Entanglement swapping and teleportation based on cavity QED method using the nonlinear atom–field interaction: Cavities with a hybrid of coherent and number states

Entanglement swapping and teleportation based on cavity QED method using the nonlinear atom–field interaction: Cavities with a hybrid of coherent and number states

Optics Communications 382 (2017) 381–385 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/o...

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Optics Communications 382 (2017) 381–385

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Entanglement swapping and teleportation based on cavity QED method using the nonlinear atom–field interaction: Cavities with a hybrid of coherent and number states R. Pakniat a, M.K. Tavassoly b,c,n, M.H. Zandi a a

Faculty of Physics, Shahid Bahonar University of Kerman, Kerman, Iran Faculty of Physics, Atomic and Molecular Group, Yazd University, Yazd, Iran c Research Group of Optics and Photonics, Yazd University, Yazd, Iran b

art ic l e i nf o

a b s t r a c t

Article history: Received 28 January 2016 Received in revised form 9 August 2016 Accepted 10 August 2016

In this paper, we outline a scheme for entanglement swapping based on the concept of cavity QED. The atom–field entangled state in our study is produced in the nonlinear regime. In this scheme, the exploited cavities are prepared in a hybrid entangled state (a combination of coherent and number states) and the swapping process is investigated using two different methods, i.e., detecting and Bellstate measurement methods through the cavity QED. Then, we make use of the atom–field entangled state obtained by detecting method to show that how the atom–atom entanglement as well as atomic and field states teleportation can be achieved with complete fidelity. & 2016 Elsevier B.V. All rights reserved.

Keywords: Entanglement swapping Nonlinear regime Hybrid entangled state Teleportation

1. Introduction Entanglement is the cornerstone of the quantum information processing and plays a crucial role in many quantum protocols, such as quantum dense coding [1], quantum cryptography [2] and quantum teleportation [3–6]. Among them, the teleportation, first suggested by Bennett et al. [3] and then experimentally reported in [7,8], whereby an unknown quantum state (not the particle or particles in such state) is transferred by a quantum channel from transmitter (Alice) to receiver (Bob) which are distinctly separated [9]. Different methods are used to explain the teleportation process such as Bell-state measurement (BSM) [10] and atom–field interaction in cavity QED [11–13]. Teleportation of atomic and field states by using of atom–field interaction in cavity QED is performed in [14,15]. One of the promising schemes to teleport a state is entanglement swapping which has been performed for the bipartite entangled state in [16,17]. The major idea of the entanglement swapping protocol is entangling two particles which do not have any interaction with each other [18–20]. Briefly, suppose that particles 1 and 2 are entangled, and separately particles 3 and 4 are also entangled. By the generation of entanglement between n Corresponding author at: Faculty of Physics, Atomic and Molecular Group, Yazd University, Yazd, Iran. E-mail addresses: [email protected] (R. Pakniat), [email protected] (M.K. Tavassoly), [email protected] (M.H. Zandi).

http://dx.doi.org/10.1016/j.optcom.2016.08.021 0030-4018/& 2016 Elsevier B.V. All rights reserved.

particles 2 and 3 via cavity QED method [21–24] or by performing a joint measurement (such as BSM) [18,25] on these particles, the particles 1 and 4 becoming entangled. In particular, very recently the effect of nonlinear Kerr medium on the entanglement swapping to a pair of three-level atomic system has been considered by one of us [26]. In this contribution, we discuss a scheme for entanglement swapping and teleportation based on cavity QED which the governing interaction between atom and field is inspired from nonlinear interaction discussed by Sivakumar [27]. As is noted above, the outcome of implementing the entanglement swapping procedure is the generation of entanglement between particles 1 and 4. To achieve this purpose, the detecting method has been used in Refs. [21–24]. In this paper, apart from our distinguishable cavity QED with nonlinear interaction, the BSM method in addition to the detecting method is employed and the results of these two methods are compared with each other. The resulting entangled state obtained from the detecting method is used in the next step as a quantum channel to teleport atomic and field states. The fidelity of teleportation in our scheme, for both atomic and field states teleportation, is shown to reach one. This paper organizes as follows: in the next section, as the first stage, we take a brief review on the nonlinear interaction of a twolevel atom with a single-mode quantized field with a specific nonlinearity function [27]. In Section 3, we suppose that two twolevel atoms (atom 1 and atom 2) have been prepared in an entangled state and two cavities (cavity 3 and cavity 4) have been

