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Scheme for entanglement concentration of atomic entangled states in cavity QED Liu Ye a,b,∗ , Guang-Can Guo a a Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, PR China b Department of Physics, Anhui University, Hefei 230039, PR China

Received 21 January 2004; received in revised form 15 May 2004; accepted 17 May 2004 Available online 31 May 2004 Communicated by P.R. Holland

Abstract We present a quantum concentration scheme for atomic entangled states. During the operations, the cavity is only virtually excited, thus our scheme is insensitive to the cavity field states and the cavity decay. The scheme can be implemented by the present cavity QED techniques. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.67.Mn; 03.65.Yz Keywords: Quantum concentration; Three-atom entangled state; Cavity QED

Quantum entanglement has been the most important resource for fundamental studies in quantum mechanics related to the Einstein–Podolsky–Rosen paradox [1] and Bell’s theorem [2] as well as in quantum information processing such as quantum teleportation [3], quantum dense coding [4] and quantum cryptography [5]. So much attention has been directed to the generation of entangled states. Two-atom and multi-particle entanglements have been demonstrated in cavity QED [6–11]. Three photon Greenberger– Horne–Zeilinger (GHZ) states have been created in linear optics [12]. In quantum information processing, most applications require the maximally entangled

* Corresponding author.

E-mail address: [email protected] (L. Ye). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.035

state for faithfully transmission of quantum data, such as teleportation, dense coding, etc. However, since an entangled state prepared for quantum information processing is open to an environment and environment is easy to interact with quantum system, the quality of entanglement will be degraded. Therefore many quantum protocols such as distillation [13,14], purification [15] and concentration [16] have been suggested. To obtain a maximally entangled state from a nonmaximally entangled state by using local operation and classical communication, Bose et al. [17] showed that entanglement swapping could be used to realize entanglement concentration. In this way, Shi et al. [18] showed that optimal entanglement concentration could be realized via entanglement swapping and a unitary transformation. In Ref. [19], we presented two schemes for nonmaximally entangled state to proba-

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bilistically produce a three-particle GHZ by POVM (Positive Operator Value Measure) measurements or a unitary transformation. Pan et al. [20] showed an entanglement purification scheme by using the polarizing beamsplitter (PBS) to replace the quantum controlled-Not. Several experimental schemes [21– 24] have also been proposed for purifying or concentrating the nonmaximally entangled photon states. In experiment, cavity QED is also one of the optimal systems in quantum communication. Recently Romero et al. [25] have presented an experimental purification protocol based on POVM in cavities. Lougovski et al. [26] proposed a scheme for probabilistically generating and purifying maximally entangled states of two atoms inside an optical cavity. In cavity QED schemes, the cavities usually act as memories, which store the information of an atom and then transfer to another atom after the conditional dynamics. Thus one of the main problems for these schemes is decoherence of the cavity field. Again within the framework of cavity QED, Zheng and Guo [27] have proposed a novel scheme for realizing a two-atom entanglement and a operation of quantum logic gate. The distinct advantage of the scheme is that during the operation, the cavity is only virtually excited and thus the effective decoherence time of the cavity is greatly prolonged. Following the idea, we suggest an entanglement concentration scheme for three-atom entangled states in cavity QED. Then we further show that the scheme can be used to concentrate a multi-atom entangled state. We consider m identical three-level atoms, whose states are denoted by |i, |e and |g. The transition frequency between |i and |e is highly detuned from the cavity frequency and thus the state |i will not be affected during the atom–cavity interaction. When two atoms simultaneously interact with a single-mode cavity field, in the interaction picture, the interaction Hamiltonian for the system is (assuming h¯ = 1) Hi = g

m

a + sj− + asj+ ,

(1)

j =1

where sj+ = |ej gj | and sj− = |gj ej | are the Pauli operators of the j th atom, a + and a are the creation and annihilation operators for the cavity mode, respectively, g is the atom–cavity coupling strength, and δ is the detuning between the atomic transition frequency

