- Email: [email protected]
Dynamics of entanglement and Bell-nonlocality for the interacting of coupled superconductor qubits and common environment
Optik 124 (2013) 1952–1956
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Dynamics of entanglement and Bell-nonlocality for the interacting of coupled superconductor qubits and common environment YingHua Ji a,b,∗ a b
Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China Key Laboratory of Photoelectronics & Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, China
a r t i c l e
i n f o
Article history: Received 12 January 2012 Accepted 30 May 2012
PACS: 03.65.Yz 03.65.Ud Keywords: Coupled qubits Non-Markovian process Concurrence Bell inequality
a b s t r a c t Utilizing the concurrence and the Bell inequality, we investigated in detail the evolution of entangled decoherence of two superconductor qubits in non-Markovian process. The results show that the evolution of entanglement dynamics can cause entanglement sudden death (ESD) and entanglement sudden birth (ESB) because of the interaction between the environment and the quantum qubits. The concurrence of a quantum system is not only related with the environmental dissipation effects, the interaction between the qubits, and the initial quantum state, but also strongly related with the coupling symmetry between qubit-environment. Under certain initial states, the effect of environmental dissipation can be avoided completely by controlling the coupling model between qubits and environments. The interaction between qubits can prolong the survival time of the entanglement, and it is the dynamic for reviving the entanglement of quantum system. The further research results indicate that, the non-Markovian memory effects of the environment are helpful to prolong the survival time of the entanglement or to prevent the outbreak of ESD. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Quantum entanglement is a nonlocal quantum correlation and is the perfect reflection of nonlocality in quantum mechanics. Quantum entanglement is not only one most difficult conception to be understood and accepted, but also the physical foundation in constructing the future quantum information science. As a physical resource, entanglement plays an important role in much quantum information, such as quantum teleportation, quantum key distribution, quantum computation, etc. [1–3] Keeping quantum entanglement of the system is the key measure in quantum information management. In the real physical world, however, the pure entanglement can degenerate the mixed state under the effect of environmental docoherence. It will bring mistakes in quantum computation and information distortion in quantum correspondence. The phenomenon, that the quantum systems fully getting rid of entanglement in a limited time due to the effect of environmental dissipation, is known as entanglement sudden death (ESD) [4–7]. In order to ensure the realization of correct calculation of quantum logic and quantum computing, people have done a lot of researches about avoiding entanglement sudden death and proposed new plans to achieve entanglement in some cases. But in
∗ Corresponding author. E-mail address: [email protected] 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.05.024
an open quantum system; there are still a lot of problems to be solved about the disentanglement. In a sense, it is one of the key technologies to implement the quantum information. Thus, searching for stable entanglement is an important issue for quantum information study. For the moment, the famous entanglement that are investigated relatively thorough and always experimentally realized in some physical systems are optical parametric transition, nuclear magnetic resonance system (NMR), thermal entanglement, atom system, cavity QED, quantum dot and superconducting Josephson junction. Indeed, The superconducting quantum circuit is now regarded as one of the most promising physical programs to test the principles of quantum mechanics and achieve scalable quantum computation at the macro scale. It distinguishes for the following features: (1) the level structure of the superconducting quantum circuit can be customized through the circuit design or regulated by the applied electromagnetic signals; (2) in superconducting quantum circuits, one can achieve the quantum coherent superposition of a large number of electronic joint degrees of freedom; (3) superconducting quantum circuit can simulate the quantum behavior of other physical systems (such as cavity quantum electrodynamics); and (4) based on the current microelectronics manufacturing processes, superconducting quantum circuit has good scalability. Based on these advantages cannot be replaced by other qubit system, superconducting quantum circuits has been developed rapidly in recent years. People have made important progress
Y. Ji / Optik 124 (2013) 1952–1956
in experiments during the operation of superconducting qubits, superposition of quantum states, coherent evolution and oscillation, the gate operation fidelity and quantum state read. Especially recently, the quantum control of superconducting quantum bit circuit has made impressive progress: people achieved the coherent entanglement manipulation of the solid quantum system including three quantum bits for the first time, which laid a solid foundation for the realization of quantum error correction [8–11]. The quantum nonlocality is also an important physical resource. This kind of nonlocality can be characterized by Bell inequality; Bell inequality has many generalized forms. The common generalized form is CHSH-type Bell inequality in the Two-body quantum system. The CHSH-type Bell inequality is a good indicator of nonlocal correlations and the relation with the entanglement has been already known. Bellomo et al., found that Bell inequality might not be violated for a state with high entanglement for a two-qubit system subject to amplitude damping [12,13]. So far, although people have done many researches on entanglement features of different quantum systems and nonlocal features, there are still many unsolved problems especially for superconducting qubits system whose memory effects of the environment influence from the dynamic evolution is sensitive. In the real physical world, the systems are referred as non-Markovian open quantum systems, where the non-Markovian dynamics is characterized by the existence of a memory time scale so that the energy or information would transfer from the system to the environment then feeds back into the system [14,15]. Therefore, it is interesting to study the impact of the coupling model between the qubits and environment on the entanglement dynamics in non-Markov process. As one essential model of describing quantum dissipation system, spin bose model has an important effect on describing broken tunneling in solids, quantum tunneling between the magnetic flux in superconducting quantum interference devices (SQUIDs), electron tunneling between quantum spots, and electron transfer in chemistry and biology reaction. It can be easily used to describe the dynamic features of qubits system. In this paper we investigate the entanglement features and nonlocality features of coupling superconducting flux qubits in non-Markovian environment based on the spin bose model. We will show that the interactions between the qubits and the qubit-environments significantly affect the decay of the entanglement. We find that, under certain initial states, we can completely avoid the environmental dissipation effects by controlling the coupling model between the qubits and environment, which guarantees the stable entanglement degree during the process of dynamic evolution.
