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Physics Letters A 361 (2007) 460–463 www.elsevier.com/locate/pla Generation of a four-atom entangled cluster state in cavity QED Liu Ye a,b,, Guang-Can Guo b a School of Physics & Material Science, Anhui University, Hefei 230039, People’s Republic of China b Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, People’s Republic of China Received 10 March 2006; received in revised form 28 July 2006; accepted 28 July 2006 Available online 7 August 2006 Communicated by B. Fricke Abstract We propose a method of generating a four-atom entangled cluster state by considering two kinds of the atoms–cavity field interaction in cavity QED. During the preparation the cavity is only virtually excited no quantum information will be transferred from the atoms to the cavity and thus the scheme is insensitive to the cavity field states and cavity decay. The scheme can also be used to generate the cluster state for the trapped ions. © 2006 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 32.80.Qk; 42.50.Dv Recently, much attention has been directed to the generation of multi-particle entangled states, such as GHZ states [1,2] not only can provide much stronger refutations of local realism and reveal a contradiction with local hidden variable theory from a single set of measurements, but also are useful in quantum information processing. In Ref. [3], Briegel et al. introduced a class of entangled states, i.e., the cluster states. The cluster states can be regarded as a resource for GHZ states and are more immune to decoherence than GHZ states. On the other hand, cluster states have been shown to constitute a universal resource for quantum computation. The proof of Bell’s the- orem without the inequality was given for cluster states, and Bell inequality is maximally violated by four-qubit cluster state and is not violated by the four-qubit GHZ state. Recently Zou et al. [4] have proposed a scheme for generation of polariza- tion entangled cluster state. As one of possible candidates for engineering quantum entanglement, the cavity quantum elec- trodynamics system has made many applications in quantum information processing [5–7]. Thus how to prepare a multi- particle entangled state in cavity QED has abstracted much attention. The generation of the GHZ state of three particles has * Corresponding author. E-mail address: [email protected] (L. Ye). been demonstrated experimentally in high-Q cavities [8]. Fidio and Vogel [9] presented a scheme for preparing W state of three trapped atoms in leaky cavities. Guo et al. [10] have proposed a scheme to generate the multi-particle entangled states for atoms interacting dispersively with a vacuum cavity. Zheng [11] also has proposed a scheme to generate N -atom GHZ states. Schön et al. [12] have introduced a method of the generation of en- tangled multiqubit states. Zou et al. [13] presented a scheme to generate the W states, GHZ states and cluster states of four dis- tant atoms trapped separately in leaky cavities with the certain probability. Recently Zou et al. [14] and Cho et al. [15] also have shown a method for the generation of cluster states, re- spectively. Here we propose a scheme to generate a cluster state of four atoms by the atom–cavity field interaction. The distinct advantage of the scheme is that during the operation, the cav- ity is only virtually excited and thus the effective decoherence time of the cavity is greatly prolonged. In addition the scheme does not require individual addressing of the atoms, the entan- gled cluster state can be generated in a simply manner with the probability of success 1. We consider m identical two-level atoms simultaneously in- teracting with a single-mode cavity field. The Hamiltonian for the system is (1) H = H 0 + H i , 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.07.070

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Page 1: Generation of a four-atom entangled cluster state in cavity QED

Physics Letters A 361 (2007) 460–463

www.elsevier.com/locate/pla

Generation of a four-atom entangled cluster state in cavity QED

Liu Ye a,b,∗, Guang-Can Guo b

a School of Physics & Material Science, Anhui University, Hefei 230039, People’s Republic of Chinab Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences,

Hefei 230026, People’s Republic of China

Received 10 March 2006; received in revised form 28 July 2006; accepted 28 July 2006

Available online 7 August 2006

Communicated by B. Fricke

Abstract

We propose a method of generating a four-atom entangled cluster state by considering two kinds of the atoms–cavity field interaction in cavityQED. During the preparation the cavity is only virtually excited no quantum information will be transferred from the atoms to the cavity and thusthe scheme is insensitive to the cavity field states and cavity decay. The scheme can also be used to generate the cluster state for the trapped ions.© 2006 Elsevier B.V. All rights reserved.

