Int J Theor Phys (2013) 52:3086–3091DOI 10.1007/s10773-013-1600-9

Preparing a Genuinely Entangled Six-Atom Statein Cavity QED

Yue Xu · Yuan-hua Li · Li-ping Nie · Xiao-lan Li

Received: 9 January 2013 / Accepted: 3 April 2013 / Published online: 20 April 2013© Springer Science+Business Media New York 2013

Abstract We propose a scheme for preparing a genuinely entangled six-atom state (in J.Phys. A 40:13407, 2007) in cavity QED, where the atoms interact simultaneously with thehighly detuned single-mode cavity and the strong classical driving field. Thus our scheme isnot sensitive to both the cavity decay and thermal field.

Keywords Cavity QED · Preparation proposal · Genuinely entangled six-atom state

1 Introduction

Quantum entanglement is the key resource for implementing quantum information process-ing protocols including quantum teleportation [1], quantum dense coding [2], geometricquantum computation [3–5], and quantum information splitting [6–12]. In recent years, dif-ferent types of multi-particle entangled states have already been proposed and extensivelyexplored in different systems [13–16], such as GHZ state [17], W state [18] and clusterstate [19].

In 2007, Borras et al. [20] first introduced a genuine six-qubit entangled state, which isgiven by

|ψ〉123456 =√

2

8

(|000000〉 + |111111〉 + |000011〉 + |111100〉 + |000101〉 + |111010〉+ |000110〉 + |111001〉 + |001001〉 + |110110〉 + |001111〉 + |110000〉

Y. Xu · Y.-h. Li (�) · L.-p. Nie · X.-l. LiDepartment of Physics, Jiangxi Normal University, Nanchang 330022, Chinae-mail: lyhua1984@163.com

Y.-h. Li · L.-p. NieKey Laboratory of Optoelectronic and Telecommunication of Jiangxi Province, Nanchang 330022,China

Present address:L.-p. NieAudit Office of Jiangxi Normal University, Jiangxi Normal University, Nanchang, China

Int J Theor Phys (2013) 52:3086–3091 3087

+ |010001〉 + |101110〉 + |010010〉 + |101101〉 + |011000〉 + |100111〉+ |011101〉 + |100010〉 − |001010〉 − |110101〉 − |001100〉 − |110011〉− |010100〉 − |101011〉 − |010111〉 − |101000〉 − |011011〉 − |100100〉− |011110〉 − |100001〉)

123456. (1)

This genuine six-qubit entangled state can not decompose into three pairs of Bell states androbust against decoherence, and its entanglement still prevails after particle loss. Moreover,the reduced single-, two-, and three-qubit density matrices of this state are all completelymixed. In addition, it has been pointed out that no other pure state of six qubits has beenfound that evolves to a mixed state with a higher amount of entanglement [21]. More im-portantly, this state has been found wide application in quantum teleportation [22], quantumstate sharing [22], and quantum secure direct communication [23]. Very recently, an ap-proach to implement this six-qubit state was proposed in ion-trap system [24].

In the present paper, a scheme is proposed to generate the genuinely entangled six-atomstate by using the atoms in cavity QED system. Furthermore, we investigate that the atomsinteract simultaneously with the thermal cavity field and the strong classical driving field,thus the scheme is not sensitive to both the cavity decay and thermal field [25, 26].

2 The Model

We first consider N identical two-level atoms in a single-mode cavity field, which in additionare driven by a classical field. In rotating-wave approximation, the Hamiltonian (in units� = 1) can be written as [27]

H = ω0Sz + ωaa†a + g

2

(a†S−

j + aS+j

) + Ω

2

(S+

j e−iωt + S−j eiωt

), (2)

where

Sz = 1

2

N∑

j=1

(|1j 〉〈1j | − |0j 〉〈0j |), S+

j =N∑

j=1

(|1j 〉〈0j |), and S−

j =N∑

j=1

(|0j 〉〈1j |),

with |0j 〉 and |1j 〉 are the ground and excited states of the j -th atom, a and a† are in turn theannihilation and creation operator of the cavity-field mode. g represents the atom-cavity-field coupling strength, and Ω denotes the Rabi frequency of classical field. ωa , ω0, ω isthe cavity frequency, the atomic transition frequency, and the frequency of the classicalfield, respectively. In the large detuning δ � g between ω0 and ωa , and strong driving fieldΩ � δ, g, the energy exchange may not take place between the atoms and the cavity. Underthe case of ω0 = ω, thus, the evolution operator of the system can be described by [27]

U(t) = e−iΩtSx−iλtS2x , (3)

where

Sx = 1

2

N∑

j=1

(S+

j + S−j

)and λ = g2/4δ.

