Entanglement evolution of a two-qubit system interacting with a quantum spin environment

Optics Communications 282 (2009) 4637–4642

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

Entanglement evolution of a two-qubit system interacting with a quantum spin environment Jin-Liang Guo *, He-Shan Song School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China

a r t i c l e

i n f o

Article history: Received 17 March 2009 Received in revised form 30 July 2009 Accepted 20 August 2009

PACS: 03.65.Ud 03.67.Mn 75.10.Jm

a b s t r a c t We investigate the entanglement dynamics and decoherence of a two-qubit system under a quantum spin environment at finite temperature in the thermodynamics limit. For the case under study, we find different initial states will result in different entanglement evolution, what deserves mentioning here is that the state jWi ¼ cos aj01i þ sin aj10i is most robust than other states when p=2 < a < p, since the entanglement remains unchanged or increased under the spin environment. In addition, we also find the anisotropy parameter D can suppress the destruction of decoherence induced by the environment, and the undesirable entanglement sudden death arising from the process of entanglement evolution can be efficiently controlled by the inhomogeneous magnetic field f. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Pairwise entanglement Dynamic evolution Concurrence

1. Introduction Entanglement is one of the most striking features of quantum mechanical systems that have no classical analog. It has been the focus of much work recently due to its key role in the topical area of quantum information processing [1]. Over the past few years, there has been considerable interest in investigating entanglement in quantum spin systems with Heisenberg interactions [2–7], since the Heisenberg model, as a simple but realistic solid-state system, not only have been used to simulate a quantum computer, as well as quantum dots [8,9], nuclear spins [10], electronic spins [11], and optical lattice [12], but also display useful applications in quantum state transfer [13]. However, in the previous studies, the detailed interaction between the system and the environment is not an essential part of the matter. On the other hand, from the practical point of views, the real quantum systems will unavoidably interact with the surrounding environments and thus leads to decoherence. This is one fundamental obstacle to perform quantum computation. Therefore, it is indispensable to take into account of the decoherence caused by the interaction between the system and the environment. Recently, the dynamic behavior of a single spin or several spins interacting with a spin bath has attracted much attention [14–18]. However, in most case it is very difficult to obtain an exact solution * Corresponding author. E-mail address: [email protected] (J.-L. Guo). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.08.036

to the evolution of the reduced density matrix with the environment modes traced over in the case of the non-Markovian process. In Ref. [19], using a novel operator technique, the authors present an exact calculation of the dynamics of the reduced density matrix of two coupled spins in a spin environment in the thermodynamics limit at finite temperature. The results show that the dynamics of the entanglement depend strongly on the initial state of the system, the coupling between the two-spin qubits, the interaction between the qubit system and environment, the interaction between the constituents of the spin environment, the environment temperature, as well as the detuning controlled by a locally applied external magnetic field. Later, the authors in Ref. [19] study the dynamics of a central spin coupling with its environment at finite temperature under the thermodynamics limit [20]. In view of the above results, we find only the initial state j00i, j11i, and p1ffiffi ðj00i þ j11iÞ are considered in Ref. [19], and other initial states 2 and parameters, such as inhomogeneous magnetic field and anisotropy are rarely included. Therefore, in this paper, considering other initial states, we study the entanglement evolution of two-qubit system interacting with a quantum spin environment at finite temperature in the thermodynamics limit with taking into account the influence of inhomogeneous magnetic field and anisotropy. To quantify the amount of entanglement between the two qubits, we consider concurrence defined by Wootters [21]

C ¼ maxfk1  k2  k3  k4 ; 0g;

ð1Þ

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where the quantities k1 P k2 P k3 P k4 are the square roots of the eigenvalues of the matrix R ¼ qðry  ry Þq ðry  ry Þ. q denotes the complex conjugate of q and ry is the normal pauli operators. The concurrence C ¼ 0 corresponds to an unentangled state and C ¼ 1 for a maximally entangled state. For the special case

0

q11

B 0 q¼B B @ 0

q14

0

q14

0

q22 q23 q23 q33 0

0

ð2Þ

q44 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q11 q44 ; jq14 j  q22 q33 g:

