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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 13440–13450

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Controllable entanglement preparations between atoms in spatially-separated cavities via quantum Zeno dynamics

Wen-An Li and Lian-Fu Wei  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 13440-13450 (2012)
http://dx.doi.org/10.1364/OE.20.013440


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Abstract

By using quantum Zeno dynamics, we propose a controllable approach to deterministically generate tripartite GHZ states for three atoms trapped in spatially separated cavities. The nearest-neighbored cavities are connected via optical fibers and the atoms trapped in two ends are tunably driven. The generation of the GHZ state can be implemented by only one step manipulation, and the EPR entanglement between the atoms in two ends can be further realized deterministically by Von Neumann measurement on the middle atom. Note that the duration of the quantum Zeno dynamics is controllable by switching on/off the applied external classical drivings and the desirable tripartite GHZ state will no longer evolve once it is generated. The robustness of the proposal is numerically demonstrated by considering various decoherence factors, including atomic spontaneous emissions, cavity decays and fiber photon leakages, etc. Our proposal can be directly generalized to generate multipartite entanglement by still driving the atoms in two ends.

© 2012 OSA

1. Introduction

Quantum entanglement, as a fundamental aspect in quantum mechanics, has occupied a central place in modern research because of its promise of enormous utility in quantum computing [1

H. K. Lo, S. Popescu, and T. Spiller, Introduction to Quantum Computation and Information (World Scientific, Singapore, 1997).

–3

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London) 404, 247–255 (2000). [CrossRef]

], cryptography [2

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

,4

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

], etc. Generally speaking, the more particles that can be entangled, the more clearly nonclassical effects are exhibited and the more useful the states are for quantum applications. Typically, tripartite GHZ state, first proposed by Greenberger, Horne, and Zeilinger, provides a possibility to test quantum mechanics against local hidden theory without inequality [5

D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990). [CrossRef]

] and has practical applications in e.g., quantum secrete sharing [6

M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999). [CrossRef]

].

Recently, many theoretical schemes [7

X. B. Zou, K. Pahlke, and W. Mathis, “Conditional generation of the Greenberger-Horne-Zeilinger state of four distant atoms via cavity decay,” Phys. Rev. A 68, 024302 (2003). [CrossRef]

–10

L. F. Wei, Y. X. Liu, and F. Nori, “Generation and control of Greenberger-Horne-Zeilinger entanglement in superconducting circuits,” Phys. Rev. Lett. 96, 246803 (2006). [CrossRef] [PubMed]

] have been proposed to generate multipartite GHZ states, and a series of experimental preparations [11

R. J. Nelson, D. G. Cory, and S. Lloyd, “Experimental demonstration of Greenberger-Horne-Zeilinger correlations using nuclear magnetic resonance,” Phys. Rev. A 61, 022106 (2000). [CrossRef]

–13

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature (London) 467, 570–573 (2010). [CrossRef]

] have already been realized. The physical systems utilized to these generations include superconducting circuits [10

L. F. Wei, Y. X. Liu, and F. Nori, “Generation and control of Greenberger-Horne-Zeilinger entanglement in superconducting circuits,” Phys. Rev. Lett. 96, 246803 (2006). [CrossRef] [PubMed]

, 13

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature (London) 467, 570–573 (2010). [CrossRef]

], trapped ions [12

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature (London) 438, 639–642 (2005). [CrossRef]

], and cavity QED systems, etc.. It is well-known that cavity QED, where atoms interact with quantized electromagnetic fields inside a cavity, is a useful platform to demonstrate fundamental quantum mechanics laws and for the implementation of quantum information processing [14

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565–582 (2001). [CrossRef]

]. Specifically, numerous proposals with cavity QED have been made for entangling atoms either inside a cavity [15

J. I. Cirac and P. Zoller, “Preparation of macroscopic superpositions in many-atom systems,” Phys. Rev. A 50, R2799–R2802 (1994). [CrossRef] [PubMed]

, 16

J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. 89, 237901 (2002). [CrossRef] [PubMed]

] or trapped individually in different cavities [17

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]

–21

A. S. Parkins and H. J. Kimble, “Position-momentum Einstein-Podolsky-Rosen state of distantly separated trapped atoms,” Phys. Rev. A 61, 052104 (2000). [CrossRef]

]. For example, Pellizzari [22

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997). [CrossRef]

] proposed an approach to realize the reliable transfer of quantum information between two atoms in distant cavities connected by an optical fiber. Based on this proposal, various schemes [23

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

–32

P.-B. Li and F.-L. Li, “Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system,” Opt. Express 19, 1207–1216 (2011) [CrossRef] [PubMed]

] have been proposed to realize the quantum manipulations of the atoms trapped in different cavities connected via optical fibers. However, these models either require strong cavity-fiber coupling [28

X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A 79, 052330 (2009). [CrossRef]

] (which is not easy to achieve in the usual experiment), or need many laser beams to implement the desirable manipulations [29

S. B. Zheng, “Generation of Greenberger-Horne-Zeilinger states for multiple atoms trapped in separated cavities,” Eur. Phys. J. D 54, 719–722 (2009). [CrossRef]

] (this increases the experimental complication), or is sensitive to the decays of cavity fields and fiber modes [30

A. Zheng and J. Liu, “Generation of an N-qubit Greenberger-Horne-Zeilinger state with distant atoms in bimodal cavities,” J. Phys. B: At. Mol. Opt. Phys. 44, 165501 (2011). [CrossRef]

] (which limits its scalability).

