Entanglement dynamics of one-dimensional driven spin systems in time-varying magnetic fields
aa r X i v : . [ qu a n t - ph ] A p r Entanglement dynamics of one-dimensional driven spin systemsin time-varying magnetic fields
Bedoor Alkurtass , Gehad Sadiek , , ∗ , Sabre Kais Department of Physics,King Saud University,Riyadh 11451, Saudi Arabia Department of Physics,Ain Shams University,Cairo 11566, Egypt Department of Chemistry and Birck Nanotechnology center,Purdue University, West Lafayette,Indiana 47907, USA ∗ Corresponding author: [email protected] bstract We study the dynamics of entanglement for a one-dimensional spin chain with a nearest neigh-bor time-dependent Heisenberg coupling J ( t ) between the spins in presence of a time-dependentexternal magnetic field h ( t ) at zero and finite temperatures. We consider different forms of timedependence for the coupling and magnetic field; exponential, hyperbolic and periodic. We exam-ined the system size effect on the entanglement asymptotic value. It was found that for a smallsystem size the entanglement starts to fluctuate within a short period of time after applying thetime dependent coupling. The period of time increases as the system size increases and disappearscompletely as the size goes to infinity. We also found that when J ( t ) is periodic the entanglementshows a periodic behavior with the same period, which disappears upon applying periodic magneticfield with the same frequency. Solving the particular case where J(t) and h(t) are proportional ex-actly, we showed that the asymptotic value of entanglement depends only on the initial conditionsregardless of the form of J ( t ) and h ( t ) applied at t > PACS numbers: 03.67.Mn, 03.65.Ud, 75.10.Jm . INTRODUCTION Quantum entanglement represents one of the corner stones of the quantum mechanicstheory with no classical analog [1]. Quantum entanglement is a nonlocal correlation be-tween two (or more) quantum systems such that the description of their states has to bedone with reference to each other even if they are spatially well separated. Understand-ing and quantifying entanglement may provide an answer for many questions regarding thebehavior of the many body quantum systems. Particularly, entanglement is considered asthe physical property responsible for the long-range quantum correlations accompanying aquantum phase transition in many-body systems at zero temperature [2–4]. Entanglementplays a crucial role in many fields of modern physics, particularly, quantum teleportation,quantum cryptography and quantum computing [5, 6]. It is considered as the physical basisfor manipulating linear superpositions of quantum states to implement the different pro-posed quantum computing algorithms. Different physical systems have been proposed aspromising candidates for the future quantum computing technology [7–15]. It is a majortask in each one of these considered systems to find a controllable mechanism to form andcoherently manipulate the entanglement between a two-qubit system, creating an efficientquantum computing gate. The coherent manipulation of entangled states has been observedin different systems such as isolated trapped ions [16], superconducting junctions [17] andcoupled quantum dots where the coupling mechanism in the latter system is the Heisenbergexchange interaction between electron spins [18–20]. One of the most interesting proposalsfor creating a controllable mechanisms in coupled quantum dot systems was introduced byD. Loss et al. [21, 22]. The coupling mechanism is a time-dependent exchange interactionbetween the two valence spins on a doubled quantum dot system, which can be pulsed overdefinite intervals resulting a swap gate. This control can be achieved by raising and loweringthe potential barrier between the two dots through controllable gate voltage. In a previouswork, a two-atom system with time dependent coupling was studied and the critical de-pendence of the entanglement and variance squeezing on the strength and frequency of thecoupling was demonstrated [23].Quantifying entanglement in the quantum states of multiparticle systems is in the focusof interest in the field of quantum information. However, quantum entanglement is veryfragile due to the induced decoherence caused by the inevitable coupling to the environ-3ent. Decoherence is considered as one of the main obstacles toward realizing an effectivequantum computing system [24]. The main effect of decoherence is to randomize the rela-tive phases of the possible states of the considered system. Quantum error correction [25]and