ð3Þ

This letter is organized as follows. In Section 2, we introduce the model and derive the time evolution by the novel operator technique. In Section 3, we present the result of entanglement dynamics for different initial states, and discuss the effects of inhomogeneous magnetic field and anisotropy parameter on the entanglement evolution. Finally, we conclude in Section 4. 2. Hamiltonian evolution Here, we extend the system Hamiltonian of the model in Ref. [19] by considering the inhomogeneous magnetic field and anisotropy while keep the spin environment unchanged. The Hamiltonian of our model is H ¼ HS þ HSB þ HB , where HS , HSB and HB denote the Hamiltonian of the system, system-bath interaction and bath, respectively. They can be written as

fÞSz01

fÞSz02

ðSþ01 S02

S01 Sþ02 Þ

DSz01 Sz02 ;

HS ¼ ðl0 þ þ ð l0  þX þ þ " # " # N N X X g0 g0 ffi ðSþ01 þ Sþ02 Þ HSB ¼ pffiffiffi Si þ pffiffiffiffi Sþi ; ðS01 þ S02 Þ N N i¼1 i¼1 HB ¼

N g X ðSþ S þ Si Sþj Þ; N i–j i j

ð4Þ ð5Þ

ð6Þ

where the external magnetic fields are assumed to be along the z-direction, l0 describes the uniformity of the field while f measures the degree of the inhomogeneity of the field. X is the coupling constant between any two-qubit spins and D is the anisotropy  parameter. Sþ 0i and S0i ði ¼ 1; 2; 3Þ are the spin–flip operators of the  spin system, respectively. Sþ i and Si are the corresponding of the ith qubit spin in the bath. N is the number of the bath atoms. g 0 is the coupling constant between the qubit system spins and bath spins, whereas g is the bath spins. Both constants pffiffiffiffithat between pffiffiffiffi are rescaled as g 0 = N and g= N [22–25]. Using the collective anguP lar momentum operators J ¼ Ni¼1 S i , the Hamiltonian Eqs. (5) and (6) turn out to be

g0 þ g0  ffi ðS01 þ Sþ02 ÞJ  þ pffiffiffi ffi ðS01 þ S02 ÞJ þ ; HSB ¼ pffiffiffi N N g HB ¼ 0 ðJ þ J  þ J  J þ Þ  g: N

ð7Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y J  ¼ ð N  b bÞb;

y

Here we have applied the approximation that b b=N tends to be vanishing, since the energy of the excitations original from the interaction between the system and the bath is very low. In the following, we are interested in the density matrix evolution of the spin system, by which we can know the entanglement dynamics under decoherence induced by the spin environment. Since the Hamiltonian is time independent, the density matrix evolves for the total system is

qðtÞ ¼ eiHt qð0ÞeiHt ;

ð9Þ

y

ð13Þ

where we assume that the initial density matrix qð0Þ is separable between the system and the bath, i.e. qð0Þ ¼ qS ð0Þ  qB . Here the initial state of the spin system is described by qS ð0Þ. The density matrix of the environment satisfies a thermal distribution, i.e. qB ¼ eHB =kT =Z, where Z is the partition function and the Boltzmann constant k is set to one in this paper. Then we can obtain the reduced system density matrix by tracing out the environment degree of freedom, i.e. qS ðtÞ ¼ TrB qðtÞ. For two-qubit system, there are two types of Bell-like states,

jUi ¼ cos aj00i þ sin aj11i;

jWi ¼ cos aj01i þ sin aj10i;

ð14Þ

with 0 6 a 6 p. In Ref. [19], the authors have studied the entanglement evolution for the initial system state j00i, j11i, and p1ffiffi2 ðj00iþ j11iÞ in the absence of inhomogeneous magnetic field and anisotropy. Here we use jUi and jWi to serve as the initial states of the system taking into account the inhomogeneous magnetic field and anisotropy. Namely, qS ð0Þ ¼ jUihUj and qS ð0Þ ¼ jWihWj. First, by taking state jWi as the initial system state, the reduced density matrix for the system can be written as

1 Z



qS ðtÞ ¼ cos2 aTrB eiHt j01ieHB =T h01jeiHt



  1 2 sin aTr B eiHt j10ieHB =T h10jeiHt Z   1 þ sin a cos aTrB eiHt j01ieHB =T h10jeiHt Z   1 þ sin a cos aTrB eiHt j10ieHB =T h01jeiHt ; Z þ

ð15Þ

and we can easily obtain

Z¼

ð8Þ

After the Holstein–Primakoff transformation:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y J þ ¼ b ð N  b bÞ;

ð12Þ

HB ¼ 2gb b:

0 C C C; 0 A

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h i y HSB ¼ g 0 ðSþ01 þ Sþ02 Þb þ ðS01 þ S02 Þb ; y