285

ω0 and cavity frequency ω, that is, δ = ω0 − ω. In the case δ g, there is no energy exchange between the atomic system and the cavity. Consider that the cavity field is initially in the vacuum state. Then the effective Hamiltonian is given by m m |ejj e| + sj+ si− , He = λ (2) j =1

i,j =1, i=j

λ = g 2 /δ.

where Now we consider a single qubit unitary operation. In cavity QED, we can finish a rotation operation by using a classical microwave pulse to address an atom. If the classical field is tuned to the transition |i ↔ |j , under the basis {|i, |j }, the transformation operator of the atomic state is cos θ −e−iϕ sin θ Uij (θ, ϕ) = iϕ , cos θ e sin θ where θ = Ωij |α|t, α = |α|eiϕ is the complex amplitude of the classical field, Ωij is the atom–field coupling constant. Now assume four atoms are initially in the state |egeg1234 and the cavity field in the vacuum state. After an interaction time t, the state evolution of the system is φ(t) = 1 e−i6λt + 3e−i2λt + 2 |egeg1234 6 1 + e−i6λt − 3e−i2λt + 2 |gege1234 6 1 + e−i6λt − 1 |egge1234 + |eegg1234 6 + |ggee1234 + |geeg1234 . With the choice of λt = π/3, we obtain the fouratom entangled state √ e−i 3 |egeg1234 + i 3 |gege1234 . 2 If we perform Ueg (θ, ϕ) on any one of four atoms, such as atom 4, no matter that the result of measurement on atom 4 is in the excited state |e4 or in the ground state |g4 , we can obtain the nonmaximally three-atom entangled state √ 1 cos θ |ege123 + i 3e−iϕ sin θ |geg123 , |φg = Ng √ 1 iϕ |φe = −e sin θ |ege123 + i 3 cos θ |geg123 , Ne π

|φ =

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where Ng , Ne are normalized factors. Without loss of generalization, the entangled state of atoms 1, 2 and 3 can be write into following state |φ123 = a|ege123 + ib|geg123 , where = 1. Here consider two triples of atoms (1, 2, 3) and (1 , 2 , 3 ) having been prepared in the following entangled states, respectively: |φ123 = α|ege123 + iβ|geg123 ,

(3)

|φ1 2 3 = a|ege1 2 3 + ib|geg1 2 3 ,

(4)

where |α| > |β|, |a| > |b|, = 1. Suppose that three atoms of the state |φ123 and atom 1 belong to Alice and atoms 2 , 3 belong to Bob. If Alice makes a joint state measurement on atoms 1, 2 and 1 , then particles 3, 2 and 3 will be collapse into one of the following states:

± Φ 121 φ 123 ⊗ |φ1 2 3 + |β|2

αa βb = √ |ege32 3 ± √ |geg32 3 , 2 2

(5)

Ψ ± 121 φ 123 ⊗ |φ1 2 3 βa αb = √ |gge32 3 ± √ |eeg32 3 , 2 2

where ± 1 Φ = √ |ege121 ± i|geg121 , 121 2 ± Ψ

121

1 = √ |egg121 ± i|gee121 . 2

(6)

(7)

(8)

For the case of Eq. (5), to get a maximally entangled state, Alice introduces an auxiliary atom with the original state |ga . Alice sends the auxiliary atom and atom 3 simultaneously through another cavity. After the interaction time t , the state of Eq. (5) will be transformed to the state |φ± 32 3 → √

1 αae−iλt cos λt |ege32 3 2N ± βb|geg32 3 |ga

|bβ| . |aα|

λt = arccos

This leads to

|a|2 + |b|2

|α|2

Choose the interaction time t , so that

− iαae−iλt sin λt |gge32 3 |ea . (9)

|φ± 32 3 →√

1

|bβ| e−iφ1 |ege32 3 ± |geg32 3 |ga 2N

− i a 2 α 2 − b2 β 2 e−iφ1 |egg32 3 |ea , (10)

|bβ| where φ1 = arccos |aβ| . Then a measurement to the auxiliary particle follows. If the measurement result is |ea , atoms 3, 2 and 3 are completely disentangled. If the result is |ga , Alice performs Ueg ( π2 , φ21 )Ueg ( π2 , π) operations on atom 3, which can discard the phase factor in above equation and change atoms 3, 2 and 3 into the state |bβ|(|ege323 ± |geg32 3 ). Obviously, here atoms 3, 2 and 3 are in a maximally entangled state with a probability |βb|2 . If Eq. (6) is obtained, two different cases should be considered: for the case of |βa| < |αb|, Alice sends the atoms 3 and an auxiliary atom with the original state |ga simultaneously through a cavity. After an interaction time τ , the state of Eq. (6) becomes