1953
assume that the correlative function of the environment decays exponentially with time t, i.e. [16] R()R(0)R = 2 e−t/ ,
(4)
where is the relaxation time of the reservoir variables. ˆ qb−B between the coupled qubits The interacting Hamiltonian H system and environment is assumed as ˆ qb−B = Sˆ B, ˆ H
(5)
with Sˆ = 1 ˆ 1z + 2 ˆ 2z .
(6)
The parameter 1,2 represents the strengths of the coupling between the qubit 1 (2) and environment. For 1 = 2 , the coupling interaction is symmetric and otherwise it is asymmetric for 1 + 2 = 0. It is known that the density matrix was introduced as a way of describing a quantum open system because the state of the open system is not completely known. The time evolution of the reduced density operator , from the integro-differential quantum-Liouville equation of motion, gives us information about the dynamics of the system. In the interaction picture, we have d I 1 I (t) = [H I (t), total (t)], dt total i where ˆ0 = H ˆ qb + H ˆB + H ˆ qb−qb H I (t) ≡ eiH0 t Hqb−B e−iH0 t , H Because the capacity of the heat storage R is very large, one may assume that, during the process of dynamic evolution, quantum system has little effect on the heat storage, i.e., total (t) = (t) ⊗ B ˆ I , (0) ⊗ B ] = 0. The involving equation of dynamics for and trB [H density operator may be rewritten as d I (t) = − dt
t
dTrB ([H I (t), [H I (), I (t) ⊗ B ]]).
(7)
0
The solutions of equation obviously depend on the initial conditions. In following, we suppose that the initial density matrix is only partially coherent, but includes an arbitrary degree of nonlocal coherence. This mixed state is easily expressed in the following form: ˚ (0) =
1−b I4 + b|˚˚|, 4
(8)
2. Model
(0) =
1−b I4 + b| |, 4
(9)
In this paper, we model the Hamiltonian of a system with two coupled superconducting flux qubits. The typical Hamiltonian for this model may be described by (kB = = 1) (1)
where b is the purity of the initial states which ranges from 0 for maximally mixed states to 1 for pure states, I4 is the 4 × 4 identity matrix, |˚ = ˛|10 + ˇ|01 and | = ˛|11 + ˇ|00 the Bell-like states and the parameter ˛ is sometimes called the degree of entan-
ω (1z + 2z ), 2
(2)
3. Entanglement dynamics
Hqb−qb = k(1+ 2− + 1− 2+ ),
(3)
ˆ total = H ˆ qb + H ˆB + H ˆ qb−qb + H ˆ qb−B , H with ˆ qb = H
where k is a strength of the qubit–qubit interaction and represents ˆ B needs not to be proa external field. The bath Hamiltonian H vided explicitly; it is sufficient to know the correlation R(0)R(t) ˆ where · · · = TrB (· · · B ) denotes a trace over of the bath operator R, the bath degrees of freedom with the bath density matrix B ; we can further assume that the fluctuations are unbiased R(t) =0. In order to simplify our calculation but without loss of generality, we
glement (here we set ˛ as real number and ˇ =
1 − ˛2 .
In order to describe the dynamic evolution of quantum entanglement, we adopt the concurrence defined by Wootters to quantify the degree of entanglement for any bi-partite system. The concurrence C(t) is defined as [17] ˜ C(t) = max{0, C(t)}, with ˜ C(t) =
1 −
2 −
(10)
3 −
4
1954
Y. Ji / Optik 124 (2013) 1952–1956
0.5
where j are the eigenvalues of W = (t)(t) ˜ in nonincreasing order (2) ⊗ y ).
