Thermal atom–atom entanglement in a bichromatic Kerr nonlinear coupler

Accepted Manuscript Thermal atom-atom entanglement in a bichromatic Kerr nonlinear coupler M.R. Abbasi PII: DOI: Reference:

S0003-4916(15)00395-4 http://dx.doi.org/10.1016/j.aop.2015.10.024 YAPHY 66998

To appear in:

Annals of Physics

Received date: 23 May 2015 Accepted date: 31 October 2015 Please cite this article as: M.R. Abbasi, Thermal atom-atom entanglement in a bichromatic Kerr nonlinear coupler, Annals of Physics (2015), http://dx.doi.org/10.1016/j.aop.2015.10.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights

   

Thermal atom-atom entanglement in a Kerr nonlinear coupler is investigated. The atom-atom, field-field and atom-field interactions are considered. The entanglement can be tuned by atom-field structure parameters. The entanglement starts from zero and terminates at a finite temperature.

Revised paper Click here to view linked References

Thermal atom-atom entanglement in a bichromatic Kerr nonlinear coupler M R Abbasi1,2 1 2

Physics Department, College of Science, Persian Gulf University, Bushehr 75168, Iran Bushehr Nuclear Power Plant Operating Co., Bushehr 75181, Iran

Corresponding author’s e-mail: [email protected]

Abstract. In this paper thermal entanglement between two identical two-level atoms within a bichromatic cavity including Kerr nonlinear coupler is investigated. In this study, besides atomfield interaction, the field–field (via linear and Kerr-type couplings) and atomic dipole-dipole interactions are also included. It is also assumed that the cavity is held at a temperature T, so that all atom-photon states with probabilities defined by Boltzmann factor are present. Using a canonical transformation, the presented model is converted to a generalized form of Jaynes– Cummings model. After introducing Casimir operators of the system, it is shown that the Hamiltonian representation is block-diagonal. Diagonalizing of each block, the thermal (Gibb’s) density matrix, written in the bases of total Hamiltonian, is obtained. The reduced atomic density matrix and consequently the concurrence, as a measure of entanglement, are obtained by partial tracing of thermal density matrix over the bichromatic photonic states. The concurrence vanishes at zero temperature, indicating that the ground state is separable, exhibits a maximal at a critical temperature and terminates at a finite temperature. The influences of coupler nonlinearities and dipole-dipole coupling on the thermal atom-atom entanglement are also addressed in detail.

Keywords: Atom-atom entanglement, Kerr nonlinear coupler, thermal states, concurrence PACS: 03.65.Ud; 03.65.Yz; 42.50.Ct; 32.80.-t 1. Introduction. Entanglement, one of the most fundamental and intriguing features of composite quantum systems, has been known as the main resource of quantum information processing (QIP)[1-5]. Therefore, it is important to explore means for creating and manipulating entanglement. In recent years different ways have been proposed to realize entangled states by means of atom-photon interaction [6-9]. The simplest way to study the atom–field interaction is the Jaynes–Cummings model (JCM). The generalizations of the JCM for the two-atom cases have also attracted considerable interest [10-12] because the two-level atom can be considered as the qubit– the basic unit of quantum information. Furthermore, the entanglement between the atoms and the field has been studied by assuming that this interaction occurs in a cavity filled with a dielectric medium [13,14]. On the other hand, Kerr nonlinear couplers can be created using optical fibers or photonic crystals and the amount of photonic couplings can be adjusted by controlling the interaction length and medium characteristics [15-17]. However, if it is assumed that the atoms are

