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Dynamic Behavior of Entanglement Between Two Spatially Separated Atoms in Two Dissipative and Driven Cavity Fields
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Commun. Theor. Phys. (Beijing, China) 51 (2009) pp. 509–513 c Chinese Physical Society and IOP Publishing Ltd
Vol. 51, No. 3, March 15, 2009
Dynamic Behavior of Entanglement Between Two Spatially Separated Atoms in Two Dissipative and Driven Cavity Fields CUI Hui-Ping,∗ LI Jian, LIU Jin, and LI Jun-Gang Department of Physics, School of Science, Beijing Institute of Technology, Beijing 100081, China
(Received April 11, 2008; Revised September 22, 2008)
Abstract We consider two two-level atoms, interacting with two independent dissipative cavities, each of which is driven by an external source. The two cavity fields are both initially prepared √ in the coherent states, and the two two-level atoms are initially prepared in the singlet state |Ψ− i = (|egi − |gei)/ 2. We investigate the influence of the damping constant κ, the intensity of the external sources F , and the relative difference of the atomic couplings r on the entanglement between the two atoms. In the dispersive approximation, we find that the entanglement between the two atoms decreases with the time evolution, and the decreasing rate of entanglement depends on the values of F/κ, κ/ω, and r. For the given small values of F/κ and κ/ω, on the one hand, the increasing of r favors entanglement decreasing of the atomic system, on the other hand, when r → 1 the entanglement decreasing becomes slower. With the increasing of the value of κ/ω, the influence of r on the decreasing rate of entanglement becomes smaller, and gradually disappears for the big value of κ/ω. PACS numbers: 42.50.Ct, 03.65.-w, 32.80.-t
Key words: entanglement, dissipation, driven field
1 Introduction Entanglement is a remarkable feature of quantum mechanics and is the key resource of quantum computation and quantum information processing.[1] There are numerous applications of spatially separated entangled pairs of particles such as quantum cryptography,[2] teleportation,[3] and superdense coding,[4] etc. Actually, real quantum systems will unavoidably interact with their surrounding environments and these interactions usually lead to decoherence and the loss of entanglement stored in this systems.[5] It is therefore of great importance to prevent or minimize the influence of environment noise in quantum information processing.[6] The problem of controlling the evolution of the entanglement and coherence between atoms (qubits), which interact with the environment, has been widely studied in recent years.[7] It has been realized that the controlled manipulation of such resources has practical importance in actual quantum information processing. Recently we have studied the influence of the dissipation on the entanglement between two remote atoms, which are located in two different environments respectively.[8] We found that when the dispersive limit is fulfilled, the entanglement between the two atoms will reach a steady value after a period of time of oscillations, and the steady entanglement between the two atoms will be stronger when the damping constant κ is larger, and will be weaker when the field intensity is bigger. The results of Ref. [8] were obtained only under the condition that the external sources are not present. Peixoto de Faria and Nemes[9] have given a fully analytical description of the dynamics of an atom dispersively coupled to a field mode in a dissipative environment fed by an ∗ E-mail:
[email protected]
external source. They found that the source will tend to compensate for the dissipation of the field intensity and to accelerate decoherence of the global and atomic states. Recently, Casagrande et al. showed that, starting from the cavity in a coherent state, the driven atom-cavity mode system can reach a maximally entangled state for long enough interaction times.[10] Also in Ref. [8] we assumed that the couplings between the two atoms and their cavity fields be equal. However, spatial and temporal variations of the couplings between the qubit and the field do arise experimentally.[11] This motivates us to study the nonsymmetric case in which the two qubits couple to the fields with different couplings. Li et al.[12] have studied the entanglement within a non-symmetric two-qubit system coupled to a coherent state field and found that in the non-symmetric case, the two atomic qubits can be entangled by the coherent state field even when they are initially prepared in the most mixed state. It is different from the symmetric case in which the two qubits cannot be entangled when they are initially in the most mixed state. In this paper, we will study the influence of the damping constant κ, the intensity of the external sources F and the relative difference of the atomic couplings r on the entanglement between two remote atoms. In the dispersive approximation, we find that for the given small value of F/κ, when the value of κ/ω is smaller, the evolution of the entanglement between the two atoms is influenced by the relative difference of the atomic couplings r, and the entanglement decreases most slowly for r → 1. With the increasing of κ/ω, the influence of r on the decreasing rate of entanglement becomes smaller, and gradually disappears for the big value of κ/ω.