As an example, the two atoms (for the case where N = 2) are initially in one of four prod-uct states |00〉, |01〉, |10〉 and |11〉. If λt = π/2 and Ωt = 2π are chosen and the two atoms

3088 Int J Theor Phys (2013) 52:3086–3091

are sent simultaneously into the single-mode cavity, the four product states will become thefollowing states, respectively,

|00〉 →√

2

2e−iπ/4

(|00〉 − i|11〉), |01〉 →√

2

2e−iπ/4

(|01〉 − i|10〉),

|10〉 →√

2

2e−iπ/4

(|10〉 − i|01〉), |11〉 →√

2

2e−iπ/4

(|11〉 − i|00〉).

3 Preparing the Genuine Six-Atom Entangled State

Our scheme can be implemented according to Fig. 1. In order to generate the genuinelyentangled six-atom state, we prepare the six atoms that are initially in the state |101110〉123456

and first let the atoms 2, 4, 5 and 6 interact with the single-mode cavity I in which are drivenby a classical field. By controlling the interaction time and the Rabi frequency, one hasλt1 = π/2 and Ωt1 = 2π . Thus the state |101110〉123456 becomes

|101110〉123456 → |ψ1〉123456 = 1√2

(|101110〉 + i|110101〉)123456

. (4)

Secondly, we make a single-atom operation X on the atom 2, and then send simultane-ously the atoms 1, 2, 3, 4 into another singe-mode cavity II with the same classical drivingfield. By choosing λt2 = π/2 and Ωt2 = 2π , the state |ψ1〉123456 would evolve into the fol-lowing state,

|ψ1〉123456 → |ψ2〉123456 = 1

2

(|101110〉 − |110101〉 + i|010010〉 − i|001001〉)123456

. (5)

In the third step, we perform the transformations Y on the atoms 1 and 4, respectively,and then let the atoms 4 and 5 interact with the third single-mode cavity III and additionallydriven by a classical field. In the situation of λt3 = π/2 and Ωt3 = 2π , the time-dependentevolving state |ψ2〉123456 is

|ψ2〉123456 → |ψ3〉123456 = 1

2√

2

[|00〉(i|1001〉 + |1111〉) + |01〉(−i|0010〉 − |0100〉)

+ |10〉(|1110〉 − i|1000〉) + |11〉(−|0101〉 + i|0011〉)]123456

.

(6)

In the fourth step, we carry out the single-atom operation X on the atom 5, then let theatoms 3 and 6 interact with the fourth single-mode cavity IV in which are driven by the sameclassical field. If we choose λt4 = π/2 and Ωt4 = 2π , the state |ψ3〉123456 will be given by

Fig. 1 Generation of the genuine six-atom entangled state. The black dots denote the atoms, the gray pansstand for the cavities, the undee lines represent the classical fields and the hollow squares denote the sin-gle-atom operations, where X = |0〉〈0| + i|1〉〈1|, Y = i|0〉〈0| + |1〉〈1| and Z = |0〉〈0| − i|1〉〈1|

Int J Theor Phys (2013) 52:3086–3091 3089

|ψ3〉123456 → |ψ4〉123456 = 1

4

[|00〉(i|1001〉 + |0000〉 + i|1111〉 + |0110〉)

+ |01〉(i|1101〉 + |0010〉 − i|1011〉 − |0100〉)

+ |10〉(i|1110〉 + |0111〉 − i|1000〉 − |0001〉)

+ |11〉(i|1100〉 + i|1010〉 − |0011〉 − |0101〉)]123456

. (7)

At last, we let the atoms 3 and 4 interact with the fifth single-mode cavity V and mean-while driven by a classical field. If we choose λt5 = π/2 and Ωt5 = 2π , the state |ψ4〉123456

will become

|ψ4〉123456 → |ψ5〉123456 = 1

4

[|00〉(i|1001〉 + |0000〉 + i|1111〉 + |0110〉)

+ |00〉(|0101〉 − i|1100〉 + |0011〉 − i|1010〉)

+|01〉(i|1101〉 + |0010〉 − i|1011〉 − |0100〉)

+ |01〉(|0001〉 − i|1110〉 − |0111〉 + i|1000〉)

+ |10〉(i|1110〉 + |0111〉 − i|1000〉 − |0001〉)

+ |10〉(|0010〉 − i|1011〉 − |0100〉 + i|1101〉)

+ |11〉(i|1100〉 + i|1010〉 − |0011〉 − |0101〉)

