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Analysis of mixed state entanglement with intrinsic decoherence
Physica E 44 (2011) 6–11
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Physica E journal homepage: www.elsevier.com/locate/physe
Analysis of mixed state entanglement with intrinsic decoherence S. Abdel-Khalek a,b,, Nour Zidan a,c, M. Abdel-Aty a,d a
Mathematics Department, Faculty of Science, Sohag University, 82524 Sohag, Egypt Mathematics Department, College of Science, Taif University, Taif, Saudi Arabia Mathematics Department, College of Science, Al-Jouf University, Saudi Arabia d College of Science, Bahrain University, 32038, Bahrain b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 December 2010 Received in revised form 20 February 2011 Accepted 5 June 2011 Available online 18 July 2011
We analyze different entanglement measures for a mixed state two-level system in the presence of intrinsic decoherence. The information about entanglement is obtained by comparing the results for the atomic Wehrl entropy and negativity with the analytical results for a simple case. For the strong decoherence case we find that a similar and long-lived maximum Wehrl entropy and negativity between atom and field are shown. The results highlight the important roles played by both the decoherence parameter and the initial state setting in determining the evolution of the atomic Wehrl entropy and negativity. & 2011 Elsevier B.V. All rights reserved.
1. Introduction The physical characteristics of the entanglement of quantum mechanical states, both pure and mixed, have been recognized as a central resource in various aspects of quantum information processing. Significant settings include quantum communication [1], cryptography [2], teleportation [3], and, to an extent that is not quite so clear, quantum computation [4]. Most of earlier studies pay much attention to the bipartite system that start in pure states, which are presently well understood [5]. While for mixed state, the entanglement is difficult to be measured, because it is not easy to define an analog of the Schmidt decomposition. However, it is more significant for studying the system in mixed states since there does not exist a system that could be decoupled perfectly from environmental influences for realistic states observed in experiments. Several entanglement measures for mixed states have been proposed, such as the entanglement of formation, the entanglement cost and the distillable entanglement [6,7], but these are not effective computational means. To overcome these shortcoming, Peres and Horodecki have proposed a new criterion for separability [8,9]. Based on this criterion, negativity is introduced, which is a good quantity for measuring the entanglement of mixed states [10,11] and is equal to absolute value of the sum of negative eigenvalues of the partial transpose od density operator with a bipartite mixed state.
Corresponding author at: Mathematics Department, Faculty of Science, Sohag University, 82524 Sohag, Egypt. Tel.: þ20 93 2905470. E-mail address: [email protected] (S. Abdel-Khalek).
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.06.007
There are a number of new papers focused on the effect of intrinsic decoherence in quantum system [12]. One explanation of why quantum coherences are deteriorated and difficult to observe is based on a hypothesis that closed quantum systems do not ¨ evolve unitarily according to the Schrodinger equation, but are governed by more generalized equations that include intrinsic decoherence. In particular, an elegant model of intrinsic decoher¨ ence based on a simple modification of a unitary Schrodinger evolution has been proposed. This model is based on an assumption that on sufficiently short time scales the closed system evolves by a random sequence of unitary phase changes generated by the system Hamiltonian. The entanglement generation and entropy growth due to intrinsic decoherence in the Jaynes–Cummings model has been investigated [13]. It is found that the intrinsic decoherence has a great effect on both of total entropy and partial entropy. Another work in this direction is the effect of Intrinsic decoherence on the entropy squeezing of Coupled field-superconducting charge qubit [14]. It is reported that the appearing and disappearing of entropy squeezing is depend on the intrinsic decoherence. Also, the intrinsic decoherence of Tripartite states Bell-nonlocality has been investigated [15]. It has been shown that the multipartite Bell-inequality violations can be destroyed in finite time in the system and the influence of the intrinsic decoherence on the tripartite states Bell-nonlocality sudden death is demonstrated. There are a number of measures of entanglement for a bipartite system. The entanglement of formation, the entanglement of distillation [16], the concurrence [17] and relative entropy [18] are some of these measures. A method using quantum mutual entropy to measure the degree of entanglement in the time development of the Jaynes–Cummings model has been adopted in Ref. [19] and the case of the two-level atom with squeezed state has
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
been studied [20]. The question of how mixed a two-level system and a field mode may be such that free entanglement arises in the course of the time evolution according to a Jaynes–Cummings type interaction has been considered [21]. Recently, the effects of atomic motion and field mode structure in a two-level atom interacting with a single mode via single photon transition have been investigated [22,23]. It has been shown that the atomic motion leads to periodic evolution of the von Neumann entropy. Also, these results have been extended to include the mixed state entanglement and analysis of the pattern entanglement between a three-level trapped ion and laser field [24,25]. Another extension is to examine the influences of the atomic motion and field-mode structure on the dynamical properties of the Wehrl entropy of the cavity field in comparison with the von Neumann entropy, for two-photon processes [26]. Experimental work has also been involved in the preparation and measurement of the notional state of a trapped ion initially laser cooled to its zero-point of motion [27]. These studies have involved the interaction of two-level systems with Fock states, coherent states, squeezed states, and Schr¨odinger cat-like states, in addition to the consideration of various side-band transitions.
2. The model In Milburn’s theory, the dynamics of the system are governed by the following equation [28]: d i b 1 b b b 2 ½H,½ b , b ðtÞ ¼ ½H, r r H, r dt _ 2_ g
ð1Þ
where g is the decoherence rate. We consider the exact solution of the Milburn equation (1) for the Hamiltonian describing that interacting a two-level atom with a single-mode cavity field via an k-photon process in the rotating-wave approximation is given by k D b ¼o a by a bþ s byk þ s bk Þ, b3 þ s b 3 þ lðs b a bþa H ð2Þ 2 2
7
b ¼H b 0 þH b 1 and ½H b 0,H b 1 ¼ 0, where [29] H ! b þk=2 n 0 b1 ¼ l 0 b0 ¼ o , H H b k=2 0 n byk a
bk a
!