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CUI Hui-Ping, LI Jian, LIU Jin, and LI Jun-Gang
This paper is organized as follows. In Sec. 2, we introduce the model of our paper. In Sec. 3 we calculate the time evolution of the initial state and obtain the reduced density operator of the atomic system. Then in Sec. 4, with the dispersive approximation we study the effect of the dissipation, the intensity of the external sources, and the relative difference of the atomic couplings on the evolution of entanglement between the two remote atoms. Finally we summarize our results in Sec. 5.
2 Model We consider two identical atoms and two spatially separated and dissipative cavities, each of which is driven by an external source. We denote the two cavities as A and B when the context requires us to differ them, but otherwise they are supposed to be identical. The atoms 1 and 2 are trapped in the cavities A and B respectively. The ground and excited states for each atom are respectively denoted by |gij and |eij (j = 1, 2). In this paper, we suppose that these atoms be coupled to the cavity fields with different coupling constants Ω1 and Ω2 . In the dispersive and rotation approximation, the effective Hamiltonian for the system consisted of two atom-cavity subsystem is (~ = 1)
− i [F a† + F ∗ a, ·] + κ[2a · a† − a† a · − · a† a], LB = −i ω2 [(b† b + 1) |ei he| − b† b |gi hg|, ·] − i [F b† + F ∗ b, ·] + κ[2b · b† − b† b · − · b† b]. (4)
Then the density operator of the total system can be expressed as ρ(t) = exp[(LA + LB )t]ρ(0), (5)
where ρ(0) is the density operator of the initial state of the total system. In Ref. [9], Peixoto de Faria et al. have studied the interaction between an atom and a dissipative cavity, which is driven by an external source. It has been found that in the limit t → ∞, the cavity field state will converge to the stationary state −i Fκ , i.e., as a result of the coupling between the field mode and the external source, a stationary coherent state is produced. In this paper we suppose that the two atoms be initially prepared √ in the singlet state |Ψ− i = (|egi − |gei)/ 2, and the two cavity fields be initially prepared in −i Fκ ⊗ −i Fκ , which can be prepared by turning on the two sources at t → −∞, and at t = 0 each cavity field reaches the stationary coherent state −i Fk . Then ρ(0) =
[9,13]
H = HA + HB ,
(1)
where
ωeg 1 σz + ω1 (a† a + 1) |ei he| − a† a |gi hg| , 2 ω eg 2 HB = ω0 b† b + σz + ω2 (b† b + 1) |ei he| − b† b |gi hg| .(2) 2 Here ωeg is the frequency between the two levels of each atom, σzj is the Pauli matrix of the atom j (j = 1, 2). We suppose that the two single mode fields in the two cavities have the same frequency ω0 . a† (b† ) and a (b) are respectively the creation and annihilation operators for the cavity field A (B). ωj = Ω2j /δ0 (j = 1, 2), is the effective coupling constant between the atom j and its cavity field, and δ0 = ωeg − ω0 , which measures how off resonance the atoms and the fields are. In our model, each cavity is dissipative and driven by an external source, and we suppose that the damping constant of the two cavities be the same and denoted by κ. Then in the interaction picture and at zero temperature, the density operator of the total system ρ(t) satisfies the following master equation,[9] d ρ(t) = −iω1 [(a† a + 1)|eihe| − a† a|gihg|, ρ(t)] dt HA = ω0 a† a +
†
†
†
∗
− i ω2 [(b b + 1)|eihe| − b b|gihg|, ρ(t)] − i[F a + F a, ρ(t)] − i[F b† + F ∗ b, ρ(t)] + κ(2aρ(t)a† − a† aρ(t) − ρ(t)a† a) + κ(2bρ(t)b† − b† bρ(t) − ρ(t)b† b), (3)
where F is the same intensity of the two external sources with the same frequency ωs . Without loss of generality, we assume that the sources be resonant with the two cavity modes, i.e., ωs = ω0 . We introduce two auxiliary superoperators LA and LB , which are defined as †
†
LA = −i ω1 [(a a + 1) |ei he| − a a |gi hg|, ·]
Vol. 51
where
1 A A B [ρ (0) ⊗ ρB gg (0) − ρeg (0) ⊗ ρge (0) 2 ee B A B − ρA ge (0) ⊗ ρeg (0) + ρgg (0) ⊗ ρee (0)],
(6)
F ED F ρspq (0) ≡ p, −i q, −i (p, q = e, g; s = A, B). κ κ Then we obtain the density operator of the total system at time t, 1 A B ρ(t) = [ρA (t) ⊗ ρB gg (t) − ρeg (t) ⊗ ρge (t) 2 ee B A B − ρA (7) ge (t) ⊗ ρeg (t) + ρgg (t) ⊗ ρee (t)],
where
ρspq (t) = e Ls t ρspq (0)
(p, q = e, g; s = A, B).
