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Quantum entanglement in the SU(1, 1)-related coherent fields interacting with a moving atom
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Physica A 365 (2006) 351–359 www.elsevier.com/locate/physa
Quantum entanglement in the SU(1, 1)-related coherent fields interacting with a moving atom Xiang-Ping Liaoa,b, Mao-Fa Fanga,, Qing-Ping Zhoua a
Department of Physics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China Department of Physics and Electronics, Zhuzhou Teacher’ College, ZhuZhou, Hunan 412007, People’s Republic of China
b
Received 13 February 2005 Available online 24 October 2005
Abstract The entanglement properties of the SU(1, 1)-related coherent fields interacting with a moving atom via the two-photon transition are investigated. We discuss the quantum entanglement between the SU(1, 1)-related coherent fields and the moving atom by using the quantum reduced entropy and that between the SU(1, 1)-related coherent fields by using the quantum relative entropy. It is shown that we can obtain the maximal entangled state between the fields and the moving atom and the high degree of entanglement between the SU(1, 1)-related coherent fields can be preserved for some choiced system parameters. r 2005 Elsevier B.V. All rights reserved. Keywords: SU(1, 1)-related coherent fields; Quantum entanglement; Reduced entropy; Relative entropy
1. Introduction Quantum entanglement (QE) is one of the most profound features of quantum mechanics and has been considered to be a valuable physical resource in the young field of quantum information science, including quantum computation and communication [1,2], quantum teleportation [3], quantum dense coding [4], and quantum information processing [5], etc. The manipulation of controlled entangled states, protected from their environment, is experimentally challenging. In real experimental situations, due to the coupling to the environment, the entangled state inevitably loses its purity; it becomes mixed. This phenomenon of decoherence is the most dangerous obstacle for all entanglement manipulations. Thus it is very important to know how the interaction with an environment influences the dynamical behavior of entanglement. Recently, unconditional quantum teleportation has been presented by exploiting a two-mode squeezed vacuum state as an entanglement resource [6]. But the influence of environment on the two-mode fields has not been considered in the scheme. In addition, a two-mode squeezed vacuum state coupled to one thermal reservoir as a model of an entangled state embedded in an environment is studied [7]. They derive a condition, which assures that the state remains entangled in spite of the interaction with the reservoir. Corresponding author. Tel.:+86 731 8871001; fax: +86 731 8872258.
E-mail address: [email protected] (M.-F. Fang). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.105
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The SU(1, 1)-related coherent state [8] is a continuous variable entangled state and is a generalization of the two-mode squeezed vacuum state. It is generated by applying the two-mode squeeze operator to a state in which there are initially q photons in one mode. It is found that this state exhibits the nonclassical properties: squeezing, sum squeezing, violations of the Cauchy–Schwarz inequality, and enhanced number-phase uncertainty fluctuations. In this paper, we investigate the entanglement properties of the SU(1, 1)-related coherent fields interacting with a moving atom via the two-photon transition by regarding the moving atom as the environment. We study the QE between the SU(1, 1)-related coherent fields and the moving atom by using the quantum-reduced entropy and that between the SU(1, 1)-related coherent fields by using the quantum relative entropy. We also examine the influence of the initial state of the system on the quantum entanglement. It is shown that we can obtain the maximal entangled state between the fields and the moving atom and the high degree of entanglement between the SU(1, 1)-related coherent fields can be preserved for some choiced system parameters. This provides a good method for preparing entangled state and realizing the entanglement preservation in the experiment. 2. The model and its solution We consider a two-level atom in motion interacting with the SU(1, 1)-related coherent fields via the twophoton transition processes. The effective Hamiltonian in the rotating-wave approximation is [9] ð_ ¼ 1Þ þ þ þ H ¼ o 0 S z þ o 1 aþ 1 a1 þ o2 a2 a2 þ gf ðzÞðS þ a1 a2 þ a1 a2 S Þ,
(1) aþ j ðaj Þ
is the photon creation where Si ði ¼ z; þ; Þ are the usual pseudo-spin operators of the two-level atom, (annihilation) operator of the field mode of frequency oj ðj 1; 2Þ and g is the atom-fields coupling constant. f ðzÞ is the shape function of the cavity field mode. We restrict our studies for the atomic motion along the zaxis so that only the z-dependence of the field-mode function would be needed to take into account. The atomic motion can be incorporated in a usual way: f ðzÞ ! f ðutÞ,
(2)
where u denotes the atomic motion velocity. In order to be specific we will define the TEMmnp modes: pput f ðzÞ ¼ sin , L
(3)
where p represents the number of half-wave lengths of the field mode inside a cavity of the length L. For simplicity, we consider the two-photon resonance case, that is o0 ¼ o1 þ o2 , and assume that the atom enters the cavity at time t ¼ 0 in the superposition state jca ð0Þi ¼ cosðg=2Þjþi þ expðijÞ sinðg=2Þji
(4)
and leaves the cavity again after passing p half-wavelengths of the electric fields. Using the standard techniques [10], it can be shown that this Hamiltonian gives rise to the following time evolution operator in the interaction picture: hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i3 2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sin aþ aþ a1 a2 gYðtÞ 1 2 þ 7 6 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ia1 a2 cos a1 a2 aþ 1 a2 gYðtÞ 7 6 þ aþ a a a 7 6 1 2 1 2 7, 6 h i (5) U I ðtÞ ¼ 6 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 þ h i sin a1 a2 aþ a gYðtÞ 7 6 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 2 5 4 iaþ aþ þ þ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a1 a2 a1 a2 gYðtÞ 1 2 þ a 1 a2 aþ 1 a2 where Z
t
f ðut0 Þ dt0 ¼
YðtÞ ¼ 0
pputi L h 1 cos . ppu L
(6)
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For a particular choice of the atomic motion velocity u ¼ gL=p, YðtÞ becomes YðtÞ ¼ ð1=pgÞ½1 cosðpgtÞ.