0
:
ð6Þ
Similarly, the square of the Hamiltonian (2) can also be expressed as a b þB b B b and ½A, b ¼ 0, where, b2 ¼ A sum of two terms H ! bb 0 b ¼ Sðn , þ Þ A , b b 0 Sðn ,Þ 0 b ¼ 2lo@ B
b k ðn b þ k=2Þ a
0 b b k=2Þa ðn
yk
0
1 A,
ð7Þ
with 2 k 2 k yk b b a b , b , þ Þ ¼ o2 n bþ Sðn þl a 2 2 k 2 yk k b b a b : b b ,Þ ¼ o2 n þl a Sðn 2
ð8Þ
Taking into account the initial condition (4) we can write down the following expression b b b 1 tÞ exp t B b r b ð0Þ ¼ expðiH b 1 ðtÞ eSt eLt r 2g t b b 1 tÞ r b 2 ðtÞ, expðiH ð9Þ exp B 2g b 1 ðtÞ is defined by where the auxiliary operator r ! jCðtÞS/CðtÞj 0 b r 1 ðtÞ ¼ , 0 j ðtÞS/ ðtÞj
ð10Þ
where t b , þ Þ jaS, Sðn jCðtÞS ¼ b1 exp b 2g t b ,Þ jaS: j ðtÞS ¼ b2 exp b Sðn 2g
ð11Þ
y
b and a b the annihilation and where D ¼ o0 ko. We denote by a b 3 is the atomic inversion operator, creation operators, respectively, s b 7 are the atomic operators, o is the frequency of and s the cavity field, o0 is the atomic transition frequency and l is the atom–field coupling constant. In this paper we will assume the exact resonance between the field and atomic frequencies (i.e. o0 ¼ ko). b ðtÞ of the Milburn We express the solution for the density operator r equation (1) applied to the Hamiltonian (2) in terms of three superoperators bJ, b S and b L as b b b
b ðtÞ ¼ eJ t eSt eLt r b ð0Þ, r
where
The superoperators are defined through their action on the density operator as
g
b b r b , b ¼ i½H, Sr
ð12Þ
ð3Þ
b ð0Þ is the density operator of the initial atom–field system. where r We assume that the field pffiffiffiffiffi is prepared in the coherent state P jaS ¼ expð 12 jaj2 Þðan = n!ÞjnS, and the atom was prepared in mixed state so that 0 1 b21 0 AjaS/aj, b2 þ b2 ¼ 1: b ð0Þ ¼ @ r ð4Þ 1 2 2 0 b2
1b b bj r b¼ H b H, r
b can be written in the form The operator expððt=2gÞBÞ t b exp B 2g 0 1 1 k b ð þ Þ ðtÞ b ðÞ ðtÞ b ffiffiffiffiffiffiffiffiffiffiffi ffi q X Y a n n B C B C bk byk a a B C B C, ¼B C ðþÞ ðÞ 1 yk b ðtÞ b ðtÞ B a C b ffiffiffiffiffiffiffiffiffiffiffi ffi q Y X n n @ A k yk b b a a
1 b b b ¼ fH b g: Lr ,r 2g 2
ð5Þ
We divide the Hamiltonian (2) into a sum of two terms as
qffiffiffiffiffiffiffiffiffiffiffiffi k b ð þ Þ ðtÞ ¼ cosh lot n bk a byk , bþ X a n 2 g qffiffiffiffiffiffiffiffiffiffiffiffi k b ðÞ ðtÞ ¼ cosh lot n byk a bk , b X a n 2 g qffiffiffiffiffiffiffiffiffiffiffiffi k b ð þ Þ ðtÞ ¼ sinh lot n bk a byk , b a þ Y n 2 g qffiffiffiffiffiffiffiffiffiffiffiffi k b ðÞ ðtÞ ¼ sinh lot n byk a bk : b a Y n 2 g b can be written in the form Similarly, the operator expðiHtÞ b expðiHtÞ
ð13Þ
8
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
0
b ð þ Þ ðtÞ C n
B B B ¼B B 1 yk bð þ Þ bðþÞ B ia @ b S n ðtÞqffiffiffiffiffiffiffiffiffiffiffiffiY n ðtÞ bk a byk a
1 ðÞ 1 bk b S n ðtÞqffiffiffiffiffiffiffiffiffiffiffiffi C ia bk C byk a a C C, C ðÞ b C X n ðtÞ A
ð14Þ
b ðmÞ b ðmÞ with the operators j n ð 7 Þ and f n ð 7 Þ are defined by, respectively, qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi k k byk , j bk , bk a byk a bþ b b n ðÞ ¼ o n b nð þ Þ ¼ o n þl a þl a j 2 2 ð24Þ
b ðþÞ ¼ o n bþ f n
where b ð þ Þ ðtÞ ¼ cos½lt C n ðþÞ b S n ðtÞ ¼ sin½lt
qffiffiffiffiffiffiffiffiffiffiffiffi byk , bk a a
qffiffiffiffiffiffiffiffiffiffiffiffi byk , bk a a
b ðÞ ðtÞ ¼ cos½lt C n ðÞ b S n ðtÞ ¼ sin½lt
qffiffiffiffiffiffiffiffiffiffiffiffi bk , byk a a
ð15Þ
qffiffiffiffiffiffiffiffiffiffiffiffi bk : byk a a
Then from Eqs. (12) and (14) it follows that 0 1 b ðÞ m V n ðtÞ b ð þ Þ ðtÞ b ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p a R B C n B bm C bym a t b a B C b ¼B expðiHtÞexp B C, ðþÞ B C 2g b ðÞ ym V n ðtÞ @ a A b b pffiffiffiffiffiffiffiffiffiffiffiffiffiffi R n ðtÞ m ym b b a a
b ðÞ ¼ o n b f n
qffiffiffiffiffiffiffiffiffiffiffiffiffi k bk : byk a l a 2
Substituting from Eq. (22) into Eq. (21), we get: 0 X 1 1 m 1 m X t t b ðmÞ ðtÞ b ðmÞ ðtÞ M M 11 12 B C g Bm¼0 g C m¼0 C, b ðtÞ ¼ B r m 1 m 1 B X C X t ðmÞ ðmÞ t @ b ðtÞ b ðtÞ A M M 21 22 m¼0
g
ð25Þ
g
m¼0
where ð16Þ
ðmÞ
ðmÞ
b 11 ðtÞ þ b ðtÞb b ðmÞ ðtÞ ¼ b f n ð þ Þ½C f n ðþÞ M 11 11 ðmÞ
b 21 ðtÞ þ b ðtÞb bk g b nðmÞ0 ðÞ½C f n ðþÞ þa 21 ðmÞ
b 12 ðtÞ þ b ðtÞgb ðmÞ0 ðÞa byk þb f n ð þ Þ½C n 12
where ð7Þ b ð 7 Þ ðtÞX b ð 7 Þ ðtÞ ¼ C b ð 7 Þ ðtÞ þ ib b ð 7 Þ ðtÞ, R S n ðtÞY n n n n
22
21
ðmÞ
b 21 ðtÞ þ b ðtÞa bk gb ðmÞ0 þb f n ðÞ½C n ðÞ 21
22
b ðtÞ ¼ j ðtÞS/ ðtÞj ij i j
ði,j ¼ 1,2Þ
b ðÞ ðtÞ yk V n b jCðtÞS, ffia jC2 ðtÞS ¼ qffiffiffiffiffiffiffiffiffiffiffi bk byk a a
b ð þ Þ ðtÞ k V n b j ðtÞS, q ffiffiffiffiffiffiffiffiffiffiffiffi a ðtÞS ¼ 1 byk bk a a
j
2 ðtÞS
ðmÞ ðmÞ b 22 ðtÞ þ b ðtÞb þb f n ðÞ½C f n ðÞ, 22
ð19Þ
b bðmÞ b ðmÞ ðtÞ ¼ ðM b ðmÞ ðtÞÞy ¼ g byk b b ðmÞ0 M 21 12 n ðÞa ½C 11 ðtÞ þ 11 ðtÞf n ð þÞ
ðmÞ
b 22 ðtÞ þ b ðtÞg byk , b nðmÞ0 ðÞa þb f n ðÞ½C 22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmÞ0 bk Þgb ðmÞ byk a where gb n ð 7 Þ ¼ ð1= a n ð 7 Þ.
