Decoherence Effect in An Anisotropic Two-Qubit Heisenberg XYZ Model with Inhomogeneous Magnetic Field

Commun. Theor. Phys. (Beijing, China) 53 (010) pp. 1053 1058 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 6, June 15, 010 Decoherence Effect in An Anisotropic Two-Qubit Heisenberg XYZ Model with Inhomogeneous Magnetic Field CHEN Tao (í ), SHAN Chuan-Jia ( ), LI Jin-Xing (Ó ), LIU Ji-Bing ( ÍÏ), LIU Tang-Kun ( ²), and HUANG Yan-Xia (á ) College of Physics and Electronic Science, Hubei Normal University, Huangshi 43500, China (Received August 3, 009; revised manuscript received October 30, 009) Abstract Taking the decoherence effect into account, the entanglement evolution of a two-qubit anisotropic Heisenberg XYZ chain in the presence of inhomogeneous magnetic field is investigated. The time evolution of concurrence is studied for the initial state cos θ 01 + sin θ 10 at zero temperature. The influences of inhomogeneous magnetic field, anisotropic parameter and decoherence on entanglement dynamic are addressed in detail, and a concurrence formula of the steady state is found. It is shown that the entanglement sudden death (ESD) and entanglement sudden birth (ESB) appear with the decoherence effect, and the stable concurrence depends on the uniform magnetic field B, anisotropic parameter and environment coupling strength γ, which is independent of different initial states and nonuniform magnetic field b. PACS numbers: 03.65.Yz, 03.67.Mn, 75.10.Jm Key words: Heisenberg model, thermal entanglement, concurrence, decoherence, inhomogeneous magnetic field 1 Introduction Quantum entanglement is a property of correlated quantum systems, and plays an important role in quantum communication and information processing, such as quantum dense coding, [1] quantum teleportation, [] quantum key distribution, and quantum computation. [3 4] It has been extensively studied for various systems including cavity-qed, [5] Ising model, [6] spin chains. [7 8] In particular, Heisenberg spin chain is the simplest spin chain in solid state systems, which is regarded as one of the natural candidates to construct a quantum computer and quantum dots. [9] The Heisenberg exchange interaction between two quantum dots can be used to implement universal one or two-qubit quantum gates. [10] In the real world, the interaction between quantum system and its environment leads to decoherence, which will destroy quantum entanglement and thus ruin the encoded quantum information. It is reported that entanglement may decrease abrupt to zero in a finite time due to the influence of quantum noise, this was termed as entanglement sudden death (ESD). [11 1] The phenomenon of ESD has motivated numerous theoretical investigation in bipartite systems. [13 17] It is found that entanglement control even can lead to entanglement sudden birth (ESB), which is the creation or rebirth of entanglement where the initially unentangled qubits can be entangled after a finite evolution time. [18 0] Although the environmental induced effect is not desired in most cases, it has been shown that entanglement between two or more subsystems may be induced by their collective interaction with a common environment. [1] A stable entanglement, in which once qubits become entangled they will never be disentangled, was also demonstrated. [ 3] On the other hand, entanglement exists naturally in the Heisenberg spin system due to the spin-spin exchange