C˜ ˚S (t) = 2
˛2 ˇ2 b2 +
1 2 1−b 2 (˛ − ˇ2 ) b2 sin2 (2kt) − 4 2
(11)
From Eq. (11) we are very easy to find the environmental dissipation has no effect on entanglement features of system and the system evolution of entanglement dynamics is related with the interaction between the initial state and the qubits. If the qubits system wants to completely avoid the ESD phenomenon, the parameter of initial state must satisfy (4˛ˇ + 1)b > 1. Especially, √ when ˛ = ˇ = 1/ 2, the system is in the Bell state (b = 1) or the mixed entangled state (b < 1) which is produced by the Bell state, the entanglement degree of the coupling qubits system keep a constant under the symmetrical coupling. On the contrary, if the coupling qubits system is initially in state (0) and when a completely asymmetrical coupling is used between the qubits and environment, the concurrence is C AS = max{0, 2˛ˇb − 0.5(1 − b)}
(12)
The environmental dissipation has no effect on entanglement features of system and the entanglement features completely depend on the initial state. At this moment, only when the parameter of initial state meets Eq. (9), the concurrence can keep a constant C˜ AS (t) = 2˛ˇb − 0.5(1 − b) and never occur ESD under the completely asymmetrical coupling. From above discussions, we can avoid the environmental dissipation effects by controlling the coupling model between qubits and environment under certain initial state ˚ (0) or (0), which can make the degree of entanglement keeps invariable with time evolution and will not occurs the ESD phenomenon. Subsequently, it is beneficial to the transmission of quantum information. Fig. 1 shows the system entanglement dynamic evolution with time from initial state ˚ (0) when a symmetrical coupling is used between the qubits and environment. The curve is equal amplitude oscillation and changes according to the sine regulation. By analyzing the concurrence C˜ ˚ (t) in Eq. (11) carefully it is not difficult to find, if a symmetrical coupling is used between the qubits and environment and the initial state meets C˜ ˚ (0) > 0, the entanglement always have C˜ ˚ (t) > 0 and would not occur ESD phenomenon during the process of the system dynamic evolve with time from initial state ˚ (0). Fig. 1(1) just describes the physical progress. We can see from Eq. (12), when the quantum system dynamic evolves with time from initial state (0), once ESD phenomenon appears, ESB would not appear because C˜ (t) decreases monotonously with time and has nothing to do with the interaction between qubits. On the contrary, when the system is initially in state ˚ (0), the concurrence C˜ ˚ (t) is closely related with the interaction between qubits. We can see from Fig. 2, the qubits system do not get entangled even if C˜ ˚ (0) ≤ 0 at the initial time. With time goes on, it will appear ESD and entanglement revives periodically depending on the interaction between qubits under certain conditions. This indicates that the interaction between qubits is the dynamic of entanglement revive indeed.
Concurrence
1-1
0.4 0.3 0.2 0.1 0.0
0
2
4
6
8
10
t/µ 0.5
1-2
0.4
Concurrence
˜ = If the sysby magnitude with (t) tem is initially in state ˚ (0) and the coupling between the qubits and environment is a symmetrical coupling, or 1 = 2 , the concurrence is
0.3 0.2 0.1 0.0
0
2
4
6
8
10
t/µ Fig. 1. Concurrence as a function of t/, when two-qubits initially in the ˚ (0) and the symmetrical √ √ coupling is used between qubits and environment (u = v = 0), where ˛ = 0.9, ˇ = 0.1, k = 1. Here panel (1) b = 06, panel (2) b = 0.45.
1.0
2-1
0.8
Concurrence
(2) (1) ⊗ y )∗ (t)(y
0.6 (c)
0.4
(b) (a)
0.2 0.0
0
2
4
t/µ
6
8
1.0
10
2-2
0.8
Concurrence
(1) (y
0.6 0.4
(c) (b)
0.2
(a)
4. The CHSH-type Bell inequality The quantum nonlocality is also an important physical resource. This kind of nonlocality can be characterized by Bell inequality; Bell inequality has many generalized forms. The common generalized form is CHSH-type Bell inequality in the Two-body quantum system. The CHSH-type Bell inequality is a good indicator of
0.0
0
2
4
t/µ
6
8
10
Fig. 2. Concurrence as a function of t/ for different rates m, when two-qubits initially in the ˚ (0), where (a) m = 1, (b) m = 0.5, (c) m = 0.1. b = 0.6, ˛ = 1/2, ˇ =
3/2.
Here panel (1) corresponds to non-Markovian dynamics and panel (2) to Markovian dynamics.
Y. Ji / Optik 124 (2013) 1952–1956
nonlocal correlations and the relation with the entanglement has been already known. It showed that the maximum of the Bell function B for a X-structured density matrix (8) can be given as
0.6
Bmax = 2
0.2
P2 + Q 2,
(13)
where
1955
3-1
0.4
(b)
0.0
P = 11 + 44 − 22 − 33 , Q = 2(|14 (t)| + |23 (t)|). For the initial state ˚ (0), the maximum of Bell function is given by
Bmax
˚ (t)
= 2b
1 + 4˛2 ˇ2 e−4xt + 4(˛2 − ˇ2 )
2
k + y 2 ˝
-0.2
-0.6
f (t), (14)
where
0
2
4
t/µ
6
8
1.0
f (t) = e−2xt sin h2 (˝t),
10
3-2
(15) 0.8
2 (1 − 2 )2 [1 − e−t/ cos(2kt) + e−t/ sin(2kt)], x= 2(1 + 4k2 2 ) y=
(a)
-0.4
(b)
(16) 0.6
2 (1 − 2 )2 [2k − 2ke−t/ cos(2kt) − e−t/ sin(2kt), 2(1 + 4k2 2 )
(a)
0.4
(17) 0.2
˝=
x2 − 4k(k + y).