(7)
We assume that the field is in the SU(1, 1)-related coherent state [11] at time t ¼ 0, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X ðn þ qÞ! n 2 ð1þqÞ=2 x jn þ q; ni jcf ð0Þi ¼ jx; qi ¼ ð1 jxj Þ n!q! n¼0 ð0ojxj2 o1; q ¼ 0; 1; 2; . . .Þ,
ð8Þ
where jx; qi, the eigenstate of operator aþ a bþ b, belongs to SU(1, 1)-related coherent state. If q ¼ 0, it reduces to two-mode squeezed vacuum state. The initial state of the system can be written as jcfa ð0Þi ¼ jcf ð0Þi jca ð0Þi.
(9)
At any time t40, the state vector of the system is given by jcfa ðtÞi ¼ U I ðtÞjcfa ð0Þi ¼ jDijþi þ jEiji
ð10Þ
with jDi ¼
jEi ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g g f n cos ðn þ 1Þðn þ 1 þ qÞgYðtÞ i sin 2 2 n¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o expðijÞf nþ1 sin ðn þ 1Þðn þ 1 þ qÞgYðtÞ jn þ q; ni, 1 n X
1 X
cos
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g expðijÞf n cos nðn þ qÞgYðtÞ 2 n¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g i cos f n1 sin nðn þ qÞgYðtÞ jn þ q; ni, 2
ð11Þ
sin
ð12Þ
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ qÞ! n x . f n ¼ ð1 jxj2 Þ1þq=2 n!q! At any time t40, the density operator for the system is given by " # jDihDj jDihEj rfa ðtÞ ¼ jEihDj jEihEj
(13)
(14)
while the reduced SU(1, 1)-related coherent-field density operator is given by rf ðtÞ ¼ Tra ðrfa ðtÞÞ ¼ jDihDj þ jEihEj.
(15)
The eigenvalues and the eigenstates of the reduced SU(1, 1)-related coherent-field density operator rf ðtÞ are given by l f ¼ hDjDi expðaÞjhDjEij ¼ hEjEi expðaÞjhDjEij, 1 1 D exp 1 ðib aÞ E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ðib aÞ , jc ðtÞi ¼ exp f 2 2 2l cosh a f
ð16Þ (17)
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where
hDjDi hEjEi a ¼ sinh1 , 2jhDjEij ImðhDjEiÞ . b ¼ tan1 ReðhDjEiÞ Similarly, we can obtain the reduced atom density operator for the system " # hDjDi hEjDi ra ðtÞ ¼ Trf ðrfa ðtÞÞ ¼ . hDjEi hEjEi It turns out that the eigenvalues of ra ðtÞ are equal to those of rf ðtÞ. That is l f ¼ la ¼ l . The eigenstates of ra ðtÞ are given by 1 1 þ exp 1 ðib aÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ðib aÞ . jf ðtÞi ¼ exp a 2 2 2l cosh a a
By using the Schmidt theorem [12], the system state vector can be rewritten as pffiffiffiffiffiffi pffiffiffiffiffi ffi þ jcfa ðtÞi ¼ lþ jfþ l jf a ijcf i þ a ijcf i.
ð18Þ
(19)
(20)
(21)
The quantum entanglement between the SU(1, 1)-related coherent fields and the moving atom and that between the SU(1, 1)-related coherent fields will be discussed in the next section by using above results. 3. The entanglement between the SU(1, 1)-related coherent fields and the moving atom The reduced entropy theory about interaction of the fields with the atom has been introduced by Knight et al. [13–15]. They have shown that the reduced entropy is very useful and is a sensitive operational measure of the purity of the quantum state, which automatically includes all the moments of the density operator. The time evolution of the reduced entropy of the field Sðrf Þ reflects the time evolution of the degree of the entanglement between the field and the atom (DEFA). The higher the reduced entropy, the greater the DEFA. Following the work by Phoenix and Knight [13], we can express the reduced entropy Sðrf Þ of the SU(1, 1)related coherent fields in terms of the eigenvalue l f given by Eq. (16) þ Sðrf Þ ¼ lþ f log2 lf lf log2 lf .