ð28Þ
ð20Þ 3. Negativity and atomic Wehrl entropy In this section: we investigate the marginal atomic Q function and atomic Wehrl entropy AWE. We start our investigation by defining the atomic Q-function as [30]
ðÞ
b ðtÞj ðtÞS: ¼R n
g
The mth power of the Hamiltonian (2) can be expressed in the form: 0 1 ðmÞ 1 k ðmÞ b b b ffiffiffiffiffiffiffiffiffiffiffi ffi q g f ð þ Þ a ðÞ n n B C B C bk byk a a B C m b B C, ð22Þ H ¼B C ðmÞ 1 yk ðmÞ b Ba C b b ffiffiffiffiffiffiffiffiffiffiffi ffi q g ð þÞ f ðÞ n n @ A byk bk a a ðmÞ
ðmÞ where the operators b f n ð 7 Þ and gb n ð 7 Þ are defined by, respectively, ðmÞ b b ðmÞ b ðmÞ f n ð 7Þ ¼ 12 ½j n ð 7Þ þ f n ð 7Þ,
ð27Þ
ðmÞ0 b 12 ðtÞ þ b ðtÞg byk ½C byk b nðmÞ0 ðÞa þ gb n ðÞa 12
Take into account the definition of the superoperator bJ to evaluate the b b 2 ðtÞ as follows: action of the operator ejt on the ‘‘density’’ operator r m 1 X t b m: b mr b 2 ðtÞH b ðtÞ ¼ r ð21Þ H m¼0
yk
ðmÞ0
ðmÞ ðmÞ b 21 ðtÞ þ b ðtÞb þb f n ðÞ½C f n ð þÞ 21
with b ð þ Þ ðtÞjCðtÞS, jC1 ðtÞS ¼ R n
ðmÞ
b 12 ðtÞ þ b ðtÞb b ½C þ gb n ðÞa f n ðÞ 12
where we have used the following symbol: b ðtÞ ¼ jC ðtÞS/C ðtÞj, C ij i j
ð26Þ
b b ðmÞ ðtÞ ¼ g byk b bk b ðmÞ0 b ðmÞ0 M n ðÞa ½C 11 ðtÞ þ 11 ðtÞa g n ðÞ 22
Substituting Eqs. (10) and (16) into Eq. (9), we obtain an explicit b 2 ðtÞ as follows: expression for the operator r 0 1 b 12 ðtÞ þ b ðtÞ b 11 ðtÞ þ b ðtÞ C C 11 12 A, b 2 ðtÞ ¼ @ ð18Þ r b ðtÞ þ b ðtÞ b ðtÞ þ b ðtÞ C C 21
b 22 ðtÞ þ b ðtÞg bk g byk b nðmÞ0 ðÞ½C b ðmÞ0 þa n ðÞa , 22
ð17Þ
ð7Þ b ð 7 Þ ðtÞY b ð 7 Þ ðtÞ þib b ð 7 Þ ðtÞ: b ð 7 Þ ðtÞ ¼ C S n ðtÞX V n n n n
j
qffiffiffiffiffiffiffiffiffiffiffiffi k byk , bk a l a 2
b ðmÞ b nðmÞ ð 7 Þ ¼ 12½j b ðmÞ g n ð 7 Þf n ð 7 Þ, ð23Þ
QA ðY, F,tÞ ¼
1 /Y, Fjr^ ðtÞjY, FS, 2p
ð29Þ
where r^ ðtÞ is the density matrix which is given in Eq. (25) and jY, FS is the atomic coherent state expressed as jY, FS ¼ cosðY=2ÞjeSþ sinðY=2ÞeiF jgS,
ð30Þ
where 0 r Y r p,0 r F r2p. The definition (29) means that two different spin coherent states overlap unless they directed into two antipodal points on the sphere [30]. The atomic Wehrl entropy can be written in terms of the atomic Q function as [30]: Z 2p Z p SAW ðtÞ ¼ QA ðY, F,tÞlnQA ðY, F,tÞ sin Y dY dF: ð31Þ 0
0
One can easily check that the QA is normalized. By integrating the atomic Q-function QA over the atomic variable Y, we obtain the marginal atomic Q-function as follows: Z 2p QF ¼ QA sin Y dY: ð32Þ 0
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
Another measure of entanglement we use here is the negativity which is based on the Peres–Horodecki [8,9] criterion for entanglement and is defined by the formula @ðrÞ ¼ maxð0,2ni Þ,
ð33Þ
where the sum is taken over the negative eigenvalues ni of the partial transposition of the density matrix r of the system.
9
4. Numerical results Based on the analytical solution in Section 3, in this section we will present some numerical results for different values of the system parameters in Figs. 1–4 to demonstrate the effects of the decoherence rate g=l on the negativity and atomic Wehrl entropy, so that we will assume that the initial state of the atom–field system accords Eq. (4), where we set ðb1 ¼ cosðp=6Þ and b2 ¼ sinðp=6ÞÞ.