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prepared in a hybrid entangled state which is a combination of coherent and number states. After the interaction of the atom 2 and cavity 3 for time t1, we use two different methods (detecting and BSM methods) to generate entanglement between atom 1 and cavity 4. In Section 4, we allow another atom (atom 5) to enter the cavity 4, to swap the entanglement from atom–field to atom–atom state and then discuss about the effect of nonlinearity which may be observed after the entrance of the atom 5. In Section 5 we investigate the teleportation in the proposed scheme for two cases of atomic and field states by making use of the created quantum channel in Section 3.1. Finally, we present a summary and conclusions in Section 6.

2. Atom–field interaction: a brief review The Hamiltonian of a two-level atom interacting with a singlemode field in the presence of nonlinear Kerr effect can be expressed as [27] † †2 2 † † ^ r ^ H = ω(a^ a^ + σ^z ) + χa^ a^ + gω( 1 + κa^ a^ a^σ^+ + a† 1 + κa^ a^ σ^−), (1) 2

with r =

ν ω

and χ = κω . The above Hamiltonian can be rewritten as

ν ^ ^ ^ ^ ^ H = ωK+K − + σ^z + g (K+σ^− + K −σ^+), 2

(2)

† † † ^ ^ by using K − = 1 + κa^ a^ a^ and K+ = a^ 1 + κa^ a^ in direct relation to the generators of su(1, 1) Lie algebra. This may be appropriately described in the nonlinear atom–field interaction [28,29] with nonlinearity function f (n) = 1 + κn^ . It should be noted that the

nonlinear atom–field interaction (with Hamiltonian (2)) has been considered in the rotating wave approximation which is valid only for adequate weak atom–field interactions (weak couplings). This regime is in comparison with ultra-strong and deep strong coupling regimes which have been investigated by Casanova et al. [30] in which the rotating wave approximation is invalid to be applied. The general form of the corresponding atom–field state is con∞ sidered as |ψ (t )〉 = ∑n = 0 Ce, n(t )|e, n〉 + Cg , n + 1(t )|g , n + 1〉 . To obtain

(

)

the probability amplitudes Ce, n(t ) and Cg , n + 1(t ), the initial values are considered as Ce, n(0) = CeCn(0) and Cg , n + 1(0) = CgCn + 1(0), where Ce,Cg are the probability coefficients of initial atomic state and similarly Cn(0) denotes to the initial field state. With the help of time-dependent Schrödinger equation in the interaction picture, we obtain the following solutions for probability coefficients:

⎧ ⎡ ⎛ Ω t⎞ ⎛ Ω t ⎞⎤ ⎪ Δ ⎢ cos⎜ n ⎟ − i n sin⎜ n ⎟⎥Ce, n(0) Ce, n(t ) = ⎨ ⎪ Ωn ⎝ 2 ⎠⎦ ⎝ 2 ⎠ ⎩⎣



Now, let us suppose that two two-level atoms have been initially prepared in the entangled state

|ϕ >12 =

⎞ 1 ⎛ ⎜ |e >1 |g >2 + |g >1 |e>2⎟, ⎠ ⎝ 2

(3)

(5)

and two cavities have been prepared in a hybrid entangled state which is a combination of coherent and specific number states as follows [31]:

|ϕ >34 =

⎞ 1 ⎛ ⎜ |0 >3 |α >4 + |1 >3 | − α>4 ⎟. ⎠ 2⎝

(6)

Then, the initial state of the whole system reads as follows: |ψ (0) >1234 = |ϕ >12 |ϕ>34 . In the following, the entanglement swapping and so the entanglement generation between atom 1 and cavity 4 is performed by cavity QED method. In this method the atom 2 is imported into the cavity 3 in which the governing interaction Hamiltonian is expressed by (2). After the interaction time t1, noticing that the atom 1 and cavity 4 remain unchanged, the atom 2 and cavity 3 evolve, therefore the state of the whole system reads as

⎛ ⎞ 1⎡ g,0 ψ ⎜ t1⎟ >1234 = ⎢ e >1 α >4 Cg( ,0 ) t1 g >2 0>3 + e >1 ⎝ ⎠ 2⎣ ⎛ g,1 ⎞ g,1 − α >4 ⎜ Cg( ,1 ) t1 g >2 1 >3 + Ce(,0 ) t1 e >2 0>3⎟ + ⎝ ⎠