(22)
If Sðrf Þ takes the minimal value zero, the fields and the moving atom are disentangled; if Sðrf Þ takes the maximal value 1, the fields and the moving atom are maximally entangled. Numerical results for the time evolution of the reduced entropy of entanglement are displayed by using the dotted line in Fig. 1 (assuming j ¼ 0 for simplicity in all figures in this paper) for x ¼ 0:5, p ¼ 2 and different q, g. From these figures, we can conclude the following: (1) The DEFA is periodic with period 2p=p. The behavior of the evolution of the reduced entropy of entanglement can be understood by Eq. (7). From Eq. (7), we have gYðtÞ ¼ 1=p½1 cosðpgtÞ. It is observed that gYðtÞ is a periodical function on the scaled time gt with period 2p=p. And when p ¼ 2, the DEFA evolves in period p. (2) No matter how the initial atomic state is, the reduced entropy of the entanglement is zero at the times gt ¼ np ðn ¼ 0; 1; 2; . . .Þ, which means that the moving atom and the SU(1, 1)-related coherent fields are disentangled. This point can be quantitatively interpreted by Eqs. (16) and (21). At the times gt ¼ np, lþ ¼ 1, l ¼ 0, the system state jcfa i described by Eq. (21) is written as þ jcfa i ¼ jfþ a ijcf i
(23)
which can be written in a direct product form. Therefore, Sðrf Þ is equal to zero, and the moving atom and the SU(1, 1)-related coherent fields are disentangled.
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(c)
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gt
18
20 (d)
gt
Fig. 1. The time evolution of the degree of entanglement. Sðrf Þ is shown by the dotted line and E R ðrf Þ is shown by the solid line with x ¼ 0:5 and p ¼ 2: (a) g ¼ 0, q ¼ 0, (b) g ¼ 0, q ¼ 65, (c) g ¼ p=4, q ¼ 65, and (d) g ¼ p=2, q ¼ 65.
(3) For the rest of the time, the system state cannot be written in the direct product form, so the SU(1, 1)related coherent fields and the moving atom are entangled. In particular, from Fig. 1(a) when g ¼ 0 (the moving atom is initially in an excited state) and q ¼ 0, we can see that the reduced entropy is approximately equal to 1 at times gt ¼ kp þ 0:3p; kp þ 0:7p ðk ¼ 0; 1; 2; . . .Þ. This means that the moving atom and the SU(1, 1)-related coherent fields are maximally entangled, and the system state is an EPR(Einstein–Podolsky–Rosen) state which is widely used in QI processing pffiffiffi þ jcfa i 1= 2½jfþ a ijcf i þ jfa ijcf i.
(24)
(4) The initial atomic state has a considerable influence on the reduced entropy of entanglement. When qa0, the case for g ¼ 0 (the excited state), g ¼ p=4 and p=2 (the moving atom is initially in a coherent trapping state) are considered in Figs. 1(b)–(d) by using the dotted line. From these figures, we can see that the amplitude of the vibration of the DEFA decreases with an increase in parameter g. Especially, when g ¼ p=2, the value of Sðrf Þ is very small during the time evolution (see Fig. 1(d)). Our results further show that, when g ¼ p=2 and both q and x are very large, Sðrf Þ tends to zero, which means that the field and the moving atom are approximately disentangled (see Fig. 2(b) the dotted line).
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10 gt
12
14
16
18
20
0
(b)
2
4
6
8
10 gt
12
14
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20
Fig. 2. The time evolution of the degree of entanglement. Sðrf Þ is shown by the dotted line and E R ðrf Þ is shown by the solid line with g ¼ p=2 and p ¼ 2: (a) x ¼ 0:5, q ¼ 40, and (b) x ¼ 0:6, q ¼ 65.
4. The entanglement between the SU(1, 1)-related coherent fields The above results show that the Hamiltonian given by Eq. (1) leads to the entanglement between the SU(1, 1)-related coherent fields and the moving atom, so that the SU(1, 1)-related coherent state may evolve into a mixed state. In this case, the degree of entanglement between the SU(1, 1)-related coherent fields cannot be measured by the reduced entropy. However, the relative entropy of entanglement is a good measure for the degree of entanglement of two subsystems in the mixed state, which is defined as [16] E R ðrÞ ¼ min SðrksÞ, s2D
(25)
where SðrksÞ ¼ Tr½rðlog2 r log2 sÞ is the relative entropy. The minimum is taken over D, the set of all disentangled states. The relative entropy of entanglement is viewed as the minimal ‘‘distance’’ between the state r and the disentangled state s. For a pure state, the relative entropy of entanglement reduces to the reduced entropy of entanglement; while for a mixed state, it is usually difficult to numerically calculate the relative entropy of entanglement except for some specific states. Recently, the following theorem on the relative entropy of entanglement has been proved [17,18], and it turns out to be quite suitable for the present analysis. If a bipartite quantum state can be expressed as X r¼ an1 ;n2 jFn1 fn1 ihFn2 fn2 j (26) n1 ;n2
the relative entropy of entanglement is given by X E R ðrÞ ¼ an;n log2 an;n þ Trðr log2 rÞ
(27)
n
and the disentangled state s that minimizes the quantum relative entropy SðrksÞ is X an;n jFn fn ihFn fn j, s ¼
(28)
n
where jFn i ðjfn iÞ is a set of orthogonal normalized states of system AðBÞ. It should be noted that the density operator of Eq. (15) can take the form of Eq. (26) X rf ¼ am;n jm þ q; mihn þ q; nj, (29) m;n
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where am;n ¼ hm þ q; mjrf jn þ q; ni. Hence, the relative entropy of entanglement of the state Eq. (15) is written as X an;n log2 an;n Sðrf Þ E R ðrf Þ ¼ n
¼
X
an;n log2 an;n þ lþ log2 lþ þ l log2 l .