0.5 0.45 N(ρ)
0.4 0.35 0.3 0.25 0.2
0
2
4
6 scaled time
8
10
12
0
2
4
6 scaled time
8
10
12
W(t)
0.5
0
−0.5
Fig. 1. Time evolution of the: (a) negativity NðrÞ, (b) atomic inversion SAW for ðn,k, yÞ ¼ ð5,1, p=6Þ and with different values of g=l.
0.5 0.45 N(ρ)
0.4 0.35 0.3 0.25 0.2
0
2
4
6 scaled time
8
10
12
0
2
4
6 scaled time
8
10
12
W(t)
0.5
0
−0.5
Fig. 2. The same as Fig. 1 but for ðn,k, yÞ ¼ ð25,2, p=6Þ.
10
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
2.52
SAW
2.515 2.51 2.505 2.5
0
5
10
15
20 scaled time
25
6 scaled time
8
30
35
40
2.52 2.518
SAW
2.516 2.514 2.512 2.51 2.508 2.506
0
2
4
10
12
Fig. 3. The atomic Wehrl entropy SAW versus the scaled time for y ¼ p=6, where (a) ðn,kÞ ¼ ð5,1Þ, (b) ðn,kÞ ¼ ð25,2Þ and with different values of g=l.
Fig. 4. The surface plot of the marginal atomic Q function QF versus the scaled time and the phase space parameter F for y ¼ p=6, where (a,b) ðn,kÞ ¼ ð5,1Þ, g/l ¼ 105, g/l ¼ 104, (c,d) ðn,kÞ ¼ ð25,2Þ, g/l ¼105, g/l ¼ 104 and with different values of g=l.
In Fig. 1, we plot the negativity and atomic inversion as functions of scaled time for different values of g=l with n ¼ 5 and one photon process (k¼1). When g=l ¼ 10 (soled curve) we see that @ðrÞ starts from minimum value and increases gradually as time goes on and the damping of oscillations occurred, while by increasing g=l we can see that more oscillations will take place. Also the atomic inversion oscillates around zero and with increasing g=l we have collapse–revival phenomenon. Fig. 2
depicts the average photon number k influences on the negativity evolution. It is shown that with increasing k (say k¼2) with small value of g=l, the negativity starts from minimum value and increases gradually as the time goes on and the damped oscillation becomes faster. By increasing the decoherence parameter the amplitude of the negativity is decreased and reaches its maximum faster. In this case a long-lived entanglement is shown once the decoherence is increased further.
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
Based on Eq. (25) we present the main results for the effect of decoherence parameter, g=l, on the evolution of the atomic Wehrl entropy SAW , in Fig. 3. We have presented the dynamics of quantum entanglement due to the atomic Wehrl entropy. The first eye-catching atomic Wehrl entropy has the same behavior as the negativity (involve typical information). The above observation highlights atomic Wehrl entropy which can successfully used an entanglement quantifier of the mixed state system in the presence of intrinsic decoherence (see Figs. 2, 3). On the other hand, when the intrinsic decoherence parameter is considered, it is noticed that the maximum value of SAW is increased and the amplitude is decreased. For the larger values of the intrinsic decoherence the phenomena of long-lived entanglement can be observed. The comparison between Fig. 3(a,b) shows that the same effect of g=l is observed with a periodic behavior, which returns to the periodicity of two-photon processes. We may refer here to the work given in Ref. [31] where an asymptotic value of the concurrence which embodies entanglement survival has been observed, independent of the interaction development but critically dependent on the environment. In Fig. 4, we consider the evolution of the marginal atomic Q-function QF ðtÞ. Our main aim in this case is find a relation between quantify the entanglement between a mixed state twolevel atom and coherent field and QF ðtÞ. Taking this fact into account, we investigate QF ðtÞ for different values of the decoherence parameter g=l. From Eq. (32) one can conclude the following remarks: QF ðtÞ has two terms with different signs due to different values of F, e.g. when F ¼ p=2 we see that QF ¼ X þY, while QF ¼ XY if F ¼ 3p=2. This in fact interprets the exchange of the maximum and minimum values of QF due to the variation of F (see Fig. 4). This also refers to the general behavior of the entanglement of the system under consideration and there is one-to-one correspondence between the behavior of QF ðtÞ and the Wehrl entropy or negativity [32]. This new observation opens the door for using QF ðtÞ as an entanglement measure. 5. Conclusion In this paper, we have developed a framework to study a mixed state two-level atomic system, in the presence of intrinsic decoherence exploring the behavior of the atomic Wehrl entropy, marginal distributions of the atomic Q-function in connection with entanglement. An exact solution of the density matrix, in terms of the intrinsic decoherence parameter g=l for the system
11
under consideration has been obtained. The results exhibit specific features, arising from variations of the adjustable parameters of the system. Namely, the intrinsic decoherence parameter g=l and initial state settings. It is shown that, the atomic Wehrl entropy exhibits behaviors that reflect a pattern of the entanglement as measured by the negativity.