Here, by using quantum Zeno dynamics we propose an alternative approach to entangle the atoms trapped in the distant cavities connected by optical fibers. It is well-known that quantum Zeno effect is an interesting phenomenon in quantum mechanics. Due to this effect the quantum system remains in its initial state via frequently measurements. Facchi et al [33

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000). [CrossRef]

–35

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009). [CrossRef]

] showed that this effect does not necessarily freeze the dynamics. Instead, by frequently projecting onto a multidimensional subspace, the system could evolve away from its initial state, although it remains in the so-called “Zeno subspace” [33

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000). [CrossRef]

,36

P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, “Zeno dynamics yields ordinary constraints,” Phys. Rev. A 65, 012108 (2002). [CrossRef]

]. Moreover, without making use of projection operators and nonunitary dynamics, the quantum Zeno effect can also be expressed in terms of a continuous coupling between the system and detector [35

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009). [CrossRef]

]. Generally, the system and its continuously coupling detector can be governed by the total Hamiltonian HK = H + KHa, with H is for the quantum system investigated and the Ha describing the additional interaction with the detector, and K the coupling constant. In the limit K → ∞, the subsystem of interest is dominated by the evolution operator U(t) = limK→∞ exp(iKHat)UK(t) which takes the form U(t) = exp(−it ∑n PnHPn) [35

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009). [CrossRef]

]. Here, Pn is the eigenprojection of Ha = ∑n λnPn corresponding to the eigenvalue λn. As a consequence, the system-detector can be described by the evolution operator UK(t) ∼ exp(−iKHat)U(t) = exp[−i∑n(KλnPn + PnHPn)t]. This result is of great importance in view of practical applications of the quantum Zeno dynamics, such as to prepare various quantum states [37

A. Luis, “Quantum-state preparation and control via the Zeno effect,” Phys. Rev. A 63, 052112 (2001). [CrossRef]

, 38

X. B. Wang, J. Q. You, and F. Nori, “Quantum entanglement via two-qubit quantum Zeno dynamics,” Phys. Rev. A 77, 062339 (2008). [CrossRef]

] and to implement the quantum gates [39

A. Beige, D. Braun, B. Tregenna, and P. L. Knight, “Quantum computing using dissipation to remain in a decoherence-free subspace,” Phys. Rev. Lett. 85, 1762–1765 (2000). [CrossRef] [PubMed]

–42

J. D. Franson, T. B. Pittman, and B. C. Jacobs, “Zeno logic gates using microcavities,” J. Opt. Soc. Am. B 24, 209–213 (2007). [CrossRef]

].

Compared with previous protocols [28

X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A 79, 052330 (2009). [CrossRef]

–30

A. Zheng and J. Liu, “Generation of an N-qubit Greenberger-Horne-Zeilinger state with distant atoms in bimodal cavities,” J. Phys. B: At. Mol. Opt. Phys. 44, 165501 (2011). [CrossRef]

] for generating GHZ states by selective absorption and emission of photons, adiabatic passages, and dispersive interactions between the atoms and cavities, etc., our approach possesses the following advantages: (i) the expected entanglement can be established by only one step operation, (ii) it is robust with respect to parameter imprecision and atomic and fibers’ dissipations, (iii) under the Zeno condition, the cavity fields are not excited really and thus is insensitive to the decays of the cavities, (iv) the generalization to N-atom entanglement is direct, and no matter how many atoms are involved, two laser beams are enough to implement the generations.

The paper is organized as follows. In section 2, we present our generic approach by using quantum Zeno dynamics to implement the GHZ entanglement of the atoms trapped in three fiber-connected cavities. The validity of the used quantum Zeno dynamics is analyzed by numerical method in detail. The direct generalization for entangling the N atoms trapped individually in N cavity is provided in Sec. 3. Finally, in Sec. 4 we discuss the feasibility of our proposal and give our conclusions.