located inside a Kerr nonlinear coupler, besides direct atomic dipole-dipole interaction [18], the photonic linear and Kerr-type couplings are also appeared. As a result, the atoms (photons) interact reciprocally as well as with both photons (atoms). These interactions strongly impress the atom-atom, atom-photon and the photon-photon entanglement. The atom-atom entanglement via coherent field at absolute zero temperature and via thermal field at finite temperature has been subject of several reports [18-20]. However, due to the interaction with the environment, as a heat reservoir, the quantum systems, as a whole, including both atoms and photons, are always in mixed states [13,21]. Therefore, temperature is of crucial importance in the field of QIP and leads disentanglement [22] or vice versa [23]. Assuming thermal equilibrium with the environment, transition tendency from an atom-photon state to another one should be counterbalanced by transition tendency in the reverse direction. This equality of tendencies is usually referred to detailed balance once thermal equilibrium is established [24,25]. In thermal equilibrium with the environment, all of atomphoton states with probabilities, specified by Boltzmann factor, are present. Furthermore, in this paper the lossless Kerr nonlinear coupler has been assumed thus the time evolution of density operator obeys von Neumann equation of motion [26]. Anyway, in this paper, a model that describes two identical two-level atoms as two qubits interacting with a bichromatic field inside a lossless Kerr nonlinear coupler in the presence of the atomic dipole-dipole coupling at an equilibrium temperature T is investigated. Apart from other new features of this work, in particular, the effect of Kerr-type coupling and atomic dipole-dipole coupling on the atom-atom entanglement will be addressed in detail. In general, this system gives more generalize and realistic features of atom-atom entanglement and introduces a better understanding of entanglement. In order to study the thermal entanglement, the concurrence, based on the entanglement of formation [27], is a useful quantity that leads to the amount of entanglement [28] especially for the two two-level atom cases. To pursue this aim, the model along with the corresponding Hamiltonian is introduced. It is then shown that the matrix representation of the Hamiltonian is block-diagonal with ever-growing dimensions whose dimensionalities depend upon the eigenvalue of the total atom-photon excitation operator,

,

which is denoted by N . Applying a canonical photonic transformation, each block is reduced to a 11 , a

3  3 and N  1 4  4 subblocks. Therefore, this transformation enables us to diagonalize the transformed Hamiltonian and calculate the eigenvalues in general. Using the eigenvalues and the eigenvectors of the Hamiltonian, the thermal (Gibb’s) density operator (matrix), as a function of temperature, is then calculated. Tracing over the bichromatic photonic states, thermal atomic density matrix and then the concurrence are obtained and consequently the roles of the atomic dipole-dipole coupling and the photonic nonlinearities are recognized. Moreover, the results show that at extremely low temperatures in which such system may be assumed to be in its ground state, the concurrence becomes zero. The concurrence then rises to a maximum value at a critical temperature and terminates at a threshold temperature as the temperature is increased. Furthermore, an increase of the atomic dipole-dipole coupling decreases the entanglement. The maximum value of concurrence also increases when the atomphoton coupling and/or the Kerr-type coupling increases. Moreover, dependence of the critical

temperature at which the entanglement is maximized and the threshold temperature at which system of atom-atom disentangles, upon the atomic dipole-dipole, atom-photon as well as Kerr-type coupling, is also addressed in detail. The remainder of this paper is organized as follows: First, in the next section, the system’s Hamiltonian along with its transformed representation is introduced. After giving the analytical approach to evaluate the thermal density operator as a function of temperature in Section 3, the reduced atomic density matrix and consequently the concurrence, as a measure of entanglement, is given in section 4. The results by some figures are presented in section 5. Concluding remarks are finally given in the last section. 2. Model’s Hamiltonian This section is devoted to describe the interaction between two-coupled two-level atoms as two identical and a ground state g , inside a bichromatic cavity involving

qubits, each having an excited state e

Kerr nonlinear coupler, taking into account the field–field interaction via photonic linear and Kerr-type couplings. The Hamiltonian describing the above system can be written as,

H  H a  H f  H af ,

(1)

where H a , H f and H af are the atom-atom, field-field and atom-field interaction Hamiltonians, respectively. The atom-atom Hamiltonian is,

Ha 

0 2



1 z







  z2    1 2   1 2 ,

(2)

where 0 is the atomic transition frequency,  is the atomic dipole-dipole coupling [18,29],

 zi   ei ei  g i

gi



2 ,  i  ei

g i and  i  gi ei where i  1, 2 . In the presence of the Kerr

nonlinear coupler, the field-filed Hamiltonian is [15,30], 2

Hf  i 1

 a a   a † i i i

†2 2 i i ai







a1†a1a2†a2   a1†a2  a1a2† ,

(3)

where i is the field frequency,  i is Kerr-self coupling,  is Kerr-cross coupling,  is the linear

 

coupling constant between the two modes, and ai ai† is the field annihilation (creation) operator. Effects of such photon-photon coupling on the atom-photons interaction are assumed negligible so that the linear Jaynes-Cummings model can be applied. Therefore, The linear atom-field interaction, in the electric dipole and rotating wave approximations, reads, 2

