Optics Communications 312 (2014) 269–274
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Generation of W state and GHZ state of multiple atomic ensembles via a single atom in a nonresonant cavity Chun-Ling Zhang, Wen-Zhang Li, Mei-Feng Chen n Laboratory of Quantum Optics, Department of Physics, Fuzhou University, Fuzhou 350002, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 June 2013 Received in revised form 28 August 2013 Accepted 18 September 2013 Available online 9 October 2013
We propose a scheme for generation of the W state and the Greenberger–Horn–Zeilinger (GHZ) state of atomic ensembles. The scheme is based on the dynamics of a single control atom and atomic ensembles interacting with a nonresonant cavity mode. By choosing the appropriate parameters, the effective Hamiltonian describing the interaction between the control atom and the atomic modes shows complete analogy with the Jaynes–Cummings Hamiltonian. The required time for preparing the W state (GHZ state) keeps unchanged (increases linearly) with the increase of the number of atomic ensembles. The effects of dissipation and the detuning between the atomic modes and the control atom on the prepared states are analyzed by numerical simulation. & 2013 Elsevier B.V. All rights reserved.
Keywords: W state GHZ state Atomic ensemble Cavity QED
1. Introduction Recently, much interest has been paid to the multipartite entanglement. For multipartite systems, there are more peculiar properties than the bipartite ones because they exhibit the contradiction between local hidden variable theories and quantum mechanics even for nonstatistical predictions, as opposed to the statistical ones for the Einstei–Podolsky–Rosen (EPR) states . Moreover, multipartite entanglements are important physical resources for quantum information processing, such as quantum cryptography , quantum teleportation  and quantum dense coding . The typical multiparticle entangled states are the W state  and the GHZ state [5,6], which have been demonstrated to be two inequivalent classes of entangled states. As we know, the W state is robust against qubit loss while the GHZ state is inequivalent to the W state in the sense that it will be reduced to the maximally mixed state when one of the qubits is decohered or to a product state when one of the qubits is measured in the logical basis. In recent years, many schemes for generation of the W state and the GHZ state have been proposed [7–12]. The physical systems utilized to generate entanglement include superconducting circuits [13–16], linear optical system , cavity quantum electrodynamics (QED) , trapped ions , and quantum dot . Among them, the cavity QED is well developed and regarded as an ideal candidate for quantum communication and quantum state engineering [21,22]. Compared with those schemes
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that use a single particle as a qubit, the schemes proposed by Lukin et al. , Xue and Guo , Duan , and Han et al.  use an atomic ensemble with a large number of identical atoms as the basic system. There are several advantages by using an atomic ensemble as a single qubit. First, the manipulation of the atomic ensemble is normally easier than the coherent control of a single atom for that the laser applied to the atomic ensemble does not separately address the individual atoms in the ensemble . Second, the atomic ensemble that contains a large number of identical atoms increases the light–matter coupling strength, which scales with the squareroot of the number of the atoms involved in the ensemble. This greatly reduces the operation time and thus suppresses the decoherence. Those advantages allow one to take a more positive view of the atomic ensemble and regard it as an essential resource for many ingenious applications