References [1] B. Schumacher, Phys. Rev. A 54 (1996) 2614. [2] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [3] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [4] M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, England, 2000. [5] C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. [6] C.H. Bennett, D.P. Divincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. [7] S. Popescu, D. Rohrlich, Phys. Rev. A 56 (1997) R3319. [8] A. Peres, Phys. Rev. Lett. 77 (1996) 1413. [9] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 210 (1996) 223. [10] G. Vidal, R.F. Werner, Phys. Rev. A 65 (2002) 032314. [11] G. Vidal, J. Mod. Opt. 47 (2000) 355. [12] B.-Q. Liu, B. Shao, J. Zou, Phys. Lett. A 374 (2010) 1970; V. Buzek, M. Konopka, Phys. Rev. A 58 (1998) 1735; X. Chena, L.-M. Ku, Phys. Lett. A 191 (1994) 18; Z. He, Z. Xiong, Y. Zhang, Phys. Lett. A 354 (2006) 79; M. Abdel-Aty, Phys. Lett. A 372 (2008) 3719. [13] A.-S.F. Obada, H.A. Hessian, J. Opt. Soc. Am. B 21 (2004) 1535. [14] Y. Xue-Qun, S. Bin, Z. Jian, Commun. Theor. Phys. 48 (2007) 63. [15] B. Shao, J. Zou, Phys. Lett. A 374 (2010) 1970. [16] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wooters, et al., Phys. Rev. A 54 (1996) 3824. [17] S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022. [18] V. Vedral, M.B. Plenio, Phys. Rev. A 57 (1998) 1619. [19] S. Furuichi, M. Ohya, Lett. Math. Phys. 49 (1999) 279. [20] M. Abdel-Aty, S. Furuichi, A.-S.F. Obada, J. Opt. B: Quant. Semiclass. Opt. 4 (2002) 37; S. Furuichi, M. Abdel-Aty, J. Phys. A: Math. Gen. 34 (2001) 6851. [21] S. Scheel, J. Eisert, P.L. Knight, M.B. Plenio, quant/ph/0207120. [22] M.F. Fang, Physica A 204 (1994) 193. [23] S. Abdalla, A.-S.F. Obada, S. Abdel-Khalek, Chaos Solitons Fractals 36 (2008) 405. [24] S. Furuichi, S. Nakamura, J. Phys. A: Math. Gen. 35 (2002) 5445. [25] M. Abdel-Aty, J. Phys. A: Math. Gen. 38 (2005) 8589. [26] S. Abdel-Khalek, Physica A 387 (2008) 779. [27] D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, D.J. Wineland, J. Mod. Opt. 44 (1997) 2485. [28] G.J. Milmurn, Phys. Rev. A 44 (1991) 5401. [29] L.-M. Kuang, X. Chen, M.-L. Ge, Phys. Rev. A 52 (1995) 1857. [30] A.-S.F. Obada, S. Abdel-Khalek, J. Phys. A: Math. Gen. 37 (2004) 6573. [31] M. Abdel-Aty, J. Phys. A: Math. Theor. 41 (2008) 185304. [32] A.-S.F. Obada, M. Abdel-Aty, Eur. Phys. J. D 27 (2003) 277.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Analysis of mixed state entanglement with intrinsic decoherence S. Abdel-Khalek a,b,, Nour Zidan a,c, M. Abdel-Aty a,d a
Mathematics Department, Faculty of Science, Sohag University, 82524 Sohag, Egypt Mathematics Department, College of Science, Taif University, Taif, Saudi Arabia Mathematics Department, College of Science, Al-Jouf University, Saudi Arabia d College of Science, Bahrain University, 32038, Bahrain b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 December 2010 Received in revised form 20 February 2011 Accepted 5 June 2011 Available online 18 July 2011
We analyze different entanglement measures for a mixed state two-level system in the presence of intrinsic decoherence. The information about entanglement is obtained by comparing the results for the atomic Wehrl entropy and negativity with the analytical results for a simple case. For the strong decoherence case we find that a similar and long-lived maximum Wehrl entropy and negativity between atom and field are shown. The results highlight the important roles played by both the decoherence parameter and the initial state setting in determining the evolution of the atomic Wehrl entropy and negativity. & 2011 Elsevier B.V. All rights reserved.
1. Introduction The physical characteristics of the entanglement of quantum mechanical states, both pure and mixed, have been recognized as a central resource in various aspects of quantum information processing. Significant settings include quantum communication [1], cryptography [2], teleportation [3], and, to an extent that is not quite so clear, quantum computation [4]. Most of earlier studies pay much attention to the bipartite system that start in pure states, which are presently well understood [5]. While for mixed state, the entanglement is difficult to be measured, because it is not easy to define an analog of the Schmidt decomposition. However, it is more significant for studying the system in mixed states since there does not exist a system that could be decoupled perfectly from environmental influences for realistic states observed in experiments. Several entanglement measures for mixed states have been proposed, such as the entanglement of formation, the entanglement cost and the distillable entanglement [6,7], but these are not effective computational means. To overcome these shortcoming, Peres and Horodecki have proposed a new criterion for separability [8,9]. Based on this criterion, negativity is introduced, which is a good quantity for measuring the entanglement of mixed states [10,11] and is equal to absolute value of the sum of negative eigenvalues of the partial transpose od density operator with a bipartite mixed state.
Corresponding author at: Mathematics Department, Faculty of Science, Sohag University, 82524 Sohag, Egypt. Tel.: þ20 93 2905470. E-mail address: [email protected] (S. Abdel-Khalek).
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.06.007
There are a number of new papers focused on the effect of intrinsic decoherence in quantum system [12]. One explanation of why quantum coherences are deteriorated and difficult to observe is based on a hypothesis that closed quantum systems do not ¨ evolve unitarily according to the Schrodinger equation, but are governed by more generalized equations that include intrinsic decoherence. In particular, an elegant model of intrinsic decoher¨ ence based on a simple modification of a unitary Schrodinger evolution has been proposed. This model is based on an assumption that on sufficiently short time scales the closed system evolves by a random sequence of unitary phase changes generated by the system Hamiltonian. The entanglement generation and entropy growth due to intrinsic decoherence in the Jaynes–Cummings model has been investigated [13]. It is found that the intrinsic decoherence has a great effect on both of total entropy and partial entropy. Another work in this direction is the effect of Intrinsic decoherence on the entropy squeezing of Coupled field-superconducting charge qubit [14]. It is reported that the appearing and disappearing of entropy squeezing is depend on the intrinsic decoherence. Also, the intrinsic decoherence of Tripartite states Bell-nonlocality has been investigated [15]. It has been shown that the multipartite Bell-inequality violations can be destroyed in finite time in the system and the influence of the intrinsic decoherence on the tripartite states Bell-nonlocality sudden death is demonstrated. There are a number of measures of entanglement for a bipartite system. The entanglement of formation, the entanglement of distillation [16], the concurrence [17] and relative entropy [18] are some of these measures. A method using quantum mutual entropy to measure the degree of entanglement in the time development of the Jaynes–Cummings model has been adopted in Ref. [19] and the case of the two-level atom with squeezed state has
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
been studied [20]. The question of how mixed a two-level system and a field mode may be such that free entanglement arises in the course of the time evolution according to a Jaynes–Cummings type interaction has been considered [21]. Recently, the effects of atomic motion and field mode structure in a two-level atom interacting with a single mode via single photon transition have been investigated [22,23]. It has been shown that the atomic motion leads to periodic evolution of the von Neumann entropy. Also, these results have been extended to include the mixed state entanglement and analysis of the pattern entanglement between a three-level trapped ion and laser field [24,25]. Another extension is to examine the influences of the atomic motion and field-mode structure on the dynamical properties of the Wehrl entropy of the cavity field in comparison with the von Neumann entropy, for two-photon processes [26]. Experimental work has also been involved in the preparation and measurement of the notional state of a trapped ion initially laser cooled to its zero-point of motion [27]. These studies have involved the interaction of two-level systems with Fock states, coherent states, squeezed states, and Schr¨odinger cat-like states, in addition to the consideration of various side-band transitions.