( ()

)

() ⎛ g > α > ⎜ C ( )( t ) e > 0 > ⎝ + g > − α > ( C ( )( t ) e > 1

4

1

e,0 e,0

4

1

e,1 e,1

2

1

3

2

() ⎞ + C ( )( t ) g > 1> ⎟ ⎠ ( ) 1> +C (t ) g >

⎫ iΔn t ⎛ Ω t⎞ ⎪ 2igω (n + 1)(1 + κn) ⎬e− 2 . sin⎜⎜ n ⎟⎟Ce, n(0)⎪ Ωn ⎝ 2 ⎠ ⎭

Here Ωn = Δn2 + 4 g 2ω2(1 + κn)(1 + n) , Δn = Δ − 2κnω and Δ = ν − ω , where ν and ω denote the frequencies of the atomic transition and the field, respectively.

1

3

e,1 g,2

2

1

)

3

2

(7)

where Ci(,jn, m) denote the amplitude probability given by Eqs. (3) and (4) with i, j = g , e and n, m = 0, 1, 2 assuming the initial state in (j, m). Now, to generate entanglement between atom 1 and cavity 4, we use two different methods, namely detecting and BSM methods. 3.1. Detecting method In this method, if the atom 2 is detected in the ground state |g >2 and the cavity 3 in the one-photon state |1>3, the state of the whole system collapses to

⎫ 1 ⎧ (g,1) ⎨C (t1)|e >1 | − α >4 + Cg(e,1,0)(t1)|g >1 |α>4 ⎬ , ⎭ 2 ⎩ g,1

(8)

at time t1, where

⎡ Ω t Δ Ω t ⎤ iΔ0 t1 Cg(g,1,1)(t1) = ⎢ cos( 0 1 ) + i 0 sin( 0 1 )⎥e− 2 , 2 2 ⎦ Ω0 ⎣

(4)

e,0 g,1

⎤ 2>3 ⎥, ⎦

|ψ (t1) >14 =

⎫ ⎛ Ω t⎞ ⎪ iΔn t 2igω (n + 1)(1 + κn) sin⎜⎜ n ⎟⎟Cg, n + 1(0)⎬ e 2 , Cg, n + 1(t ) − ⎪ 2 Ωn ⎠ ⎝ ⎭ ⎧ ⎡ ⎛ Ω t⎞ ⎛ Ω t ⎞⎤ ⎪ Δ ⎢ cos⎜ n ⎟ + i n sin⎜ n ⎟⎥Cg, n + 1(0) =⎨ ⎪ Ωn ⎝ 2 ⎠⎦ ⎝ 2 ⎠ ⎩⎣

3. Model

⎡ 2igω Ω t ⎤ iΔ0 t1 Cg(e,1,0)(t1) = ⎢ − sin( 0 1 )⎥e− 2 . 2 ⎦ ⎣ Ω0

(9)

(10)

The success probability to detect the atom 2 in its ground state and cavity 3 in the one-photon state is 0.6 and 0.3, respectively, in the π time t1 = Ω and Δ0 = − 2 gω . It is clear from (8) that the state 0

|ψ (t1)>14 indicates the “atom 1–cavity 4″ entangled state. Now we are able to present our numerical results. It should be mentioned that the effect of nonlinearity function characterized by χ and so κ is not so visible in many of our results. So, its values in the plotted figures in the continuation are not referred. Indeed they will be

R. Pakniat et al. / Optics Communications 382 (2017) 381–385

for α ≳ 1, i.e., for enough intensities of initial field, which indicates the maximum entanglement between atom 1 and cavity 4.

1

Fidelity

0.8

3.2. The Bell-state measurement (BSM) method

0.6

The state of the system after the interaction between the atom 2 and cavity 3 in the time duration t1 has been introduced in (7). Now, we define a Bell-state as

0.4 0.2 0

383

|ψ >Bell =

0

0.1

0.2

0.3

0.4

1 (|g >2 |1 >3 + |e >2 |0 >3 ) 2

(12)

and perform the BSM on the state (7). Then, the state of the atom 2 and cavity 3 is projected onto the introduced Bell-state and the state of the atom 1 and cavity 4 results in

0.5

gt1 Fig. 1. Fidelity versus scaled time gt1, with ω = 100 g and Δ0 = − 2 gω .

