Optics Communications 284 (2011) 1094–1098

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Optics Communications

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Preparation of Greenberger–Horne–Zeilinger and W states of three atoms trapped inone cavity through cavity output process

Yan Xia a,b,⁎, Jie Song b, Pei-Min Lu a, He-Shan Song b

a Department of Physics, Fuzhou University, Fuzhou 350002, Chinab School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China

⁎ Corresponding author. Department of Physics, FuzhChina. Tel.:+86 591 22865133.

E-mail address: xia-208@163.com (Y. Xia).

0030-4018/$ – see front matter. Crown Copyright © 20doi:10.1016/j.optcom.2010.10.036

a b s t r a c t

a r t i c l e i n f o

Article history:Received 9 July 2010Received in revised form 17 September 2010Accepted 9 October 2010

Keywords:Entangled statesThree-level ∧-type atomsLinear optical elements

We propose a protocol to generate Greenberger–Horne–Zeilinger (GHZ) andW states of three atoms trappedin only one cavity. The setup involves one cavity and linear optical elements. The quantum information of eachqubit is skillfully encoded on the degenerate ground states of the three different atoms, hence theentanglement between them is relatively stable against spontaneous emission. The advantages of the protocolare their robustness against detection inefficiency and asynchronous emission of the photons. We discuss theissue related to the practical implementation and show that the protocol is accessible within the currentcavity QED technology and linear optical technology.

Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction

Quantum entanglement is a resource for quantum informationprocessing and quantum computing. Entangled states of two or manyqubits not only give the possibility to test quantummechanics against alocal hidden variable theory [1,2], but also have practical applications inrealizing quantum information processing protocols, such as quantumteleportation [3,4], quantum secret sharing [5], quantum cryptography[6], quantum secure direct communication [7,8], quantum cloningmachine [9], and so on. These concepts motivated and intensiveresearch in the generation and the manipulation of entangled states.

For tripartite systems, it has been known that there exist at leasttwo different types of multipartite entanglement: namely, theGreenberger–Horne–Zeilinger-type (GHZ-type) entanglement [10]and the W-type entanglement [11]. These two different types ofentanglement are not equivalent and cannot be converted to eachother by local unitary operations combined with classical communi-cation. As two classes of important quantum resources, great efforthas been taken to studying of entangled state generation in the pastyears [12–24]. For example, in Refs. [12], the authors have proposedprotocols to generate multi-particle entangled state in cavity. In Ref.[13], the authors have proposed a protocol to generate N distantphotons GHZ state with linear optical elements. On the other hand,many works have been proposed to generate entangled states usingthe combination of separate cavities and optics elements [17–23]. Forexample, in Ref. [17], the authors have proposed protocols to generate

ou University, Fuzhou 350002,

10 Published by Elsevier B.V. All rig

of multi-particle entangled state with separate cavities and linearoptical elements. Very recently, Su et al. [22] have presented the firstexperimental results generating continuous variable quadripartiteGHZ entangled state of electromagnetic fields. The approach based onindistinguishability was shown to have a lot of advantages, amongthem the robustness is the most distinct one.

Experimentally, it is difficult to generate multi-atom entangledstates with only cavity technology and the inevitable interactionbetween system and environment will destroy the system quantumcoherence, i.e., causing decoherence. In particular, the partial quantuminformation of a qubit is encoded on the excited state of atoms (ions) insome protocols, which means that the entanglement of the qubits isfragile (not stable). If two- or more atoms trapped in only one multi-mode cavity, and we want to generate atoms entangled states with thehelp of linear optical elements, how can we do this? This problem hasnot been addressed.

We propose a simple protocol to generate stable maximallyentangled GHZ andW states of three atoms trapped in only onemulti-mode cavity with the help of linear optical elements. The realization ofthis protocol is appealing due to the fact that quantum state of light isrobust against the decoherence and photons are ideal carriers fortransmitting quantum information over long distances. The protocolis based on the combination of the atom-cavity interaction and linearoptics elements. The success of the protocol depends upon thedetection of a photon leaking out of the cavity, and thus the fidelity isalso not affected by the imperfection of the photon detectors.

The model we are considering consists of three different Λ-typethree-level atoms (Fig. 1), with the three atoms (1,2,3) are trapped inone three-mode optical cavity A, as shown in Fig. 2. All the three atomshave one degenerate excited states |e⟩j (j=1,2,3), two degenerate

hts reserved.

a1l a1

r

|gl>1 |gr>1

|e>1

|e>3

|e>2

|gl>3|gl>2 |gr>3|gr>2

a3la2

la3

ra2r

Fig. 1. Atomics level structures.