1

the concurrence can be easily obtained

C ¼ 2 maxfjq23 j 

In the thermodynamic limit (i.e., N ! 1) at finite temperature, we have

1 : 1  e2g=T

ð16Þ

Following the idea of operator technique introduced in Ref. [19], we first need to convert the time evolution equation of the qubit system under the action of the total Hamiltonian into a set of coupled non-commuting operator variable equations. Considering the constituents of Hamiltonian H, we can write

with ½b; b  ¼ 1, the Hamiltonian Eqs. (7) and (8) can be written as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y b b bb y b þ g 0 ðS01 þ S02 Þb 1  ; HSB ¼ g 0 ðSþ01 þ Sþ02 Þ 1  N N ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s s 2 3 ! y y y bb bb y b b5 y H B ¼ g 4b 1  bþ 1 bb 1   g: N N N

eiHt j01i ¼ Aj00i þ Bj01i þ Cj10i þ Dj11i; ð10Þ

ð11Þ

ð17Þ y

where A; B; C and D are functions of operators b, b , and time t. Using the Schrödinger equation identity

i

   d  iHt e j01i ¼ H eiHt j01i ; dt

ð18Þ

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According to Eq. (3), the concurrence C ¼ 2 maxf0; jq23 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q11 q44 g.

and Eq. (17), we can obtain

  dA D y y i ¼ l0 þ þ 2gb b A þ g 0 b ðB þ CÞ; dt 4   dB D y y ¼ f  þ 2gb b A þ g 0 bA þ g 0 b D þ XC; i dt 4   dC D y y i ¼ f  þ 2gb b A þ g 0 bA þ g 0 b D þ XB; dt 4   dD D y ¼ l0 þ þ 2gb b D þ g 0 bðB þ CÞ; i dt 4

3. Entanglement evolution and decoherence

ð19Þ

with initial condition from Eq. (17) being Að0Þ ¼ Cð0Þ ¼ Dð0Þ ¼ 0, and Bð0Þ ¼ 1. It is obvious that Eq. (19) cannot be solved by using conventional methods for ordinary number variables, since Eq. (19) is composed of coupled differential equations of non-commuting operator variables. In order to solve this problem, we introduce the following transformation y

y

A ¼ b ei2gb bt A1 ; i2gby bt

D ¼ be

y

B ¼ ei2gb bt B1 ;

y

C ¼ ei2gb bt C 1 ;

D1 :

ð20Þ

Then Eq. (19) turns out to be

  dA1 D ¼ l0 þ  2g A1 þ g 0 ðB1 þ C 1 Þ; dt 4   dB1 D ¼ f   2g B1 þ g 0 ðn þ 1ÞA1 þ g 0 nD1 þ XC 1 ; i dt 4   dC 1 D ¼ f   2g C 1 þ g 0 ðn þ 1ÞA1 þ g 0 nD1 þ XB1 ; i dt 4   dD1 D ¼ l0 þ  2g D1 þ g 0 ðB1 þ C 1 Þ; i dt 4 i

ð21Þ

y

with n ¼ b b and initial condition B1 ð0Þ ¼ 1, and A1 ð0Þ ¼ C 1 ð0Þ ¼ D1 ð0Þ ¼ 0. At this stage, we find the coefficients of Eq. (19) after the transformation of Eq. (20) involves only the operator n. As a result, A1 ; B1 ; C 1 and D1 are functions of n and t, and commute with each other. We can then treat Eq. (21) as coupled complex-number differential equations and solve them in a usual way. Following the similar calculation above, when the initial system state is j10i, we have

eiHt j10i ¼ Ej00i þ Fj01i þ Gj10i þ Kj11i:

ð22Þ

After the similar transformation of Eq. (20), we can also obtain the equation similar to the Eq. (21) that can be solved in a usual way with initial condition C 1 ð0Þ ¼ 1, and A1 ð0Þ ¼ B1 ð0Þ ¼ D1 ð0Þ ¼ 0. Then the density matrix for the system qS ðtÞ can be expressed in the basis fj1i ¼ jggi; j2i ¼ jgei; j3i ¼ jegi; j4i ¼ jeei, 1 1X 2 cos2 aA1 Ay1 þ sin aE1 Ey1 þ sin a cos aA1 Ey1 Z n¼0

þ sin a cos aE1 Ay1 ðn þ 1Þe2gn=T ; 1 1X 2 ¼ cos2 aB1 By1 þ sin aF 1 F y1 þ sin a cos aB1 F y1 Z n¼0  þ sin a cos aF 1 By1 e2gn=T ; 1 1X 2 ¼ cos2 aC 1 C y1 þ sin aG1 Gy1 þ sin a cos aC 1 Gy1 Z n¼0