(8)
3 The Evolution of Initial State First we consider the interaction between the atom 1 and the dissipative and driven cavity A. In order to solve Eq. (8) we introduce several Liouvillian operators[9,13] L± = ∓ i ω1 (M − P ) + κ(2J − M − P ) + L0 , Leg = −i ω1 (M + P + 1) + κ(2J − M − P ) + L0 , Lge = i ω1 (M + P + 1) + κ(2J − M − P ) + L0 , (9)
A where L+ , L− , Leg , and Lge will act on ρA ee (0), ρgg (0), A A † ρeg (0), and ρge (0) respectively, and M = a a·, P = ·a† a, J = a·a† , and L0 ≡ −i[F a† +F ∗ a, ·] = −iF [(a† ·)−(·a† )]− iF ∗ [(a·) − (·a)]. After some calculations we can get[9]
ρA ee (t) = |e, βe (ω1 , t)i he, βe (ω1 , t)| , A ρgg (t) = |g, βg (ω1 , t)i hg, βg (ω1 , t)| ,
ρA eg (t) = exp[Φ(ω1 , t)]|e, βe (ω1 , t)ihg, βg (ω1 , t)|, ∗ ρA ge (t) = exp[Φ (ω1 , t)]|g, βg (ω1 , t)ihe, βe (ω1 , t)|,
where
(10)
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Dynamic Behavior of Entanglement Between Two Spatially Separated Atoms in Two Dissipative and Driven Cavity Fields
βe (ω1 , t) = and
F F [ e −(κ+iω1 )t − 1] − i e −(κ+iω1 )t , ω1 − iκ κ
βg (ω1 , t) = −
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F F [ e −(κ−iω1 )t − 1] − i e −(κ−iω1 )t , (11) ω1 + iκ κ
Φ(ω1 , t) = − iω1 t + z(ω1 , t) + |F |2 (p2 − q 2 + 2pq + |p + q|2 )(ω1 , t) + iΘ(ω1 , t) + Γ(ω1 , t) 2
|F | {2iRe[(p + q)(ω1 , t) e −κt cos ω1 t] − 2iIm[(p + q)(ω1 , t) e −κt sin ω1 t] κ − 4 e −(κ+iω1 )t [Imq(ω1 , t) + iRep(ω1 , t)]} ,
+
(12)
where the functions in Φ(ω1 , t) are 2 o 2iω1 |F | n 4[ e −(κ+iω1 )t − 1] − e −2(κ+iω1 )t + 1 ω1 2 t + + i {cosh[(κ + iω )t] − 1} , z(ω1 , t) = − 1 (κ + iω1 )2 2(κ + iω1 ) (κ + iω1 )2 k sinh[(κ + iω1 )t] ω1 {cosh[(κ + iω1 )t] − 1} − i , q(ω1 , t) = − {cosh[(κ + iω1 )t] − 1}, p(ω1 , t) = i (κ + iω1 )2 κ + iω1 (κ + iω1 )2 Θ(ω1 , t) =
|F |2 [ e −2κt (κ sin 2ω1 t + ω1 cos 2ω1 t) − ω1 ] , κ(κ2 + ω12 )
(13)
and 2
Γ(ω1 , t) = −
2
|F | |F | (1 − e −2κt ) − [ e −2κt (κ cos 2ω1 t − ω1 sin 2ω1 t) − κ]. κ2 κ(κ2 + ω12 )
(14)
Similar results can be obtained for the atom 2 and the cavity B. To make our presentation clearer we let ω = (ω1 + ω2 )/2 denote the average effective coupling, and r = (ω1 − ω2 )/(ω1 + ω2 ) express the relative difference of the atomic effective couplings. The value of r ranges from zero to one. When r = 0, ω1 = ω2 , which is the symmetry case; when r → 1, we have ω1 ω2 . Then from Eq. (7) we can get the density operator of the total system 1 ρ(t) = {|eg, βe (ω(1 + r), t), βg (ω(1 − r), t)i heg, βe (ω(1 + r), t), βg (ω(1 − r), t)| 2 + |ge, βg (ω(1 + r), t), βe (ω(1 − r), t)i hge, βg (ω(1 + r), t), βe (ω(1 − r), t)| + exp[Φ(ω(1 + r), t) + Φ∗ (ω(1 − r), t)] × |eg, βe (ω(1 + r), t), βg (ω(1 − r), t)i hge, βg (ω(1 + r), t), βe (ω(1 − r), t)| + exp[Φ∗ (ω(1 + r), t) + Φ(ω(1 − r), t)] × |ge, βg (ω(1 + r), t), βe (ω(1 − r), t)i heg, βe (ω(1 + r), t), βg (ω(1 − r), t)|}.
(15)