1095Y. Xia et al. / Optics Communications 284 (2011) 1094–1098

ground states |gl⟩j and |gr⟩j, as shown in Fig. 1. The quantuminformation is encoded on the states |gl⟩ and |gr⟩. The three atoms(1,2,3) transition |e⟩j→ |gl⟩j and |e⟩j→ |gr⟩j are strongly coupled withleft and right circularly polarizing cavity modes Fj, respectively. Thefrequencies of the cavity modes are different but they are no betterthan each other, that is F1≈F2≈F3. The atomic level structure can beachieved by Zeeman sublevels [25] and has been realized to entangletwo atoms [26]. We suppose the three atoms are all initially preparedin their excited states and cavity in the vacuum state. We require herethat the cavity is one-sided such that the photons leakage occursthrough the side of the cavity facing the linear optical elements. TheHamiltonian governing the evolution of the atom-cavity systems canbe given in the interaction picture by (setting ħ=1)

H = ∑j=1;2;3

λjLa

jL je⟩jj⟨gl j + λj

RajR je⟩jj⟨gr j + h:c:

� �; ð1Þ

where L, R denote the left- and right-circularly polarizing cavity modesF1, F2, and F3, ak

j+, akj (k=R,L) are the creation and annihilation

operators of the jmode in the cavity A and λkj are the coupling constants(atom 1 vsmode F1, and atom 2 vsmode F2, and atom 3 vsmode F3, Thefrequencies of the cavitymodes are different but they are no better thaneach others, that is F1≈F2≈F3. The atoms and the cavity are preparedinitially in its excited states |eee⟩123 and vacuum states |00⟩lr

j ,respectively. The upper levels |eee⟩123 can decay to the two degenerateground states |g⟩lj and |g⟩rj with the rates 2γl

j and 2γrj, respectively, and

the cavity has a leakage rate 2κ. Hence, the master equation describingthe evolution of density operator ρ (atom and cavity) is given by

ρ̇ = −i Heffρ−ρH†eff

� �+ 2κ ∑

j=1;2;3ajLρa

j†L + ajRρa

j†R

� �+ 2 ∑

j=1;2;3γjl jgl⟩jj⟨e jρ je⟩jj⟨gl j + γj

r jgr⟩jj⟨e jρ je⟩jj⟨gr j� �

;ð2Þ

Fig. 2. Experimental setup for generation of three atoms GHZ state. The three atoms aretrapped in only one cavity. PBS is polarizing beam splitter, HWP is half-wave plate, QWPis quarter-wave plate, PNS is photon number splitter (PNS :50/50) and D is detector.

where

Heff = H−iκ ∑j=1;2;3

aj†L a

jL + aj†

R ajR−i γ j

l + γ jr

� �je⟩jj⟨e j

h i: ð3Þ

If thewhole state of the initial state is given by |eee⟩123|000⟩M1M2M3,and after a enough long time t, the whole state of the system willbecome (assumed pure state for simplicity)

jψ tð Þ⟩j = ∏j=1;2;3

xj je⟩j j0l⟩j j0r⟩j + yj jgl⟩j j1l0r⟩j + zj jgr⟩j j0l1r⟩jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijxj2 + jyj2 + jzj2

q0B@

1CA;

ð4Þ

where

xj = e−γ j

L + γ jR + κ

2t ½cos t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ jr

� �24

vuut0BB@

1CCA

+κ−γ j

l−γ jr

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γj

l−γjr

� �24

vuutsin t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ jr

� �24

vuut0BB@

1CCA�;ð5Þ

yj = −e−γ jL

+ γ jR

+ κ

2 t

sin t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

r !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γj

l−γ j

rð Þ24

r λjL; ð6Þ

zj = −e−γ jL

+ γ jR

+ κ

2 t

sin t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

r !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

r λjR: ð7Þ

After the transformation the atom–cavity interaction is frozensince Heffj|ψ(t)⟩j=0. Now we wait for the photodetectors to click. Wesuppose the evolution time of every subsystems (atom 1 and cavitymode F1, atom 2 and cavitymode F2 and atom 3 and cavitymode F3) tobe τj (τ1=τ2=τ3). So, in such an interval of time, one can obtain thestate |ψτ⟩j given by Eq. (4) with the probability