⎛ ⎞ Ψ ⎜ t1⎟ ⎝ ⎠ 1 ⎡ ( g,1) ( g,1) ⎢ Cg,1 t1 + Ce,0 t1 2 2⎣

( () ( )) e > − α> + ( C ( )( t ) + C ( )( t )) g >

>14 =

4

e,0 g,1

1

e,0 e,0

1

1

1

⎤ α>4 ⎥, ⎦

(13)

where

Fig. 2. Linear entropy versus real α, with g = 1, ω = 100 g, Δ0 = − 2 gω and t1 =

π . Ω0

⎡ ⎛Ω t ⎞ Δ Ω t ⎤ iΔ0 t1 e,0) Ce(,0 (t1) = ⎢ cos⎜ 0 1 ⎟ − i 0 sin( 0 1 )⎥e 2 , 2 ⎦ Ω0 ⎝ 2 ⎠ ⎣

(14)

⎡ 2igω Ω t ⎤ iΔ0 t1 g,1) Ce(,0 sin( 0 1 )⎥e 2 , (t1) = ⎢ − 2 ⎦ ⎣ Ω0

(15)

C g(g,1,1)(t1)

C g(e,1,0)(t1)

and the coefficients and have been introduced in (9) and (10), respectively. The fidelity of the state (13) relative to (11) is plotted in Fig. 3. As is observed, the fidelity can become unity or approximately unity at some proper values of time. In comparison between Figs. 1 and 3, it can be observed that the fidelity obtained by the detecting method has larger mean-time value and its diagram has more regular oscillations than the one obtained from the BSM. Also, the minimum value in the detecting method is larger than the minimum value in BSM. We end this section with mentioning the fact that, whereas the entangled states with photons are difficult to be stored for future use, instead, we can generate entanglement between two atoms which can be easily stored for future use. This will be discussed in the next section.

Fig. 3. Fidelity versus scaled time gt1, with ω = 100 g and Δ0 = − 2 gω .

4. The entanglement swapping from atom–field to atom–atom true for arbitrary values of k such that 0 ≤ κ ≤ 1. The fidelity of the latter state relative to the following maximally entangled state:

|ψ ′ >14 =

1 (|e >1 | − α >4 + |g >1 |α >4 ), 2

is plotted in Fig. 1 (the fidelity 2

14

(11) in this case reads as

< ψ ′|ψ (t1)>14 ). As is seen, in the specific time

t1 =

π Ω0

and

Δ0 = − 2 gω the fidelity will be equal to unity and this is repeated regularly with period τ =

2π . Ω0

It is worth to notice that the chosen

values for the parameters have been used in numerical analysis in this work are extracted from the main Refs. [23,27,30], so that the most favorable results can be obtained. Also, we used the value of α ≤ 5 or equivalently |α|2 ≤ 25 (low radiation intensities) since we aimed that the quantumness of the initial field state well-preserved. Moreover, as is clearly deduced from Fig. 2, the linear entropy after the initial increment, remains at its maximum value

Now, after the entanglement generation between atom 1 and cavity 4, we can send another atom in cavity 4 in order to generate entanglement between two atoms instead of atom and field. To do this, after the entanglement swapping process which we explained in the detecting method (Section 3.1), we allow the atom 51 to enter the cavity 4 in its ground state, therefore the initial state of the system is

⎡ 1⎧ ⎫⎤ |ψ (t1) >145 = ⎢ ⎨ Cg(g,1,1)(t1)|e >1 | − α >4 + Cg(e,1,0)(t1)|g >1 |α>4 ⎬⎥ ⎭⎦ ⎣ 2⎩ |g >5 .

(16)

We allow atom 5 to interact with cavity 4 and after the time of 1 Since ”atom 5″ is used after utilizing the ”atom 1″, ”atom 2″, ”cavity 3″ and ”cavity 4″, we labeled it by number 5; i.e., the fifth subsystem.

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Fig. 4. Fidelity versus scaled time gt2, with ω = 100 g , Δ0 = 10 g , t1 =

π Ω0

and β = 0.1.