2. Generation of GHZ state of atoms trapped in different cavities by quantum Zeno dynamics

We consider the physical configuration shown in the Fig. 1, three Λ-type atoms are trapped in three distant optical cavities coupled by two short optical fibers. Each atom has one excited state |e〉 and two dipole-transition forbidden ground states |0〉, |1〉. The first and third atomic transitions |1〉 ↔ |e〉 are driven resonantly by classical lasers with the couplings coefficient Ω1 and Ω3, respectively. The other atomic transition is resonantly coupled to the corresponding cavity mode with coupling constant gi,r(l) (i = 1, 2, 3). The subscript r(l) denotes the right (left) circularly polarization. In the short fiber limit, (2Lv̄)/(2πc) ≪ 1, where L is the length of the fiber and v̄ is the decay rate of the cavity fields into a continuum of fiber modes [23

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

]. In the interaction picture, the Hamiltonian of the whole system can be written as
H total= Hl+ H a−c−f,
(1)
Hl= Ω1 |e〉1 〈 1 | + Ω3|e〉3 〈1|+h.c.,
(2)
H a−c−f = g 1,r a 1,r |e〉1 〈 0 | + g 2,r a 2,r|e〉2 〈 0 | + g 2,l a 2,l|e〉2 〈 1 | + g 3,l a 3,l|e〉3 〈0| + v1 b1† ( a 1,r+ a 2,r)+ v2 b2† ( a 2,l+ a 3,l)+h.c.,
(3)
where a and b are the annihilation operators associated with the modes of cavity and fiber respectively and vi(i = 1, 2) is the corresponding cavity-fiber coupling constant. We assume that g1,r = g2,r(l) = g3,l = g and v1 = v2 = v for convenience. If the initial state of the whole system is |1,0,0〉a |0〉c1 |0〉f1 |0,0〉c2 |0〉f2 |0〉c3, then the evolution of system will be restricted in the subspace spanned by
| ϕ1〉= | 1,0,0〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3, | ϕ2〉= | e,0,0〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3, | ϕ3〉= | 0,0,0〉a |1〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3, | ϕ4〉= | 0,0,0〉a |0〉 c1 |1〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3, | ϕ5〉= | 0,0,0〉a |0〉 c1 |0〉 f1 | 1,0〉 c2 |0〉 f2 |0〉 c3, | ϕ6〉= | 0,e,0〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3, | ϕ7〉= | 0,1,0〉a |0〉 c1 |0〉 f1 | 0,1〉 c2 |0〉 f2 |0〉 c3, | ϕ8〉= | 0,1,0〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |1〉 f2 |0〉 c3, | ϕ9〉= | 0,1,0〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |1〉 c3, | ϕ 10〉= | 0,1,e〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3, | ϕ 11〉= | 0,1,1〉a |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3,
(4)
where |i, j,k〉a (i, j, k = 0, 1, e) denotes the state of atoms in each cavity, n in |n〉s (s = c1, f1, c2, f2, c3) denotes the photon number in cavities or fibers.

Fig. 1 Zeno manipulations of atoms in spatially-separated cavities connected via optical fibers. Here, Ω1 (Ω3) is the classical field coupled to the first (third) atom, and b1 and b2 are the bosonic operators in fibers and couple to the corresponding cavity modes.

Under the Zeno condition g,v ≫ Ω1, Ω3, the above Hilbert subspace is split into nine invariant Zeno subspaces [35

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009). [CrossRef]

