The Retarding Effect of Noise on Entanglement Sudden Death

Vol. 76 (2015)

REPORTS ON MATHEMATICAL PHYSICS

No. 2

THE RETARDING EFFECT OF NOISE ON ENTANGLEMENT SUDDEN DEATH ¨ H UNKAR K AYHAN Department of Physics, Abant Izzet Baysal University, Bolu–14280, Turkey (e-mail:hunkar− [email protected]) (Received November 28, 2014 – Revised August 1, 2015) In this paper, we consider a system of two atoms in which one atom is in a JC cavity under the influence of a random phase telegraph noise and the other is an isolated atom. We obtain an exact solution to the time evolution of this system to investigate the effects of noise on the entanglement dynamics of the atoms. We show that the noise causes entanglement sudden death without recovery in a finite time interval. The time for this is independent of the initial state of the pure entangled atomic state. Moreover, an intensive noise delays the entanglement sudden death. Keywords: entanglement sudden death, noise, Jaynes–Cummings model.

1.

Introduction

Entanglement is one of the most striking features of quantum mechanics. It has a central role in quantum information processing and quantum computation [1]. The main difficulty of entanglement is its very fragility due to the decoherence effects. Yu and Eberly [2] observed that entanglement of two noninteracting qubits in their own cavities under the influence of the local noisy environments can abruptly decay to zero irreversibly in a finite time interval. And this time interval is much shorter than the exponential decay time of the coherence of the qubits. They termed this phenomenon as entanglement sudden death (ESD). ESD phenomenon was observed experimentally [3, 4]. The Jaynes–Cummings (JC) model [5–7] is an idealized model for describing the atom–field interactions in a lossless cavity. This model produces entanglement between atom and field which was studied extensively [8–11]. ESD phenomenon was also observed in the systems of the JC atoms. Y¨onac¸ et al. [12] investigated the entanglement properties of two isolated atoms which were in their own JC cavity. They showed the appearance of the ESD between the atoms which depends on the initial pure entangled atomic state. Li et al. [13] studied another system in which they assumed that one of the atoms was in the JC cavity and the other was isolated completely. The authors revealed the ESD between the atoms for the nonzero number cavity state which is independent of the initial pure entangled [197]

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atomic state. For the initially vacuum state of the cavity, the ESD between atoms does not occur. However, in the real-world situations, these aforementioned JC systems are influenced unavoidably by their environment which constructs a decoherence mechanism and destructs entanglement. One needs some generalizations which enable to consider the effects of decoherence on entanglement. One generalization was modeled by Joshi et al. [14, 15]. They generalized the JC model to include the effects of the random telegraph noise. The authors addressed some physical situations for the emergence of this noise in the atom–field interactions as follows. This noise might arise from the source of the field. Another possible mechanism might be instability in the atomic vapor production or in the dye laser system in a superconducting cavity the Rydberg atoms enter into. Another reason may be due to a stray electric field generated by ribidium deposits at the cavity coupling holes or the electric field between adjacent crystal domain in the cavity walls made of niobium. Also, the motion of an ion in a harmonic trap interacting with a standing or traveling wave can lead the JCM with this noise. In this type of intrinsic decoherence mechanism, the noise influences the dipole or the transverse relaxation of the interaction. Consequently, the decoherence mechanism conserves the energy of the system, but destructs the quantum coherence. In this sense, it has the similar features with the intrinsic decoherence mechanism model proposed by Milburn [16]. But, the origins of these two intrinsic decoherence mechanisms are completely different. In our previous works [17–20], we studied the entanglement dynamics between atom(s) and field(s) under the random phase telegraph noise in the model described in [14, 15]. In this paper, we consider the system described in [13] by taking into account the random phase telegraph noise modeled in [14, 15]. We explore the effects of noise on the entanglement between atoms. We show that noise causes the ESD without recovery in a finite time interval. The time for this is independent of the initial state of the pure entangled atomic state. Moreover, an intensive noise delays the ESD. We organize this paper as follows: In Section 2, we give the model and the system. In Section 3, we obtain an exact solution to the system in the presence of a random phase telegraph noise. Thereafter in Section 4, we study the influence of a noise on the entanglement properties of the atoms. In Section 5, we summarize our results. 2.