PACS: 03.67.-a; 32.80.Qk; 42.50.Dv

Recently, much attention has been directed to the generationof multi-particle entangled states, such as GHZ states [1,2] notonly can provide much stronger refutations of local realism andreveal a contradiction with local hidden variable theory froma single set of measurements, but also are useful in quantuminformation processing. In Ref. [3], Briegel et al. introduceda class of entangled states, i.e., the cluster states. The clusterstates can be regarded as a resource for GHZ states and aremore immune to decoherence than GHZ states. On the otherhand, cluster states have been shown to constitute a universalresource for quantum computation. The proof of Bell’s the-orem without the inequality was given for cluster states, andBell inequality is maximally violated by four-qubit cluster stateand is not violated by the four-qubit GHZ state. Recently Zouet al. [4] have proposed a scheme for generation of polariza-tion entangled cluster state. As one of possible candidates forengineering quantum entanglement, the cavity quantum elec-trodynamics system has made many applications in quantuminformation processing [5–7]. Thus how to prepare a multi-particle entangled state in cavity QED has abstracted muchattention. The generation of the GHZ state of three particles has

* Corresponding author.E-mail address: [email protected] (L. Ye).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.07.070

been demonstrated experimentally in high-Q cavities [8]. Fidioand Vogel [9] presented a scheme for preparing W state of threetrapped atoms in leaky cavities. Guo et al. [10] have proposed ascheme to generate the multi-particle entangled states for atomsinteracting dispersively with a vacuum cavity. Zheng [11] alsohas proposed a scheme to generate N -atom GHZ states. Schönet al. [12] have introduced a method of the generation of en-tangled multiqubit states. Zou et al. [13] presented a scheme togenerate the W states, GHZ states and cluster states of four dis-tant atoms trapped separately in leaky cavities with the certainprobability. Recently Zou et al. [14] and Cho et al. [15] alsohave shown a method for the generation of cluster states, re-spectively. Here we propose a scheme to generate a cluster stateof four atoms by the atom–cavity field interaction. The distinctadvantage of the scheme is that during the operation, the cav-ity is only virtually excited and thus the effective decoherencetime of the cavity is greatly prolonged. In addition the schemedoes not require individual addressing of the atoms, the entan-gled cluster state can be generated in a simply manner with theprobability of success 1.

We consider m identical two-level atoms simultaneously in-teracting with a single-mode cavity field. The Hamiltonian forthe system is

(1)H = H0 + Hi,

Page 2: Generation of a four-atom entangled cluster state in cavity QED

L. Ye, G.-C. Guo / Physics Letters A 361 (2007) 460–463 461

where

(2)H0 = ωa+a + ω0Sz,

(3)Hi = g

m∑j=1

(a+s−

j + as+j

),

where Sz = 12

∑mj=1(|ej 〉〈ej | − |gj 〉〈gj |), s+

j = |ej 〉〈gj | and

s−j = |gj 〉〈ej |, with |ej 〉 and |gj 〉 being the excited and ground

states of the j th atom, a+ and a are, respectively, the creationand annihilation operators for the cavity mode, g is the atom–cavity coupling strength, and δ is the detuning between theatomic transition frequency ω0 and cavity frequency ω. In thecase δ � g, there is no energy exchange between the atomicsystem and the cavity. The Rabi frequency λ for the transitionsbetween these states, mediated by |gjgkn + 1〉 and |ej ekn − 1〉is given by [11,16]

λ = 〈ejgkn|Hi |gjgkn + 1〉〈gjgkn + 1|Hi |gj ekn〉δ

+ 〈ejgkn|Hi |ej ekn − 1〉〈ej ek − 1|Hi |gj ekn〉−δ

= g2

δ.

Then the effective Hamiltonian is

He = λ

[(n + 1)

m∑j=1

|e〉j j 〈e| − n

m∑j=1

|g〉j j 〈g|

(4)+m∑

i,j=1,i �=j

s+j s−

i

],

where λ = g2/δ, n is the photon number of the cavity field ini-tially in the Fock state |n〉.