Fig. 5. Fidelity versus the nonlinearity parameter κ, with g ¼ 1, ω = 100 g , Δ0 = 10 g , t1 =

π , t Ω0 2

= 103 and β = 0.1.

R. Pakniat et al. / Optics Communications 382 (2017) 381–385

somewhat be observed. For instance, the effect of the nonlinearity parameter on the fidelity (the fidelity of the state (18) relative to the maximally entangled state (23)) has been investigated in Fig. 5, after fixing t1 at π with regard to the different values of α. As is

interaction t2, the resulted state reads as

⎛ ψ ⎜⎜ t1, ⎝

⎞ ⎛ ⎞ 1 g,1 t2⎟⎟ >145 = Cg( ,1 )⎜⎜ t1⎟⎟ 2 ⎠ ⎝ ⎠ ⎡ ∞ ⎛ ⎞ g, −α e >1 ⎢ ∑ Cg( , n )⎜⎜ t2⎟⎟ n >4 g >5 + ⎢⎣ n = 0 ⎝ ⎠



∑ n= 0

Ω0

⎛ ⎞

g, −α Ce(, n )⎜⎜ t2⎟⎟

⎝ ⎠

⎤ ⎛ ⎞ 1 e,0 n >4 e>5⎥ + Cg( ,1 )⎜⎜ t1⎟⎟ ⎥⎦ 2 ⎝ ⎠ ⎡ ∞ ⎛ ⎞ g, α g >1 ⎢ ∑ Cg( , n )⎜⎜ t2⎟⎟ n >4 g >5 + ⎢⎣ n = 0 ⎝ ⎠



⎛ ⎞

n= 0

⎝ ⎠

∑ Ce(,gn, α)⎜⎜ t2⎟⎟

n >4

⎤ e>5⎥, ⎥⎦

(17)

,k) Cv(w ,n

where are the amplitude probabilities given by Eqs. (3) and (4) with v, w = g , e and k = α , −α assuming the initial state in (w, k). Now we take a measurement on the final state to obtain the coherent state |β >4 . The reduced state can be obtained as

ψ t1, t2 >15 = N ⎡⎣ a1( t1, t2, α, β ) g >5 + a2( t1, t2, α, β ) e>5⎤⎦

(

)

g >1 ,

(18)

where N is a normalization factor and 2

|β| 1 (g,1) C (t1)e− 2 2 g,1

n= 0

2

a2(t1, t2, α, β ) =

|β| 1 (g,1) C (t1)e− 2 2 g,1

2

a3(t1, t2, α, β ) =

|β| 1 (e,0) Cg,1 (t1)e− 2 2



∑ n= 0



∑ n= 0

2

a4(t1, t2, α, β ) =





|β| 1 (e,0) C (t1)e− 2 2 g,1



∑ n= 0

β n (g , − α ) Cg, n (t2), n!

(19)

(20)

β n (g , α ) Cg, n (t2), n!

(21)

β n (g , α ) Ce, n (t2). n!

(22)

⎤ 1⎡ ⎢⎣ ( |g >5 + |e>5)|e >1 + ( |g >5 − |e>5)|g >1⎥⎦, 2

is plotted in Fig. 4 after fixing t1 at

π Ω0

As previously stated, in teleportation protocol, Alice intends to transmit an unknown state to Bob. For this purpose, she requires a quantum channel. So, our proposal allows her to use the entangled state (8) in Section 3.1 to transmit an unknown atomic or field state to Bob. In this paper, the teleportation scheme of both atomic and field states is presented by making use of the BSM method.

5.1. Atomic state teleportation

β n (g , − α ) Ce, n (t2), n!

Let us assume the atom 2 initially has been prepared in the following state:

|ϕ >2 = η|e >2 + ζ |g >2 ,

The fidelity of the latter state relative to the maximally entangled state

|ψ ′ >15 =

observed, by considering constant values of β and t2, the maximum value of fidelity decreases as α increases. It is worth mentioning that the decrease in fidelity with increasing in α was obtained in Section 4, too. The investigation of the effects of other forms of nonlinearity functions on the results of the considered quantities in the paper is the subject of our next works which will be presented elsewhere. We should finally point out that by the proposed model (Sections 2 and 3) in the paper, two nonlinearity effects are taken into account, i.e., the intensity-dependent atom– field coupling and Kerr medium. However, their effects have been connected to each other via χ = κω , therefore decreasing (increasing) of κ results in decreasing (increasing) χ. So they cannot be considered distinctly.