2





H af    i ai † j  ai j , i 1 j 1

(4)

where i is atom-field coupling constant. 2

It is evident from Eqs. (1)-(4) that the Casimir operator

  i  i  ai † ai (with eigenvalue N and i 1

eigenvector i , j , n1, n 2  i , j  n1, n 2 ; i , j  e , g where n i

is the eigenstate of i-th photonic number

operator), commutes with the total Hamiltonian. As a result, if the atom-photon states are labeled respecting to the eigenvalues of the Casimir operator, the matrix representation of the total Hamiltonian

becomes

(irreducible)

ever-growing

block-diagonal

each

of

dimensions

 N  3   N  1  N  1    2    4 N , for N  1 , according to the inclusion and exclusion principal [31].  3   3   3  At this point, the unitary Bogoliubov–Valatin transformation [32-34] of photonic operators, a1  b1 cos  b2 sin  a2  b2 cos  b1 sin 

,

(5)

is then applied. Under this transformation the number of photonic excitations remains invariant thus 2

2

i 1

i 1

 ai†ai   bi†bi

. Substituting the canonical transformations in (5) into the whole Hamiltonian in Eq.(1)

and choosing tan   1 2 and assuming 1  2   2 and   2  1  tan 2 2 , it is not difficult to show that the field-field and atom-field Hamiltonian become, Hf  

 cos    sin    sin 2  b b     b b  b b  2b b b b  2

2

1

† 1 1

2

†2 2 1 1

†2 2 2 2

2 cos

2



  1 sin 2    sin 2 b2†b2

(6)

† † 1 1 2 2

and,

H af 

2





12  22  b2† i  b2 i , i 1

(7)

where in this representation the evanescent wave term related to the linear field–field interaction along with the first type of the atom–photon interaction are absent. The assumptions following Eq.(5) in fact have been used in recent papers related to co-directional nonlinear couplers [35-37]. Moreover, the total 2

  i  i  bi †bi (with eigenvalue N

“dressed” excitation number operator,

and eigenvector

i 1

i, j, n1, n2  i, j  n1, n2 ; i, j  e, g where ni

operator) and

1

is the eigenstate of i-th “dressed” photonic number

 b1†b1 (with eigenvalue n1 ), commute with the total transformed Hamiltonian given in

Eqs. (2), (6) and (7) thus they are Casimir operators of transformed Hamiltonian. Again the representation of the total transformed Hamiltonian, in the bases of

is also block diagonal. As a result, this

observation leads to the fact that if the states are grouped for fixed n1 , each 4N -dimensional block is reduced

to

a 1 1

block

with

{ g , g , n1,1 , e, g , n1,0 , g , e, n1,0 } and

one

basis

N 1

g , g , n1,0 ,

4 4

a 3 3 blocks

block in

with the

three

basis

manifold

{ g , g , n1, n2 , e, g , n1, n2  1 , g , e, n1, n2  1 , e, e, n1, n2  2 } . In this manner the transformed Hamiltonian may be written as direct sum of these blocks,

H  H 11  H 33  H 44 ,

(8)

where,

H 11 

  (n1,0)  0  g , g , n1,0

n1 0

g , g , n1 ,0 ,

(9)

H 33 

  (n1,1)  0 

g , g ; n1,1 g , g ; n1,1

n10

 (n1 ,0)  e, g ; n1,0 e, g ; n1,0  g , e; n1,0 g , e; n1,0 

12

 22

 g , g ; n1,1



e, g ; n1 ,0  g , g ; n1 ,1 g , e; n1 ,0  H .C.

,

(10)

 e, g ; n1 ,0 g , e; n1,0  H .C.

and, H 4 4 

   (n1, n2 )  0 

n1  0 n2  2

g , g ; n1, n2 g , g ; n1, n2

 (n1 , n2  1)  e, g ; n1, n2  1 e, g ; n1, n2  1  g , e; n1, n2  1 g , e; n1, n2  1 





 (n1 , n2  2)  0 e, e; n1, n2  2 e, e; n1, n2  2  g , e; n1, n2  1 e, g ; n1, n2  1  H .C. ,  

 

 n

2 1

 22 n2  g , g ; n1 , n2 e, g ; n1, n2  1  g , g ; n1, n2 g , e; n1, n2  1  H .C.