(18)
And for the initial state (0), the maximum of Bell function is
Bmax
(t)
=2
b2
+ 4˛2 ˇ2 b2
exp
2(1 + 2 )2 − (1 − e−t/ )t . Tph (19)
After comparing Bmax ˚ (t) with C˜ ˚ and Bmax (t) with C˜ under the same parameters, we can find, because of the environmental decoherence effects, the survival time for entanglement should be much longer than the Bell inequality violation, which agrees with the research results of the literature. Based on above comparing results about the influence on the entanglement dynamic process from the non-Markovian and Markovian, we can further know that non-Markovian effect is not helpful to the violation of the CHSH inequality to judge nonlocal entanglement features. When the symmetrical coupling model is used between the qubits system in initial state ˚ (0) and environment, the dynamics of concurrence and Bell nonlocality as a function of the time are displayed in Fig. 3. All the curves are equal amplitude oscillation and changes according to the sine regulation. Fig. 3(1) shows explicitly that when C˜ > 0, the CHSH-type Bell inequality is not violated (Bmax ≤ 2). In these regions it is therefore not possible to say with certainty that quantum correlations are not reproducible by a classical local model. Fig. 3 indicates that under certain initial state, when the symmetrical model is used between the quantum system and environment, the concurrence C˜ and the Maximum of Bell function Bmax have the same regular evolution with time. When the initial state of the quantum system is in (0) and the completely asymmetrical coupling model is used between the system and environment, the maximum of Bell function is
B AS = 2
b2 + 4˛2 ˇ2 b2
√ From above equation, we can see that, when ˛ = ˇ and b ≥ 1/ 2, it always violates the inequality. On the contrary, it always does not violate the inequality.
0.0
0
2
4
t/µ
6
8
10
Fig. 3. Concurrence and Maximum of Bell function, Bmax − 2 as a function of t/, when two-qubits initially in the ˚ (0) and the symmetrical coupling is used between qubits and environment, where ˛ = 1/2, ˇ =
3/2, 2 2 = 0.25, k = 1,
|1 | = |2 |=0.5. Here panel (1), b = 0.6, not violate the Bell inequality. Panel (2), b = 0.9, violate the Bell inequality.
5. Conclusions In this paper, utilizing the concurrence and the Bell inequality, we investigate in detail the evolution of entangled decoherence of two superconductor qubits in non-Markovian process. The results show that under certain initial states, we can completely avoid the environmental dissipation effects by controlling the coupling model between qubits and environments, which guarantees the stable entanglement degree and avoids the outbreak of ESD during the process of dynamic evolution. The further research results indicate that, the non-Markovian memory effects of the environment are helpful to prolong the survival time of the entanglement or to prevent the outbreak of ESD. Acknowledgement This work is supported by the National Natural Science Foundation of China under Grant No. 11164009. References [1] S. Etaki, M. Poot, I. Mahboob, K. Onomitsu, H. Yamaguchi, H.S.J. van der Zant, Motion detection of a micromechanical resonator embedded in a d.c. SQUID, Nat. Phys. 4 (2008) 785. [2] Y. Yu, S.L. Zhu, G.Z. Sun, X.D. Wei, N. Dong, J. Chen, P.H. Wu, Quantum jumps between macroscopic quantum states of a superconducting qubit coupled to a microscopic two-level system, Phys. Rev. Lett. 101 (2008) 157001. [3] T. Yu, J.H. Eberly, Sudden death of entanglement, Science 323 (2009) 598. [4] Y.H. Ji, J.J. Hu, Entanglement and decoherence of coupled superconductor qubits in a non-Markovian environment, Chin. Phys. B 19 (2010) 060304.
1956
Y. Ji / Optik 124 (2013) 1952–1956
[5] W. Cui, Z.R. Xi, Y. Pan, Optimal decoherence control in non-Markovian open dissipative quantum systems, Phys. Rev. A 77 (2008) 032117. [6] Y.H. Ji, Y.Y. Jiang, Investigation of decoherence of superconducting flux qubit entangled with environment, Mod. Phys. Lett. B 22 (2008) 581. [7] S.K. Wang, J.S. Jin, X.Q. Li, Continuous weak measurement and feedback control of a solid-state charge qubit: a physical unravelling of non-Lindblad master equation, Phys. Rev. B 75 (2007) 155304. [8] Y.H. Ji, H.F. Lai, S.H. Cai, Z.S. Wang, Controlled decoherence of three-level rfSQUID qubit with asymmetric potential, Chin. Phys. B 19 (2010) 030310. [9] H. Bartolf, A. Engel, A. Schilling, K. ll’in, M. Siegel, H.-W. Hubers, A. Semenov, Current-assisted thermally activated flux liberation in ultrathin nanopatterned NbN superconducting meander structures, Phys. Rev. B 81 (2010) 024520. [10] S. Gustavsson, J. Bylander, F. Yan, D. William, F. Oliver, Y. Yoshihara, Nakamura, Noise correlations in flux qubit with tunable tunnel coupling, Phys. Rev. B 84 (2011) 014525. [11] I. Kamleitner, P. solinas, C. Muller, A. Ahnirman, M. Mottonen, Geometric quantum gates with superconducting qubits, Phys. Rev. B 83 (2011) 214518.