þ sin a cos aG1 C y1 e2gn=T ; 1 1X 2 ¼ cos2 aD1 Dy1 þ sin aK 1 K y1 þ sin a cos aD1 K y1 Z n¼0  þ sin a cos aK 1 Dy1 ne2gn=T ; 1 1X 2 ¼ cos2 aB1 C y1 þ sin aF 1 Gy1 þ sin a cos aB1 Gy1 Z n¼0

þ sin a cos aF 1 C y1 e2gn=T :

Since the explicit expressions of solutions of Eq. (21) are very complicated, here we skip the details and give our results in terms of figures. When a ¼ 0, the two qubits are initially in the separate state j01i, from Fig. 1 we can find the initial unentangled system evolves into an entangled state. This can be easily understood that when the temperature is low, the decoherence effect induced by the environment is very weak, and the two qubits are coupled to the same environment that in turn generates some effective interaction between any two qubits initially uncoupled. As a result, a considerable entanglement appears and the state will remain entangled for all time at low temperature, which is different from the result of Ref. [19] for the initial system state j00i or j11i under the same environment, where the entanglement not only nonsmoothly becomes and stay zero for a finite interval of time, but also gets a less value than that of state j01i. This implies that the initial system state j01i is more constructive for the generation of entanglement than that state j00i or j11i under the system-bath interaction. In Figs. 2 and 3, we give the plot of the entanglement evolution when the two qubits are initially in the state jUi ¼ cos aj00iþ sin aj11i and jWi ¼ cos aj01i þ sin aj10i for different values of a. When 0 < a < p=2, one can see from Fig. 2 that there are two interesting features in the process of entanglement evolution. One is that the entanglement decays to zero and will remain zero for a period of time before entanglement recovers no matter what the value of a, this striking phenomenon is the so-called entanglement sudden death (ESD) first proposed by Yu and Eberly [26]. The other is that the death time depends on the initial states strongly. For the state jUi ¼ cos aj00i þ sin aj11i, the death time is shorter than that of state jWi ¼ cos aj01i þ sin aj10i, that is to say, jUi is more robust against the decoherence of environment than that of state jWi. However, it is interesting to see from Fig. 3 that for the initial state jWi ¼ cos aj01i þ sin aj10i when p=2 < a < p, such as a ¼ 3p=4, namely the initial state is the maximally entangled state jWi ¼ p1ffiffi2 ðj01i  j10iÞ, the environment has no effect on the entanglement evolution between the two qubits, though the initial entanglement is the same as the state jWi ¼ p1ffiffi2 ðj01i þ j10iÞ. This is because initial state jWi ¼ p1ffiffi2 ðj01i  j10iÞ is in the decoherence-free space(DFS), which is defined first by Duan and Lidar, respectively [27,28]. What’s more, for a ¼ 5p=6 and a ¼ 11p=12,

q11 ¼

q33

q44

q23

0.8 0.7 0.6

C(t)

q22

0.9

ð23Þ

0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40

50

60

70

80

90

100

t Fig. 1. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the separate state j01i. The parameter T ¼ 1, l0 ¼ 2, g 0 ¼ g ¼ 1, X ¼ D ¼ f ¼ 0.

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α=π/4

1

α=π/6

1

0.8

0.8

0.6

0.6

α=π/12

0.7 0.6

0.4

C(t)

C(t)

C(t)

0.5

0.4

0.4 0.3 0.2

0.2

0.2

0

0

1

2

3

4

0

5

0.1 0

1

2

3

4

0

5

0

1

2

t

t

3

4

5

t

Fig. 2. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the state jUi ¼ cos aj00i þ sin aj11i (solid line) and jWi ¼ cos aj01i þ sin aj10i (dotted line) for different values of a. The parameter T ¼ 2, l0 ¼ 2, g 0 ¼ g ¼ 1, X ¼ D ¼ f ¼ 0.

α=3π/4

α=5π/6

1

α=11π/12

0.8

1 0.8 0.6

0.4

0.4

0.4

0.2

0.2

0.2 0

0.6

C(t)

0.6

C(t)

C(t)

0.8

0

1

2

t

3

4

0

5

0

1

2

t

3

4

0

5

0

1

2

t

3

4

5

Fig. 3. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the state jUi ¼ cos aj00i þ sin aj11i and jWi ¼ cos aj01i þ sin aj10i for different values of a. The parameter T ¼ 2, l0 ¼ 2, g 0 ¼ g ¼ 1, X ¼ D ¼ f ¼ 0.