1 |ψ± βa|gge323 32 3 → √ 2N ± αbe−iλτ cos λτ |eeg32 3 |ga ∓ iαbe−iλt sin λτ |geg32 3 |ea . (11) With the choice of λτ = arccos |βa| |αb| . The state (11) can be changed into |ψ± 32 3 →√

|βa| |gge32 3 ± e−iφ2 |eeg32 3 |ga 2N

∓ ie−iφ2 |αb|2 − |βa|2 × |geg32 3 |ea , (12) 1

|aβ| where φ2 = arccos |αb| . Likewise, after the measurement on the auxiliary atom, Alice makes a rotation operation on atom 3 to discard the phase factor. Atoms 3, 2 and 3 will

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be projected into a maximally entangled state with probability |βa|2 . In the case of |βa| > |αb|, Alice introduces an auxiliary atom with an original state |ea and sends the atom and atom 3 into a cavity. Choose an interaction |αb| time τ = λ1 arccos |βa| , Eq. (6) can be transformed to the result 1 e−iφ3 |ψ± 32 3 → √ 2N × |αb| |gge32 3 ± e−iφ3 |eeg32 3 |ea

− i |βa|2 − |αb|2 |ege32 3 |ga , (13) |αb| |βa| .

where φ3 = arccos Similarly, Alice makes a measurement on the auxiliary atom. When |ea is obtained, Alice makes Ueg ( π2 , φ23 )Ueg ( π2 , π) operations on atom 3. Atoms 3, 2 and 3 will be projected into a maximally entangled state with |αb|2 . So the total probability of obtaining a maximally entangled state from the original less entangled state is 2β 2 or 2b2 . Now we consider Alice how to perform her measurement on her atoms 1, 2 and 1 . Firstly Alice performs Uef ( π2 , π)Ueg ( π4 , π) operations on atom 1 . Then she sends atoms 2 and 1 simultaneously enter a single-mode cavity for an interaction time π/λ. Afterward Alice performs Ueg ( π4 , 0)Uef ( π2 , 0) operations on atom 1 , and then she sends atoms 1 and 2 simultaneously through another cavity for an interaction time π/4λ, thus Eqs. (7) and (8) will be transformed into ± Ψ → |geg121 , (14) −i|egg121 , ± Φ → |ege121 , (15) −i|gee121 , where the common phase factor e−iπ/4 is discarded. Hence, Alice’s joint measurement can be achieved by detecting atoms 1, 2 and 1 separately. With the outcome of the measurement on Alice’s three atoms, we can obtain the state (5) or (6). Alice can introduce an auxiliary atom, send the auxiliary atom and atom 3 into another cavity. Choosing an appropriate interaction time, Alice measures the state of the auxiliary atom. Then she can perform singleatom rotation operations to discard a phase factor, and

287

make her atom and Bob’s two atoms into a three-atom GHZ state. The proposed scheme of three-particle entanglement concentration can be used to the case of n-atom entanglement in a straightforward way. We should consider two different cases: (1) Consider two (2n + 1)-particle entangled states of the form |φ2n+1 = α|eg . . . e12...2n+1 + iβ|ge . . . g12...2n+1 ,