The model and the system

Hamiltonian of a system with the resonance between atoms and cavity is given by (h¯ = 1) [13] ω ω (1) H = σzA + σzB + ωa † a + g(a † σ−A + aσ+A ), 2 2 where σzA,B and σ±A are the spin–1/2 operators of atom A and atom B and a, a † are the field annihilation and creation operators. g denotes the strength of interaction

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between atom A and cavity c. (The schematic diagram of the model is given in [13].) The Hamiltonian in Eq. (1) tells that there is no interaction between atom B and cavity c, and between the atom B and the atom A. Rewriting the Hamiltonian with the random phase telegraph noise [15], the Hamiltonian of the noisy system becomes ω ω (2) H = σzA + σzB + ωa † a + g0 (eiφ(t) a † σ−A + e−iφ(t) aσ+A ), 2 2 where g0 corresponds to the strength of the non-noisy interaction in Eq. (1). φ(t) is a stochastic function that describes the random phase telegraph noise. It jumps randomly and continuously between different phases as Markov–type. So, φ(t) is not correlated to its past in the neighbouring intervals. In this case, phases of the noise φ(t) are equally probable. The average separation time between these random jumps is called the mean dwell time during which φ(t) is constant. The random telegraph noise was introduced in quantum optics first by Burshtein and Oseledchik [21] as an alternative model to some other more popular noise models such as Gaussian process and Wiener–Levy process. These noise models in the atom–field interaction were considered elsewhere [22, 23]. The random telegraph noise has some advantages. It gives, in a nonperturbative manner, the exact algebraic equations which have finite terms [24, 25]. These algebraic equations describe the average reactive response of the system to the noise. For the Gaussian noise, analytical solution can be obtained by cumulant approximation only for short coherence times. The telegraph noise has no such restriction. As the initial state of the system, we consider that the cavity is in the number state |ψc i = |ni, where n is the initial photon number in the cavity and atoms are in an entangled pure Bell state |ψAB i = cos θ |ei ⊗ |gi + sin θ|gi ⊗ |ei, where |ei and |gi are the excited and the ground states of each atom. In this case, the initial pure state of the total system becomes |ψABc (0)i = |ni ⊗ (cos θ|ei ⊗ |gi + sin θ|gi ⊗ |ei) = cos θ |negi + sin θ |ngei.

(3)

Then, the initial condition for the density matrix of the system becomes ρABc (0) = |ψABc (0)ihψABc (0)| = cos2 θ|negihneg| + sin2 θ|ngeihnge| + cos θ sin θ (|negihnge| + |ngeihneg|). 3.

(4)

Exact solution We obtain an exact solution to the system under the random phase telegraph noise by the master equation described in [15]. Z τ ρ(τ ) exp = U (φ; τ ; 0)ρ(0)U −1 (φ; τ ; 0)dQ(φ) τ0  Z Z 1 t + exp U (φ; τ ; t)ρ(t)U −1 (φ; τ ; t)dQ(φ)dt, (5) τ0 τ0

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where ρ(τ ) denotes the noise–averaged density matrix of the system at time τ and τ0 denotes the mean dwell time. Physically, intensity of the noise is proportional to the factor τ1 . The smaller τ0 , the more intensive noise. In this integral equation, the 0 statistical average over the noise is realized by the factor dQ(φ). This factor gives the probability of a given φ(t) at any instant. It is defined as dQ(φ) = dφ/2π, because the probability of all phases is the same. The term U (φ; τ ; t) describes a unitary evolution of the system from the instants t to τ . U −1 (φ; τ ; t) is the inverse of U (φ; τ ; t). This unitary transformation is given by U (φ; τ ; t) = exp(−iH (τ − t)). In order to solve the system, we express these aforementioned operators in the matrix form. In the standard basis |negi, |ngei, |n + 1ggi and |n − 1eei, the initial state of the system becomes   cos2 θ cos θ sin θ 0 0   2  cos θ sin θ  sin θ 0 0   ρABc (0) =  (6)    0 0 0 0   0 0 0 0 and the state of the system at any instant t is 