Pj = e− γjL + γj

R + κð Þτjf½cos τj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ jr

� �24

vuut0BB@

1CCA

+κ−γ j

l−γ jr

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ jr

� �24

vuutsin τj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ jr

� �24

vuut0BB@

1CCA�2

+

sin2 τj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ jr

� �24

vuut0BB@

1CCA

λj2

L + λj2

R−κ−γ j

l−γ jr

� �24

λj2

L + λj2

R

� �g:ð8Þ

Thus, the joint state of the three atoms and three modes can begiven by P1=∏Pj. Because the vacuum state has no contribution tothe click of the photon-detectors, so the term |e⟩j|0l⟩j|0r⟩j in Eq. (4) can

1096 Y. Xia et al. / Optics Communications 284 (2011) 1094–1098

be safely post selected for simplification. Therefore, after evolutiontime τj, the total state of the photons and the atoms can be written by(unnormalized)

jϕ⟩123 = λ1L jgl⟩1 j1l0r⟩1 + λ1

R jgr⟩1 j0l1r⟩1� �

λ2L jgl⟩2 j1l0r⟩2 + λ2

R jgr⟩2 j0l1r⟩2� �

× λ3L jgl⟩3 j1l0r⟩3 + λ3

R jgr⟩3 j0l1r⟩3� �

;

ð9Þ

with the total probability

P2 = ∏j

e− γjL + γj

R + κð Þτjsin2 τj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

r !

λj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

λj2

L + λj2

R

� �266664

377775:

ð10Þ

From Fig. 1 we can see that the transition frequencies (f) of thethree atoms are different, so the photons' frequencies of the threeatoms spontaneous emission are different, but they are no better thaneach others, that is f1≈ f2≈ f3.

GHZ states [see Fig. 2]. One can find that photons leaking out of thecavity will first pass through the photon number splitters (PNSs)(PNS :50/50) (for example, grating and prism), which can split theinput pulses, according to the photon numbers, in different modes(b1, b2, and b3), respectively. We can see from Fig. 2 that photons inmodes b1, b2, and b3 will pass through quarter-wave plates (QWP),respectively. QWP can change circularly polarizing light into linearlypolarizing light. That is, left-polarized and right-polarized photonsbecome vertically (V) and horizontally (H) polarized photons,respectively. That is, |1l0r⟩j→ |V⟩j and |0l1r⟩j→ |H⟩j, so Eq. (9) will bechanged into

jϕ⟩123 = λ1L jgl⟩1 jV⟩1 + λ1

R jgr⟩1 jH⟩1� �

λ2L jgl⟩2 jV⟩2 + λ2

R jgr⟩2 jH⟩2� �

× λ3L jgl⟩3 jV⟩3 + λ3

R jgr⟩3 jH⟩3� �

;

ð11Þ

after normalization, the state in Eq. (11) will be

jϕ⟩123 =1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λ12L + λ12

R

q 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ22L + λ22

R

q 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ32

L + λ32

R

q× λ1

L jgl⟩1 jV⟩1 + λ1R jgr⟩1 jH⟩1

� �λ2L jgl⟩2 jV⟩2 + λ2

R jgr⟩2 jH⟩2� �

× λ3L jgl⟩3 jV⟩3 + λ3

R jgr⟩3 jH⟩3� �

:

ð12Þ

Photons in modes b1 and b2, (suppose photon 1 in mode b1 andphoton 2 in mode b2) will first meet polarizing beam splitter 1 (PBS1),which always transmit H polarizing photons and reflects V polarizingphotons. Thus the joint states including atoms 1, 2 and photons 1, 2will become

12

jgl⟩1 jV⟩1 + jgr⟩1 jH⟩1ð Þ jgl⟩2 jV⟩2 + jgr⟩2 jH⟩2ð Þ

→12ð jgl⟩1 jgl⟩2 jV⟩d jV⟩c1 + jgl⟩1 jgr⟩2 jV⟩d jH⟩d

+ jgr⟩1 jgl⟩2 jH⟩c1 jV⟩c1 + jgr N ⟩1 jgr⟩2 jH⟩c1 jH⟩dÞ;

ð13Þ

where we suppose λL=λR [17]. If we want to prepare the GHZ state,both of the modes c1 and d are not idle, otherwise one of the requireddetectors cannot be clicked, hencewe have discarded the two photonsin one mode outcomes in Eq. (13) and preserved the two photons intwo modes outcomes with the probability P3=50%. This procedure

can be realized by post-selection method. Thus, after passing throughthe PBS1, the state given by Eq. (13) becomes

1ffiffiffi2

p jgl⟩1 jgl⟩2 jV⟩d jV⟩c1 + jgr⟩1 jgr⟩2 jH⟩c1 jH⟩d� �

: ð14Þ

Then the photon in mode d and that photon in mode b3 will meetPBS2. As a result, the joint state of the whole system follows that