5. Atomic and field state teleportation

e >1 + N ⎡⎣ a3( t1, t2, α, β ) g >5 + a4( t1, t2, α, β ) e>5⎤⎦

a1(t1, t2, α, β ) =

385

(23)

where the unknown coefficients η and ζ satisfy the normalization condition. We are going to teleport the atomic state (24) to atom 1 through the quantum channel which we expressed in (8). To achieve this purpose, we first observe that the initial state of the total system reads as

⎛ ⎞ ⎛ ⎞ Φ⎜ t1⎟ >124 = ϕ >2 ψ ⎜ t1⎟ ⎝ ⎠ ⎝ ⎠ 1 ⎡ ( g,1) e,0 >14 = η⎢ Cg,1 t1 e >1 e >2 − α >4 + Cg( ,1 ) t1 2 ⎣

()

()

⎤ g >1 e >2 α>4 ⎥ ⎦

with regard to different

values of α. As is seen, by considering constant value of β, the maximum value of fidelity (approximately 1) decreases as the value of α increases. Also, the maximum value of fidelity (F ≃ 1) is achieved with α = 1. A typical collapse-revival phenomenon is observed in all plots especially in Fig. 4a. Now it is useful to outline a discussion on the effect of nonlinearity in our presented results. At first, it is clear that, in the provided model by Sivakumar [27], which we reviewed in Section 2, if κ = 0 then the Hamiltonian (1) and coefficients (3) and (4) convert to their equivalent forms in the standard Jaynes–Cummings model [32]. In this study, according to our computations the effect of the nonlinearity parameter κ (0 < κ < 1), which guarantees the nonlinearity, is not visible before the entrance of atom 5. Indeed none of Figs. 1–4 did not critically altered. However, after the entrance of atom 5 in cavity 4 and its interaction in time t2, the effect of this parameter may

(24)

1 ⎡ ( g,1) e,0 t1 e >1 g >2 − α >4 + Cg( ,1 ) t1 g >1 ζ⎢ C 2 ⎣ g,1 ⎤ g >2 α>4 ⎥, ⎦

+

()

()

(25)

where the coefficient C g(g,1,1)(t1) and C g(e,1,0)(t1) have been explained in (9) and (10). Now, we introduce the quasi Bell-state as [33]

|ψ >Bell =

⎞ 1 ⎛ ⎜ |e >2 | − α >4 + |g >2 |α>4 ⎟. ⎠ 2⎝

(26)

By performing a BSM on the state (29), the state of atom 2 and cavity 4 is projected onto the introduced Bell-state and the state of the atom 1 reads as

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⎛ ⎞ Φ⎜⎜ t1⎟⎟ ⎝ ⎠ >1 =

2 ⎡ ⎛ ⎞ N′ ⎢ ( g,1)⎛ ⎞ e,0 ηCg,1 ⎜⎜ t1⎟⎟ e >1 + ηe−2 α Cg( ,1 )⎜⎜ t1⎟⎟ 2 2 ⎢⎣ ⎝ ⎠ ⎝ ⎠

2 ⎤ ⎛ ⎞ ⎛ ⎞ g,1 e,0 g >1+ζ e−2 α Cg( ,1 )⎜⎜ t1⎟⎟ e >1 + ζCg( ,1 )⎜⎜ t1⎟⎟ g >1⎥, ⎥⎦ ⎝ ⎠ ⎝ ⎠

(27)

where N′ is a normalization factor. The fidelity of state (27) relative to the ideal state in teleportation processing, i.e., π |ϕ >1 = η|e >1 + ζ |g >1 for large values of α, at t1 = Ω and with 0

Δ0 = − 2 gω is exactly 1. 5.2. Field state teleportation Let us assume the cavity 2 initially has been prepared in the state

|ϕ′ >2 = γ |α >2 + λ| − α >2 ,

(28)

where the unknown coefficients γ and λ satisfy the normalization condition. We intend to teleport the cavity state (28) to the cavity 4 through the quantum channel which is expressed in (8). To achieve this goal, we first observe that the initial state of the whole system reads as