ð30Þ
n
The numerical results of Eq. (30) are shown by the solid lines in Figs. 1 and 2 for p ¼ 2 and different g; x and q, which reflect the time evolution of the degree of entanglement between the SU(1, 1)-related coherent fields (DESRCF). The DESRCF increases monotonically with the relative entropy of entanglement. Like the DEFA, the DESRCF exhibits the periodic evolution with period 2p=p. And when p ¼ 2, the DESRCF evolves in period p. Comparing the solid line with the dotted line in Figs. 1(a)–(d), we can find that the time evolution of the DESRCF is just the opposite to the DEFA. That is, when the latter increases with time, the former decreases with time. This comes from the fact that the DEFA can impair the DESRCF due to the SU(1, 1)-related coherent fields interacting with the moving atom. In particular, in Fig. 1(a) when q ¼ 0; g ¼ 0 and gt ¼ npðn ¼ 0; 1; 2 . . .Þ, it can be obtained that the states of the system can be written in a direct form which means that the moving atom is disentangled from the SU(1, 1)-related coherent fields. However, at the same time, the relative entropy of entanglement achieves its maximal value. At the time gt ¼ np, the field state is X jcf ðgt ¼ npÞi ¼ f n jn; ni (31) n
the relative entropy of entanglement of the above state is equal to the relative entropy of entanglement of initial field state for q ¼ 0 1 log2 ð1 jxj2 Þ 1 jxj2 jxj2 jxj2 log2 . 2 1 jxj 1 jxj2
E R ðgt ¼ npÞ ¼
ð32Þ
Our results further show that when x ¼ 0:5; q ¼ 65 and g ¼ p=2 (the moving atom is initially in a coherent trapping state), the sustained high degree of entanglement between the SU(1, 1)-related coherent fields can be obtained, as shown in Fig. 1(d). Moreover, when g ¼ p=2, an increase in parameter q or x results in an increase of the amount of the relative entropy of entanglement which is independent of the evolution time. This can be obtained by comparing Fig. 1(d), 2(a) and (b).
5. Conclusion In this paper, we have investigated the entanglement properties of the SU(1, 1)-related coherent state interacting with a moving atom. We can conclude as follows: (1) Both the QE between the SU(1, 1)-related coherent fields and the moving atom and that between the two modes of the fields evolve periodically with period 2p=p. (2) The entanglement between the SU(1, 1)-related coherent fields and the moving atom is just the opposite to the entanglement between the two modes of the fields due to the fields interacting with the moving atom. At the time gt ¼ npðn ¼ 0; 1; 2 . . .Þ, the moving atom and the fields are disentangled, but the relative entropy of entanglement is equal to the entanglement of initial field state. (3) When the moving atom is initially in an excited state (g ¼ 0), by choosing q ¼ 0 and gt ¼ kp þ 0:3p; kp þ 0:7p ðk ¼ 0; 1; 2; . . .Þ, the maximal entangled state between the SU(1, 1)-related coherent fields and the moving atom can be produced. (4) When q and x are very large and the moving atom is initially in a coherent trapping state (g ¼ p=2), the high degree of entanglement between the SU(1, 1)-related coherent fields can be preserved.
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Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 10374025) and by the Young Scientific Research Foundation of Hunan Provincial Education Department, China (Grand No. 04B070). Appendix In this appendix, with regard to the referee’s concern of significance of this work, we discuss the rationality of the entropy of entanglement for quantification of the degree of entanglement of quantum system from the following three aspects: (I) The relation between the measures of entanglement and the quantum entropy. As we know, several kinds of measures of entanglement have been proposed at present, which include entanglement of formation [19], entanglement of distillation [20], relative entropy of entanglement [16] and partial entropy of entanglement [13–15]. The entanglement of formation is defined as X E F ðrÞ ¼ min pi E p ðjcii Þ, (A.1) fpi ;jcii g
i
where E p ðjcii Þ is partial entropy of entanglement of state jcii . So, entanglement of formation is related to partial entropy of entanglement. And the concurrence introduced by Wootters [19] is also the special case of entanglement of formation. Thus, we can see that most of the measures of entanglement are related to the quantum entropy. (II) The rationality of the entropy of entanglement. The partial entropy of entanglement for a bipartite pure state is defined as Von Neumann entropy of the reduced state. Bennett et al. [21] have shown that it is a good entanglement measure for a bipartite pure state. In Refs. [22], Vedral et al. have proved the rationality of the relative entropy of entanglement for quantification of the degree of entanglement of quantum system in cases of mixed states. (III) The determination of the entropy of entanglement. The partial entropy of entanglement is defined as E p ðjciAB Þ ¼ SðrA Þ ¼ SðrB Þ,
(A.2)
where SðrÞ ¼ Trðr ln rÞ is the Von Neumann entropy. And, the quantum relative entropy of entanglement is defined as E R ðrÞ ¼ min SðrksÞ, s2D
(A.3)
where SðrksÞ ¼ Tr½rðlog2 r log2 sÞ is the relative entropy. From these equations, we can see that the entropy of entanglement is associated with the density matrix of quantum state. By using the quantum state tomography [23], we can experimentally determine what the density matrix of quantum state is. So the entropy of entanglement can be determined indirectly by experiment. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Physica A 365 (2006) 351–359 www.elsevier.com/locate/physa
Quantum entanglement in the SU(1, 1)-related coherent fields interacting with a moving atom Xiang-Ping Liaoa,b, Mao-Fa Fanga,, Qing-Ping Zhoua a
Department of Physics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China Department of Physics and Electronics, Zhuzhou Teacher’ College, ZhuZhou, Hunan 412007, People’s Republic of China
b