(16)

|φ2n +1 = a|eg . . . e1 2 ...2n +1 + ib|ge . . . g1 2 ...2n +1 ,

(17)

where |a| > |b|, |α| > |β|, all atom of |φ12...2n+1 and atom 1 belongs to Alice and other atoms belong to Bob. After Alice makes a joint measurement on the atoms 1, 2, . . . , 2n, 1 , the remaining atoms will be projected to one of following states αa √ |eg . . . e(2n+1)2 ...(2n +1) 2 βb ± √ |ge . . . g(2n+1)2 ...(2n +1) , (18) 2 βa √ |gge . . . ge(2n+1)2 3 ...(2n +1) 2 αb ± √ |eeg . . . eg(2n+1)2 ...(2n +1) , (19) 2 where ± Ψ = √1 |ge . . . gee12...(2n)1 2 (20) ± i|eg . . . egg12...(2n)1 , ± 1 Φ = √ |eg . . . ege12...(2n)1 2 (21) ± i|ge . . . geg12...(2n)1 . (2) Suppose two 2n-atom entangled states be in the following forms, respectively, |φ2n = α|eg . . . eg12...2n + iβ|ge . . . ge12...2n , (22) |φ2n = a|eg . . . eg1 2 ...2n + ib|ge . . . ge1 2 ...2n .

(23)

Similarly Alice makes a joint state measurement on atoms 1, 2, . . . , (2n − 1), 1 , the remaining

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atoms will be projected to one of following states αa √ |gge . . . geg(2n)2 ...(2n ) 2 βb ± √ |eeg . . . ege(2n)2 ...(2n ) , 2 βa √ |eg . . . eg(2n)2 ...(2n ) 2 αb ± √ |ge . . . ge(2n)2 ...(2n ) , 2 where ± 1 Ψ = √ |ge . . . ge12...(2n−1)1 2 ± i|eg . . . eg12...(2n−1)1 , ± Φ = √1 |eg . . . egee12...(2n−1)1 2 ± i|ge . . . gegg12...(2n−1)1 .

(24)

(25)

(26)

(27)

For the joint measurement, start from the last atom 1 , Alice can perform Uef ( π2 , π)Ueg ( π4 , π) operations on this atom, and send this atom and its previous atom simultaneously through a cavity for an interaction time π/λ. Then Alice performs Ueg ( π4 , 0)Uef ( π2 , 0) on this atom. After (2n−1) such operations for Eqs. (20), (21) and (2n − 2) such operations for Eqs. (26), (27), and an atom–cavity interaction on atoms 1, 2 for a time π/4λ, the Eqs. (20), (21) and (26), (27) will be transformed into ± Ψ → |gee . . . eg12...(2n)1 , (28) −i|ege . . . eg12...(2n)1 , ± Φ → |ege . . . ee12...(2n)1 , (29) −i|gee . . . ee12...(2n)1 , ± |gee . . . ee12...(2n−1)1 , Ψ (30) → −i|ege . . . ee12...(2n−1)1 , ± |ege . . . eg12...(2n−1)1 , Φ ← (31) −i|gee . . . eg12...(2n−1)1 . From Eqs. (28)–(31), we can find that the joint state measurement can be finished only via detecting atoms 1, 2 and 1 separately. According to the result of Alice’s measurement, Alice can introduce an auxiliary atom, then send her atom and the auxiliary atom into another cavity. Like the transformation of Eqs. (5) or (6), after choosing an appropriate interaction time Alice can measure the state of the auxiliary atom. In

this way, a multi-atom maximally entangled state can be obtained from a less entangled state with a certain probability. In conclusion, we have proposed a simple protocol to realize quantum concentration of an atomic entangled state in cavity QED. A pair of three-atom entangled states can be readily prepared by atom–cavity field interaction. The joint state-measurement can be finished via detecting atom separately. Thus we provide a way to achieve all operations of concentrating a three-atom entangled state by using a pair of nonmaximally three-atom entangled states. By a straightforward generalization of the schemes, the concentration of a multi-atom entangled state can be realized. During the operations, our scheme only involves atom–field interaction with a large detuning and does not require the transfer of quantum information between the atoms and cavity. Thus the requirement on the quality factor of the cavities is greatly loosened and the scheme can be implemented by the present cavity QED techniques.

Acknowledgements This work was funded by National Fundamental Research Program (2001CB309300), and innovation funds from the Chinese Academy of Sciences and Educational Developing Project Facing the Twentyfirst Century, and also by the Program of the Education Department of Anhui Province (2003kj029) and the Talent Foundation of Anhui University.

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