]
Γ P1= { | ϕ1〉, | ψ1〉, | ϕ 11〉},      Γ P2= { | ψ2〉}, Γ P3= { | ψ3〉},      Γ P4= { | ψ4〉},      Γ P5= { | ψ5〉},      Γ P6= { | ψ6〉}, Γ P7= { | ψ7〉},      Γ P8= { | ψ8〉},      Γ P9= { | ψ9〉},
(5)
corresponding to the projections
Piα= |α〉 〈α|, ( |α〉∈ Γ Pi)
(6)
with eigenvalues λ1 = 0, λ2=− ( g2+2 v2−A)/2, λ3= ( g2+2 v2−A)/2, λ4=− ( 3 g2+2 v2−A)/2, λ5= ( 3 g2+2 v2−A)/2, λ6=− ( 3 g2+2 v2+A)/2, λ7= ( 3 g2+2 v2+A)/2, λ8=− ( g2+2 v2+A)/2, λ9= ( g2+2 v2+A)/2, where A= g4+4 v4. Here,
| ψ1〉= N1 ( | ϕ2〉− gv | ϕ4〉+ | ϕ6〉− gv | ϕ8〉+ | ϕ 10〉) | ψ2〉= N2 ( − | ϕ2〉+ ε1 | ϕ3〉− η1 | ϕ4〉− χ1 | ϕ5〉+ χ1 | ϕ7〉+ η1 | ϕ8〉− ε1 | ϕ9〉+ | ϕ 10〉) | ψ3〉= N3 ( − | ϕ2〉− ε1 | ϕ3〉− η1 | ϕ4〉+ χ1 | ϕ5〉− χ1 | ϕ7〉+ η1 | ϕ8〉+ ε1 | ϕ9〉+ | ϕ 10〉) | ψ4〉= N4 ( | ϕ2〉− μ1 | ϕ3〉− ζ1 | ϕ4〉+ δ1 | ϕ5〉− θ1 | ϕ6〉+ δ1 | ϕ7〉− ζ1 | ϕ8〉− μ1 | ϕ9〉+ | ϕ 10〉) | ψ5〉= N5 ( | ϕ2〉+ μ1 | ϕ3〉− ζ1 | ϕ4〉− δ1 | ϕ5〉− θ1 | ϕ6〉− δ1 | ϕ7〉− ζ1 | ϕ8〉+ μ1 | ϕ9〉+ | ϕ 10〉) | ψ6〉= N6 ( − | ϕ2〉+ ε2 | ϕ3〉− η2 | ϕ4〉+ χ2 | ϕ5〉− χ2 | ϕ7〉+ η2 | ϕ8〉− ε2 | ϕ9〉+ | ϕ 10〉) | ψ7〉= N7 ( − | ϕ2〉− ε2 | ϕ3〉− η2 | ϕ4〉− χ2 | ϕ5〉+ χ2 | ϕ7〉+ η2 | ϕ8〉+ ε2 | ϕ9〉+ | ϕ 10〉) | ψ8〉= N8 ( | ϕ2〉− μ2 | ϕ3〉+ ζ2 | ϕ4〉− δ2 | ϕ5〉+ θ2 | ϕ6〉− δ2 | ϕ7〉+ ζ2 | ϕ8〉− μ2 | ϕ9〉+ | ϕ 10〉) | ψ9〉= N9 ( | ϕ2〉+ μ2 | ϕ3〉+ ζ2 | ϕ4〉+ δ2 | ϕ5〉+ θ2 | ϕ6〉+ δ2 | ϕ7〉+ ζ2 | ϕ8〉+ μ2 | ϕ9〉+ | ϕ 10〉)
(7)
with
ε1= g2+2 v2−A 2g,      η1= − g2+2 v2−A 2gv,      χ1= g2+2 v2−A( g2+A) 2 2g v2, μ1= 3 g2+2 v2−A 2g,      ζ1= − g2−2 v2+A 2gv,      δ1= 3 g2+2 v2−A(− g2+A) 2 2g v2, θ1= − g2+A v2,      ε2= g2+2 v2+A 2g,      η2= − g2+2 v2+A 2gv, χ2= g2+2 v2+A(− g2+A) 2 2g v2,      μ2= 3 g2+2 v2+A 2g,      ζ2= g2+2 v2+A 2gv, δ2= 3 g2+2 v2+A( g2+A) 2 2g v2,      θ2= g2+A v2,
and Ni (i = 1, 2, 3,..., 9) being the normalization factor for the eigenstate |ψi〉. Under above condition, the system can be effectively described by the following Hamiltonian
H total ≃ ∑ i,α,β λi Piα+ Piα Hl Piβ = ∑ i=29 λi | ψi〉 〈 ψi|+ N1 ( Ω1 | ϕ1〉 〈 ψ1|+ Ω3 | ϕ 11〉 〈 ψ1|+h.c.).
(8)
It reduces to
H eff= N1 ( Ω1 | ϕ1〉 〈 ψ1|+ Ω3 | ϕ 11〉 〈 ψ1|+h.c.),
(9)
if the initial state is |ϕ1〉. The effective Hamiltonian Heff implies that the evolution of system is restricted in the subspace, wherein the cavity modes are kept in the vacuum. Consequently, after the evolution time t the state of the system becomes
| Ψ(t)〉 = cos ( N1t Ω12+ Ω32) Ω12+ Ω32 Ω12+ Ω32 | ϕ1〉− i Ω1 Ω12+ Ω32sin ( N1t Ω12+ Ω32) Ω12+ Ω32 | ψ1〉 + [ cos ( N1t Ω12+ Ω32)−1] Ω1 Ω3 Ω12+ Ω32 | ϕ 11〉.
(10)
Obviously, if Ω1= ( 2+1) Ω3 and the evolution time is set as t=τ=π/ N1 Ω12+ Ω32, then the triatomic GHZ state |Ψ〉a can be generated, i.e.,
| Ψ(τ)〉=− 1 2 ( | ϕ1〉+ | ϕ 11〉)= |Ψ〉a⊗ |0〉 c1 |0〉 f1 | 0,0〉 c2 |0〉 f2 |0〉 c3,
(11)
where |Ψ〉a=− ( | 1,0,0〉a+ | 0,1,1〉a)/ 2. It is emphasized that the “free” Hamiltonian (2), that couples atomic states |1〉 and |e〉 in the first and third atom, would provoke a slow evolution of the state away from GHZ for t > τ. Therefore, once the GHZ state has been generated, one must switch the lasers off. Since Hl in Eq. (2) becomes zero and Ha–c–f |ψ(τ)〉 = 0, the GHZ state is preserved. Furthermore, if we perform a single-qubit rotation Rx(π/4)(= exp(iσx · π/4)) on the middle atom, then the tripartite GHZ state reduces to the state: [(|1,0〉1,3 + i|0,1〉1,3)|0〉2 + (i|1,0〉1,3 + |0,1〉1,3)|1〉2]/2. Consequently, the EPR entanglement between two distant atoms can be deterministically obtained by projective measurement on the middle atom (no matter it is found at which state) [43

L. F. Wei, Yu-xi Liu, and F. Nori, “Testing Bell’s inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B 72, 104516 (2005). [CrossRef]

], without taking any operation on the first and third atom directly.