In this paper we are interested in the evolution of entanglement between the two atoms. Then from Eq. (15) by tracing over the field variables we obtain the density operator of the atomic system ∞ X 1 ρa (t) = hmn|ρ(t)|mni = (|egi heg| − X |egi hge| − X ∗ |gei heg| + |gei hge|), (16) 2 m,n=0 where X = exp[Φ(ω(1 + r), t) + Φ∗ (ω(1 − r), t)]exp[βe (ω(1 + r), t)βg∗ (ω(1 + r), t) + βe∗ (ω(1 − r), t)βg (ω(1 − r), t)] h 1 i 2 2 2 2 × exp − (|βe (ω(1 + r), t)| + |βg (ω(1 + r), t)| + |βe (ω(1 − r), t)| + |βg (ω(1 − r), t)| ) . 2 From Eqs. (16) and (17) we know that the entanglement between the two atoms is influenced by the presence of the reservoir and the external sources, even if the atoms are not directly coupled to them.
4 Entanglement Evolution Between the Two Atoms In this paper, we adopt Wotters’ concurrence[14] to quantify the entanglement of a two-qubit system. The concurrence Ca for the density operator ρa of the atomic system is defined as p p p p (18) Ca = max{0, λ1 − λ2 − λ3 − λ4 },
where λi are the eigenvalues of the matrix ζ = ρa (σy1 ⊗
(17) 1(2)
σy2 )ρ∗a (σy1 ⊗ σy2 ) arranged in decreasing order. Here σy are the y-Pauli matrix acting on atom 1(2), and ρ∗a is the complex conjugation of ρa in the standard basis. For the separable state Ca = 0, whereas Ca = 1 for maximally entangled state. For the density matrix (16) of the atomic system, the concurrence can be derived as Ca = max{0, |X|}.
(19)
In what follows, we will study the effect of the dissipation, the intensity of the external sources, and the relative difference of the atomic couplings on the evolution of entanglement between the two remote atoms. For simplicity, we assume F is real in this paper. In Fig. 1(a) we plot Ca as a function of r and ωt for F/κ = 0.5 and κ/ω = 0.2.
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CUI Hui-Ping, LI Jian, LIU Jin, and LI Jun-Gang
Vol. 51
smaller, and the influence of r on the entanglement evolution disappears, i.e., at this time the decreasing rate of entanglement between the two atoms is independent of the value of r. Then we can conclude that for the given small values of F/κ and κ/ω, on the one hand, the increasing of r favors entanglement decreasing of the atomic system, on the other hand, when r → 1 the entanglement decreasing becomes slower. In this case, by controlling the couplings between the two atoms and their cavity fields we can maintain the strong entanglement between the two atoms for a long time. With the increasing of the value of κ/ω, the effect of r on the entanglement evolution becomes smaller and gradually disappears. While in the symmetric case and without the driven fields considered, we have found that the entanglement between the two atoms will reach a steady value after a period of time of oscillations, and the steady entanglement between the two atoms will be stronger when the damping constant κ is larger.[8]
Fig. 1 (a) Ca as a function of r and ωt for F/κ = 0.5 and κ/ω = 0.2; (b) Ca as a function of ωt for κ/ω = 0.2, F/κ = 0.5 and different values of r.