Received 13 February 2005 Available online 24 October 2005
Abstract The entanglement properties of the SU(1, 1)-related coherent fields interacting with a moving atom via the two-photon transition are investigated. We discuss the quantum entanglement between the SU(1, 1)-related coherent fields and the moving atom by using the quantum reduced entropy and that between the SU(1, 1)-related coherent fields by using the quantum relative entropy. It is shown that we can obtain the maximal entangled state between the fields and the moving atom and the high degree of entanglement between the SU(1, 1)-related coherent fields can be preserved for some choiced system parameters. r 2005 Elsevier B.V. All rights reserved. Keywords: SU(1, 1)-related coherent fields; Quantum entanglement; Reduced entropy; Relative entropy
1. Introduction Quantum entanglement (QE) is one of the most profound features of quantum mechanics and has been considered to be a valuable physical resource in the young field of quantum information science, including quantum computation and communication [1,2], quantum teleportation [3], quantum dense coding [4], and quantum information processing [5], etc. The manipulation of controlled entangled states, protected from their environment, is experimentally challenging. In real experimental situations, due to the coupling to the environment, the entangled state inevitably loses its purity; it becomes mixed. This phenomenon of decoherence is the most dangerous obstacle for all entanglement manipulations. Thus it is very important to know how the interaction with an environment influences the dynamical behavior of entanglement. Recently, unconditional quantum teleportation has been presented by exploiting a two-mode squeezed vacuum state as an entanglement resource [6]. But the influence of environment on the two-mode fields has not been considered in the scheme. In addition, a two-mode squeezed vacuum state coupled to one thermal reservoir as a model of an entangled state embedded in an environment is studied [7]. They derive a condition, which assures that the state remains entangled in spite of the interaction with the reservoir. Corresponding author. Tel.:+86 731 8871001; fax: +86 731 8872258.
E-mail address: [email protected] (M.-F. Fang). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.105
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The SU(1, 1)-related coherent state [8] is a continuous variable entangled state and is a generalization of the two-mode squeezed vacuum state. It is generated by applying the two-mode squeeze operator to a state in which there are initially q photons in one mode. It is found that this state exhibits the nonclassical properties: squeezing, sum squeezing, violations of the Cauchy–Schwarz inequality, and enhanced number-phase uncertainty fluctuations. In this paper, we investigate the entanglement properties of the SU(1, 1)-related coherent fields interacting with a moving atom via the two-photon transition by regarding the moving atom as the environment. We study the QE between the SU(1, 1)-related coherent fields and the moving atom by using the quantum-reduced entropy and that between the SU(1, 1)-related coherent fields by using the quantum relative entropy. We also examine the influence of the initial state of the system on the quantum entanglement. It is shown that we can obtain the maximal entangled state between the fields and the moving atom and the high degree of entanglement between the SU(1, 1)-related coherent fields can be preserved for some choiced system parameters. This provides a good method for preparing entangled state and realizing the entanglement preservation in the experiment. 2. The model and its solution We consider a two-level atom in motion interacting with the SU(1, 1)-related coherent fields via the twophoton transition processes. The effective Hamiltonian in the rotating-wave approximation is [9] ð_ ¼ 1Þ þ þ þ H ¼ o 0 S z þ o 1 aþ 1 a1 þ o2 a2 a2 þ gf ðzÞðS þ a1 a2 þ a1 a2 S Þ,
(1) aþ j ðaj Þ
is the photon creation where Si ði ¼ z; þ; Þ are the usual pseudo-spin operators of the two-level atom, (annihilation) operator of the field mode of frequency oj ðj 1; 2Þ and g is the atom-fields coupling constant. f ðzÞ is the shape function of the cavity field mode. We restrict our studies for the atomic motion along the zaxis so that only the z-dependence of the field-mode function would be needed to take into account. The atomic motion can be incorporated in a usual way: f ðzÞ ! f ðutÞ,
(2)
where u denotes the atomic motion velocity. In order to be specific we will define the TEMmnp modes: pput f ðzÞ ¼ sin , L
(3)
where p represents the number of half-wave lengths of the field mode inside a cavity of the length L. For simplicity, we consider the two-photon resonance case, that is o0 ¼ o1 þ o2 , and assume that the atom enters the cavity at time t ¼ 0 in the superposition state jca ð0Þi ¼ cosðg=2Þjþi þ expðijÞ sinðg=2Þji
(4)
and leaves the cavity again after passing p half-wavelengths of the electric fields. Using the standard techniques [10], it can be shown that this Hamiltonian gives rise to the following time evolution operator in the interaction picture: hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i3 2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sin aþ aþ a1 a2 gYðtÞ 1 2 þ 7 6 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ia1 a2 cos a1 a2 aþ 1 a2 gYðtÞ 7 6 þ aþ a a a 7 6 1 2 1 2 7, 6 h i (5) U I ðtÞ ¼ 6 ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 þ h i sin a1 a2 aþ a gYðtÞ 7 6 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 2 5 4 iaþ aþ þ þ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a1 a2 a1 a2 gYðtÞ 1 2 þ a 1 a2 aþ 1 a2 where Z
t
f ðut0 Þ dt0 ¼
YðtÞ ¼ 0
pputi L h 1 cos . ppu L
(6)
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For a particular choice of the atomic motion velocity u ¼ gL=p, YðtÞ becomes YðtÞ ¼ ð1=pgÞ½1 cosðpgtÞ.