2 1

 22

2

(11)

 1  e, g ; n1 , n2  1 e, e; n1, n2  2  g , e; n1, n2  12 e, e; n1, n2  2  H .C.  

where,









(n1, n2 )  1 cos 2   2 sin 2    sin 2 n1  2 cos 2   1 sin 2    sin 2 n2    n1  n1  1  n2  n2  1  2n1n2 

The eigenvalues of the 11 blocks (see Eq.(9)) are readily

.

(12)

 (n1,0)  0  . As for the 3  3 blocks (see

Eq. (10)) the corresponding eigenvalues and eigenvectors are calculated giving,   n1 ,0     n1 ,1  0 2

E1,2 





2     n1 ,0     n1,1  0   4   n1,1  0    n1,0    8 12  22  , 

(13)

   n1,0 

E3 

with the respective eigenstates,

 1   2   3

  1     2     3

1  1   g , g ; n1 ,1   2  2   e, g ; n1 ,0 3  3   g , e; n1 ,0

  ,  

(14)

where, 

1

2   E1,2  (n,1)  0   2 E  (n,1)  0   ,  1,2  1,2  1,2 1,2  1  2  1,2 2 2     2    2 12  22 . 1 2      3  1 2   2 1 , 3   1 2   21 ,  3   21  1 2

(15)

To calculate the thermal density matrix, one also needs to evaluate eigenvalues and eigenstates of the

4  4 blocks by a lengthy but straightforward procedure as,

E1  n1 , n2   (n1 , n2  1)   E2  n1, n2   S  T  E3,4  n1 , n2  

a 3

,

(16)

S T a 3  i (S  T ) 2 3 2

where,

S  3 R  Q3  R 2 ,T  3 R  Q3  R 2 ,

(17)

with, Q

3b  a 2 9ab  27c  2a 3 , ,R  9 54

(18)

and, 4 4 4 4 4 4 a   ( H )11  ( H ) 22  ( H ) 44    

b







2

2

4 4 4 4 4 4 4 4 4 4  4 4   ( H )4224 ( H )11  ( H )44  ( H )11 ( H ) 44  2 ( H )12  2 ( H ) 24     ’ 2

(19)

2

4 4 4 4  4 4 4 4 4 4  c  2 ( H ) 424 4  ( H )11  2 ( H )12 ( H ) 44  ( H )11 ( H ) 44  ( H ) 422 4       

where the basis 1  g , g; n1, n2 , 2  e, g; n1, n2  1 , 3  g , e; n1, n2  1 and 4  e ,e ; n1 , n 2  2 have been used for labeling Hamiltonian subscripts. The corresponding “dressed” eigenstates are,

  1;n ,n 1 2    2;n1 ,n2    3;n1 ,n2    4;n ,n 1 2 

   1;n1 ,n2      2;n1 ,n2     3;n1 ,n2    4;n ,n 1 2   

1;n1 ,n2

 1;n1 ,n2

 2;n1 ,n2

 2;n1 ,n2

3;n1 ,n2

 3;n1 ,n2

 4;n1 ,n2

 4;n1 ,n2

1;n1 ,n2   g , g ; n , n  1 2   2;n1 ,n2   e, g ; n1 , n2  1 

  3;n1 ,n2   g , e; n1 , n2  1  ,

(20)

   4;n1 ,n2   e, e; n1 , n2  2 

where,

 i;n1 ,n2

 44  2  44 2   Ei;n ,n  ( H )11 ( H )4244 Ei;n1 ,n2  ( H )11    1 2     1  2 44 44 44       ( H )12 2( H )12 Ei;n1 ,n2  ( H ) 44      

i;n1 ,n2   i;n1 ,n2 

44 Ei;n1 ,n2  ( H )11 44 2( H )12

 i;n1 ,n2 ;  i;n1 ,n2 

44

44

44

1 2

44 ( H )4244 Ei;n1 ,n2  ( H )11 44 44 ( H )12 Ei;n1 ,n2  ( H )44

 12   H 21   H 13   H 31

where the facts that H



44

and  H 

44 24

 i;n1 ,n2 ,

(21)