1

0.12 0.1

Δ=1

0.8

Δ=2

C(t)

0.08

C(t)

Δ=0

0.9

0.06

Δ=0

0.7

Δ=2

0.6

Δ=5

0.5 0.4

0.04

0.3 0.2

0.02

0.1

0

0

5

10

15

20

25

t

30

35

40

45

50

Fig. 4. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the separate state j00i for different values of anisotropy parameter D. The parameter T ¼ 1, l0 ¼ 2, g 0 ¼ g ¼ 1, X ¼ f ¼ 0.

compared to the initial states for a ¼ p=6 and a ¼ p=12 presented in Fig. 2, the entanglement not only does not decrease under the influence of the environment but also has a slight increase with time evolves. However, for the initial state jUi ¼ cos aj00iþ sin aj11i, the above conclusion is not valid, since the entanglement evolution exhibits the same behavior for 0 < a < p=2 and p=2 < a < p. On the basis of the result, we can say that the state

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t Fig. 5. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the maximally entangled state jWi ¼ p1ffiffi2 ðj00i þ j11iÞ for different values of anisotropy parameter D. The parameter T ¼ 5, l0 ¼ 2, g 0 ¼ g ¼ 1, X ¼ f ¼ 0.

cos aj01i þ sin aj10i for p=2 < a < p is most robust than other states under such spin environment, so this state is the appropriate for the information carrier under the system-bath interaction. In Ref. [19], the authors have revealed that the decoherence can be suppressed by controlling the parameters related to the system and environment, such as increasing the coupling constants g, and

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X, and decreasing g 0 . But the above discussion only concentrates on the entanglement evolution in the absence of the inhomogeneous magnetic field and anisotropy. So in the following, take the initial state jUi ¼ cos aj00i þ sin aj11i as an example, we will investigate the effects of inhomogeneous magnetic field and anisotropy on the entanglement dynamics and decoherence. In Figs. 4 and 5, the time evolution of the entanglement when the two-qubit are initially in the separate state j00i and the maximally entangled state jUi ¼ p1ffiffi2 ðj00i þ j11iÞ are shown. One can find that the entanglement generation of the two qubits is very sensitive to the anisotropy parameter when the initial state is the separate state j00i. Increasing the value of D will cause the entanglement collapse which is opposite to the case for the initial state jUi ¼ p1ffiffi2 ðj00i þ j11iÞ, where the larger the value of D, the more slowly the entanglement decays, that is to say, the anisotropy parameter D can moderate the destruction of decoherence induced by the environment and benefit the entanglement. In Fig. 6a and b, we give the plot of entanglement evolution for different values of uniform and inhomogeneous magnetic fields when the two qubits are initially in the separate state j00i, respectively. We find the effects of uniform and inhomogeneous magnetic fields on the evolution of entanglement are completely opposite. Increasing the value of uniform magnetic field l0 results in a stronger decoherence effect, as a result, in most time regions the entanglement is weakened or disappears. On the other hand, we also find that introducing the inhomogeneous magnetic field

(b)

0.12

0.1

0.1

0.08

0.08

C(t)

C(t)

(a) 0.12

f is very helpful for the generation of the entanglement, which can be seen from that in most time regions the entanglement is not only induced but also increased with the increasing f. Moreover, the inhomogeneous magnetic field in the construction of the entanglement is more obvious when the initial state is entangled. Fig. 7 shows that the time evolution of the entanglement for different values of inhomogeneous field f when the system is initially in the maximally entangled state jUi ¼ p1ffiffi2 ðj00i þ j11iÞ. One can see that when the temperature is very low, the entanglement oscillates with time continuously and the minimal value of the entanglement is increased with increasing f. But as the temperature increases (T ¼ 5), since the more the temperature the stronger effect of decoherence induced by the environment, as a consequence, the entanglement sudden death occurs. At the same time, we also observe that increasing f will weaken the effects of ESD, and when f is strong enough the ESD disappears completely, that is to say, the inhomogeneous magnetic field may actually suppress the decoherence and help generate continuous-in-time entanglement between the two qubits. So we can conclude that we may efficiently control the occurrence of ESD by using the inhomogeneous magnetic field f. This may be useful for the quantum information process based on the entanglement. Since the abrupt disappearance of entanglement may be a bad thing for most of quantum information processes based on entanglement. For example, in the Bennett’s quantum teleportation scheme, the quantum channel between Alice and Bob is one entanglement pair. When

0.06

0.06

0.04

0.04

0.02

0.02

0

0

10

20

t

30

40

0

50

0

10

20

t

30

40

50

Fig. 6. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the separate state j00i. (a) for different values of uniform magnetic field l0 . l0 ¼ 2 (solid line), l0 ¼ 3 (dotted line), f ¼ 0. (b) for different values of inhomogeneous magnetic field f. f ¼ 0 (solid line), f ¼ 1 (dotted line), l0 ¼ 2. The parameter T ¼ 1, g 0 ¼ g ¼ 1, X ¼ D ¼ 0.