interaction at zero temperature, and the external magnetic field can affect the entanglement of the quantum system. So it is necessary to include external magnetic field in the studies of entanglement evolution with decoherence. In this paper, we present an exact calculation of the entanglement dynamic between two qubits coupling with a common environment at zero temperature. The two qubits interact via a Heisenberg XYZ interaction and are subject to an external inhomogeneous magnetic field. We examine the evolution dynamic of different initial quantum states including the unentangled state and the maximally entangled states, and find that inhomogeneous magnetic field has a notable effect on the time evolution of concurrence, and the stable concurrence depends on the uniform magnetic field B, anisotropic parameter, and decay rate of qubits γ, which is independent of different initial states and nonuniform magnetic field b. Heisenberg XYZ Model with Inhomogeneous Magnetic Field We note that the Hamiltonian of an anisotropic twoqubit Heisenberg XYZ chain with inhomogeneous magnetic field is Supported by National Natural Science Foundation of China under Grant No. 10904033, Natural Science Foundation of Hubei Province under Grant No. 009CDA145, Educational Commission of Hubei Province under Grant No. D00904 and Natural Science Foundation of Hubei Normal University under Grant No. 007D1 E-mail: taochen46@163.com E-mail: scj11@163.com

1054 CHEN Tao, SHAN Chuan-Jia, LI Jin-Xing, LIU Ji-Bing, LIU Tang-Kun, and HUANG Yan-Xia Vol. 53 H = 1 [J xσ 1x σ x + J y σ 1y σ y + J z σ 1z σ z + (B + b)σ 1z + (B b)σ z ] = J(σ 1+ σ + σ 1 σ + ) + (σ 1+ σ + + σ 1 σ ) + J z σ 1zσ z + B (σ 1z + σ z ) + b (σ 1z σ z ), (1) where J i (i = x, y, z) are the spin-spin coupling coefficients, J i > 0 corresponds to the antiferromagnetic case, and J i < 0 corresponds to the ferromagnetic case. B is the z-axis uniform external magnetic field, b is the z-axis nonuniform external magnetic field, and σ i (i = x, y, z) are the Pauli matrices. J = (J x + J y )/, = (J x J y )/, and σ ± = (1/)(σ x ± iσ y ) are the spin raising and lowering operators. In the standard basis { 11, 10, 01, 00 }, the eigenvalues and eigenvectors can be derived as ( H Ψ ± = J ) z ± v Ψ ±, ( Jz ) H Φ ± = ± u Φ ±, () where the eigenstates Ψ ± = Φ ± = ( ±v + b v ± bv 10 + ( ±u + B u ± Bu 11 + and v = b + J, u = B +. J ) v ± bv 01, ) u ± Bu 00, (3) 3 Entanglement Dynamic at Zero Temperature The description of a time evolution open system at zero temperature is provided by the master equation, which can be written most generally in the Lindblad form with the assumption of weak system-reservoir coupling and Born Markov approximation. [4] The Lindblad equation for our case is given as follows dρ = i[h, ρ] + γ j [σ j ρσ j+ 1 ] {σ j+σ j, ρ}, (4) j=1, where the terms γ 1 (γ ) are the decay rate of qubit 1() to the environment, and here we consider γ 1 = γ = γ, {} means anticommutator. ρ is the time-varying density matrix. For our Hamiltonian, ρ can be represented as ρ = ρ 11 0 0 ρ 14 0 ρ ρ 3 0 0 ρ 3 ρ 33 0 ρ 41 0 0 ρ 44. (5) Combining the Eqs. (1), (4), and (5), we can obtain the first-order differential equations of the density matrix elements in the standard basis { 11, 10, 01, 00 }: [5] dρ 11 (t) = γρ 11 (t) i [ρ 41 (t) ρ 14 (t)], dρ 14 (t) = γρ 14 (t) i[ ρ 44 (t) ρ 11 (t) + Bρ 14 (t)], dρ (t) = γ[ρ 11 (t) ρ (t)] ij[ρ 3 (t) ρ 3 (t)], dρ 3 (t) = i[jρ 33 (t) Jρ (t) + bρ 3 (t)] γρ 3 (t), dρ 33 (t) = γ[ρ 11 (t) ρ 33 (t)] ij[ρ 3 (t) ρ 3 (t)], dρ 44 (t) = γ[ρ (t) + ρ 33 (t)] i [ρ 14 (t) ρ 41 (t)], (6) where ρ 41 (t) = ρ 14 (t), ρ 3(t) = ρ 3 (t). The solution of Eqs. (6) depends on the initial state of the two qubits, and we assume that the initial state of the system is in general form cos θ 01 + sin θ 10. On solving the master equation in Eqs. (6), we obtain the time evolution of the reduced density matrix elements in the standard basis { 11, 10, 01, 00 } as ρ 11 (t) = u(4u + γ ) [u ue γt γe γt sinh(ut)], ρ 14 (t) = 4u 3 (4u + γ ) [γe γt sinh(ut)( B 3 iγu Bγ ) e γt Bγ u 8e γt Bu 3 + e γt cosh(ut)(bγ u iγu 3 ) + 8Bu 3 + 4iγu 3 ], 1 ρ (t) = 4u v (4u + γ ) {v [8e γt B 4 + (e γt γ cosh(ut) + 4 e γt + 4 ) + e γt B (γ + 4 ) + 4e γt B + 4B ] + u (4u + γ )[ e γt b cos(θ) e γt J cosh(ut)cos(θ) e γt bj(e 4ut e ut + 1)sin(θ)]}, 1 ρ 33 (t) = 4u v (4u + γ ) {v [8e γt B 4 + (e γt γ cosh(ut) + 4 e γt + 4 ) + e γt B (γ + 4 ) + 4e γt B + 4B ] + u (4u + γ )[e γt b cos(θ) + e γt J cosh(ut)cos(θ) + e γt bj(e 4ut e ut + 1)sin(θ)]}, ρ 3 (t) = 1 4v 3 {J cos(θ)[ ie γt v sinh(vt) + e γt bv(cosh(vt) 1)]

No. 6 Decoherence Effect in An Anisotropic Two-Qubit Heisenberg XYZ Model with Inhomogeneous Magnetic Field 1055 + sin(θ)[e γt b v cosh(vt) + e γt J v ie γt bv sinh(vt)]}, 1 ρ 44 (t) = u 3 (4u + γ ) {u[8b4 (1 e γt ) (e γt γ cosh(ut) + e γt (γ + ))] + e γt γ (B + )sinh(ut) }, ρ 41 (t) = ρ 14(t), ρ 3 (t) = ρ 3(t). (7) From Eq. (7), we know that θ should vary between 0 and π/, and the coupling coefficient J z can not influence the time evolution of concurrence. In order to investigate the effect of interaction among the two qubits on decoherence we need to study the dynamic of two-qubit entanglement. Since decoherence process leads the pure quantum system states to mixed states, we use the concurrence as a measure of entanglement. [6 7] For a system described by the density matrix ρ, the concurrence is C = max( λ 1 λ λ 3 λ 4, 0), (8) where λ 1, λ, λ 3, and λ 4 are the eigenvalues in a decreasing order of the density operator R defined by R = ρ(σ 1y σ y )ρ (σ 1y σ y ), denotes the complex conjugate, σ y is the usual Pauli matrix. Then the concurrence of density matrix ρ in Eq. (5) can be expressed as C = max[( ρ 3 ρ 3 ρ 11 ρ 44 ), ( ρ 14 ρ 41 ρ ρ 33 ), 0]. (9) The concurrence ranges from 0 to 1, C = 0 and C = 1 indicate the vanishing entanglement and the maximal entanglement respectively. In the following, we investigate the entanglement dynamic under different system parameters, such as the anisotropic parameter, uniform magnetic field B, nonuniform magnetic field b, coefficient J, and environment coupling strength γ, for several different initial cases: the unentangled state (θ = 0), non-maximally entangled state (θ = π/8, θ = 3π/8), and maximally entangled state (θ = π/4). If there is no decoherence, the entanglement evolves periodically with the nonuniform magnetic field b, and the uniform magnetic field B has no effect in the entanglement evolution. However, it is contrary when we take the initial sate cosθ 00 +sin θ 11 into account. [8 9] Fig. 1 Time evolution of the concurrence is plotted as a function of b and t with γ = 0, = 0., J = 1, and θ = π/4. Fig. Time evolution of the concurrence for various values of the uniform magnetic field parameter B with γ = 0.5, = 0., J = 1, and b = 0.5 for different initial state: (a) θ = 0; (b) θ = π/8; (c) θ = π/4; (d) θ = 3π/8.