 42   H 34   H 43

 H

44

44

44

have

been used (see Eq. (11)). In the next section Eqs. (13)-(21) are used to form the thermal density operator. 3. Thermal Density Operator The thermal density operator of the present system in equilibrium with the environment at a temperature T, is given by,

 (T )  Z 1e



H k BT

  Z 1   exp( ((n1 ,0)  0 ) / k BT ) g , g , n1 ,0 g , g , n1,0  n1 3

  exp( Ei / k BT )  i  i

,

(22)

n1 i 1

4



  exp( Ei;n ,n 1

n1 n2  2 i 1

2

 / k BT )  i;n1 ,n2  i;n1 ,n2  

where k B is the Boltzmann’s constant and Z , the partition function, shall be defined shortly. When  i s and  i;n1 ,n2 s from Eqs. (14) and (20) are substituted into Eq.(22) the thermal density operator, describing the two-coupled two-level atoms and bichromatic photonic mode in the Kerr nonlinear coupler, becomes,



 (T )  Z 1   exp( ((n1,0)  0 ) / k BT ) g , g , n1,0 g , g , n1,0  n1

3

  exp( Ei / k BT )  i2 g , g , n1,1 g , g , n1,1   i i g , g , n1,1 e, g , n1,0  H .C.  n1 i 1

 i2  e, g , n1 ,0 e, g , n1,0  g , e, n1,0 g , e, n1,0    i i e, g , n1,0 g , e, n1,0  H .C .   i i g , g , n1,1 g , e, n1,0  H .C. 

4

   exp( Ei;n ,n 1

n1  0 n2  2 i 1

2

/ k BT )  i2;n1 , n2 g , g , n1, n2 g , g , n1, n2 

  i2;n1 , n2 e, e, n1, n2  2 e, e, n1, n2  2   i; n1 , n2  i;n1 , n2  g , g , n1, n2 e, e, n1, n2  2  H .C.   i; n1 , n2 i; n1 , n2  g , g , n1, n2 e, g , n1, n2  1  g , g , n1, n2 g , e, n1, n2  1  H .C.

 i; n1 , n2  i;n1 , n2  e, g , n1 , n2  1 e, e, n1, n2  2  g , e, n1, n2  1 e, e, n1, n2  2  H .C.  i2;n1 , n2  e, g , n1 , n2  1 e, g , n1, n2  1  g , e, n1, n2  1 g , e, n1, n2  1



  e, g , n1, n2  1 g , e, n1, n2  1  H .C    

(23)

where i;n1 ,n2 , i;n1,n2 ,  i;n1,n2 and  i;n1 ,n2 are defined from Eq.(21). The explicit expression for the canonical partition function is, 3

Z   exp( ((n1 ,0)  0 ) / k BT )   exp( Ei / k BT )  i2  2 i2    n1 i 1

n1



4

  exp( Ei;n ,n

n1 n2  2 i 1

1

2

.

(24)

/ k BT )  i2;n1 ,n2  2i2;n1 ,n2  i2;n1 ,n2   

In the next section Eqs. (23) and (24) are used to calculate the thermal atomic density operator, its eigenvalues and, consequently, the concurrence as a measure of entanglement. 4. The thermal atomic density operator and concurrence Entanglement, as a fragile phenomenon, can be deteriorated when the quantum system interacts with the environment. Therefore, quantifying the entanglement of a quantum system is of crucial importance for the QIP. In recent years several measures have been proposed to study Entanglement of composite system [28]. Among of these measures, the Wootters concurrence is more acceptable measure for the problem in

hand. The concurrence, based on the atomic density matrix  AA , is given by, C (T )  max{0, 1(T )  2 (T )  3 (T )  4 (T )} ,

1(T )  2 (T )  3 (T )  4 (T )  0

where







 AA  1y   y2  AA  1y   y2



where  AA 

are



(25) the

eigenvalues

n1 , n2  (T ) n1 , n2

of

the

matrix

is the reduced density matrix

n1 ,n2

representing the quantum state of the two atoms,  AA is its complex conjugate and  yi is the ycomponent of Pauli operator for i-th two-level atom. When the value of C (T ) is positive, the two atoms are entangled. Concurrence equals 1 corresponds to the maximally entangled state, while concurrence equals zero indicates that the atom A and the atom B are in separable (unentangled) state. If the thermal density operator, Eq.(23), is traced over respecting to the photonic states and the result is regrouped respecting the atomic bases, { g , g , e, g , g , e , e, e } , the atomic density operator reads,  11  0  AA (T )    0   0

0

0

 22  23

 23 33

0

0

  ,    44  0 0 0

(26)

where,   

3

11  Z 1   exp( ((n1 ,0)  0 ) / k BT )   exp( Ei / k BT ) i2 

n1 i 1

n1 4

   exp( Ei;n ,n 1

n1 0 n2  2 i 1

  

2

3

22  33  Z 1   exp( Ei / k BT ) i2 

44  Z 1 

n1 i 1

4

  exp( Ei;n ,n 1

n1 0 n2  2 i 1

,

 / k BT ) i2;n1 ,n2   

2

 / k BT )i2;n1 ,n2  ,  

4

   exp( Ei;n ,n

n1 0 n2  2 i 1

1

2

/ k BT ) i2;n1 ,n2 ,

(27)

(28)

(29)

and,

  

3

 23  32  Z 1   exp( Ei / k BT ) i i  n1 i 1

4

   exp( Ei;n ,n

n1 0 n2  2 i 1

1

2

 / k BT )i;n1 ,n2  i;n1 ,n2  .  

(30)

For thermal reduced atomic density matrix defined in Eq. (26) the concurrence (see Eq. (25)) turns out to be [38], C (T )  2max{0, 23  1144 } .

(31)

In the next section use is made of Eqs. (26)-(31) to investigate the effect of atom-photon coupling and Kerr-type coupling on the thermal atom-atom entanglement. 5. Results and discussion In the previous section, an infrastructure for developing the calculation of concurrence as a measure of atom-atom entanglement was presented. Thereby, the concurrence, as a function of temperature, can be plotted. In order to obtain the concurrence numerically, the density matrix (see Eq.(26)) should be

truncated.

Anyway,

for

truncating

the

density

matrix,

  Trn1 ,n2   exp  Ei;n1 ,n2 k BT  i;n1 ,n2  i;n1 ,n2  104 has been taken as truncation criterion. Since this i  1   4





cut-off introduces a reasonable error, it is concluded that the density matrix and consequently concurrence do converge at any temperatures. To meet this criterion, in the numerical calculation of the concurrence, the density matrix is terminated when the number of total excitations is 6. This point is further addressed in the insets of figures (1)-(3). Generally for the case in hand one can expect: (i) the concurrence starts at absolute zero temperature since the combined system at this temperature collapses into its ground state which is a separable (untangled) state. (ii) When the temperature increases up to a critical point, some excited states are generated and participate in the entanglement formation thus the concurrence increases [39]. (iii) At very high temperatures, the state of system goes toward the fully mixed state so the entanglement becomes zero [40]. In the next three subsections, the effects of the atom-photon, dipoledipole and Kerr-type couplings on the thermal atom-atom entanglement are examined in datail. 5.1. Effect of the atom-photon coupling The atom-photon couplings indirectly couple atomic states. Accordingly, since a larger atom-photon coupling gives rise to stronger atom-atom coupling, it takes less thermal energy to involve the excited atom-photon states in the entanglement thus the maximal entanglement occurs at a lower temperature when the atom-photon coupling is increased. To examine the effect of atom-photon coupling on the thermal atom-atom entanglement, the concurrence, as a function of scaled dimensionless temperature,

kBT

1 , for different values of atom-photon couplings is depicted in figure 1. Moreover, it is evident

that for lower atom-photon couplings the concurrence goes towards zero. Furthermore, as the atomphoton coupling increases the threshold temperature at which the atom-atom entanglement terminates, rises.

Figure 1. Concurrence versus scaled temperature. The parameters are, 2 / 1  1.1, 0 / 1  1.15,  / 1  1,

1 / 1  0.5, / 1  0.01, for different values of 2 : 2 / 1  0.1 (dotted line), 2 / 1  0.3 (dashed line) 

4







and 2 / 1  0.4 (solid line). The insets depict Trn1 ,n2   exp  Ei;n1 ,n2 k BT  i;n1 ,n2  i;n1 ,n2  for (a)

 i 1



2 / 1  0.1 , (b) 2 / 1  0.3 and (c) 2 / 1  0.4 to show minimum value of photonic number for which the concurrence converges at different temperatures.