12

jgl⟩3 jV⟩b3 + jgr⟩3 jH⟩b3� �

jgl⟩1 jgl⟩2 jV⟩d jV⟩c1 + jgr⟩1 jgr⟩2 jH⟩c1 jH⟩d� �

→1ffiffiffi2

p jgl⟩1 jgl⟩2 jgl⟩3 jV⟩c1 jV⟩c2 jV⟩c3 + jgr⟩1 jgr⟩2 jgr⟩3 jH⟩c1 jH⟩c2 jH⟩c3� �

;

ð15Þ

where we only preserve the antibunching outcomes with theprobability P4=50%. At last, the three photons with different modeswill, respectively, meet three HWPs and three PBSs. After passingthrough the HWPs and PBSs, if three detectors of the different modesof c1, c2, and c3 are clicked, the atoms 1, 2, and 3 will collapse to oneof the GHZ sates 1ffiffi

2p jgr⟩1 jgr⟩2 jgr⟩3F jgl⟩1 jgl⟩2 jgl⟩3ð Þ, where an odd

number of Dx, 1 clicked corresponds to “+”, otherwise, “–”. The totalprobability of getting the state is given by

PGHZ =14∏j

e− γ jL + γ j

R + κð Þτjsin2 τj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

r !

λj2

L + λj2

R−κ−γ j

l−γ j

rð Þ24

λj2

L + λj2

R

� �266664

377775:

ð16Þ

We have successfully realized three atoms GHZ state preparationwith the help of linear optical elements.

W state [see Fig. 3]. After photons leaking out of the cavity, theywill meet QWP. As a result, when the three photons are transmittedout of the output port of QWP, the joint state of the whole system is

jϕ⟩123 =1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λ12L + λ12

R

� �λ22

L + λ22

R

� �λ32L + λ32

R

� �r

× λ1L jgl⟩1 jV⟩1 + λ1

R jgr⟩1 jH⟩1� �

λ2L jgl⟩2 jV⟩2 + λ2

R jgr⟩2 jH⟩2� �

× λ3L jgl⟩3 jV⟩3 + λ3

R jgr⟩3 jH⟩3� �

:

ð17Þ

Then the photonswill meet at PBS1, whichwill transform the inputmodes to the output modes as (if we set λL=λR [17])

12ffiffiffi2

p jgl⟩1 jV⟩1 + jgr⟩1 jH⟩1ð Þ jgl⟩2 jV⟩2 + jgr⟩2 jH⟩2ð Þ jgl⟩3 jV⟩3 + jgr⟩3 jH⟩3ð Þ

=1

2ffiffiffi2

p ½jgl⟩1 jgl⟩2 jgl⟩3 jV1⟩a jV2⟩a jV3⟩a + jgr⟩1 jgr⟩2 jgr⟩3 jH1⟩b jH2⟩b jH3⟩b

+ jgl⟩1 jgl⟩2 jgr⟩3 jV1⟩a jV2⟩a jH3⟩b + jgl⟩1 jgr⟩2 jgl⟩3 jV1⟩a jH2⟩b jV3⟩a

+ jgl⟩1 jgr⟩2 jgr⟩3 jV1⟩a jH2⟩b jH3⟩b + jgr⟩1 jgl⟩2 jgl⟩3 jH1⟩b jV2⟩a jV3⟩a

+ jgr⟩1 jgr⟩2 jgl⟩3 jH1⟩b jH2⟩b jV3⟩b + jgr⟩1 jgl⟩2 jgr⟩3 jH1⟩b jV2⟩a jH3⟩b�

=1

2ffiffiffi2

p ½jgl⟩1 jgl⟩2 jgl⟩3 jVVV⟩a + jgr⟩1 jgr⟩2 jgr⟩3 jHHH⟩b

+ jgl⟩1 jgl⟩2 jgr⟩3 + jgl⟩1 jgr⟩2 jgl⟩3 + jgr⟩1 jgl⟩2 jgl⟩3ð Þ jH⟩b jVV⟩a

+ jgl⟩1 jgr⟩2 jgr⟩3 + jgr⟩1 jgr⟩2 jgl⟩3 + jgr⟩1 jgl⟩2 jgr⟩3ð Þ jHH⟩b jV⟩a�:ð18Þ

After passing through the PBS1, paths a and b will be detectedby two polarization analyzers (PA) [27], respectively. Polarizationanalyzers (PA, single photon detectors followed PBS1 shown in the

Fig. 3. Schematic diagram for generating of three atoms W state.