1056 CHEN Tao, SHAN Chuan-Jia, LI Jin-Xing, LIU Ji-Bing, LIU Tang-Kun, and HUANG Yan-Xia Vol. 53 Fig. 3 Time evolution of the concurrence for various values of the nonuniform magnetic field parameter b with γ = 0.5, = 0., J = 1, and B = 1 for different initial states: (a) θ = 0; (b) θ = π/8; (c) θ = π/4; (d) θ = 3π/8. Fig. 4 Time evolution of the concurrence for various values of the coefficient J with γ = 0.5, = 0., B = 1, and b = 0.5 for different initial states: (a) θ = 0; (b) θ = π/8; (c) θ = π/4; (d) θ = 3π/8. In Fig. 1, the time evolution of concurrence is plotted as a function of b and time t without the decoherence when the qubits are initially in the maximally entangled state. One can find that the entanglement evolves periodically, and the amplitude and frequency change with the varied b. When the nonuniform magnetic field b = 0, the initial maximally entangled state does not evolve. With the decoherence, Figs., 3, 4, and 5 illustrate the time evolution of concurrence for the different uniform magnetic field B, nonuniform magnetic field b, coefficient J,anisotropic parameter respectively. From Figs. (a), 3(a), 4(a), and 5(a), we can find that the concurrence of the initial unentangled state increases from zero to a certain value with increasing time, the ESB appears. Then the entanglement may decrease to zero, the ESB can occur again. Finally, the quantum state evolves into a stable entangled state from initial unentangled state. That is, decoherence

No. 6 Decoherence Effect in An Anisotropic Two-Qubit Heisenberg XYZ Model with Inhomogeneous Magnetic Field 1057 can drive the two qubits into a stable entangled state instead of completely destroying the entanglement. Therefore we can manipulate and control the entanglement by applying the uniform magnetic field B. Figures (c), 3(c), 4(c), and 5(c) show that the maximally entanglement of two qubits can fall abruptly to zero with increasing time, this is the ESD and it occurs more rapidly with the increasing nonuniform magnetic field b. From Figs., 3, 4, and 5, we see that the stable concurrences are different with different values of the uniform magnetic field B and anisotropic parameter, but the different values of the nonuniform magnetic field b and coefficient J just generate the same stable concurrence, and this has no relation to the initial states. According to the Eq. (7), the corresponding steady concurrence is as follow: ( 4B + γ C steady = max ) 4(B + ) + γ, 0. (10) The stable concurrence depends on the uniform magnetic field parameter B, anisotropic parameter and decay rate of qubits γ, does not lie on the nonuniform magnetic field b and coefficient J. If = 0, the stable concurrence is zero all the time. Fig. 5 Time evolution of the concurrence for various values of the anisotropic parameter with γ = 0.5, B = 1, J = 1, and b = 0.5 for different initial states: (a) θ = 0; (b) θ = π/8; (c) θ = π/4; (d) θ = 3π/8. γ = 0.5, = 0., J = 1, B = 0.5, and b = 0.5 for different initial states, it is observed that the steady concurrence is fixed with the given system parameters. Fig. 6 Time evolution of the concurrence with γ = 0.5, = 0., J = 1, B = 0.5, and b = 0.5 for different initial states. Figure 6 shows the time evolution of concurrence with 4 Conclusions In this paper, we investigate the entanglement dynamic of a two-qubit anisotropic Heisenberg XYZ chain in the presence of inhomogeneous magnetic field for different initial system states. The system papameters, such as inhomogeneous magnetic field, anisotropic parameter, decay rate of qubits to the environment, can influence the entanglement evolution. It is shown that the ESD and ESB appear under the spin-spin interaction, external magnetic field and decoherence, and by controlling the values of the uniform magnetic field, and anisotropic parameter, we can obtain a certain stable concurrence.

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