[12] B. Bellomo, R. Lo Franco, G. Compagno, Entanglement dynamics of two independent qubits in environments with and without memory, Phys. Rev. A 77 (2008) 032342. [13] L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B.M. Garraway, Sudden death and sudden birth of entanglement in common structured reservoirs, Phys. Rev. A 79 (2009) 042302. [14] Y.H. Ji, Y.M. Liu, Z.S. Wang, Avoiding the decay of entanglement for coupling two-qubit system interacting with a non-Markov environment, Chin. Phys. B 20 (2011) 070304. [15] X.Y. Zhao, J. Jing, B. Corn, T. Yu, Dynamics of interacting qubits coupled to a common bath: Non-Markovian quantum-state-diffusion approach, Phys. Rev. A 84 (2011) 032101. [16] M. Ban, Dephasing of two interacting qubits under the influence of thermal reservoirs, Phys. Rev. A 80 (2009) 032114. [17] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Dynamics of entanglement and Bell-nonlocality for the interacting of coupled superconductor qubits and common environment YingHua Ji a,b,∗ a b
Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China Key Laboratory of Photoelectronics & Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, China
a r t i c l e
i n f o
Article history: Received 12 January 2012 Accepted 30 May 2012
PACS: 03.65.Yz 03.65.Ud Keywords: Coupled qubits Non-Markovian process Concurrence Bell inequality
a b s t r a c t Utilizing the concurrence and the Bell inequality, we investigated in detail the evolution of entangled decoherence of two superconductor qubits in non-Markovian process. The results show that the evolution of entanglement dynamics can cause entanglement sudden death (ESD) and entanglement sudden birth (ESB) because of the interaction between the environment and the quantum qubits. The concurrence of a quantum system is not only related with the environmental dissipation effects, the interaction between the qubits, and the initial quantum state, but also strongly related with the coupling symmetry between qubit-environment. Under certain initial states, the effect of environmental dissipation can be avoided completely by controlling the coupling model between qubits and environments. The interaction between qubits can prolong the survival time of the entanglement, and it is the dynamic for reviving the entanglement of quantum system. The further research results indicate that, the non-Markovian memory effects of the environment are helpful to prolong the survival time of the entanglement or to prevent the outbreak of ESD. © 2012 Elsevier GmbH. All rights reserved.
1. Introduction Quantum entanglement is a nonlocal quantum correlation and is the perfect reflection of nonlocality in quantum mechanics. Quantum entanglement is not only one most difficult conception to be understood and accepted, but also the physical foundation in constructing the future quantum information science. As a physical resource, entanglement plays an important role in much quantum information, such as quantum teleportation, quantum key distribution, quantum computation, etc. [1–3] Keeping quantum entanglement of the system is the key measure in quantum information management. In the real physical world, however, the pure entanglement can degenerate the mixed state under the effect of environmental docoherence. It will bring mistakes in quantum computation and information distortion in quantum correspondence. The phenomenon, that the quantum systems fully getting rid of entanglement in a limited time due to the effect of environmental dissipation, is known as entanglement sudden death (ESD) [4–7]. In order to ensure the realization of correct calculation of quantum logic and quantum computing, people have done a lot of researches about avoiding entanglement sudden death and proposed new plans to achieve entanglement in some cases. But in
∗ Corresponding author. E-mail address: [email protected] 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.05.024
an open quantum system; there are still a lot of problems to be solved about the disentanglement. In a sense, it is one of the key technologies to implement the quantum information. Thus, searching for stable entanglement is an important issue for quantum information study. For the moment, the famous entanglement that are investigated relatively thorough and always experimentally realized in some physical systems are optical parametric transition, nuclear magnetic resonance system (NMR), thermal entanglement, atom system, cavity QED, quantum dot and superconducting Josephson junction. Indeed, The superconducting quantum circuit is now regarded as one of the most promising physical programs to test the principles of quantum mechanics and achieve scalable quantum computation at the macro scale. It distinguishes for the following features: (1) the level structure of the superconducting quantum circuit can be customized through the circuit design or regulated by the applied electromagnetic signals; (2) in superconducting quantum circuits, one can achieve the quantum coherent superposition of a large number of electronic joint degrees of freedom; (3) superconducting quantum circuit can simulate the quantum behavior of other physical systems (such as cavity quantum electrodynamics); and (4) based on the current microelectronics manufacturing processes, superconducting quantum circuit has good scalability. Based on these advantages cannot be replaced by other qubit system, superconducting quantum circuits has been developed rapidly in recent years. People have made important progress
Y. Ji / Optik 124 (2013) 1952–1956
in experiments during the operation of superconducting qubits, superposition of quantum states, coherent evolution and oscillation, the gate operation fidelity and quantum state read. Especially recently, the quantum control of superconducting quantum bit circuit has made impressive progress: people achieved the coherent entanglement manipulation of the solid quantum system including three quantum bits for the first time, which laid a solid foundation for the realization of quantum error correction [8–11]. The quantum nonlocality is also an important physical resource. This kind of nonlocality can be characterized by Bell inequality; Bell inequality has many generalized forms. The common generalized form is CHSH-type Bell inequality in the Two-body quantum system. The CHSH-type Bell inequality is a good indicator of nonlocal correlations and the relation with the entanglement has been already known. Bellomo et al., found that Bell inequality might not be violated for a state with high entanglement for a two-qubit system subject to amplitude damping [12,13]. So far, although people have done many researches on entanglement features of different quantum systems and nonlocal features, there are still many unsolved problems especially for superconducting qubits system whose memory effects of the environment influence from the dynamic evolution is sensitive. In the real physical world, the systems are referred as non-Markovian open quantum systems, where the non-Markovian dynamics is characterized by the existence of a memory time scale so that the energy or information would transfer from the system to the environment then feeds back into the system [14,15]. Therefore, it is interesting to study the impact of the coupling model between the qubits and environment on the entanglement dynamics in non-Markov process. As one essential model of describing quantum dissipation system, spin bose model has an important effect on describing broken tunneling in solids, quantum tunneling between the magnetic flux in superconducting quantum interference devices (SQUIDs), electron tunneling between quantum spots, and electron transfer in chemistry and biology reaction. It can be easily used to describe the dynamic features of qubits system. In this paper we investigate the entanglement features and nonlocality features of coupling superconducting flux qubits in non-Markovian environment based on the spin bose model. We will show that the interactions between the qubits and the qubit-environments significantly affect the decay of the entanglement. We find that, under certain initial states, we can completely avoid the environmental dissipation effects by controlling the coupling model between the qubits and environment, which guarantees the stable entanglement degree during the process of dynamic evolution.