Especially consider that the cavity field is initially in the vac-uum state, the Hamiltonian reduces to [11]

(5)He = λ

(m∑

j=1

|e〉j j 〈e| +m∑

i,j=1,i �=j

s+j s−

i

).

Assume that four atoms are initially in the state |eeee〉, eachatom crosses a classical field and undergoes a single-qubit ro-tation |ej 〉 → (1/

√2 )(|gj 〉 + i|ej 〉). In this case four atoms are

sent into an initially vacuum cavity. After an interaction time t ,the state evolution of the system is

(6)∣∣ψ(t)

⟩ = 1

4

4∑k=0

ike−ik(5−k)λt |φk〉,

where

(7)|φ0〉 = |gggg〉1234,

(8)|φ1〉 = |eggg〉1234 + |gegg〉1234 + |ggeg〉 + |ggge〉,

(9)

|φ2〉 = |eegg〉1234 + |egeg〉1234 + |egge〉 + |geeg〉+ |gege〉 + |ggee〉,

(10)|φ3〉 = |eeeg〉1234 + |geee〉1234 + |egee〉 + |eege〉.

With the choice λt = π/2, we can obtain the state

(11)|Ψ 〉1234 = 1

4√

2

[4∏

j=1

(|g〉j + |e〉j) − i

4∏j=1

(|g〉j − |e〉j)]

,

where a common phase factor eiπ/4 is discarded.After a single-qubit rotation, we can rewrite Eq. (11) into a

four-atom GHZ state

(12)|Ψ 〉1234 = 1√2

(|gggg〉 − i|eeee〉).Then we consider another model of the atoms–cavity field

interaction in cavity QED. Assume two identical two-levelatoms simultaneously interacting with a single-mode cavityfield, at the same time the atoms are driven by a classical field.In the rotating-wave approximation, the Hamiltonian for thesystem is [17–20]

H = ω0Sz + ωa+a +2∑

j=1

[g(a+S−

j + aS+j

)

(13)+ Ω(S+

j e−iωat + S−j eiωat

)],

where ωa is the frequency of the classical field. Assume ω0 =ωa , in the interaction picture, the interaction Hamiltonian is

(14)HI = Ω∑

j=1,2

(S+

j + S−j

) + g

2∑j=1

(e−iδt a+S−

j + eiδt aS+j

).

We assume in the strong driving regime Ω � δ, g and canneglect the terms oscillating fast. In the case δ � g, there isno energy exchange between the atomic system and the cavity.Then in the interaction picture, the effective interaction Hamil-tonian reads [21]

He = χ

2

[2∑

j=1

(|e〉j j 〈e| + |g〉j j 〈g|)

(15)+2∑

i,j=1,i �=j

(S+

i S−j + S+

i S+j + H.c.

)],

where χ = g2/2δ. We note that the effective Hamiltonian is in-dependent of the cavity field state, allowing it to be in a thermalstate.

Then the evolution operator of the system is given by

U(t) = e−iH0t e−iHet ,

where

H0 = Ω∑

j=1,2

(S+

j + S−j

).

In order to generate a four-atom entangled cluster state. Wesend any two of four atoms, such as the atoms 3, 4 of Eq. (12)interact simultaneously with a single-mode cavity, at the sametime the atoms are driven by a classical field. After an interac-

Page 3: Generation of a four-atom entangled cluster state in cavity QED

462 L. Ye, G.-C. Guo / Physics Letters A 361 (2007) 460–463

tion time τ we have

|Ψ 〉1234 = e−i2χτ

√2

(|gg〉12{cos(χτ)

[cos(Ωτ)|g〉3

− i sin(Ωτ)|e〉3][

cos(Ωτ)|g〉4 − i sin(Ωτ)|e〉4]

− i sin(χτ)[cos(Ωτ)|e〉3 − i sin(Ωτ)|g〉3

]× [

cos(Ωτ)|e〉4

− i sin(Ωτ)|g〉4]} − i|ee〉12

{cos(χτ)

[cos(Ωτ)|e〉3

− i sin(Ωτ)|g〉3][

cos(Ωτ)|e〉4 − i sin(Ωτ)|g〉4]

− i sin(χτ)[cos(Ωτ)|g〉3 − i sin(Ωτ)|e〉3

](16)× [

cos(Ωτ)|g〉4 − i sin(Ωτ)|e〉4]})

.