From Fig. 1(a) we can see that the entanglement between the two atoms is decreasing with the time evolution. And it is interesting to find that the decreasing rate of entanglement between the two atoms strongly depends on the value of r: when r < 0.7 it increases with the increasing of r, while it will decrease with the increasing of r for r > 0.7. It can be clearly seen from Fig. 1(b) that Ca decreases most rapidly when r = 0.7 (the dashed line), and when r → 1 (the solid line), Ca decreases most slowly, i.e., when ω1 ω2 , the strong entanglement between the two atoms exists for a long time. This may be explained that for the small values of F/ω and κ/ω, when r = 1, only atom 1 is coupled to the cavity field and atom 2 is not, hence the entanglement between the two atoms can be maintained greatly. For the given small value of F/κ, when the value of κ/ω is increased, the influence of r on the decreasing rate of entanglement between the two atoms is quite different from the case in which the value of κ/ω is smaller. In Fig. 2 we plot Ca as a function of r and ωt for F/κ = 0.5 and κ/ω = 1. In this case, the decreasing rate of entanglement is decreasing with the increasing of r monotonously, i.e., when r → 1 the entanglement also decreases most slowly. When the value of κ/ω is increased further, e.g., κ/ω > 2, it has been found that Ca decreases more rapidly than the case when κ/ω is
Fig. 2 Ca as a function of r and ωt for F/κ = 0.5 and κ/ω = 1.
Fig. 3 Ca as a function of ωt for κ/ω = 0.2, r = 0.95 and different values of F/κ.
In Fig. 3 we plot Ca as a function of ωt for κ/ω = 0.2, r = 0.95 and different values of F/κ. It is noted that the larger the intensity of the external sources, the more rapid
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Dynamic Behavior of Entanglement Between Two Spatially Separated Atoms in Two Dissipative and Driven Cavity Fields
the entanglement loss between the two atoms is. Since the intensity of the injected fields by the sources is a measure of their “classicality”, the entanglement loss of the atomic system becomes faster as the intensity increases. Moreover, it is obvious from Eqs. (11), (16), and (17) that with the increasing of the value of F , the nondiagonal elements of ρa (t) decays more quickly, so the entanglement between the two atoms loses more rapidly. In addition, in absence of the external sources, i.e., making F = 0 in equations of this paper, we find that the dissipation and the relative difference of the atomic coupling have no influence on the atomic system, and the two atoms are always entangled maximally. When r = 0 we recover the results obtained in Ref. [8] with the initial vacuum state field |00i.
5 Summary and Discussion In this paper, we consider two two-level atoms, which are respectively located in two independent dissipative cavities, each of which is driven by an external source. We have studied the influence of the damping constant κ, the intensity of the external sources F , and the relative difference of the atomic couplings r on the entanglement between the two atoms. Initially, the two atoms
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√ are prepared in the singlet state |Ψ− i = (|egi − |gei)/ 2, and are prepared in the coherent states Fcavities Fthe two −i . In the dispersive approximation, we find −i ⊗ k k that for the given small value of F/κ, when the value κ/ω is smaller, the decreasing rate of entanglement between the two atoms strongly depends on the value of r, and the entanglement decreases most slowly for r → 1. In this case, by controlling the couplings between the two atoms and their cavity fields we can maintain the strong entanglement between the two atoms for a long time. With the increasing of the value of κ/ω, the effect of r on the entanglement evolution becomes smaller, and gradually disappears for the big value of κ/ω. And, the larger the intensity of the external sources is, the more rapid the entanglement loss between the two atoms is. Besides, when the two cavities are initially prepared in the vacuum state field |00i, the dissipation and the relative difference of the atomic coupling have no influence on the atomic system, which will be in the singlet state always. Although we √ focus on the singlet state |Ψ− i = (|egi − |gei)/ 2 in this paper, it can be easily proved that all the conclusions in this paper can still be valid when the two atoms are initially prepared in the other three Bell states.
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