such as subshot noise spectroscopy and atom interferometry , secure cryptography protocols , and generation of squeezed states for atomic ensembles . For the generation of entangled states of atomic ensembles, the schemes in Refs. [23–26] are based on single-photon detection, thus the success probabilities of getting the desired states are very small. The scheme in Ref.  for preparing the W state and that in Refs. [24,26] for preparing the GHZ state requires the operation time polynomially and exponentially with the number of atomic ensembles. Thus they are also sensitive to the photon loss. Recently, Zheng  has found that the dynamics of an atomic system which contains a single control atom and an atomic ensemble can be described as an effective Jaynes–Cummings model (JCM). In his proposal, the atomic ensemble acts as the bosonic mode, and the single control atom and the atomic ensemble are dispersively coupled to a cavity while the control atom is also illuminated by a highly detuned auxiliary
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classical ﬁeld. Stimulated by this idea, we present here a new scheme to generate the W state and the GHZ state. The scheme has the following advantages: (i) it does not depend on the photon detection, which simpliﬁes the experimental equipment. (ii) The high-ﬁdelity W state and GHZ state can be achieved even in the presence of the decoherence arising from atomic spontaneous emission and photon leakage. (iii) The required time for preparing the GHZ state increases linearly with the number of atomic ensembles and that for the W state is unchanged
2. Dynamical model of the atomic system Let us ﬁrst brieﬂy describe the dynamical model of the system under consideration. A single control atom and an atomic ensemble which contain N identical atoms are trapped in a single-mode cavity. The atomic level conﬁguration and the corresponding transitions are shown in Fig. 1. Each atom has an excited state je〉 and two ground states jf 〉 and jg〉. The atomic transition je〉2jg〉 of the control atom (the atoms in the ensemble) is coupled to the cavity with coupling coefﬁcient gc (ge) with detuning Δg. Meanwhile, the atoms are driven by two classical ﬁelds with the Rabi frequencies Ω1 and Ω2 and detunings Δ1 and Δ2. In the interaction picture, the Hamiltonian describing the system is ðℏ ¼ 1Þ H I ¼ ðΩ1 eiΔ1 t jec 〉〈f c j þ Ω2 eiΔ2 t j f c 〉〈ec jþ g c eiΔg t ajec 〉〈g c jÞ N
þ ∑ ðΩ1 eiΔ1 t jei 〉〈f i jþ Ω2 eiΔ2 t j f i 〉〈ei j þ g e eiΔg t ajei 〉〈g i jÞþ H:c:;
Then the Hamiltonian can be written as H ′I ¼
ð1Þ where a is the annihilation operator for the cavity mode. Under the large detunings condition, i.e., Δ1 ; Δ2 ; Δg ⪢g c ; g e ; Ω1 ; Ω2 , the upperlevel je〉 can be adiabatically eliminated. Moreover, we set the parameters Ω1 ¼ Ω2 ¼ Ω and Δ1 ¼ Δ2 ¼ Δ to eliminate the Stark shift induced by the classical pulses. Furthermore, choose the detunings appropriately so that the dominant Raman transition is induced by the classical ﬁeld Ω1 and the cavity mode a, while the other Raman transitions are far off-resonant and can be neglected.
a þ ajg c 〉〈g c jþ ∑
i ¼ 1 Δg
a þ ajg i 〉〈g i j
þ ðλc e iδt aj f c 〉〈g c jþ ∑ λ1 e iδt aj f i 〉〈g i jþ H:c:Þ;
δ ¼ Δg Δ, λc ¼ ðgc Ω=2Þð1=Δg þ 1=ΔÞ, and where λ1 ¼ ðge Ω=2Þð1=Δg þ 1=ΔÞ. In the case that δ⪢λ1 ; λc , the atoms cannot exchange energy with the ﬁeld. However, the atoms can exchange energy with each other via the virtual excitation of ﬁeld mode. Then the Hamiltonian of Eq. (2) can be replaced by the effective Hamiltonian ! 2 λ2 g 2c λc H eff ¼ c aa þ j f c 〉〈f c j þ a þ ajg c 〉〈g c j δ
2 λ þ g 2e λ1 aa j f i 〉〈f i j þ þ ∑ a þ ajg i 〉〈g i j Δg δ i¼1 δ N
N λ2 λc λ1 þ ðSc Sl þ Sc Slþ Þ þ ∑ 1 Sjþ Sk l¼1 δ j;k ¼ 1 δ N
ðj a kÞ