(7)
We assume that the field is in the SU(1, 1)-related coherent state [11] at time t ¼ 0, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X ðn þ qÞ! n 2 ð1þqÞ=2 x jn þ q; ni jcf ð0Þi ¼ jx; qi ¼ ð1 jxj Þ n!q! n¼0 ð0ojxj2 o1; q ¼ 0; 1; 2; . . .Þ,
ð8Þ
where jx; qi, the eigenstate of operator aþ a bþ b, belongs to SU(1, 1)-related coherent state. If q ¼ 0, it reduces to two-mode squeezed vacuum state. The initial state of the system can be written as jcfa ð0Þi ¼ jcf ð0Þi jca ð0Þi.
(9)
At any time t40, the state vector of the system is given by jcfa ðtÞi ¼ U I ðtÞjcfa ð0Þi ¼ jDijþi þ jEiji
ð10Þ
with jDi ¼
jEi ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g g f n cos ðn þ 1Þðn þ 1 þ qÞgYðtÞ i sin 2 2 n¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o expðijÞf nþ1 sin ðn þ 1Þðn þ 1 þ qÞgYðtÞ jn þ q; ni, 1 n X
1 X
cos
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g expðijÞf n cos nðn þ qÞgYðtÞ 2 n¼0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g i cos f n1 sin nðn þ qÞgYðtÞ jn þ q; ni, 2
ð11Þ
sin
ð12Þ
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ qÞ! n x . f n ¼ ð1 jxj2 Þ1þq=2 n!q! At any time t40, the density operator for the system is given by " # jDihDj jDihEj rfa ðtÞ ¼ jEihDj jEihEj
(13)
(14)
while the reduced SU(1, 1)-related coherent-field density operator is given by rf ðtÞ ¼ Tra ðrfa ðtÞÞ ¼ jDihDj þ jEihEj.
(15)
The eigenvalues and the eigenstates of the reduced SU(1, 1)-related coherent-field density operator rf ðtÞ are given by l f ¼ hDjDi expðaÞjhDjEij ¼ hEjEi expðaÞjhDjEij, 1 1 D exp 1 ðib aÞ E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ðib aÞ , jc ðtÞi ¼ exp f 2 2 2l cosh a f
ð16Þ (17)
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where
hDjDi hEjEi a ¼ sinh1 , 2jhDjEij ImðhDjEiÞ . b ¼ tan1 ReðhDjEiÞ Similarly, we can obtain the reduced atom density operator for the system " # hDjDi hEjDi ra ðtÞ ¼ Trf ðrfa ðtÞÞ ¼ . hDjEi hEjEi It turns out that the eigenvalues of ra ðtÞ are equal to those of rf ðtÞ. That is l f ¼ la ¼ l . The eigenstates of ra ðtÞ are given by 1 1 þ exp 1 ðib aÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ðib aÞ . jf ðtÞi ¼ exp a 2 2 2l cosh a a
By using the Schmidt theorem [12], the system state vector can be rewritten as pffiffiffiffiffiffi pffiffiffiffiffi ffi þ jcfa ðtÞi ¼ lþ jfþ l jf a ijcf i þ a ijcf i.
ð18Þ
(19)
(20)
(21)
The quantum entanglement between the SU(1, 1)-related coherent fields and the moving atom and that between the SU(1, 1)-related coherent fields will be discussed in the next section by using above results. 3. The entanglement between the SU(1, 1)-related coherent fields and the moving atom The reduced entropy theory about interaction of the fields with the atom has been introduced by Knight et al. [13–15]. They have shown that the reduced entropy is very useful and is a sensitive operational measure of the purity of the quantum state, which automatically includes all the moments of the density operator. The time evolution of the reduced entropy of the field Sðrf Þ reflects the time evolution of the degree of the entanglement between the field and the atom (DEFA). The higher the reduced entropy, the greater the DEFA. Following the work by Phoenix and Knight [13], we can express the reduced entropy Sðrf Þ of the SU(1, 1)related coherent fields in terms of the eigenvalue l f given by Eq. (16) þ Sðrf Þ ¼ lþ f log2 lf lf log2 lf .