1953
assume that the correlative function of the environment decays exponentially with time t, i.e. [16] R()R(0)R = 2 e−t/ ,
(4)
where is the relaxation time of the reservoir variables. ˆ qb−B between the coupled qubits The interacting Hamiltonian H system and environment is assumed as ˆ qb−B = Sˆ B, ˆ H
(5)
with Sˆ = 1 ˆ 1z + 2 ˆ 2z .
(6)
The parameter 1,2 represents the strengths of the coupling between the qubit 1 (2) and environment. For 1 = 2 , the coupling interaction is symmetric and otherwise it is asymmetric for 1 + 2 = 0. It is known that the density matrix was introduced as a way of describing a quantum open system because the state of the open system is not completely known. The time evolution of the reduced density operator , from the integro-differential quantum-Liouville equation of motion, gives us information about the dynamics of the system. In the interaction picture, we have d I 1 I (t) = [H I (t), total (t)], dt total i where ˆ0 = H ˆ qb + H ˆB + H ˆ qb−qb H I (t) ≡ eiH0 t Hqb−B e−iH0 t , H Because the capacity of the heat storage R is very large, one may assume that, during the process of dynamic evolution, quantum system has little effect on the heat storage, i.e., total (t) = (t) ⊗ B ˆ I , (0) ⊗ B ] = 0. The involving equation of dynamics for and trB [H density operator may be rewritten as d I (t) = − dt
t
dTrB ([H I (t), [H I (), I (t) ⊗ B ]]).
(7)
0
The solutions of equation obviously depend on the initial conditions. In following, we suppose that the initial density matrix is only partially coherent, but includes an arbitrary degree of nonlocal coherence. This mixed state is easily expressed in the following form: ˚ (0) =
1−b I4 + b|˚˚|, 4
(8)
2. Model
(0) =
1−b I4 + b| |, 4
(9)
In this paper, we model the Hamiltonian of a system with two coupled superconducting flux qubits. The typical Hamiltonian for this model may be described by (kB = = 1) (1)
where b is the purity of the initial states which ranges from 0 for maximally mixed states to 1 for pure states, I4 is the 4 × 4 identity matrix, |˚ = ˛|10 + ˇ|01 and | = ˛|11 + ˇ|00 the Bell-like states and the parameter ˛ is sometimes called the degree of entan-
ω (1z + 2z ), 2
(2)
3. Entanglement dynamics
Hqb−qb = k(1+ 2− + 1− 2+ ),
(3)
ˆ total = H ˆ qb + H ˆB + H ˆ qb−qb + H ˆ qb−B , H with ˆ qb = H
where k is a strength of the qubit–qubit interaction and represents ˆ B needs not to be proa external field. The bath Hamiltonian H vided explicitly; it is sufficient to know the correlation R(0)R(t) ˆ where · · · = TrB (· · · B ) denotes a trace over of the bath operator R, the bath degrees of freedom with the bath density matrix B ; we can further assume that the fluctuations are unbiased R(t) =0. In order to simplify our calculation but without loss of generality, we
glement (here we set ˛ as real number and ˇ =
1 − ˛2 .
In order to describe the dynamic evolution of quantum entanglement, we adopt the concurrence defined by Wootters to quantify the degree of entanglement for any bi-partite system. The concurrence C(t) is defined as [17] ˜ C(t) = max{0, C(t)}, with ˜ C(t) =
1 −
2 −
(10)
3 −
4
1954
Y. Ji / Optik 124 (2013) 1952–1956
0.5
where j are the eigenvalues of W = (t)(t) ˜ in nonincreasing order (2) ⊗ y ).