where Siþ ¼ j f i 〉〈g i j and Si ¼ jg i 〉〈f i j ði ¼ c; 1; 2; …; NÞ. Since ½a þ a; H eff ¼ 0, the photon number is conserved during the interaction. If the cavity is initially in the vacuum state, it will remain in this state and the effective Hamiltonian reduces to H ′eff ¼
λ2c λ2 N λc λ1 N þ j f c 〉〈f c j þ 1 ∑ Sjþ Sk þ ∑ ðS S þ Sc Siþ Þ: δ δ j;k ¼ 1 δ i¼1 c i
pﬃﬃﬃﬃ pﬃﬃﬃﬃ þ þ b ¼ ð1= NÞ∑N Setting b ¼ ð1= N Þ∑N i ¼ 1 Si , i ¼ 1 Si , þ N nb ¼ ∑i ¼ 1 j f i 〉〈f i j, then we have ½b; b ¼ 1 ð2=NÞnb . Suppose that the average number of atoms in the state jf 〉 is much smaller than the total atomic number, i.e., nb ⪡N, then b and b þ can be regarded as the bosonic operators. In this case, the Hamiltonian can be rewritten as þ
H 1 ¼ νSzc þ ɛb b þ μðScþ b þ Sc b Þ;
ν ¼ λc =δ, ɛ ¼ N λ1 =δ, where Szc ¼ 12 ðj f c 〉〈f c j jg c 〉〈g c jÞ, pﬃﬃﬃﬃ μ ¼ N ðλc λ1 Þ=δ, and we have discarded the constant energy λ2c =δ. The Hamiltonian H1 shows complete analogy with the Jaynes–Cummings Hamiltonian. Under the resonant condition 2
ν ¼ ɛ;
the Hamiltonian describes the resonant coupling between the control atom and the atomic mode and leads to the transitions j f c 〉j0〉e -e iðɛtÞ=2 ½ cos ðμtÞj f c 〉j0〉e i sin ðμtÞjg c 〉j1〉e ; jg c 〉j1〉e -e iðɛtÞ=2 ½ cos ðμtÞjg c 〉j1〉e i sin ðμtÞj f c 〉j0〉e ;
Fig. 1. A single control atom and an atomic ensemble which contain N identical atoms are trapped in a single-mode cavity with different coupling coefﬁcients.
where jx〉e (x¼0,1) denotes the Fock state of the atomic ensemble, with x¼0 denoting that all the atoms in the ensemble are in the state jg〉, while x¼1 denoting that there is only one atom in the state jf 〉 and the others in the state jg〉. In order to validate the feasibility of the above theoretical analysis, we perform a direct numerical simulation of the Schrödinger equation with the full Hamiltonian in Eq. (1) and the effective Hamiltonian in Eq. (5). To satisfy pﬃﬃﬃﬃ the resonant condition ν ¼ ɛ, we set the parameters ge ¼ g= N and gc ¼ g. We should mention that the coupling coefﬁcient between the atoms and the cavity mode is dependent on the waist of the cavity and p the ﬃﬃﬃﬃ position of the atoms in the cavity. Hence, the relations g e ¼ g= N and g c ¼ g could be reachable. In the following simulation, we calculate the temporal evolution of the system with the initial state j f c 〉j0〉e . We plot the time-dependent populations of the basic states j f c 〉j0〉e (P1) and jg c 〉j1〉e (P2) governed by the full Hamiltonian in Eq. (1) (green lines in Fig. 2) and the effective Hamiltonian in Eq. (5) (red lines in Fig. 2), where Ω ¼ g, N ¼ 104 and (a) Δ ¼ 11g, Δg ¼ 12g; (b) Δ ¼ 49g, Δg ¼ 50g. We can see that the effective and full dynamics exhibit excellent agreement when the
C.-L. Zhang et al. / Optics Communications 312 (2014) 269–274
effective Hamiltonian will be written as
þ H eff W n ¼ νSzc þ ɛ ∑ ∑ bj bk þ μ Sc ∑ bj þ Sc ∑ bj j¼1k¼1
P1(Eq. 5 )
P2(Eq. 5 ) P2(Eq. 1 ) 0.4
P1(Eq. 1 )
1 jW〉 ¼ pﬃﬃﬃðj1〉1 j0〉2 …j0〉n þ j0〉1 j1〉2 …j0〉n þ …j0〉1 j0〉2 …j1〉n Þ; n
0.8 P1(Eq.5) 0.6
P2(Eq.5) P2(Eq.1) P1(Eq.1)
where bi and bi ði ¼ 1; 2; …; nÞ are the creation and annihilation operators for the collective of atomic modes. Suppose that the control atom is initially in the ground state j f c 〉 and the n atomic ensembles are initially in the vacuum state j0〉1 j0〉2 …j0〉n . In the case with ν ¼ nɛ, the evolution of the system is pﬃﬃﬃ jϕðtÞ〉 ¼ e inɛt=2 ½ cos ð nμtÞj f c 〉j0〉1 j0〉2 …j0〉n pﬃﬃﬃ i sin ð nμtÞjg c 〉ðj1〉1 j0〉2 …j0〉n pﬃﬃﬃ þ j0〉1 j1〉2 …j0〉n þ…j0〉1 j0〉2 …j1〉n Þ= n: ð10Þ pﬃﬃﬃ If the interaction time t W n is set to be π =2 nμ, the atomic modes will evolve into the W state
gt Fig. 2. The populations of the basic states jf 〉c j0〉e and jg〉c j1〉e governed by the Hamiltonian in Eq. (1) (red lines) and Eq. (5) (green lines), where (a) Δ ¼ 11g; Δg ¼ 12g; (b) Δ ¼ 49g; Δg ¼ 50g. Other common parameters: Ω ¼ g, N ¼ 104 , ge ¼ 0:01g and g c ¼ g.
