Birnbaum, K. M.
Blakestad, R. B.
Blatt, R.
Boca, A.
Boozer, A. D.
Bose, S.
Brassard, G.
Briegel, J.
Britton, J.
Browaeys, A.
Brune, M.
Buzek, V.
Cao, Z. L.
Chang, M. S.
Chapman, M. S.
Chen, J. J.
Chen, Q.
Cho, J.
Cirac, J. I.
Colombe, Y.
Crepeau, C.
Darqui, B.
Dayan, B.
Deng, F. G.
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