⎛ ⎞ ⎛ ⎞ Φ′⎜ t1⎟ >124 = ϕ′ >2 ψ ⎜ t1⎟ ⎝ ⎠ ⎝ ⎠ 1 ⎡ ( g,1) e,0 >14 = γ ⎢ Cg,1 t1 e >1 α >2 − α >4 + Cg( ,1 ) t1 2 ⎣

()

()

⎤ g >1 α >2 α>4 ⎥ ⎦ 1 ⎡ ( g,1) e,0 t1 e >1 − α >2 − α >4 + Cg( ,1 ) t1 λ⎢ C 2 ⎣ g,1 ⎤ g >1 − α >2 α>4 ⎥. ⎦

+

()

() (29)

Now, we introduce the quasi Bell-state as

|ψ ′ >Bell =

⎞ 1 ⎛ ⎜ |e >1 | − α >2 + |g >1 |α>2⎟. ⎠ 2⎝

(30)

By performing a BSM on the state (29), the state of atom 1 and cavity 2 is projected onto the introduced Bell-state and the state of cavity 4 arrives at

⎛ ⎞ Φ′⎜⎜ t1⎟⎟ ⎝ ⎠ >4 =

swapping based on the obtained atom–field nonlinear entangled states. In this scheme, the field of cavities is assumed to be prepared in a hybrid entangled state (a combination of coherent and number states). The entanglement swapping process has been then investigated by two different methods, i.e., detecting and BSM methods, both in the cavity QED framework. In this line, using the atom–field entangled state obtained by detecting method, we have illustrated that one can generate the atom–atom entanglement and atomic and field states teleportation with complete fidelity. In comparison with Refs. [21–24], we have employed a hybrid entangled state (see (6))) for two cavities instead of some entangled states combing from two number states (such as the state |0 >3 |1 >4 + |1 >3 |0>4 ). In addition, we have used the detecting as well as BSM methods to generate entanglement between atom 1 and cavity 4 and compared them, while in Refs. [21,22,24], for instance, only the detecting method is used for this purpose. In the conclusion, we have presented a well-working scheme for entanglement swapping from atom–field to atom–atom state (which is easier to be stored compared with entangled states with photons) by using cavity QED interaction and then performing a measurement on the cavity 4. Then, we have investigated the effect of the nonlinearity parameter on the fidelity, after the entrance of atom 5 and entanglement swapping from atom–field to atom–atom. As is clear in Fig. 5, the maximum value of fidelity decreases as the value of the intensity of initial field increases. Finally, we have employed the ”atom 1–cavity 4″ entangled state, which is generated in detecting method, to teleport atomic and field states with the help of BSM method. Also, what distinguish our work with Refs. [14,15] are as follows: (i) our quantum channel is an atom–field entangled state, which is obtained via entanglement swapping process, instead of atom–field entangled state which was obtained via the interaction of an atom with a cavity in Jaynes–Cummings model in the mentioned Refs. (i) Our method in both atomic and field states teleportation is BSM-based method, but in the mentioned Refs. the cavity QED method was employed. (iii) The fidelity of teleportation in our scheme, for both atomic and field states teleportation, is precisely 1, while in the mentioned Refs. the value of fidelity is close to 1.

2 ⎡ ⎛ ⎞ N″ ⎢ ( e,0)⎛ ⎞ e,0 γCg,1 ⎜⎜ t1⎟⎟ α >4 + λe−2 α Cg( ,1 )⎜⎜ t1⎟⎟ 2 2 ⎢⎣ ⎝ ⎠ ⎝ ⎠

2 ⎤ ⎛ ⎞ ⎛ ⎞ g,1 g,1 α>4 +γ e−2 α Cg( ,1 )⎜⎜ t1⎟⎟ − α >4 + λCg( ,1 )⎜⎜ t1⎟⎟ − α>4 ⎥ ⎥⎦ ⎝ ⎠ ⎝ ⎠

(31)

where N″ is a normalization factor. The fidelity of the state (31) relative to the ideal state in teleportation process, i.e., π |ϕ′ >4 = γ |α >4 + λ| − α>4 for large values of α, at t1 = Ω and with 0

Δ0 = − 2 gω is exactly 1.

6. Summary and conclusion In this paper, we first presented a brief review on the nonlinear atom–field interaction through the Jaynes–Cummings model in cavity QED approach. Then, we studied the entanglement

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