It is clear that the validity of our scheme mainly relies on if the so-called Zeno condition g,v ≫ Ω1, Ω3 is satisfied robustly. Now, we discuss how the ratio Ω3/g influences the fidelity F = |〈Ψ(τ)|φ(τ)〉|2, with |φ(τ)〉 being the relevant final state evolved by the original Htotal defined in Eq. (1). On the other hand, the ratio v/g is another important factor affecting the fidelity [23

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

–25

Z.-Q. Yin and F.-L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007). [CrossRef]

, 28

X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A 79, 052330 (2009). [CrossRef]

]. Indeed, from Fig. 2 we see that the smaller the ratio Ω3/g corresponds to the higher fidelity. We also note that, even at the relatively-low ratio v/g = 0.5, the fidelity is still high, i.e., above 97%. This is very important for the practical application of our scheme, as the large cavity-fiber coupling is not easy to be satisfied in the realistic experiments. Furthermore, in order to obtain high fidelity in moderate time, we can choose typically the ratios: Ω3/g = 0.04 and v/g = 1.

Fig. 2 The influence of the ratios: Ω3/g and v/g, on the fidelity of the prepared GHZ state.

Our proposal is also robust for the imprecision of experimental parameters. Indeed, our previous discussions are based on certain ideal assumptions, e.g., gi,r(l) ≡ g, vi ≡ v. Practically, the coupling strengths gi,r(l) depend on the positions of the atoms in the cavities, and thus certain deviations are unavoidable. To numerically consider these deviations, we typically set g2,r(l) = g, g1,r = g + δg1 and g3,l = g + δg3 for simulations. In Fig. 3(a), the fidelity of the prepared state versus the variation δg1 and δg3 is plotted. It is seen that, even under a deviation |δg1,3| = 10%g, the fidelity is still sufficiently high, e.g., larger than 94%. Similarly, we plot the fidelity versus the deviation of the cavity-fiber coupling (v) in Fig. 3(b), and find also that the great robustness against the errors in the selected parameters.

Fig. 3 The fidelity of the three-atom GHZ state versus various parameter errors: (a) δg1 and δg3; (b) δv1 and δv2.

We now numerically verify that the above effective dynamics restricted in the subspace without exciting the cavity modes is also robust. In fact, considering all possible states of the system, the state at the time t reads |φtotal〉 = ∑i ci(t)|ϕi〉 within the subspace spanned by the basic state vectors in Eq. (4). The occupation probability for each state vector |ϕi〉 during the evolution is Pi(t) = |ci(t)|2, and satisfies the condition ∑i Pi(t) = 1. By solving the Schrödinger equation, the variation of the occupation probability Pc(t) (≡ ∑i=3,5,7,9 Pi(t)) is portrayed in Fig. 4 (red line). As shown in Fig. 4, the occupation probability of the cavity photonic state is less than 0.04. Therefore, our claim that the scheme is immune to the cavity decay seems justifiable. Meanwhile, Fig. 4 also describes the occupation probabilities Pf (t)(≡ ∑i=4,8 Pi(t), green line) and Pe(t)(≡ ∑i=2,6,10 Pi(t), black line) for those states, wherein the photon is in the fiber and one of the atoms in its excited state, respectively. Since Pe > Pf in Fig. 4, our protocol is more sensitive to atomic spontaneous emission than the fiber loss. To check this, let us investigate the evolution of the system governed by the following non-Hermitian Hamiltonian
H dec+ H total− iγ2 ∑ i=13 |e〉i 〈e|− i κc2 ( ∑ i=1,2 a i,r† a i,r+ ∑ i=2,3 a i,l† a i,l)− i κf2 ∑ j=1,2 bi† bi,
(12)
where γ is the spontaneous emission rate for atoms and κc(f) denotes the decay rate of the cavity modes (fiber modes). It is seen from Fig. 5 that the atomic spontaneous emission is the dominant factor of degrading the fidelity of the generated GHZ state. This can also be seen directly in the explicit form of the state |ψ1〉, where the population probability of the atomic excited state is larger than that of the photon in fiber.

Fig. 4 The population probabilities of cavity photonic state Pc, fiber photonic state Pf and the atomic excited state Pe versus the dimensionless parameter gt by exactly solving the evolution equation of the system without any approximation.
Fig. 5 The influences of atomic spontaneous emission γ/g, cavity field decay κc/g and fiber photonic leakage κf/g on the fidelity of the triatomic GHZ state for the typical ratios: Ω3/g = 0.04 and v/g = 1.