C˜ ˚S (t) = 2
˛2 ˇ2 b2 +
1 2 1−b 2 (˛ − ˇ2 ) b2 sin2 (2kt) − 4 2
(11)
From Eq. (11) we are very easy to find the environmental dissipation has no effect on entanglement features of system and the system evolution of entanglement dynamics is related with the interaction between the initial state and the qubits. If the qubits system wants to completely avoid the ESD phenomenon, the parameter of initial state must satisfy (4˛ˇ + 1)b > 1. Especially, √ when ˛ = ˇ = 1/ 2, the system is in the Bell state (b = 1) or the mixed entangled state (b < 1) which is produced by the Bell state, the entanglement degree of the coupling qubits system keep a constant under the symmetrical coupling. On the contrary, if the coupling qubits system is initially in state (0) and when a completely asymmetrical coupling is used between the qubits and environment, the concurrence is C AS = max{0, 2˛ˇb − 0.5(1 − b)}
(12)
The environmental dissipation has no effect on entanglement features of system and the entanglement features completely depend on the initial state. At this moment, only when the parameter of initial state meets Eq. (9), the concurrence can keep a constant C˜ AS (t) = 2˛ˇb − 0.5(1 − b) and never occur ESD under the completely asymmetrical coupling. From above discussions, we can avoid the environmental dissipation effects by controlling the coupling model between qubits and environment under certain initial state ˚ (0) or (0), which can make the degree of entanglement keeps invariable with time evolution and will not occurs the ESD phenomenon. Subsequently, it is beneficial to the transmission of quantum information. Fig. 1 shows the system entanglement dynamic evolution with time from initial state ˚ (0) when a symmetrical coupling is used between the qubits and environment. The curve is equal amplitude oscillation and changes according to the sine regulation. By analyzing the concurrence C˜ ˚ (t) in Eq. (11) carefully it is not difficult to find, if a symmetrical coupling is used between the qubits and environment and the initial state meets C˜ ˚ (0) > 0, the entanglement always have C˜ ˚ (t) > 0 and would not occur ESD phenomenon during the process of the system dynamic evolve with time from initial state ˚ (0). Fig. 1(1) just describes the physical progress. We can see from Eq. (12), when the quantum system dynamic evolves with time from initial state (0), once ESD phenomenon appears, ESB would not appear because C˜ (t) decreases monotonously with time and has nothing to do with the interaction between qubits. On the contrary, when the system is initially in state ˚ (0), the concurrence C˜ ˚ (t) is closely related with the interaction between qubits. We can see from Fig. 2, the qubits system do not get entangled even if C˜ ˚ (0) ≤ 0 at the initial time. With time goes on, it will appear ESD and entanglement revives periodically depending on the interaction between qubits under certain conditions. This indicates that the interaction between qubits is the dynamic of entanglement revive indeed.
Concurrence
1-1
0.4 0.3 0.2 0.1 0.0
0
2
4
6
8
10
t/µ 0.5
1-2
0.4
Concurrence
˜ = If the sysby magnitude with (t) tem is initially in state ˚ (0) and the coupling between the qubits and environment is a symmetrical coupling, or 1 = 2 , the concurrence is
0.3 0.2 0.1 0.0
0
2
4
6
8
10
t/µ Fig. 1. Concurrence as a function of t/, when two-qubits initially in the ˚ (0) and the symmetrical √ √ coupling is used between qubits and environment (u = v = 0), where ˛ = 0.9, ˇ = 0.1, k = 1. Here panel (1) b = 06, panel (2) b = 0.45.
1.0
2-1
0.8
Concurrence
(2) (1) ⊗ y )∗ (t)(y
0.6 (c)
0.4
(b) (a)
0.2 0.0
0
2
4
t/µ
6
8
1.0
10
2-2
0.8
Concurrence
(1) (y
0.6 0.4
(c) (b)
0.2
(a)
4. The CHSH-type Bell inequality The quantum nonlocality is also an important physical resource. This kind of nonlocality can be characterized by Bell inequality; Bell inequality has many generalized forms. The common generalized form is CHSH-type Bell inequality in the Two-body quantum system. The CHSH-type Bell inequality is a good indicator of
0.0
0
2
4
t/µ
6
8
10
Fig. 2. Concurrence as a function of t/ for different rates m, when two-qubits initially in the ˚ (0), where (a) m = 1, (b) m = 0.5, (c) m = 0.1. b = 0.6, ˛ = 1/2, ˇ =
3/2.
Here panel (1) corresponds to non-Markovian dynamics and panel (2) to Markovian dynamics.
Y. Ji / Optik 124 (2013) 1952–1956
nonlocal correlations and the relation with the entanglement has been already known. It showed that the maximum of the Bell function B for a X-structured density matrix (8) can be given as
0.6
Bmax = 2
0.2
P2 + Q 2,
(13)
where
1955
3-1
0.4
(b)
0.0
P = 11 + 44 − 22 − 33 , Q = 2(|14 (t)| + |23 (t)|). For the initial state ˚ (0), the maximum of Bell function is given by
Bmax
˚ (t)
= 2b
1 + 4˛2 ˇ2 e−4xt + 4(˛2 − ˇ2 )
2
k + y 2 ˝
-0.2
-0.6
f (t), (14)
where
0
2
4
t/µ
6
8
1.0
f (t) = e−2xt sin h2 (˝t),
10
3-2
(15) 0.8
2 (1 − 2 )2 [1 − e−t/ cos(2kt) + e−t/ sin(2kt)], x= 2(1 + 4k2 2 ) y=
(a)
-0.4
(b)
(16) 0.6
2 (1 − 2 )2 [2k − 2ke−t/ cos(2kt) − e−t/ sin(2kt), 2(1 + 4k2 2 )
(a)
0.4
(17) 0.2
˝=
x2 − 4k(k + y).