(22)
If Sðrf Þ takes the minimal value zero, the fields and the moving atom are disentangled; if Sðrf Þ takes the maximal value 1, the fields and the moving atom are maximally entangled. Numerical results for the time evolution of the reduced entropy of entanglement are displayed by using the dotted line in Fig. 1 (assuming j ¼ 0 for simplicity in all figures in this paper) for x ¼ 0:5, p ¼ 2 and different q, g. From these figures, we can conclude the following: (1) The DEFA is periodic with period 2p=p. The behavior of the evolution of the reduced entropy of entanglement can be understood by Eq. (7). From Eq. (7), we have gYðtÞ ¼ 1=p½1 cosðpgtÞ. It is observed that gYðtÞ is a periodical function on the scaled time gt with period 2p=p. And when p ¼ 2, the DEFA evolves in period p. (2) No matter how the initial atomic state is, the reduced entropy of the entanglement is zero at the times gt ¼ np ðn ¼ 0; 1; 2; . . .Þ, which means that the moving atom and the SU(1, 1)-related coherent fields are disentangled. This point can be quantitatively interpreted by Eqs. (16) and (21). At the times gt ¼ np, lþ ¼ 1, l ¼ 0, the system state jcfa i described by Eq. (21) is written as þ jcfa i ¼ jfþ a ijcf i
(23)
which can be written in a direct product form. Therefore, Sðrf Þ is equal to zero, and the moving atom and the SU(1, 1)-related coherent fields are disentangled.
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(c)
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2
4
6
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10
12
14
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gt
18
20 (d)
gt
Fig. 1. The time evolution of the degree of entanglement. Sðrf Þ is shown by the dotted line and E R ðrf Þ is shown by the solid line with x ¼ 0:5 and p ¼ 2: (a) g ¼ 0, q ¼ 0, (b) g ¼ 0, q ¼ 65, (c) g ¼ p=4, q ¼ 65, and (d) g ¼ p=2, q ¼ 65.
(3) For the rest of the time, the system state cannot be written in the direct product form, so the SU(1, 1)related coherent fields and the moving atom are entangled. In particular, from Fig. 1(a) when g ¼ 0 (the moving atom is initially in an excited state) and q ¼ 0, we can see that the reduced entropy is approximately equal to 1 at times gt ¼ kp þ 0:3p; kp þ 0:7p ðk ¼ 0; 1; 2; . . .Þ. This means that the moving atom and the SU(1, 1)-related coherent fields are maximally entangled, and the system state is an EPR(Einstein–Podolsky–Rosen) state which is widely used in QI processing pffiffiffi þ jcfa i 1= 2½jfþ a ijcf i þ jfa ijcf i.
(24)
(4) The initial atomic state has a considerable influence on the reduced entropy of entanglement. When qa0, the case for g ¼ 0 (the excited state), g ¼ p=4 and p=2 (the moving atom is initially in a coherent trapping state) are considered in Figs. 1(b)–(d) by using the dotted line. From these figures, we can see that the amplitude of the vibration of the DEFA decreases with an increase in parameter g. Especially, when g ¼ p=2, the value of Sðrf Þ is very small during the time evolution (see Fig. 1(d)). Our results further show that, when g ¼ p=2 and both q and x are very large, Sðrf Þ tends to zero, which means that the field and the moving atom are approximately disentangled (see Fig. 2(b) the dotted line).
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4
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3.5
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3.5 3 2.5 2 1.5 1 0.5
0
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10 gt
12
14
16
18
20
0
(b)
2
4
6
8
10 gt
12
14
16
18
20
Fig. 2. The time evolution of the degree of entanglement. Sðrf Þ is shown by the dotted line and E R ðrf Þ is shown by the solid line with g ¼ p=2 and p ¼ 2: (a) x ¼ 0:5, q ¼ 40, and (b) x ¼ 0:6, q ¼ 65.
4. The entanglement between the SU(1, 1)-related coherent fields The above results show that the Hamiltonian given by Eq. (1) leads to the entanglement between the SU(1, 1)-related coherent fields and the moving atom, so that the SU(1, 1)-related coherent state may evolve into a mixed state. In this case, the degree of entanglement between the SU(1, 1)-related coherent fields cannot be measured by the reduced entropy. However, the relative entropy of entanglement is a good measure for the degree of entanglement of two subsystems in the mixed state, which is defined as [16] E R ðrÞ ¼ min SðrksÞ, s2D
(25)
where SðrksÞ ¼ Tr½rðlog2 r log2 sÞ is the relative entropy. The minimum is taken over D, the set of all disentangled states. The relative entropy of entanglement is viewed as the minimal ‘‘distance’’ between the state r and the disentangled state s. For a pure state, the relative entropy of entanglement reduces to the reduced entropy of entanglement; while for a mixed state, it is usually difficult to numerically calculate the relative entropy of entanglement except for some specific states. Recently, the following theorem on the relative entropy of entanglement has been proved [17,18], and it turns out to be quite suitable for the present analysis. If a bipartite quantum state can be expressed as X r¼ an1 ;n2 jFn1 fn1 ihFn2 fn2 j (26) n1 ;n2
the relative entropy of entanglement is given by X E R ðrÞ ¼ an;n log2 an;n þ Trðr log2 rÞ
(27)
n
and the disentangled state s that minimizes the quantum relative entropy SðrksÞ is X an;n jFn fn ihFn fn j, s ¼
(28)
n
where jFn i ðjfn iÞ is a set of orthogonal normalized states of system AðBÞ. It should be noted that the density operator of Eq. (15) can take the form of Eq. (26) X rf ¼ am;n jm þ q; mihn þ q; nj, (29) m;n
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where am;n ¼ hm þ q; mjrf jn þ q; ni. Hence, the relative entropy of entanglement of the state Eq. (15) is written as X an;n log2 an;n Sðrf Þ E R ðrf Þ ¼ n
¼
X
an;n log2 an;n þ lþ log2 lþ þ l log2 l .