ρ11 (t) ρ12 (t) ρ13 (t) ρ14 (t)



   ρ (t) ρ (t) ρ (t) ρ (t)  22 23 24  21  ρABc (t) =  .  ρ31 (t) ρ32 (t) ρ33 (t) ρ34 (t)    ρ41 (t) ρ42 (t) ρ43 (t) ρ44 (t)

(7)

In the same way, U (φ; τ ; t) and U −1 (φ; τ ; t) can be constructed. Putting these matrices into the master equation (5) and by using the Laplace transformation technique (τ → s), one can obtain the following expressions for the elements of the transformed density matrix ρABc (s) of the total system:
s 2 + τs + 2αn2 cos2 θ 0 , (8) ρ11 (s) = 2 s 3 + sτ + 4αn2 s 0
2 s 2 2 s + τ + 2αn−1 sin θ 0 ρ22 (s) = , (9) 2 2 s 3 + sτ + 4αn−1 s 0

ρ33 (s) =

2αn2 cos2 θ , 2 s 3 + sτ + 4αn2 s 0

(10)

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THE RETARDING EFFECT OF NOISE ON ENTANGLEMENT SUDDEN DEATH

ρ44 (s) =

2 2αn−1 sin2 θ

s3 +

s2 τ0

2 + 4αn−1 s

(11)

,

ρ12 (s) 2

s 3 + 3 sτ + s 0

= s4

3 3 sτ 0

s 2 ( 32 τ0

3 τ02


2 + 2αn−1 +1 + 1 τ03

2 4αn−1

1 τ0

+ 3 τ0

1 τ03

+

2
2αn−1 cos θ τ0

2
6αn−1 τ0

1 τ02

sin θ 2 2αn−1

,

(12)

+ 2) + s + + 2 +1 + + + + τ0 √ where αn = g0 n + 1. The inverse Laplace transformation (s → t) of these elements gives an exact time–dependent solution to the system as
δ 3 X δj2 + τj + 2αn2 cos2 θ 0 Q ρ11 (t) = exp(δj t), (13) k6=j (δj − δk ) j =1 +

ρ33 (t) =

3 X j =1

2α 2 cos2 θ Q n exp(δj t), k6=j (δj − δk )

(14)

where δj s are roots of the equation δj2

+ 4αn2 δj = 0,
2 µ 2 3 X µ2j + τ j + 2αn−1 cos θ 0 Q ρ22 (t) = exp(µj t), k6=j (µj − µk ) j =1 δj3 +

ρ44 (t) =

3 X j =1

(15)

τ0

(16)

2α 2 sin2 θ Q n−1 exp(µj t), k6=j (µj − µk )

(17)

where µj s are roots of the equation µ3j + νj2

ρ12 (t) =

3 4  νj + 3 τ + νj X 0 j =1

 × exp(νj t) ,

µ2j

2 + 4αn−1 µj = 0,
2 + 2αn−1 + 1 + τ1 + 0 Q k6=j (νj − νk )

(18)

τ0

3 τ02

1 τ03

+

2
2αn−1 cos θ τ0

sin θ

(19)

and where νj s are roots of the equation    2  νj3 3 1 3 6αn−1 1 2α 2 4 2 2 νj + 3 + νj 2 + 4αn−1 + 2 + νj 3 + + + 2 + n−1 + 1 = 0. (20) τ0 τ0 τ0 τ0 τ02 τ0 τ0

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Eqs. (13)–(20) describe the time evolution of the system, ρABc (t), under the effect of the random phase telegraph noise. In this case, the noise–averaged density matrix of the system at time t will evolve into the state ρABc (t) = ρ11 (t)|negihneg| + ρ12 (t)|negihnge| + ρ21 (t)|ngeihneg| + ρ33 (t)|n + 1ggihn + 1gg| + ρ22 (t)|ngeihnge| + ρ44 (t)|n − 1eeihn − 1ee|.

(21)

The noise–averaging in the master equation washes out most elements of ρABc (t). 4.