+ |11〉(i|1001〉 + |0000〉 + i|1111〉 + |0110〉)]123456

, (8)

where we have discarded the overall phase factor. By performing the single-atom operationZ on the atom 3, we find

|ψ5〉123456 → |ψ6〉123456 = 1

4

[|00〉(|1001〉 + |0000〉 + |1111〉 + |0110〉)

+ |00〉(|0101〉 − |1100〉 + |0011〉 − |1010〉)

+ |01〉(|1101〉 + |0010〉 − |1011〉 − |0100〉)

+ |01〉(|0001〉 − |1110〉 − |0111〉 + |1000〉)

+ |10〉(|1110〉 + |0111〉 − |1000〉 − |0001〉)

+ |10〉(|0010〉 − |1011〉 − |0100〉 + |1101〉)

+ |11〉(|1100〉 + |1010〉 − |0011〉 − |0101〉)

+ |11〉(|1001〉 + |0110〉 + |1111〉 + |1001〉)]123456

. (9)

It is interest to note that the state |ψ6〉123456 can be written as

|ψ〉123456 =√

2

8

(|000000〉 + |111111〉 + |000011〉 + |111100〉 + |000101〉 + |111010〉+ |000110〉 + |111001〉 + |001001〉 + |110110〉 + |001111〉 + |110000〉+ |010001〉 + |101110〉 + |010010〉 + |101101〉 + |011000〉 + |100111〉+ |011101〉 + |100010〉 − |001010〉 − |110101〉 − |001100〉 − |110011〉− |010100〉 − |101011〉 − |010111〉 − |101000〉 − |011011〉 − |100100〉− |011110〉 − |100001〉)

123456, (10)

which is just the genuinely entangled six-atom state [20]. Thus, an approach to generate thegenuinely entangled six-atom state has been realized.

3090 Int J Theor Phys (2013) 52:3086–3091

4 Some Brief Discussions

The proposed scheme to generate a genuinely entangled six-atom state may be realized bythe experiment. Let us consider the simultaneous interactions between the cavity decay andthe atom radiation. With an open-cavity-setup, the flying atoms in a cavity can be directly il-luminated by an external classical driving field. Furthermore, the rubidium atoms in circularRydberg states have been controlled to interact with only one of the two cavity modes withdifferent detuning (from zero detuning to large detuning) to realize given quantum tasks aspointed out by Refs. [28–30].

It is known that under the large-detuning condition (δ � g), the Rabi frequency Ω canbe controlled by adjusting the intensity of the classical driving field to satisfy the conditionΩ � δ. In this situation the cavity decay can be safely neglected.

For the radiations of atoms. Typically the radiation time for the Rydberg atoms withprincipal quantum numbers 49, 50 and 51 is about Tr = 3×10−2 s, and the coupling constantis g = 2π × 24 kHz [31]. The required atom-cavity-field interaction time is on an order ofT = 1×10−4 s. Then the time needed to complete the whole procedure is much shorter thanTr . Thus the influence of the atom radiation may be negligible.

In our scheme, it is necessary to control two and more atoms interact simultaneouslywith the thermal cavity field and the strong classical driving field, which is a large challengefor the experiment. Fortunately, the infidelity is not sensitive to the temporal fluctuationof the neighboring atoms and thus can be neglected as pointed out by the refs. [16, 32–36]. The main experimental challenge for our scheme is how to control the non-neighboredatoms (e.g., the atoms 2, 4, 5, 6, and the atoms 3, 6) in a cavity at the same time. Withinmy knowledge, it is difficult to implement such the operation. Fortunately, this problemhas been studied in cavity QED system [35, 36] and an ion-trap system [37], and it hasbeen shown that, we can change the position of the non-neighbored atoms in the entangledstate, therefore it is possible to experimentally solve the difficulty about operating the non-neighbored atoms in a not long future. In addition the two-atom entangled state with such anatom–cavity field interaction has been experimentally realized [30]. Therefore our schememight be realized in current experimental setups based on cavity QED technique.

5 Conclusion

In conclusion, we have proposed a different scheme to generate the genuinely entangledsix-atom state in cavity QED system. Because two and more atoms interact with a thermalcavity and additionally driven by a classical field, especially, our scheme is not sensitive toboth the cavity decay and the thermal field. Based on the experiments reported in Refs. [30,38], our scheme is realizable with techniques presently available.

Acknowledgements This work is supported by the National Natural Science Foundation of China (GrantNo. 61265001), the Natural Science Foundation of Jiangxi Province, China (Grant No. 20122BAB202005),the Research Foundation of state key laboratory of advanced optical communication systems and networks,Shanghai Jiao Tong University, China (2011GZKF031104), and the Research Foundation of the EducationDepartment of Jiangxi Province.

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