(18)
And for the initial state (0), the maximum of Bell function is
Bmax
(t)
=2
b2
+ 4˛2 ˇ2 b2
exp
2(1 + 2 )2 − (1 − e−t/ )t . Tph (19)
After comparing Bmax ˚ (t) with C˜ ˚ and Bmax (t) with C˜ under the same parameters, we can find, because of the environmental decoherence effects, the survival time for entanglement should be much longer than the Bell inequality violation, which agrees with the research results of the literature. Based on above comparing results about the influence on the entanglement dynamic process from the non-Markovian and Markovian, we can further know that non-Markovian effect is not helpful to the violation of the CHSH inequality to judge nonlocal entanglement features. When the symmetrical coupling model is used between the qubits system in initial state ˚ (0) and environment, the dynamics of concurrence and Bell nonlocality as a function of the time are displayed in Fig. 3. All the curves are equal amplitude oscillation and changes according to the sine regulation. Fig. 3(1) shows explicitly that when C˜ > 0, the CHSH-type Bell inequality is not violated (Bmax ≤ 2). In these regions it is therefore not possible to say with certainty that quantum correlations are not reproducible by a classical local model. Fig. 3 indicates that under certain initial state, when the symmetrical model is used between the quantum system and environment, the concurrence C˜ and the Maximum of Bell function Bmax have the same regular evolution with time. When the initial state of the quantum system is in (0) and the completely asymmetrical coupling model is used between the system and environment, the maximum of Bell function is
B AS = 2
b2 + 4˛2 ˇ2 b2
√ From above equation, we can see that, when ˛ = ˇ and b ≥ 1/ 2, it always violates the inequality. On the contrary, it always does not violate the inequality.
0.0
0
2
4
t/µ
6
8
10
Fig. 3. Concurrence and Maximum of Bell function, Bmax − 2 as a function of t/, when two-qubits initially in the ˚ (0) and the symmetrical coupling is used between qubits and environment, where ˛ = 1/2, ˇ =
3/2, 2 2 = 0.25, k = 1,
|1 | = |2 |=0.5. Here panel (1), b = 0.6, not violate the Bell inequality. Panel (2), b = 0.9, violate the Bell inequality.
5. Conclusions In this paper, utilizing the concurrence and the Bell inequality, we investigate in detail the evolution of entangled decoherence of two superconductor qubits in non-Markovian process. The results show that under certain initial states, we can completely avoid the environmental dissipation effects by controlling the coupling model between qubits and environments, which guarantees the stable entanglement degree and avoids the outbreak of ESD during the process of dynamic evolution. The further research results indicate that, the non-Markovian memory effects of the environment are helpful to prolong the survival time of the entanglement or to prevent the outbreak of ESD. Acknowledgement This work is supported by the National Natural Science Foundation of China under Grant No. 11164009. References [1] S. Etaki, M. Poot, I. Mahboob, K. Onomitsu, H. Yamaguchi, H.S.J. van der Zant, Motion detection of a micromechanical resonator embedded in a d.c. SQUID, Nat. Phys. 4 (2008) 785. [2] Y. Yu, S.L. Zhu, G.Z. Sun, X.D. Wei, N. Dong, J. Chen, P.H. Wu, Quantum jumps between macroscopic quantum states of a superconducting qubit coupled to a microscopic two-level system, Phys. Rev. Lett. 101 (2008) 157001. [3] T. Yu, J.H. Eberly, Sudden death of entanglement, Science 323 (2009) 598. [4] Y.H. Ji, J.J. Hu, Entanglement and decoherence of coupled superconductor qubits in a non-Markovian environment, Chin. Phys. B 19 (2010) 060304.
1956
Y. Ji / Optik 124 (2013) 1952–1956
[5] W. Cui, Z.R. Xi, Y. Pan, Optimal decoherence control in non-Markovian open dissipative quantum systems, Phys. Rev. A 77 (2008) 032117. [6] Y.H. Ji, Y.Y. Jiang, Investigation of decoherence of superconducting flux qubit entangled with environment, Mod. Phys. Lett. B 22 (2008) 581. [7] S.K. Wang, J.S. Jin, X.Q. Li, Continuous weak measurement and feedback control of a solid-state charge qubit: a physical unravelling of non-Lindblad master equation, Phys. Rev. B 75 (2007) 155304. [8] Y.H. Ji, H.F. Lai, S.H. Cai, Z.S. Wang, Controlled decoherence of three-level rfSQUID qubit with asymmetric potential, Chin. Phys. B 19 (2010) 030310. [9] H. Bartolf, A. Engel, A. Schilling, K. ll’in, M. Siegel, H.-W. Hubers, A. Semenov, Current-assisted thermally activated flux liberation in ultrathin nanopatterned NbN superconducting meander structures, Phys. Rev. B 81 (2010) 024520. [10] S. Gustavsson, J. Bylander, F. Yan, D. William, F. Oliver, Y. Yoshihara, Nakamura, Noise correlations in flux qubit with tunable tunnel coupling, Phys. Rev. B 84 (2011) 014525. [11] I. Kamleitner, P. solinas, C. Muller, A. Ahnirman, M. Mottonen, Geometric quantum gates with superconducting qubits, Phys. Rev. B 83 (2011) 214518.
[12] B. Bellomo, R. Lo Franco, G. Compagno, Entanglement dynamics of two independent qubits in environments with and without memory, Phys. Rev. A 77 (2008) 032342. [13] L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B.M. Garraway, Sudden death and sudden birth of entanglement in common structured reservoirs, Phys. Rev. A 79 (2009) 042302. [14] Y.H. Ji, Y.M. Liu, Z.S. Wang, Avoiding the decay of entanglement for coupling two-qubit system interacting with a non-Markov environment, Chin. Phys. B 20 (2011) 070304. [15] X.Y. Zhao, J. Jing, B. Corn, T. Yu, Dynamics of interacting qubits coupled to a common bath: Non-Markovian quantum-state-diffusion approach, Phys. Rev. A 84 (2011) 032101. [16] M. Ban, Dephasing of two interacting qubits under the influence of thermal reservoirs, Phys. Rev. A 80 (2009) 032114. [17] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245.