1097Y. Xia et al. / Optics Communications 284 (2011) 1094–1098

dashed box in Fig. 3) on each path fulfill the task of polarizationmeasurement on photons 1, 2, and 3. It is clear from Eq. (18) that aW-state can always be obtained if D1 register two V polarizingphotons an one H polarizing photon or D2 register two H polarizingphotons and one V polarizing photon. All other measurement resultscorrespond to inconclusive events.

The protocol given here is similar in spirit to that of Feng et al. [17].Both protocols are based on the interference effect of light to generateentangled states rather than on the effective interaction between theatoms. There are two differences between the two protocols: (1) inpresent protocol, all different atoms are trapped in only one cavity,but in Ref. [17], atoms are trapped in distant cavities. (2) The detectorsare different. In Ref. [17], they need conventional photon detectors,but in our protocol, polarization analyzers are used. The realization ofour protocol is appealing due to the fact that photons are ideal carriersfor transmitting quantum information over long distances, and theatoms are good memorizers for storing information long time. So webelieve that the combination of the atom-cavity interaction and linearoptics elements is a good idea to generate entangled states.

In an experimental scenario, the atomic level structure can beachieved by Zeeman sublevels [25]. The similar optical setups havebeen used to successfully prepare W (GHZ) states of photons inexperiment [28,29]. Experimental techniques for single-photondetection have made tremendous progress [30]. A photon detectorbased on a visible light photon counter has been reported, which candistinguish between single-photon incidence and two-photon inci-dence with high quantum efficiency, good time resolution, and lowbit-error rate [31]. In Lamb–Dicke limit it is not necessary to require asimultaneous click of the detectors, which will relax the requirementon the physical realization. Therefore, the setup in our protocol arefeasible by current technologies. But we must point out that thereexist some negative effects. For example: (1) in three-mode cavity,not all the atoms are initially prepared in their excited states, and it isdifficult to realize three atoms are coupled with three modes in onecavity, respectively. (2) Some photons are absorbed by cavity walls,even optical elements. (3) The imperfection of interferometers anddetectors. (4) It is difficult to trap three atoms in one cavity. Althoughthere has been no report about the strong interaction betweenmultimode field and the multilevel atoms experimentally yet, manyschemes were proposed for kinds of quantum information processingby using the cavity supporting three modes [32,33]. Meanwhile, withthe development of the cavity techniques, strong coupling can berealized by many ways [34,35]. As a consequence, it's a promisingscheme in the future.

For GHZ state, in the protocol we set λL=λR [17], the fidelity of thefinal state |ψ⟩ is F=|⟨ψ|GHZ⟩ |2≃0.99. But in fact, if λL

λR= 1:1, the

fidelity is |⟨ψ|GHZ⟩ |2N0.98, which shows slight influence. For Wstate, one can find that different λL and λR only reduce the efficiencybut the fidelity is not influenced at all. What is more, the inefficientdetections leading to less clicks of the detectors only reduce thesuccess probability instead of fidelity, so does the failure ofinitialization of the initial states of atoms and cavities. As mentioned

in Ref. [18], because the photons from spontaneous emissions to freemodes run with random directions, they cannot be registered by thedetectors. Thus the fidelity is not influenced too.

In summary, we have presented a simple protocol to generatethree atoms GHZ and W states with only one cavity and linear opticalelements. Since the quantum information is encoded on thedegenerate ground states of the three atoms, the entanglement ofthe three atoms is relatively stable. The fidelity is independent of theinefficient detections, spontaneous emissions and so on. They canreadily be generalized to entangle multipartite distant atoms inprinciple, but the efficiency would be reduced. We hope that with thedevelopment of technology in experiment it may be possible toimplement the protocol with ease.

Acknowledgements

The authors are grateful to anonymous referees for his/hervaluable comments. This work was supported by the Natural ScienceFoundation of Fuzhou University of China under Grant No. 022264and No. 2010-XQ-28, the National Natural Science Foundation ofFujian Province of China under Grant No. 2010J01006, the NationalNatural Science Foundation of China under Grant No. 10875020 and10974028, Doctoral Foundation of the Ministry of Education of Chinaunder Grant No. 20070386002, the fund from State Key LaboratoryBreeding Base of Photocatalysis, Fuzhou University, the funds fromEducation Department of Fujian Province of China under Grant No.JB08010, No. JA10009 and No. JA10039, and China PostdoctoralScience Foundation under Grant No. 20100471450.

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