detunings are large enough. However, the deviations decrease at the cost of the long evolution time. That is, the larger the values of Δ and Δg are the longer the evolution time is. Eventually, the simulation result of Eq. (1) is almost the same as that of Eq. (5) when Δ ¼ 49g, Δg ¼ 50g. Thus, the above approximation for the Hamiltonian is reliable as long as the detunings are large enough.
with the control atom left in the ground state jg c 〉. The above discussion indicates that, pﬃﬃﬃﬃﬃﬃﬃto satisfy the resonant condition ν ¼ nɛ, we set g c ¼ g, g e ¼ g= nN . That is to say, the coupling coefﬁcient ge must vary with the number of atomic ensembles if we set other parameters unchanged. We can obtain the time for preparing 2 the n atomic-ensemble W state is t W n ¼ 2πδ=g 2 Ω ð1=Δg þ 1=ΔÞ2 , which means that the time required to prepare the W state keeps unchanged as the number of atomic ensembles increases. Next, we show how to prepare the n atomic-ensemble GHZ state by using this model. In order to do so, we have to introduce another atomic level jhc 〉 for the control atom, the level structure of which is depicted in Fig. 3. The transition between jhc 〉2jec 〉 is highly detuned from the cavity mode, so the energy level jhc 〉 will not be affected when the control atom interacts with the cavity mode. Suppose that the control atom and n atomic ensembles are trapped in the vacuum cavity. The scheme for preparation of the GHZ class of maximally entangled states between atomic ensembles works in the following way (see Fig. 4). (i) The ﬁrst step is to apply different nonresonant classical ﬁelds separately to the atomic ensemble 1 and the control atom, as shown in Section 2. The interaction between the other atomic ensembles and the cavity mode is frozen due to large detuning. Then the atomic ensemble 1 and the control atom will evolve according to Eq. (5). Suppose that the control atom is initially in the ground state p1ﬃﬃ2ðj f c 〉 þ jhc 〉Þ and all the atomic ensembles are initially in the vacuum state j0〉k ; ðk ¼ 1; 2; 3Þ. With the interaction time t 1 ¼ π =2μ, the system evolves into 1 jψ ðt 1 Þ〉 ¼ pﬃﬃﬃðjhc 〉j0〉1 ijg c 〉j1〉1 Þj0〉2 j0〉3 …j0〉n : 2
3. Generation of multipartite entanglement of atomic ensembles We note that the above idea can be used to generate the W state and the GHZ state of atomic ensembles. We ﬁrst focus on the generation of the W state. Suppose that a control atom and the n atomic ensembles are trapped in the vacuum cavity and driven by two classical laser ﬁelds at the same time. The full Hamiltonian describing the system is
(ii) The single-qubit operationpjgﬃﬃﬃc 〉-ij f c 〉 is performed to transfer the system to the state 1= 2ðjhc 〉j0〉1 þ j f c 〉j1〉1 Þj0〉2 j0〉3 .
H W n ¼ ðΩeiΔt jec 〉〈f c j þ ΩeiΔt j f c 〉〈ec j þ g c eiΔg t ajec 〉〈g c jÞ n
þ ∑ ∑ ðΩeiΔt jei 〉jj 〈f i j þ ΩeiΔt j f i 〉jj 〈ei jþ g e eiΔg t ajei 〉jj 〈g i jÞ þ H:c: j¼1i¼1
ð8Þ If the parameters satisfy the conditions mentioned above, the atomic ensembles can be regarded as the bosonic modes, the
Fig. 3. The level structure and the responding transitions of the control atom for generating the GHZ state.
C.-L. Zhang et al. / Optics Communications 312 (2014) 269–274
Fig. 4. Schematic experimental setup and the steps for generation of the n atomic-ensemble GHZ state.