5.2. Effect of the dipole-dipole coupling In order to demonstrate the influence of the dipole-dipole coupling, the concurrence as a function of scaled temperature, kBT

1 , for different values of the dipole-dipole couplings, is depicted in figure 2.

From this figure it is evident that the maximal value of entanglement reduces when the dipole-dipole coupling is increased. It is also not difficult to find that the maximal entanglement occurs at higher temperatures for larger dipole-dipole couplings. Moreover, the threshold temperature at which the concurrence terminates occurs at higher temperatures for lower dipole-dipole coupling.

Figure 2. Concurrence versus scaled temperature. The parameters are, 2 / 1  1.1, 0 / 1  1.15,  / 1  1,

1 / 1  0.5, 2 / 1  0.4, for different values of  : / 1  0.1 (dotted line), / 1  0.05 (dashed line) and  4  / 1  0.01 (solid line). The insets show Trn1 ,n2   exp  Ei;n1 ,n2 k BT  i;n1 ,n2  i;n1 ,n2   i 1 





for (a)

/ 1  0.01 , (b) / 1  0.05 and (c) / 1  0.1 for illustrating the minimum value of photonic number for which the concurrence converges at different temperatures.

5.3. Effect of the Kerr-type coupling From the field–field Hamiltonian, Eqs. (3) and (6), it is observed that as the Kerr-type couplings are increased, the virtual energy level separations may increase or decrease. In order to study the effect of Kerr-type coupling on the concurrence, the concurrence as a function of scaled dimensionless temperature, kBT

1 , for different values of Kerr-type couplings, is illustrated in Figure 3. It is

observed that the maximal value of entanglement be enhanced when the Kerr-type coupling is increased. From this figure it is also noted that for low temperatures the concurrence is not sensitive to the Kerr-type coupling and maximum entanglement occurs at higher temperatures for larger nonlinearities. Furthermore, when the Kerr-type coupling increase, the threshold temperature for which atomic entanglement terminates is also enhanced.

Figure 3. Concurrence versus scaled temperature. The parameters are, 2 / 1  1.1, 0 / 1  1.15,/ 1  0.01,

1 / 1  0.5, 2 / 1  0.4, for different values of  :  / 1  0.15 (dotted line),  / 1  0.5 (dashed line) and  4   / 1  1 (solid line). The insets depict the behavior of Trn1 ,n2   exp  Ei;n1 ,n2 k BT  i;n1 ,n2  i;n1 ,n2  for  i 1 





(a)  / 1  0.15 , (b)  / 1  0.5 and (c)  / 1  1 to show minimum value of photonic number for which the concurrence converges at different temperatures.

6. Conclusion In this paper the thermal entanglement between two-coupled two-level identical atoms in the Kerr nonlinear coupler has been investigated. As a result of the presented model, the linear and nonlinear field– field interaction was taken into account. Using a particular canonical transformation and assuming the whole system, atoms and coupler, is in thermal equilibrium with a heat reservoir, an analytical and also exact solution for Gibb’s thermal density operator and consequently the concurrence have been presented. Although the results are thoroughly presented in section 5, in what follows the main results of the paper are listed: 1- At zero temperature the system is expected to collapse into its ground separable state. Therefore, the entanglement, measured by concurrence, becomes zero at extremely low temperatures, reaches to its maximum value at a critical temperature and finally terminates at a threshold finite temperature.

2- The degree of atom-atom entanglement, at any temperature, may be controlled by adjusting the controlling parameters; the atom-atom, atom-photon, Kerr-type couplings. 3- An enhancement of the atom-photon coupling increases the atom-atom entanglement and the maximum value of concurrence occurs at a lower critical temperature. 4- The maximal of entanglement also increases when Kerr-type coupling increases and occurs at higher critical temperatures. 5- An enhancement of the atomic dipole-dipole coupling decreases the thermal entanglement. The maximum value of entanglement occurs at higher temperature when this coupling increases. 6- When the atom-photon and the Kerr-type couplings are increased, the threshold temperature at which atom-atom entanglement vanishes is also increased while increasing the atom-atom coupling decreases the threshold temperature.

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