3. Generalization to N-atom entanglement

We now generalize the present scheme to generate N-atom GHZ states. Let us consider the configuration shown in Fig. 6, where N atoms are individually trapped in N cavities connected by N − 1 short fibers. The level configuration of the atoms between two ends are chosen the same as that of the middle atom in the above 3-atom case. The Hamiltonian of the present system reads
H total′= Hl′+ H a−c−f′,
(13)
where
Hl′= Ω1 |e〉1 〈 1 | ΩN|e〉N 〈1|+h.c.,
(14)
H a−c−f′ = g1 a 1,r |e〉1 〈0|+ ∑ j=1k ( g 2j,r a 2j,r |e〉 2j 〈0|+ g 2j,l a 2j,l |e〉 2j 〈1|) + ∑ j=1 k−1 ( g 2j+1,r a 2j+1,r |e〉 2j+1 〈1|+ g 2j+1,l a 2j+1,l |e〉 2j+1 〈0|)+ gN a N,l |e〉1 〈0| + ∑ j=1k [ v 2j−1 b 2j−1+ ( a 2j−1,r+ a 2j,r)+ v 2j b 2j+ ( a 2j,l+ a 2j+1,l)]+h.c..
(15)
Without loss of generality, we consider the case where N is odd number (N = 2k + 1, k = 1,2,3,...), and gi,r(l) = g, vi = v. If the initial state of the whole system is prepared at the state |1,0,0,...,0〉a|0〉all, then the system will evolve within the subspace Γfull spanned by the vectors: {|ϕ′1〉, |ϕ′2〉, |ϕ′3〉,..., |ϕ′8k+3〉}:
| ϕ1′〉= | 1,0,0,…,0〉a |0〉 all,    | ϕ2′〉= | e,0,0,…,0〉a |0〉 all,    | ϕ3′〉= | 0,0,0,…,0〉a |1〉 c1, | ϕ4′〉= | 0,0,0,…,0〉a |1〉 f1,    | ϕ5′〉= | 0,0,0,…,0〉a | 1,0〉 c2,    | ϕ6′〉= | 0,e,0,…,0〉a |0〉 all, | ϕ7′〉= | 0,1,0,…,0〉a | 0,1〉 c2,    | ϕ8′〉= | 0,1,0,…,0〉a |1〉 f2,    | ϕ9′〉= | 0,1,0,…,0〉a | 0,1〉 c3, | ϕ 10′〉= | 0,1,e,…,0〉a |0〉 all,    …    | ϕ 8k+3′〉= | 0,1,1,…,1〉a |0〉 all,
(16)
where |0〉all means that all boson modes are in the vacuum state, n (n1, n2) in |n〉s or |n1, n2〉s (s = ci, fi) denotes the photon number in the corresponding resonator while other cavities or fibers are in the vacuum state. Similar to the above procedure, we get the effective Hamiltonian
H eff′= N1′ ( Ω1 | ϕ1′〉 〈 ψ1′|+ ΩN | ϕ 8k+3′〉 〈 ψ1′|+h.c.),
(17)
where
| ψ1 ′〉= N1′ ( ∑ i=1N | ϕ 4i−2′〉− ∑ i=1 N−1 gv | ϕ 4i′〉).
(18)
Set Ω1= ( 2+1) ΩN and the interaction time τ′=π/ N1′ Ω12+ ΩN2, the desirable N-atomic GHZ state
| Ψ( τ′)〉=− 1 2 ( | ϕ1′〉+ | ϕ 8k+3′〉)=− 1 2 ( | 1,0,0,…,0〉a+ | 0,1,1,…,1〉a)⊗ |0〉 all
(19)
can be generated. In order to test the effectiveness of our proposal, we consider specifically, for example, the case of five atoms. In Fig. 7, we plot the time-evolution behaviors of the occupation probability in the initial state |ϕ′1〉 governed by total Hamiltonian (Eq. (13)) and by the effective Hamiltonian (Eq. (17)). It is shown that the numerical results under these two Hamiltonians agree with each other reasonably well. Therefore, our effective model is valid.

Fig. 6 N atom trapped in distant cavities.
Fig. 7 The occupation probabilities of the state |ϕ′1〉 versus the dimensionless parameter gt under the total Hamiltonian (solid-line) and effective Hamiltonian (dotted-line).

4. Discussions and conclusions

The experimental feasibility of the proposed scheme is briefly analyzed as follows. Practically, the atomic configuration involved in our scheme can be implemented with the 87Rb atom, whose relevant atomic levels are shown in Fig. 8. Where the first (third) atom is coupled resonantly to an external π-polarized classical field and a π+ (π−) polarized photon modes of the cavity, and the middle atom is coupled resonantly to a π+ and a π− polarized modes. Based on the above discussions, in order to obtain fidelity larger than 90%, one should keep the decay rate γ < 0.0055g. This condition can be satisfied in recent experiments [44

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]

, 45

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67, 033806 (2003). [CrossRef]

], wherein for an optical cavity with the wavelength about 850 nm the parameters are set as: g/2π = 750 MHz, γ/2π = 2.62 MHz, κc/2π = 3.5 MHz. Also, a near perfect fiber-cavity coupling with an efficiency larger than 99.9% can be realized using fiber-taper coupling to high-Q silica microspheres [46

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]

]. The fiber loss at the 852 nm wavelength is about 2.2 dB/km [47

K. J. Gordon, V. Fernandez, P. D. Townsend, and G. S. Buller, “A short wavelength GigaHertz clocked fiber-optic quantum key distribution system,” IEEE J. Quantum Electron. 40, 900–908 (2004). [CrossRef]

], which corresponds to the fiber decay rate 1.52 × 105 Hz. With these parameters, it seems that the present entanglement-generation scheme with a high fidelity larger than 93% could be feasible with the present experimental technique. The duration of the Zeno pulses Ω1 and Ω3 utilized to generate the desirable three-atom GHZ state can be estimated as π/ N1 Ω12+ Ω32≈1.43× 10 −8s, which is easily implemented for the present laser technology.