(iii) Repeat the ﬁrst step to make the atomic ensemble 2 and the control atom evolve according to Eq. (5) successfully by applying classical ﬁelds to the atomic ensemble 2 and the control atom. After the interaction time t 2 ¼ π =2μ, the system evolves into 1 jψ ðt 2 Þ〉 ¼ pﬃﬃﬃðjhc 〉j0〉1 j0〉2 ijg c 〉j1〉1 j1〉2 Þj0〉3 …j0〉n : 2
Similarly, make the control pﬃﬃﬃ atom to have ﬂip jg c 〉-ij f c 〉, then the system will be into 1= 2ðjhc 〉j0〉1 j0〉2 þ j f c 〉j1〉1 j1〉2 Þj0〉3 …j0〉n . (iv) Similarly, repeat steps (i) and (ii) for the control atom and the atomic ensemble k ðk ¼ 3; 4; …; nÞ. We then get 1 jψ ðt n Þ〉 ¼ pﬃﬃﬃðjhc 〉j0〉1 j0〉2 …j0〉n þ j f c 〉j1〉1 j1〉2 …j1〉n Þ: 2
Then pﬃﬃﬃ we measure pﬃﬃﬃ the control atom in the basis f1= 2ðj f c 〉 þjhc 〉Þ; 1= 2ðj f p jhc 〉Þg. If the control atom is c 〉 ﬃﬃﬃ measured in the state 1= 2ðj f c 〉 jhc 〉Þ, the n atomic ensembles will collapse to a GHZ class state 1 jGHZ′〉 ¼ pﬃﬃﬃðj1〉1 j1〉2 …j1〉n j0〉1 j0〉2 …j0〉n Þ: 2
Otherwise, if the control atom is measured in the state pﬃﬃﬃ 1= 2ðj f c 〉 þ jhc 〉Þ, the n atomic ensembles will collapse to a GHZ state 1 jGHZ〉 ¼ pﬃﬃﬃðj1〉1 j1〉2 …j1〉n þ j0〉1 j0〉2 …j0〉n Þ: 2
4. Analysis of the experimental feasibility Now we investigate the ﬁdelity of the prepared states. For a pure state, the ﬁdelity is deﬁned by F ¼ 〈ψ ðtÞjψ ideal 〉2 , where jψ ðtÞ〉 is the ﬁnal state which is governed by the full Hamiltonian, and jψ ideal 〉 is the ideal prepared state. For the W state, we take the three-atomicensemble W state as an example. Fig. 5 plots the ﬁdelity of the three4 atomic-ensemble pﬃﬃﬃ W state versus Δg =g with N ¼ 10 , Ω ¼ g, g c ¼ g, g 1 ¼ 0:01g= 3 and δ ¼ g. We see that, in a speciﬁc range, the larger the detuning is, the higher the ﬁdelity of W state is. However, there would be no coupling between the atoms and the cavity mode if the detuning is too large. It can make sure the coupling of atom-cavitymode even when Δg =g ¼ 100. On the other hand, the large Δg can ensure the whole system that evolves in accordance with that governed by the effective Hamiltonian in Eq. (9). However, the large detuning prolongs the evolution time which will lead to the worse impacts of decoherence. It would be interesting to perform a numerical analysis taking into account the inﬂuence of atomic spontaneous emission and photon leakage on our protocol. Under the condition that no photon is detected either by the atomic spontaneous emission or by the leakage of a photon from the cavity, the evolution of the system is governed by the Hamiltonian
ρ_ ¼ i½HW 3 ; ρ þ
It is clear that, for the generation of the GHZ state are, it is not necessary to change the control strategy as the number of atomic ensembles increases. On the other hand, it demonstrates the step-by-step engineered entanglement for n atomic ensembles. The more atomic ensembles for preparing the GHZ state, the more steps there are, then the longer it takes. That is, the required time for preparing the GHZ state increases linearly with the number of atomic ensembles.