Results and discussion We now analyze numerically the effects of a random phase telegraph noise on the entanglement properties of the atoms. For this, we need the reduced atomic density matrix ρAB (t) which is obtained by tracing out ρABc (t) over the cavity basis ρAB (t) = Tr{c} ρABc (t) = ρ11 (t)|egiheg| + ρ12 (t)|egihge| + ρ21 (t)|geiheg| + ρ22 (t)|geihge| + ρ33 (t)|ggihgg| + ρ44 (t)|eeihee|.

(22)

The dimension of ρAB (t) in Eq. (22) is 2 ⊗ 2. In this dimension, the Wootters’ concurrence [26] is a convenient entanglement measure. It is defined for a density matrix ρ as C = max(0, λ1 − λ2 − λ3 − λ4 ) where λi s are square roots of eigenvalues of the matrix R = ρ(σy ⊗ σy ρ ∗ σy ⊗ σy ), in the decreasing order. The concurrence is equal to zero for the separable states and unity for the maximally entangled states. Computation of the concurrence from ρAB (t) gives p (23) CAB = 2 max{0, |ρ12 (t)| − ρ33 (t)ρ44 (t)}. The effects of noise on the entanglement properties of atoms are given by the following figures (we assume that g0 = 1). Fig. 1 shows that when the effect of noise is removed (τ0 → ∞), we obtain the same results [13]. The ESD appears. Entanglement abruptly falls to zero and remains so for a period of time before its recovery. The death time of the entanglement is independent of the initial entanglement degree of atoms. When the noise is involved, as shown in Figs. 2–4, a strong damping effect of the noise obviously causes the decay of entanglement to zero rapidly in a finite time interval. Strong noise prevents recovery of entanglement after its death. The atomic state will remain separable after the death of entanglement at any further time. Moreover, if intensity of the noise increases (the value of τ0 decreases), the complete destruction of entanglement takes longer time. Intensive noise delays the time for ESD. Therefore, intensive noise in the interaction has a retarding effect on the ESD phenomenon, which can be very useful in quantum information processing and quantum computation, since the maintenance of entanglement of two qubits is essential. This situation can be attributed to that; when τ0 is very small, the phase jumps of the telegraph noise become more fast. Accordingly, the system faces difficulty in perceiving all of these

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1 0.9 0.8 0.7

CAB

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

t

Fig. 1. Concurrence of atoms CAB is plotted as a function of time t with the mean dwell time τ0 → ∞, the initial photon number n = 3 and the atomic state parameter θ = π/4 (solid line), θ = π/6 (dashed line), θ = π/12 (dotted line). 1 0.9 0.8 0.7

CAB

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

t

Fig. 2. Concurrence of atoms CAB is plotted as a function of time t with the mean dwell time τ0 = 1, the initial photon number n = 3 and the atomic state parameter θ = π/4 (solid line), θ = π/6 (dashed line), θ = π/12 (dotted line).

phase jumps. In this case, the system cannot react on all these jumps individually but only in an average way. As a result, the system feels relatively less the influence

204

H. KAYHAN 1 0.9 0.8 0.7

CAB

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

t

Fig. 3. Concurrence of atoms CAB is plotted as a function of time t with the mean dwell time τ0 = 0.1, the initial photon number n = 3 and the atomic state parameter θ = π/4 (solid line), θ = π/6 (dashed line), θ = π/12 (dotted line). 1 0.9 0.8 0.7

CAB

0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

12

14

16

18

20

t

Fig. 4. Concurrence of atoms CAB is plotted as a function of time t with the mean dwell time τ0 = 0.01, the initial photon number n = 3 and the atomic state parameter θ = π/4 (solid line), θ = π/6 (dashed line), θ = π/12 (dotted line).

of the noise. It is also clear that, the death time of entanglement is independent of the initial state of the pure entangled atomic state as determined by θ. The

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1 0.8

CAB

0.6 0.4 0.2 0

0.2

0.4 0

τ

0 0.6

5

t

0.8

10

15 1

20

Fig. 5. Concurrence of atoms CAB is plotted as a function of time t and the mean dwell time τ0 with the initial photon number n = 3 and the atomic state parameter θ = π/4.

entanglement will vanish in the same time for any degree of the initial entanglement between the atoms. Fig. 5 shows the evolution of entanglement of atoms as a function of time and the mean dwell time. We see again that if the noise becomes more intense (τ0 becomes smaller), the ESD takes longer time. We have also considered the initial pure entangled state of atoms as another Bell state |ψAB i = cos θ|eei + sin θ|ggi. Our results presented above remain the same. 5.