2. The model In Milburn’s theory, the dynamics of the system are governed by the following equation [28]: d i b 1 b b b 2 ½H,½ b , b ðtÞ ¼ ½H, r r H, r dt _ 2_ g
ð1Þ
where g is the decoherence rate. We consider the exact solution of the Milburn equation (1) for the Hamiltonian describing that interacting a two-level atom with a single-mode cavity field via an k-photon process in the rotating-wave approximation is given by k D b ¼o a by a bþ s byk þ s bk Þ, b3 þ s b 3 þ lðs b a bþa H ð2Þ 2 2
7
b ¼H b 0 þH b 1 and ½H b 0,H b 1 ¼ 0, where [29] H ! b þk=2 n 0 b1 ¼ l 0 b0 ¼ o , H H b k=2 0 n byk a
bk a
!
0
:
ð6Þ
Similarly, the square of the Hamiltonian (2) can also be expressed as a b þB b B b and ½A, b ¼ 0, where, b2 ¼ A sum of two terms H ! bb 0 b ¼ Sðn , þ Þ A , b b 0 Sðn ,Þ 0 b ¼ 2lo@ B
b k ðn b þ k=2Þ a
0 b b k=2Þa ðn
yk
0
1 A,
ð7Þ
with 2 k 2 k yk b b a b , b , þ Þ ¼ o2 n bþ Sðn þl a 2 2 k 2 yk k b b a b : b b ,Þ ¼ o2 n þl a Sðn 2
ð8Þ
Taking into account the initial condition (4) we can write down the following expression b b b 1 tÞ exp t B b r b ð0Þ ¼ expðiH b 1 ðtÞ eSt eLt r 2g t b b 1 tÞ r b 2 ðtÞ, expðiH ð9Þ exp B 2g b 1 ðtÞ is defined by where the auxiliary operator r ! jCðtÞS/CðtÞj 0 b r 1 ðtÞ ¼ , 0 j ðtÞS/ ðtÞj
ð10Þ
where t b , þ Þ jaS, Sðn jCðtÞS ¼ b1 exp b 2g t b ,Þ jaS: j ðtÞS ¼ b2 exp b Sðn 2g
ð11Þ
y
b and a b the annihilation and where D ¼ o0 ko. We denote by a b 3 is the atomic inversion operator, creation operators, respectively, s b 7 are the atomic operators, o is the frequency of and s the cavity field, o0 is the atomic transition frequency and l is the atom–field coupling constant. In this paper we will assume the exact resonance between the field and atomic frequencies (i.e. o0 ¼ ko). b ðtÞ of the Milburn We express the solution for the density operator r equation (1) applied to the Hamiltonian (2) in terms of three superoperators bJ, b S and b L as b b b
b ðtÞ ¼ eJ t eSt eLt r b ð0Þ, r
where
The superoperators are defined through their action on the density operator as
g
b b r b , b ¼ i½H, Sr
ð12Þ
ð3Þ
b ð0Þ is the density operator of the initial atom–field system. where r We assume that the field pffiffiffiffiffi is prepared in the coherent state P jaS ¼ expð 12 jaj2 Þðan = n!ÞjnS, and the atom was prepared in mixed state so that 0 1 b21 0 AjaS/aj, b2 þ b2 ¼ 1: b ð0Þ ¼ @ r ð4Þ 1 2 2 0 b2
1b b bj r b¼ H b H, r
b can be written in the form The operator expððt=2gÞBÞ t b exp B 2g 0 1 1 k b ð þ Þ ðtÞ b ðÞ ðtÞ b ffiffiffiffiffiffiffiffiffiffiffi ffi q X Y a n n B C B C bk byk a a B C B C, ¼B C ðþÞ ðÞ 1 yk b ðtÞ b ðtÞ B a C b ffiffiffiffiffiffiffiffiffiffiffi ffi q Y X n n @ A k yk b b a a
1 b b b ¼ fH b g: Lr ,r 2g 2
ð5Þ
We divide the Hamiltonian (2) into a sum of two terms as
qffiffiffiffiffiffiffiffiffiffiffiffi k b ð þ Þ ðtÞ ¼ cosh lot n bk a byk , bþ X a n 2 g qffiffiffiffiffiffiffiffiffiffiffiffi k b ðÞ ðtÞ ¼ cosh lot n byk a bk , b X a n 2 g qffiffiffiffiffiffiffiffiffiffiffiffi k b ð þ Þ ðtÞ ¼ sinh lot n bk a byk , b a þ Y n 2 g qffiffiffiffiffiffiffiffiffiffiffiffi k b ðÞ ðtÞ ¼ sinh lot n byk a bk : b a Y n 2 g b can be written in the form Similarly, the operator expðiHtÞ b expðiHtÞ
ð13Þ
8
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
0
b ð þ Þ ðtÞ C n
B B B ¼B B 1 yk bð þ Þ bðþÞ B ia @ b S n ðtÞqffiffiffiffiffiffiffiffiffiffiffiffiY n ðtÞ bk a byk a
1 ðÞ 1 bk b S n ðtÞqffiffiffiffiffiffiffiffiffiffiffiffi C ia bk C byk a a C C, C ðÞ b C X n ðtÞ A
ð14Þ
b ðmÞ b ðmÞ with the operators j n ð 7 Þ and f n ð 7 Þ are defined by, respectively, qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi k k byk , j bk , bk a byk a bþ b b n ðÞ ¼ o n b nð þ Þ ¼ o n þl a þl a j 2 2 ð24Þ
b ðþÞ ¼ o n bþ f n
where b ð þ Þ ðtÞ ¼ cos½lt C n ðþÞ b S n ðtÞ ¼ sin½lt
qffiffiffiffiffiffiffiffiffiffiffiffi byk , bk a a
qffiffiffiffiffiffiffiffiffiffiffiffi byk , bk a a
b ðÞ ðtÞ ¼ cos½lt C n ðÞ b S n ðtÞ ¼ sin½lt
qffiffiffiffiffiffiffiffiffiffiffiffi bk , byk a a
ð15Þ
qffiffiffiffiffiffiffiffiffiffiffiffi bk : byk a a