∑ ð2s ρs þ sclþ scl ρ ρsclþ scl Þ 2 l ¼ f ;g cl cl
þ ∑ ∑ ∑ ð2s ρs þ sijlþ sijl ρ ρsijL sijl Þ 2 i ¼ 1 j ¼ 1 l ¼ f ;g ijl ijl
þ ð2aρa þ a þ aρ ρa þ aÞ; 2
where scl ¼ jlc 〉〈ec j, sclþ ¼ jec 〉〈lc j, sijl ¼ jli 〉jj 〈ei j, sijlþ ¼ jei 〉jj 〈li j. Here, Γ and κ denote the atomic spontaneous emission rate and the decay rate of the cavity, respectively. For simplicity, we have assumed that the atomic spontaneous emission rate of jei 〉-jg i 〉 ði ¼ c; 1; 2; …; NÞ is equal to that of jei 〉-j f i 〉, and being labeled Γ . Fig. 6 plots the ﬁdelity
C.-L. Zhang et al. / Optics Communications 312 (2014) 269–274
0.99 0.985 0.98 0.975 0.97 0.965 10
Δg/g Fig. 5. The ﬁdelity of the three-atomic-ensemble W state versus Δg =g. 1 Γ/g κ/g
The atomic conﬁguration involved in our scheme can be achieved with a cesium atom . Four atomic states je〉, jf 〉, jg〉, and jh〉 correspond to jF ¼ 5; M F ¼ 5〉 hyperﬁne state of 62 P 3=2 , jF ¼ 4; M F ¼ 4〉 hyperﬁne state of 62 S1=2 , jF ¼ 4; M F ¼ 4〉 hyperﬁne state of 62 S1=2 , and jF ¼ 3; M F ¼ 2〉 hyperﬁne levels of 62 S1=2 , respectively. In recent experiments, a set of parameters g ¼ 2π 750 MHz, Γ ¼ 2π 2:62 MHz, κ ¼ 2π 3:5 MHz is predicted achievable by using fused-silica microspheres [33,34]. In such a case, the ﬁdelity of the three-atomic-ensemble W state is 0.98 with Δg ¼ 50g. For the generation of the multi-atomicensemble GHZ state, the ﬁdelity also depends on the number of atomic ensembles. With the above parameters, the ﬁdelity of the n atomic-ensemble GHZ state is over 0.9 when n r 6. Additionally, a single cesium atom can be localized at a ﬁxed position in the cavity with a high precision for a long time . All the atoms in the cavity collectively interact with the cavity ﬁeld with the nearly same coupling strength has been reported . Furthermore, it should be noted that our scheme for generation of the GHZ state requires the atomic ensembles alternately interact with the cavity mode and the classical ﬁeld. As all the atoms are initially in the ground state, they are decoupled with the cavity mode if no classical ﬁelds are applied on them. Thus, the requirement for separately interacting is achievable with currently available technology.
5. Conclusion 0.7 0.65
Fig. 6. The ﬁdelity of the three-atomic-ensemble W state versus Γ=g with κ=g ¼ 0 and κ=g with Γ=g ¼ 0 with Δg ¼ 50g. 1
In summary, we have proposed a scheme for generating the W state and the GHZ state of multiple atomic ensembles trapped in a vacuum cavity. Compared with Refs. [24–26], our scheme does not depend on the photon detection. It is based on nothing else than the atom-cavity-mode dispersive interaction and the application of laser pulses. The required time for preparing the W state (the GHZ state) keeps unchanged (increases linearly) as the number of ensembles increases. Thus, our scheme is advantageous in terms of preparing entangled state of large-scale atomic ensemble, especially the W state among multiple atomic ensembles. The inﬂuence of the atomic spontaneous emission and the cavity decay on the ﬁdelities of the prepared states is also analyzed, which shows that the W state and the GHZ state can be generated with a high ﬁdelity even in the presence of the decoherence. Also, we have brieﬂy discussed the experimental feasibility of the proposed scheme.
Acknowledgments 0.9975 −0.1
−0.08 −0.06 −0.04 −0.02
ς Fig. 7. The ﬁdelity of the three-atomic-ensemble W state versus the deviations ς ¼ ðν 3ɛÞ=ν.
versus Γ =g and κ =g with Δg ¼ 50g. We see that the atomic spontaneous emission dominates the reduction of ﬁdelity, while the decay rates of the cavity inﬂuence the ﬁdelity slightly, which can be understood by the virtual excitation of the ﬁeld mode. In the above calculation, we have assumed that the resonant condition is satisﬁed by selecting the appropriate parameters, i.e., ν ¼ nɛ ðν ¼ ɛÞ for the n atomic-ensemble W state (GHZ state). But this assumption is difﬁcult to achieve in practice because it involves in many experimental parameters, i.e., Δ, Δg, Ω, ge, gc. Fig. 7 illustrates the ﬁdelity of the three-atomic-ensemble W state versus the deviation ς ¼ ðν 3ɛÞ=ν. We see that the ﬁdelity is always larger than 0.99 even ς is up to 0.1. Thus, our scheme is robust against moderate deviations of the resonant condition.