Summary In summary, we have considered the system of two atoms, one of them in a JC cavity under influence of the random phase telegraph noise and the other completely isolated. There is no interaction of the isolated atom with the cavity and with the atom inside the cavity. We have obtained an exact solution to the system to investigate the effects of noise on the entanglement dynamics of the atoms. We have shown that strong noise caus the ESD without recovery in a finite time–interval. The time for this is independent of the initial state of the pure entangled atomic state. Moreover, if intensity of the noise increases, the complete destruction of entanglement takes longer time. Intensive noise delays the time for

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ESD. Therefore, an intensive noise in interaction has a retarding effect on the ESD phenomenon, which can be very useful in quantum information processing and quantum computation, since the maintenance of entanglement of two qubits is essential. Acknowledgements I am grateful to the referee for valuable comments and recommendations for improving this paper. REFERENCES [1] M. A. Nielsen and I. L. Chuang: Quantum Computation and Quantum Information, Cambridge University Press, Cambridge 2000. [2] T. Yu and J. H. Eberly: Phys. Rev. Lett. 93 (2004), 140404. [3] M. P. Almeida, F. de Melo, M. Hor–Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro and L. Davidovich: Science 316 (2007), 579. [4] A. Salles, F. de Melo, M. P. Almeida, M. Hor–Meyll, S. P. Walborn, P. H. Souto Ribeiro and L. Davidovich: Phys. Rev. A 78 (2008), 022322. [5] E. T. Jaynes and F. W. Cummings: Proc. IEEE 51 (1963), 89. [6] H.–I. Yoo and J. H. Eberly: Phys. Rep. 118 (1985), 239. [7] B. W. Shore and P. L. Knight: J. Mod. Opt. 40 (1993), 1195. [8] S. J. D. Phoenix and P. L. Knight: Phys. Rev. A 44 (1991), 6023. [9] S. Bose, I. Fuentes–Guridi, P. L. Knight and V. Vedral: Phys. Rev. Lett. 87 (2001), 050401. [10] S. Furuichi and M. Abdel–Aty: J. Phys. A 34 (2001), 6851. [11] H. Kayhan: Phys. Scr. 83 (2011), 025402. [12] M. Y¨onac¸, T. Yu and J. H. Eberly: J. Phys. B 39 (2006), S621. [13] Z.–J. Li, J.–Q. Li, Y.–H. Jin and Y.–H. Nie: J. Phys. B 40 (2007), 3401. [14] S. V. Lawande, A. Joshi and Q. V. Lawande: Phys. Rev. A 52 (1995), 619. [15] A. Joshi: J. Mod. Optics 42 (1995), 2561. [16] G. J. Milburn: Phys. Rev. A 44 (1991), 5401. [17] H. Kayhan: Eur. Phys. J. D 48 (2008), 443. [18] H. Kayhan: Braz. J. Phys. 38, no 3A (2008), 329. ¨ [19] C. Ozel, E. Yılmaz, H. Kayhan and A. Aktaˇg: Int. J. Theor. Phys. 47 (2008), 3101. [20] A. Aktaˇg and H. Kayhan: Phys. Scr. 79 (2009), 065015. [21] A. I. Burshtein and Y. S. Oseledchik: Zh. Eksp. Teor. Fiz. 51 (1966), 1071 [Sov. Phys. JETP 24 (1967), 716]. [22] A. Joshi and S. V. Lawande: Phys. Lett. A 184 (1994), 390. [23] S. V. Lawande and A. Joshi: Phys. Rev. A 50 (1994), 1692. [24] J. H. Eberly, K. W´odkiewicz and B. W. Shore: Phys. Rev. A 30 (1984), 2381. [25] K. W´odkiewicz, B. W. Shore and J. H. Eberly: Phys. Rev. A 30 (1984), 2390. [26] W. K. Wootters: Phys. Rev. Lett. 80 (1998), 2245.