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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 4093–4105
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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.

© 2013 OSA

1. Introduction

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

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

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

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

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

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

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

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

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

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

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

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

].

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 79064304 (2009). [CrossRef]

], QIP in decoherence-free subspace [57

57. Q. Chen and M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A 82052329 (2010). [CrossRef]

], quantum teleportation [58

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

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

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

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

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.

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

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

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

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

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 ain, 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

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 ω0ωpγ2, 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
r1(ωp)=[i(ωcωp)κ2][i(ω0ωp)+γ2]+g2[i(ωcωp)+κ2][i(ω0ωp)+γ2]+g2,
(1)
and
r0(ωp)=i(ωcωp)κ2i(ωcωp)+κ2,
(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〉. ωc and ωp 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 12(|0+|1). The evolution of the whole state with the cavity-assisted transformation can be described as:
12(|0+|1)|αin12(|0|αeiθ0out+|1|αeiθ1out),
(3)
here θi = arg(ri) (i = 0, 1) are controlled by ω0, ωc, ωp and g in the case of low-Q cavity (κγ). Obviously, with the input-output process, a phase shift corresponds to the atomic state is generated on the output coherent state. By adjusting ω0ωp=ωcωp=κ2 and g=κ2 in the atom-cavity intermediate coupling region, we could get huge nonlinearity phase shifts θ1π2 and θ0π2, respectively [61

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 |αβ2|α+β2. Di (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>12, 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
ρab=F|ϕ+abϕ+|+(1F)|ψ+abψ+|,
(4)
here |ϕ+ab=12(|0a|0b+|1a|1b) and |ψ+ab=12(|0a|1b+|1a|0b). 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 |2α and lets it pass through a 50 : 50 beam splitter (BS) to generate |α1 = |α2 = |α〉. Then the coherent state |α2 interacts with cavity A1 and cavity A2 sequentially as quantum channel. We have adjusted the parameters of the QED systems as ω0ωp=ωcωp=κ2 and g=κ2 in the case of low-Q cavity beforehand. After that, the coherent states in two arms of the PCM |α1 and |α2′ (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
|ϕ+a1b1|ϕ+a2b2|2αa|2αb12(|0a1|0b1|0a2|0b2+|1a1|1b1|1a2|1b2)|2α1|02|2α3|04+12(|0a1|0b1|1a2|1b2+|1a1|1b1|0a2|0b2)|01|2α2|03|2α4.
(5)
If Alice and Bob get the result |2α1|02|2α3|04 by detecting the outgoing coherent states, the atomic entangled state will collapse to 12(|0a1|0b1|0a2|0b2+|1a1|1b1|1a2|1b2). 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 |01|2α2|03|2α4, they will obtain 12(|0a1|0b1|1a2|1b2+|1a1|1b1|0a2|0b2), 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:
|ϕ+a1b1|ψ+a2b2|2αa|2αb12(|0a1|0b1|0a2|1b2+|1a1|1b1|1a2|0b2)|2α1|02|03|2α4+12(|0a1|0b1|1a2|0b2+|1a1|1b1|0a2|1b2)|01|2α2|2α3|04,
(6)
|ψ+a1b1|ϕ+a2b2|2αa|2αb12(|0a1|1b1|0a2|0b2+|1a1|0b1|1a2|1b2)|2α1|02|03|2α4+12(|0a1|1b1|1a2|1b2+|1a1|0b1|0a2|0b2)|01|2α2|2α3|04,
(7)
and
|ψ+a1b1|ψ+a2b2|2αa|2αb12(|0a1|1b1|0a2|1b2+|1a1|0b1|1a2|0b2)|2α1|02|2α3|04+12(|0a1|1b1|1a2|0b2+|1a1|0b1|0a2|1b2)|01|2α2|03|2α4.
(8)

Alice and Bob discard all the items only if both their detected results on outgoing coherent states are |2α|0 (or |0|2α). 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>12.

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

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]

], 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

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]

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

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

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:
|ϕ+a1b1=m|0a1|0b1+n|1a1|1b1,|ϕ+a2b2=m|0a2|0b2+n|1a2|1b2,
(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 |2α to input the PCMs, respectively, and the evolution of the whole state is
|ϕ+a1b1|ϕ+a2b2|2αa|2αb(m2|0a1|0b1|0a2|0b2+n2|1a1|1b1|1a2|1b2)|2α1|02|2α3|04+mn(|0a1|0b1|1a2|1b2+|1a1|1b1|0a2|0b2)|01|2α2|03|2α4.
(10)
Alice and Bob can compare their detected results on outgoing coherent states with classical communication and only keep the instance corresponding to |01|2α2|03|2α4. Then the four atoms are projected to the state 12(|0a1|0b1|1a2|1b2+|1a1|1b1|0a2|0b2) 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

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]

].

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 |αβ2|α+β2Di (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 γg=0.05, 0.1 0.15 and 0.2 when ηD = 0.7, η=13, 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 Pe1 and Pe2, 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 Pe1 and Pe2 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 ω0ωp=ωcωp=κ2. Thus, the frequency of the input coherent state ωp 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(r)=g0cos(kzc)exp[r2/ωc2],
(13)
here g0 is the peak coupling strength, r is the radial distance of the atoms with respect to the cavity axis, ωc and kc are the width and the wave vector of the Gaussian cavity mode, respectively. In 2005, Nuβmann et al.[63

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

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

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(kzc) should be matched with the cavity decay rate κ as g=κ2. Suppose the 87Rb atom resonant frequency ω0=2πcλ at λ= 780nm. Parameters of the cavity are same as in Ref. [65

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]

] 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=nλ2+173nm. As the input coherent optical pulse in the state |2α 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.

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 (ωcω0). The variation of the atomic position in the cavity may lead to the mismatch of the coupling strength ( g~κ2). 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

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

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 ηD, the error probability coming from the detections can be defined to Pe1 = exp[−2ηD(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 Pe2=γg[55

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

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 79064304 (2009). [CrossRef]

,62

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 ηD = 0.7, η=13, 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 γg 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 Pe1 and Pe2, 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.

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.

References and links

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2.

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A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (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. A 69, 052307 (2004). [CrossRef]

6.

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

7.

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys.Rev. Lett. 68, 557–559 (1992). [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]

9.

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

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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]

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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]

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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).

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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]

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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]

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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]

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

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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]

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

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X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010). [CrossRef]

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

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

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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]

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

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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

  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]
  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]
  3. M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A59, 1829–1834 (1999). [CrossRef]
  4. A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A59, 162–168 (1999). [CrossRef]
  5. L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A69, 052307 (2004). [CrossRef]
  6. A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett.67, 661–663 (1991). [CrossRef] [PubMed]
  7. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys.Rev. Lett.68, 557–559 (1992). [CrossRef] [PubMed]
  8. 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]
  9. G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A65, 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. A68, 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. A71, 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. A74, 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,” Nature414, 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,” Nature423, 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. A77, 042308 (2008). [CrossRef]
  21. Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A82, 044305 (2010). [CrossRef]
  22. X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A82, 044304 (2010). [CrossRef]
  23. F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A83, 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. A57, R4075–R4078 (1998). [CrossRef]
  26. F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A84, 052312 (2011). [CrossRef]
  27. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A53, 2046 (1996). [CrossRef] [PubMed]
  28. S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A60, 194–197 (1999). [CrossRef]
  29. B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A62, 054301 (2000). [CrossRef]
  30. Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A64, 014301 (2001). [CrossRef]
  31. T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A64, 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. A77, 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. A85, 012307 (2012). [CrossRef]
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