This work is supported by the Major State Basic Research Development Program of China (Grant no. 2012CB921601), National Natural Science Foundation of China (Grant no. 10974028), the Doctoral Foundation of the Ministry of Education of China (Grant no. 20093514110009). References      
J.S. Bell, Physics (Long Island City, NY) 1 (1965) 195. A.K. Ekert, Physical Review Letters 67 (1991) 661. C.H. Bennett, et al., Physical Review Letters 70 (1993) 1895. J.C. Hao, C.F. Li, G.C. Guo, Physical Review A 63 (2001) 054301. W. Dür, G. Vidal, J.I. Cirac, Physical Review A 62 (2000) 062314. D.M. Greenberger, et al., in: M. Kafatos (Ed.), Bells Theorem, Quantum Theory, and Conceptions of the Universe, Kluwer, Dordrecht, 1989; D.M. Greenberger, et al., American Journal of Physics 58 (1990) 1131.  S.B. Zheng, Physical Review Letters 87 (2001) 230404.  S.B. Zheng, Physics Letters A 372 (2008) 591.  W. Xiong, L. Ye, Optics Communications 284 (2011) 3659.  D. Gonta, S. Fritzsche, T. Radtke, Physical Review A 77 (2008) 062312.  C.P. Yang, Shih-I. Chu, S. Han, Physical Review A 86 (2012) 022329.
           
C.-L. Zhang et al. / Optics Communications 312 (2014) 269–274
C.P. Yang, Q.P. Su, S.B. Zheng, S. Han, Physical Review A 87 (2013) 022320. C.P. Yang, Shih-I. Chu, S. Han, Physical Review A 67 (2003) 042311. J.Q. You, F. Nori, Physical Review B 68 (2003) 064509. C.P. Yang, Shih-I. Chu, S. Han, Physical Review Letters 92 (2004) 117902. A. Blais, et al., Physical Review A 69 (2004) 062320. A. Zeilinger, Reviews of Modern Physics 71 (1999) 288. J.M. Raimond, M. Brtule, S. Haroehe, Reviews of Modern Physics 73 (2004) 565. D. Leibfried, et al., Nature 438 (2005) 639. G. Chen, et al., Science 289 (2000) 1906. S.Y. Ye, Z.R. Zhong, S.B. Zheng, Physical Review A 77 (2008) 014303. J. Song, Y. Xia, H.S. Song, Journal of Physics B 41 (2008) 065507. M.D. Lukin, M. Fleischhauer, R. Cote, Physical Review Letters 87 (2001) 037901.  L.M. Duan, Physical Review Letters 87 (2002) 170402.  P. Xue, G.C. Guo, Physical Review A 67 (2003) 034302.
 C. Han, P. Xue, G.C. Guo, Physical Review A 72 (2005) 034301.  L.M. Duan, J.I. Cirac, P. Zoller, Physical Review A 66 (2002) 023818.  J. Bollinger, et al., Physical Review A 54 (1996) 4649; S.F. Huelga, et al., Physical Review Letters 79 (1997) 3865.  Deterministic single-photon sources are crucial for physical implementation of a secure BB84 quantum cryptography protocol; see e.g., N. Lütkenhaus, Physical Review A 61 (2000) 052304.  S.B. Zheng, Z.B. Yang, Y. Xia, Physical Review A 81 (2010) 015804.  S.B. Zheng, Physical Review A 77 (2008) 045802.  A. Boca, et al., Physical Review Letters 93 (2004) 233603.  J.R. Buck, H.J. Kimble, Physical Review A 67 (2003) 033806.  S.M. Spillane, et al., Physical Review A 71 (2005) 013817.  J. McKeever, et al., Physical Review Letters 90 (2003) 133602.  Z.Q. Yin, F.L. Li, Physical Review A 75 (2007) 012324.