ð30Þ
n
The numerical results of Eq. (30) are shown by the solid lines in Figs. 1 and 2 for p ¼ 2 and different g; x and q, which reflect the time evolution of the degree of entanglement between the SU(1, 1)-related coherent fields (DESRCF). The DESRCF increases monotonically with the relative entropy of entanglement. Like the DEFA, the DESRCF exhibits the periodic evolution with period 2p=p. And when p ¼ 2, the DESRCF evolves in period p. Comparing the solid line with the dotted line in Figs. 1(a)–(d), we can find that the time evolution of the DESRCF is just the opposite to the DEFA. That is, when the latter increases with time, the former decreases with time. This comes from the fact that the DEFA can impair the DESRCF due to the SU(1, 1)-related coherent fields interacting with the moving atom. In particular, in Fig. 1(a) when q ¼ 0; g ¼ 0 and gt ¼ npðn ¼ 0; 1; 2 . . .Þ, it can be obtained that the states of the system can be written in a direct form which means that the moving atom is disentangled from the SU(1, 1)-related coherent fields. However, at the same time, the relative entropy of entanglement achieves its maximal value. At the time gt ¼ np, the field state is X jcf ðgt ¼ npÞi ¼ f n jn; ni (31) n
the relative entropy of entanglement of the above state is equal to the relative entropy of entanglement of initial field state for q ¼ 0 1 log2 ð1 jxj2 Þ 1 jxj2 jxj2 jxj2 log2 . 2 1 jxj 1 jxj2
E R ðgt ¼ npÞ ¼
ð32Þ
Our results further show that when x ¼ 0:5; q ¼ 65 and g ¼ p=2 (the moving atom is initially in a coherent trapping state), the sustained high degree of entanglement between the SU(1, 1)-related coherent fields can be obtained, as shown in Fig. 1(d). Moreover, when g ¼ p=2, an increase in parameter q or x results in an increase of the amount of the relative entropy of entanglement which is independent of the evolution time. This can be obtained by comparing Fig. 1(d), 2(a) and (b).
5. Conclusion In this paper, we have investigated the entanglement properties of the SU(1, 1)-related coherent state interacting with a moving atom. We can conclude as follows: (1) Both the QE between the SU(1, 1)-related coherent fields and the moving atom and that between the two modes of the fields evolve periodically with period 2p=p. (2) The entanglement between the SU(1, 1)-related coherent fields and the moving atom is just the opposite to the entanglement between the two modes of the fields due to the fields interacting with the moving atom. At the time gt ¼ npðn ¼ 0; 1; 2 . . .Þ, the moving atom and the fields are disentangled, but the relative entropy of entanglement is equal to the entanglement of initial field state. (3) When the moving atom is initially in an excited state (g ¼ 0), by choosing q ¼ 0 and gt ¼ kp þ 0:3p; kp þ 0:7p ðk ¼ 0; 1; 2; . . .Þ, the maximal entangled state between the SU(1, 1)-related coherent fields and the moving atom can be produced. (4) When q and x are very large and the moving atom is initially in a coherent trapping state (g ¼ p=2), the high degree of entanglement between the SU(1, 1)-related coherent fields can be preserved.
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Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 10374025) and by the Young Scientific Research Foundation of Hunan Provincial Education Department, China (Grand No. 04B070). Appendix In this appendix, with regard to the referee’s concern of significance of this work, we discuss the rationality of the entropy of entanglement for quantification of the degree of entanglement of quantum system from the following three aspects: (I) The relation between the measures of entanglement and the quantum entropy. As we know, several kinds of measures of entanglement have been proposed at present, which include entanglement of formation [19], entanglement of distillation [20], relative entropy of entanglement [16] and partial entropy of entanglement [13–15]. The entanglement of formation is defined as X E F ðrÞ ¼ min pi E p ðjcii Þ, (A.1) fpi ;jcii g
i
where E p ðjcii Þ is partial entropy of entanglement of state jcii . So, entanglement of formation is related to partial entropy of entanglement. And the concurrence introduced by Wootters [19] is also the special case of entanglement of formation. Thus, we can see that most of the measures of entanglement are related to the quantum entropy. (II) The rationality of the entropy of entanglement. The partial entropy of entanglement for a bipartite pure state is defined as Von Neumann entropy of the reduced state. Bennett et al. [21] have shown that it is a good entanglement measure for a bipartite pure state. In Refs. [22], Vedral et al. have proved the rationality of the relative entropy of entanglement for quantification of the degree of entanglement of quantum system in cases of mixed states. (III) The determination of the entropy of entanglement. The partial entropy of entanglement is defined as E p ðjciAB Þ ¼ SðrA Þ ¼ SðrB Þ,
(A.2)
where SðrÞ ¼ Trðr ln rÞ is the Von Neumann entropy. And, the quantum relative entropy of entanglement is defined as E R ðrÞ ¼ min SðrksÞ, s2D
(A.3)
where SðrksÞ ¼ Tr½rðlog2 r log2 sÞ is the relative entropy. From these equations, we can see that the entropy of entanglement is associated with the density matrix of quantum state. By using the quantum state tomography [23], we can experimentally determine what the density matrix of quantum state is. So the entropy of entanglement can be determined indirectly by experiment. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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