Then from Eqs. (12) and (14) it follows that 0 1 b ðÞ m V n ðtÞ b ð þ Þ ðtÞ b ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p a R B C n B bm C bym a t b a B C b ¼B expðiHtÞexp B C, ðþÞ B C 2g b ðÞ ym V n ðtÞ @ a A b b pffiffiffiffiffiffiffiffiffiffiffiffiffiffi R n ðtÞ m ym b b a a
b ðÞ ¼ o n b f n
qffiffiffiffiffiffiffiffiffiffiffiffiffi k bk : byk a l a 2
Substituting from Eq. (22) into Eq. (21), we get: 0 X 1 1 m 1 m X t t b ðmÞ ðtÞ b ðmÞ ðtÞ M M 11 12 B C g Bm¼0 g C m¼0 C, b ðtÞ ¼ B r m 1 m 1 B X C X t ðmÞ ðmÞ t @ b ðtÞ b ðtÞ A M M 21 22 m¼0
g
ð25Þ
g
m¼0
where ð16Þ
ðmÞ
ðmÞ
b 11 ðtÞ þ b ðtÞb b ðmÞ ðtÞ ¼ b f n ð þ Þ½C f n ðþÞ M 11 11 ðmÞ
b 21 ðtÞ þ b ðtÞb bk g b nðmÞ0 ðÞ½C f n ðþÞ þa 21 ðmÞ
b 12 ðtÞ þ b ðtÞgb ðmÞ0 ðÞa byk þb f n ð þ Þ½C n 12
where ð7Þ b ð 7 Þ ðtÞX b ð 7 Þ ðtÞ ¼ C b ð 7 Þ ðtÞ þ ib b ð 7 Þ ðtÞ, R S n ðtÞY n n n n
22
21
ðmÞ
b 21 ðtÞ þ b ðtÞa bk gb ðmÞ0 þb f n ðÞ½C n ðÞ 21
22
b ðtÞ ¼ j ðtÞS/ ðtÞj ij i j
ði,j ¼ 1,2Þ
b ðÞ ðtÞ yk V n b jCðtÞS, ffia jC2 ðtÞS ¼ qffiffiffiffiffiffiffiffiffiffiffi bk byk a a
b ð þ Þ ðtÞ k V n b j ðtÞS, q ffiffiffiffiffiffiffiffiffiffiffiffi a ðtÞS ¼ 1 byk bk a a
j
2 ðtÞS
ðmÞ ðmÞ b 22 ðtÞ þ b ðtÞb þb f n ðÞ½C f n ðÞ, 22
ð19Þ
b bðmÞ b ðmÞ ðtÞ ¼ ðM b ðmÞ ðtÞÞy ¼ g byk b b ðmÞ0 M 21 12 n ðÞa ½C 11 ðtÞ þ 11 ðtÞf n ð þÞ
ðmÞ
b 22 ðtÞ þ b ðtÞg byk , b nðmÞ0 ðÞa þb f n ðÞ½C 22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmÞ0 bk Þgb ðmÞ byk a where gb n ð 7 Þ ¼ ð1= a n ð 7 Þ.
ð28Þ
ð20Þ 3. Negativity and atomic Wehrl entropy In this section: we investigate the marginal atomic Q function and atomic Wehrl entropy AWE. We start our investigation by defining the atomic Q-function as [30]
ðÞ
b ðtÞj ðtÞS: ¼R n
g
The mth power of the Hamiltonian (2) can be expressed in the form: 0 1 ðmÞ 1 k ðmÞ b b b ffiffiffiffiffiffiffiffiffiffiffi ffi q g f ð þ Þ a ðÞ n n B C B C bk byk a a B C m b B C, ð22Þ H ¼B C ðmÞ 1 yk ðmÞ b Ba C b b ffiffiffiffiffiffiffiffiffiffiffi ffi q g ð þÞ f ðÞ n n @ A byk bk a a ðmÞ
ðmÞ where the operators b f n ð 7 Þ and gb n ð 7 Þ are defined by, respectively, ðmÞ b b ðmÞ b ðmÞ f n ð 7Þ ¼ 12 ½j n ð 7Þ þ f n ð 7Þ,
ð27Þ
ðmÞ0 b 12 ðtÞ þ b ðtÞg byk ½C byk b nðmÞ0 ðÞa þ gb n ðÞa 12
Take into account the definition of the superoperator bJ to evaluate the b b 2 ðtÞ as follows: action of the operator ejt on the ‘‘density’’ operator r m 1 X t b m: b mr b 2 ðtÞH b ðtÞ ¼ r ð21Þ H m¼0
yk
ðmÞ0
ðmÞ ðmÞ b 21 ðtÞ þ b ðtÞb þb f n ðÞ½C f n ð þÞ 21
with b ð þ Þ ðtÞjCðtÞS, jC1 ðtÞS ¼ R n
ðmÞ
b 12 ðtÞ þ b ðtÞb b ½C þ gb n ðÞa f n ðÞ 12
where we have used the following symbol: b ðtÞ ¼ jC ðtÞS/C ðtÞj, C ij i j
ð26Þ
b b ðmÞ ðtÞ ¼ g byk b bk b ðmÞ0 b ðmÞ0 M n ðÞa ½C 11 ðtÞ þ 11 ðtÞa g n ðÞ 22
Substituting Eqs. (10) and (16) into Eq. (9), we obtain an explicit b 2 ðtÞ as follows: expression for the operator r 0 1 b 12 ðtÞ þ b ðtÞ b 11 ðtÞ þ b ðtÞ C C 11 12 A, b 2 ðtÞ ¼ @ ð18Þ r b ðtÞ þ b ðtÞ b ðtÞ þ b ðtÞ C C 21
b 22 ðtÞ þ b ðtÞg bk g byk b nðmÞ0 ðÞ½C b ðmÞ0 þa n ðÞa , 22
ð17Þ
ð7Þ b ð 7 Þ ðtÞY b ð 7 Þ ðtÞ þib b ð 7 Þ ðtÞ: b ð 7 Þ ðtÞ ¼ C S n ðtÞX V n n n n
j
qffiffiffiffiffiffiffiffiffiffiffiffi k byk , bk a l a 2
b ðmÞ b nðmÞ ð 7 Þ ¼ 12½j b ðmÞ g n ð 7 Þf n ð 7 Þ, ð23Þ
QA ðY, F,tÞ ¼
1 /Y, Fjr^ ðtÞjY, FS, 2p
ð29Þ
where r^ ðtÞ is the density matrix which is given in Eq. (25) and jY, FS is the atomic coherent state expressed as jY, FS ¼ cosðY=2ÞjeSþ sinðY=2ÞeiF jgS,
ð30Þ
where 0 r Y r p,0 r F r2p. The definition (29) means that two different spin coherent states overlap unless they directed into two antipodal points on the sphere [30]. The atomic Wehrl entropy can be written in terms of the atomic Q function as [30]: Z 2p Z p SAW ðtÞ ¼ QA ðY, F,tÞlnQA ðY, F,tÞ sin Y dY dF: ð31Þ 0
0
One can easily check that the QA is normalized. By integrating the atomic Q-function QA over the atomic variable Y, we obtain the marginal atomic Q-function as follows: Z 2p QF ¼ QA sin Y dY: ð32Þ 0
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
Another measure of entanglement we use here is the negativity which is based on the Peres–Horodecki [8,9] criterion for entanglement and is defined by the formula @ðrÞ ¼ maxð0,2ni Þ,
ð33Þ
where the sum is taken over the negative eigenvalues ni of the partial transposition of the density matrix r of the system.