With the choice of Ωτ = π , χτ = π/4, we obtain the maxi-mally four-atom entangled state

(17)|Ψ 〉1234 = 1

2

(|gggg〉 − i|ggee〉 − |eegg〉 − i|eeee〉),where a common phase factor e−iπ/2 is discarded.

Using local operation, we can transformed the state (17) intothe entangled cluster state

(18)|Ψ 〉1234 = 1

2

(|gggg〉 + |ggee〉 + |eegg〉 − |eeee〉).We note that the idea can also be applied to the ion trap

system. Sackett et al. [22] have produced the maximally en-tangled state (1/

√2 )(|gggg〉 − i|eeee〉) with four ions. Then

we consider that N ions are confined in a linear trap. The ionsare simultaneously excited with two lasers. In the Lamb–Dickeregime, the interaction Hamiltonian in the interaction picture is[16,19–21]

(19)Hi = iηΩe−iφ2∑

j=1

S+j

(a+e−iδt + aeiδt

) + H.c.,

where η is the Lamb–Dicke parameter. Here assume that thelasers have the same Rabi frequencies Ω . In the case δ � ηΩ ,there is no energy exchange between the external and internaldegrees of freedom. The effective Hamiltonian has same formas Eq. (15), with χ = 2η2Ω2/δ. Thus we can generate the en-tangled cluster states of four trapped ions using the proceduresimilar to that for cavity QED.

Next we give a brief discussion on the experimental mat-ters. The coupling strength between the atoms and the cavitydepends on the atomic positions g = Ωe−r2/ω2

, where Ω is thecoupling strength at the cavity center, ω is the waist of the cav-ity mode, and r is the distance between the atom and the cavitycenter [6]. If we assume that all the atoms go along the r direc-tion through antinodes, then the atom–cavity coupling strengthfor each atom is identical [23]. In addition the two-atom en-tangled state has also been prepared by such an atom–cavityfield interaction and has been experimentally realized [6]. Forthe influence of spontaneous atomic emission, its effect may benegligible if we choose long lived atomic levels as consideredby Munhoz et al. [24]. For the Rydberg atoms, the radiativetime is about Tr = 3 × 10−2 s, and the coupling constant is

g = 2π × 24 kHz [5]. The required atom–cavity–field interac-tion time is on the order of T ≈ 10−4 s. Then the time needed tocomplete the whole procedure is much shorter than Tr . Mean-while it is noted that the atomic state evolution is independentof the cavity field state, thus the cavity decay will not affect thegeneration of the entangled states.

In conclusion, we have proposed a simple protocol to realizethe generation of the four-atom entangled cluster states in cav-ity QED. In the scheme a maximally four-atom entangled stateis firstly prepared by simultaneously interacting with a single-mode cavity for the four atoms. Then the arbitrary two atomsof the four-atom GHZ state again simultaneously interact withanother single-mode cavity with the help of a strong classicaldriving field. In this case by using appropriate local operationthe generation of a four-atom entangled cluster state is finishedwith the success probability 1. During the whole preparationour scheme only involves atom–field interaction with a large de-tuning and does not require the transfer of quantum informationbetween the atoms and cavity. The requirement on the qualityfactor of the cavities is greatly loosened. In addition the schemedoes not require individual addressing of the atoms. Thus it canbe implemented by the present cavity QED techniques.

Acknowledgements

This work was supported by the National Science Foun-dation of China under Grant No. 10574001, and also by theProgram of the Education Department of Anhui Province(2004kj029) and the Talent Foundation of Anhui University.

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