Fig. 8 Experimental atomic configuration used to generate tripartite GHZ state. Here, Ω1 and Ω3 are the classical fields, and π+ and π− are the quantized cavity modes with different polarizations, respectively.

In conclusion, we have proposed an approach to achieve three-atom GHZ states by taking advantage of quantum Zeno dynamics. In present proposal, only one step operation is required to complete the generation. The generation of the GHZ states is controllable, as the proposed quantum Zeno dynamics can be switched on/off by controlling the classical drivings of the atoms at two ends. We note also that, the generated GHZ state is no longer evolving, at least theoretically, after switching off the classical drivings of the atoms at two-side cavities. Additionally, the evolution of system is always kept in the subspace with null-excitation of cavity fields. So, the scheme is immune to the decay of cavity modes. Moreover, the scheme loosens the requirement of strong cavity-fiber coupling. We also show that the proposal can be easily generalized to prepared the N-atom entanglement without increasing the number of the applied classical fields.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China, under Grants No. 90921010 and No. 11174373, and the National Fundamental Research Program of China, through Grant No. 2010CB923104.

References and links

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M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

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M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999). [CrossRef]

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X. B. Zou, K. Pahlke, and W. Mathis, “Conditional generation of the Greenberger-Horne-Zeilinger state of four distant atoms via cavity decay,” Phys. Rev. A 68, 024302 (2003). [CrossRef]

8.

K. Pahlke, X. B. Zou, and W. Mathis, “The generation of the Greenberger-Horne-Zeilinger state of four distant atoms conditioned on cavity decay,” J. Opt. B Quantum Semiclass. Opt. 6, S142–S146 (2004). [CrossRef]

9.

D. Gonta, S. Fritzsche, and T. Radtke, “Generation of four-partite Greenberger-Horne-Zeilinger and W states by using a high-finesse bimodal cavity,” Phys. Rev. A 77, 062312 (2008). [CrossRef]

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L. F. Wei, Y. X. Liu, and F. Nori, “Generation and control of Greenberger-Horne-Zeilinger entanglement in superconducting circuits,” Phys. Rev. Lett. 96, 246803 (2006). [CrossRef] [PubMed]

11.

R. J. Nelson, D. G. Cory, and S. Lloyd, “Experimental demonstration of Greenberger-Horne-Zeilinger correlations using nuclear magnetic resonance,” Phys. Rev. A 61, 022106 (2000). [CrossRef]

12.

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature (London) 438, 639–642 (2005). [CrossRef]

13.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature (London) 467, 570–573 (2010). [CrossRef]

14.

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565–582 (2001). [CrossRef]

15.

J. I. Cirac and P. Zoller, “Preparation of macroscopic superpositions in many-atom systems,” Phys. Rev. A 50, R2799–R2802 (1994). [CrossRef] [PubMed]

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J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. 89, 237901 (2002). [CrossRef] [PubMed]

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18.

S. van Enk, J. Cirac, and P. Zoller, “Ideal quantum communication over noisy channels: A Quantum optical implementation,” Phys. Rev. Lett. 78, 4293–4296 (1997). [CrossRef]

19.

S. Bose, P. L. Knight, M. B. Plenio, and V. Vedral, “Proposal for teleportation of an atomic state via cavity decay,” Phys. Rev. Lett. 83, 5158–5161 (1999). [CrossRef]

20.

S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long Distance, Unconditional Teleportation of Atomic States via Complete Bell State Measurements,” Phys. Rev. Lett. 87, 167903 (2001). [CrossRef] [PubMed]

21.

A. S. Parkins and H. J. Kimble, “Position-momentum Einstein-Podolsky-Rosen state of distantly separated trapped atoms,” Phys. Rev. A 61, 052104 (2000). [CrossRef]

22.

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997). [CrossRef]

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A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef] [PubMed]

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P. Peng and F.-L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007). [CrossRef]

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Z.-Q. Yin and F.-L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007). [CrossRef]

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S.-Y. Ye, Z.-R. Zhong, and S.-B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77, 014303 (2008). [CrossRef]

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Z.-B. Yang, S.-Y. Ye, A. Serafini, and S.-B. Zheng, “Distributed coherent manipulation of qutrits by virtual excitation processes,” J. Phys. B: At. Mol. Opt. Phys. 43, 085506 (2010). [CrossRef]

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S. B. Zheng, “Generation of Greenberger-Horne-Zeilinger states for multiple atoms trapped in separated cavities,” Eur. Phys. J. D 54, 719–722 (2009). [CrossRef]

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A. Zheng and J. Liu, “Generation of an N-qubit Greenberger-Horne-Zeilinger state with distant atoms in bimodal cavities,” J. Phys. B: At. Mol. Opt. Phys. 44, 165501 (2011). [CrossRef]

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X.-Y. Lv, P.-J. Song, J.-B. Liu, and X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2009). [CrossRef]

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44.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]

45.