(a)

1

(b)

0.9

0.8

0.8

0.7 0.6

0.7

C(t)

C(t)

1 0.9

0.6

0.5 0.4 0.3

0.5

0.2

0.4

0.1 0

2

4

t

6

8

10

0

0

2

4

t

6

8

10

Fig. 7. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the maximally entangled state jWi ¼ p1ffiffi2 ðj00i þ j11iÞ for different values of inhomogeneous magnetic field f. The parameters l0 ¼ 2, g 0 ¼ g ¼ 1, X ¼ D ¼ 0, f ¼ 0 (solid line), f ¼ 2 (dashed line), f ¼ 4 (dotted line). (a) T ¼ 0:1, and (b) T ¼ 5.

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J.-L. Guo, H.-S. Song / Optics Communications 282 (2009) 4637–4642

(b)

1

1

0.8

0.8

0.6

0.6

C(t)

C(t)

(a)

0.4

0.4

0.2

0.2

0

0

2

4

t

6

8

0

10

0

2

4

t

6

8

10

Fig. 8. The time evolution of the concurrence CðtÞ when the two-qubit are initially in the maximally entangled state jWi ¼ p1ffiffi2 ðj00i þ j11iÞ for different values of uniform magnetic field l0 . The parameters f ¼ 0, g 0 ¼ g ¼ 1, X ¼ D ¼ 0, l0 ¼ 2 (solid line), l0 ¼ 4 (dashed line), l0 ¼ 6 (dotted line). (a) T ¼ 0:1, and (b) T ¼ 5.

the entanglement between Alice and Bob disappears abruptly, the quantum channel between them is also cut off, which may spoil the quantum information transferring from Alice to Bob. Therefore, from the viewpoint that the entanglement is a resource, it would be a meaningful work to have some methods to control the occurrence of ESD. Now we may ask weather the ESD can be controlled by the uniform magnetic field. In order to answer this question, in Fig. 8 we give the plot of the time evolution of the entanglement for different values of uniform magnetic field l0 when the system is initially in the maximally entangled state jUi ¼ p1ffiffi2 ðj00i þ j11iÞ. We can easily see that when the uniform magnetic field l0 increases, the continuous non-zero entanglement evolution turns to be intermittent at low temperature, and the ESD effect is enlarged for a higher temperature. But going on increasing l0 , the entanglement can be enhanced and the ESD effect is weakened. This implies that increasing uniform magnetic field l0 can not only induce ESD but also suppress ESD, which depends on the value of l0 . So using the uniform magnetic field to control the ESD is more difficult than that of inhomogeneous magnetic field in practice. 4. Conclusion In conclusion, we have investigated the entanglement dynamics and decoherence of a two-qubit system under a quantum spin environment at finite temperature in the thermodynamics limit. By using the novel operator technique, we obtain analytical results of entanglement evolution and decoherence with employing the entanglement measure of concurrence. Our results show that when the initial state is unentangled, the state j01i is more constructive for the generation of entanglement than that state j00i or j11i under the system-bath interaction. Moreover, when the initial state is entangled, for the initial state jUi ¼ cos aj00iþsin aj11i and jWi ¼ cos aj01i þ sin aj10i, we find the character of the entanglement evolution for both initial states behavior differently. When 0 < a < p=2, we can say the state jUi is stronger than state jWi as the death time is shorter for the state jUi that of state jUi. But when p=2 < a < p, we may conclude that the state jWi is most robust than other states, since the entanglement remains unchanged or

increased under such spin environment. In addition, we also find the anisotropic parameter D can moderate the destruction of decoherence induced by the environment and benefit the entanglement, which is also depends on the initial states. Finally, introducing the inhomogeneous magnetic field f, we find the entanglement sudden death can be controlled by f, which may be useful for the quantum information process based on the entanglement. Acknowledgement We would like to acknowledge National Natural Science Foundation of China under Grant No. 10875020. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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