9
4. Numerical results Based on the analytical solution in Section 3, in this section we will present some numerical results for different values of the system parameters in Figs. 1–4 to demonstrate the effects of the decoherence rate g=l on the negativity and atomic Wehrl entropy, so that we will assume that the initial state of the atom–field system accords Eq. (4), where we set ðb1 ¼ cosðp=6Þ and b2 ¼ sinðp=6ÞÞ.
0.5 0.45 N(ρ)
0.4 0.35 0.3 0.25 0.2
0
2
4
6 scaled time
8
10
12
0
2
4
6 scaled time
8
10
12
W(t)
0.5
0
−0.5
Fig. 1. Time evolution of the: (a) negativity NðrÞ, (b) atomic inversion SAW for ðn,k, yÞ ¼ ð5,1, p=6Þ and with different values of g=l.
0.5 0.45 N(ρ)
0.4 0.35 0.3 0.25 0.2
0
2
4
6 scaled time
8
10
12
0
2
4
6 scaled time
8
10
12
W(t)
0.5
0
−0.5
Fig. 2. The same as Fig. 1 but for ðn,k, yÞ ¼ ð25,2, p=6Þ.
10
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
2.52
SAW
2.515 2.51 2.505 2.5
0
5
10
15
20 scaled time
25
6 scaled time
8
30
35
40
2.52 2.518
SAW
2.516 2.514 2.512 2.51 2.508 2.506
0
2
4
10
12
Fig. 3. The atomic Wehrl entropy SAW versus the scaled time for y ¼ p=6, where (a) ðn,kÞ ¼ ð5,1Þ, (b) ðn,kÞ ¼ ð25,2Þ and with different values of g=l.
Fig. 4. The surface plot of the marginal atomic Q function QF versus the scaled time and the phase space parameter F for y ¼ p=6, where (a,b) ðn,kÞ ¼ ð5,1Þ, g/l ¼ 105, g/l ¼ 104, (c,d) ðn,kÞ ¼ ð25,2Þ, g/l ¼105, g/l ¼ 104 and with different values of g=l.
In Fig. 1, we plot the negativity and atomic inversion as functions of scaled time for different values of g=l with n ¼ 5 and one photon process (k¼1). When g=l ¼ 10 (soled curve) we see that @ðrÞ starts from minimum value and increases gradually as time goes on and the damping of oscillations occurred, while by increasing g=l we can see that more oscillations will take place. Also the atomic inversion oscillates around zero and with increasing g=l we have collapse–revival phenomenon. Fig. 2
depicts the average photon number k influences on the negativity evolution. It is shown that with increasing k (say k¼2) with small value of g=l, the negativity starts from minimum value and increases gradually as the time goes on and the damped oscillation becomes faster. By increasing the decoherence parameter the amplitude of the negativity is decreased and reaches its maximum faster. In this case a long-lived entanglement is shown once the decoherence is increased further.
S. Abdel-Khalek et al. / Physica E 44 (2011) 6–11
Based on Eq. (25) we present the main results for the effect of decoherence parameter, g=l, on the evolution of the atomic Wehrl entropy SAW , in Fig. 3. We have presented the dynamics of quantum entanglement due to the atomic Wehrl entropy. The first eye-catching atomic Wehrl entropy has the same behavior as the negativity (involve typical information). The above observation highlights atomic Wehrl entropy which can successfully used an entanglement quantifier of the mixed state system in the presence of intrinsic decoherence (see Figs. 2, 3). On the other hand, when the intrinsic decoherence parameter is considered, it is noticed that the maximum value of SAW is increased and the amplitude is decreased. For the larger values of the intrinsic decoherence the phenomena of long-lived entanglement can be observed. The comparison between Fig. 3(a,b) shows that the same effect of g=l is observed with a periodic behavior, which returns to the periodicity of two-photon processes. We may refer here to the work given in Ref. [31] where an asymptotic value of the concurrence which embodies entanglement survival has been observed, independent of the interaction development but critically dependent on the environment. In Fig. 4, we consider the evolution of the marginal atomic Q-function QF ðtÞ. Our main aim in this case is find a relation between quantify the entanglement between a mixed state twolevel atom and coherent field and QF ðtÞ. Taking this fact into account, we investigate QF ðtÞ for different values of the decoherence parameter g=l. From Eq. (32) one can conclude the following remarks: QF ðtÞ has two terms with different signs due to different values of F, e.g. when F ¼ p=2 we see that QF ¼ X þY, while QF ¼ XY if F ¼ 3p=2. This in fact interprets the exchange of the maximum and minimum values of QF due to the variation of F (see Fig. 4). This also refers to the general behavior of the entanglement of the system under consideration and there is one-to-one correspondence between the behavior of QF ðtÞ and the Wehrl entropy or negativity [32]. This new observation opens the door for using QF ðtÞ as an entanglement measure. 5. Conclusion In this paper, we have developed a framework to study a mixed state two-level atomic system, in the presence of intrinsic decoherence exploring the behavior of the atomic Wehrl entropy, marginal distributions of the atomic Q-function in connection with entanglement. An exact solution of the density matrix, in terms of the intrinsic decoherence parameter g=l for the system
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under consideration has been obtained. The results exhibit specific features, arising from variations of the adjustable parameters of the system. Namely, the intrinsic decoherence parameter g=l and initial state settings. It is shown that, the atomic Wehrl entropy exhibits behaviors that reflect a pattern of the entanglement as measured by the negativity.
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