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67, 033806 (2003). [CrossRef]

46.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]

47.

K. J. Gordon, V. Fernandez, P. D. Townsend, and G. S. Buller, “A short wavelength GigaHertz clocked fiber-optic quantum key distribution system,” IEEE J. Quantum Electron. 40, 900–908 (2004). [CrossRef]

OCIS Codes
(270.5580) Quantum optics : Quantum electrodynamics
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: March 26, 2012
Revised Manuscript: May 12, 2012
Manuscript Accepted: May 15, 2012
Published: May 31, 2012

Citation
Wen-An Li and Lian-Fu Wei, "Controllable entanglement preparations between atoms in spatially-separated cavities via quantum Zeno dynamics," Opt. Express 20, 13440-13450 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13440


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References

  1. H. K. Lo, S. Popescu, and T. Spiller, Introduction to Quantum Computation and Information (World Scientific, Singapore, 1997).
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
  3. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London)404, 247–255 (2000). [CrossRef]
  4. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67, 661–663 (1991). [CrossRef] [PubMed]
  5. D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys.58, 1131–1143 (1990). [CrossRef]
  6. M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A59, 1829–1834 (1999). [CrossRef]
  7. X. B. Zou, K. Pahlke, and W. Mathis, “Conditional generation of the Greenberger-Horne-Zeilinger state of four distant atoms via cavity decay,” Phys. Rev. A68, 024302 (2003). [CrossRef]
  8. K. Pahlke, X. B. Zou, and W. Mathis, “The generation of the Greenberger-Horne-Zeilinger state of four distant atoms conditioned on cavity decay,” J. Opt. B Quantum Semiclass. Opt.6, S142–S146 (2004). [CrossRef]
  9. D. Gonta, S. Fritzsche, and T. Radtke, “Generation of four-partite Greenberger-Horne-Zeilinger and W states by using a high-finesse bimodal cavity,” Phys. Rev. A77, 062312 (2008). [CrossRef]
  10. L. F. Wei, Y. X. Liu, and F. Nori, “Generation and control of Greenberger-Horne-Zeilinger entanglement in superconducting circuits,” Phys. Rev. Lett.96, 246803 (2006). [CrossRef] [PubMed]
  11. R. J. Nelson, D. G. Cory, and S. Lloyd, “Experimental demonstration of Greenberger-Horne-Zeilinger correlations using nuclear magnetic resonance,” Phys. Rev. A61, 022106 (2000). [CrossRef]
  12. D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature (London)438, 639–642 (2005). [CrossRef]
  13. M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature (London)467, 570–573 (2010). [CrossRef]
  14. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys.73, 565–582 (2001). [CrossRef]
  15. J. I. Cirac and P. Zoller, “Preparation of macroscopic superpositions in many-atom systems,” Phys. Rev. A50, R2799–R2802 (1994). [CrossRef] [PubMed]
  16. J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett.89, 237901 (2002). [CrossRef] [PubMed]
  17. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett.78, 3221–3224 (1997). [CrossRef]
  18. S. van Enk, J. Cirac, and P. Zoller, “Ideal quantum communication over noisy channels: A Quantum optical implementation,” Phys. Rev. Lett.78, 4293–4296 (1997). [CrossRef]
  19. S. Bose, P. L. Knight, M. B. Plenio, and V. Vedral, “Proposal for teleportation of an atomic state via cavity decay,” Phys. Rev. Lett.83, 5158–5161 (1999). [CrossRef]
  20. S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, “Long Distance, Unconditional Teleportation of Atomic States via Complete Bell State Measurements,” Phys. Rev. Lett.87, 167903 (2001). [CrossRef] [PubMed]
  21. A. S. Parkins and H. J. Kimble, “Position-momentum Einstein-Podolsky-Rosen state of distantly separated trapped atoms,” Phys. Rev. A61, 052104 (2000). [CrossRef]
  22. T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett.79, 5242–5245 (1997). [CrossRef]
  23. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett.96, 010503 (2006). [CrossRef] [PubMed]
  24. P. Peng and F.-L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A75, 062320 (2007). [CrossRef]
  25. Z.-Q. Yin and F.-L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A75, 012324 (2007). [CrossRef]
  26. S.-Y. Ye, Z.-R. Zhong, and S.-B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A77, 014303 (2008). [CrossRef]
  27. Z.-B. Yang, S.-Y. Ye, A. Serafini, and S.-B. Zheng, “Distributed coherent manipulation of qutrits by virtual excitation processes,” J. Phys. B: At. Mol. Opt. Phys.43, 085506 (2010). [CrossRef]
  28. X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A79, 052330 (2009). [CrossRef]
  29. S. B. Zheng, “Generation of Greenberger-Horne-Zeilinger states for multiple atoms trapped in separated cavities,” Eur. Phys. J. D54, 719–722 (2009). [CrossRef]
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