decoherence free subspace [26, 27] have been proposed to protect the quantum propertyduring the computation process. Nevertheless, offering a potentially ideal protection againstenvironmentally induced decoherence is a difficult task. Moreover, a spin-pair entanglementis a reasonable measure for decoherence between the considered two-spin system and its envi-ronment constituted by the rest of spins on the chain. The coupling between the system andits environment leads to decoherence in the system and sweeping out entanglement betweenthe two spins. Therefore, monitoring the entanglement dynamics in the considered systemhelps us to understand the behavior of the decoherence between the considered two spinsand their environment. Particularly, the effect of the environment size on the coherence ofquantum states of the system can be considered by watching the spin pair entanglementevolution versus the the number of sites N in the chain.Developing new experimental techniques enabled the generation and control of multipar-ticle entanglement [28–33] as well as the fabrication of one dimensional spin chains [34–36].This progress in the experimental arena sparked an intensive theoretical research over themultiparticle systems and particularly the one dimensional spin chains [37–46]. The dynam-ics of entanglement in an XY and Ising spin chains has been studied considering a constantnearest neighbor exchange interaction, in presence of a time varying magnetic field repre-sented by a step, exponential and sinusoidal functions of time [47, 48]. Furthermore, thedynamics of entanglement in a one dimensional Ising spin chain at zero temperature wasinvestigated numerically where the number of spins was seven at most [49]. The generationand transportation of the entanglement through the chain, which irradiated by a weak res-onant field under the effect of an external magnetic field were investigated. Recently, theentanglement in anisotropic XY model with a small number of spins, with a time dependentnearest neighbor coupling at zero temperature was studied too [50]. The time-dependentspin-spin coupling was represented by a dc part and a sinusoidal ac part. It was observedthat there is an entanglement resonance through the chain whenever the ac coupling fre-quency is matching the Zeeman splitting. Very recently, we have studied the time evolutionof entanglement in a one dimensional spin chain in presence of a time dependent magneticfield h ( t ) considering a time dependent coupling parameter J ( t ) where both h ( t ) and J ( t )4ere assumed to be of a step function form [51]. Solving the problem exactly, we found thatthe system undergoes a nonergodic behavior. At zero temperature we found that the asymp-totic value of the entanglement depends only on the ratio λ = J/h . However, at nonzerotemperatures it depends on the individual values of h and J . Also we have demonstratedthat the quantum effects dominate within certain regions of the temperature- λ space thatvary significantly depending on the degree of the anisotropy of the system.In this work, we investigate the time evolution of quantum entanglement in a one di-mensional XY spin chain system coupled through nearest neighbor interaction under theeffect of an external magnetic field at zero and finite temperature. We consider both time-dependent nearest neighbor Heisenberg coupling J ( t ) between the spins on the chain andmagnetic field h ( t ), where the function forms are exponential, periodic and hyperbolic intime.This paper is organized as follows. In Sec. II, we present our model and discuss thenumerical solution for the the XY spin chain for a general form of the coupling and magneticfield. Then, we present an exact solution for the system for the special case J ( t ) = λh ( t ),where λ is a constant. In Sec. III, we evaluate the entanglement using the magnetizationand the spin-spin correlation functions of the system. We present our results and discussthem in sec. IV. Finally, in Sec. V we conclude and discuss future directions. II. THE TIME DEPENDENT XY MODELA. A Numerical Solution
In this section, we present a numerical solution for the XY model of a spin chain with N sites in the presence of a time-dependent external magnetic field h ( t ). We consider a time-dependent coupling J ( t ) between the nearest neighbor spins on the chain. The Hamiltonianfor such a system is given by H = − J ( t )2 (1 + γ ) N X i =1 σ xi σ xi +1 − J ( t )2 (1 − γ ) N X i =1 σ yi σ yi +1 − N X i =1 h ( t ) σ zi , (1)where σ i ’s are the Pauli matrices and γ is the anisotropy parameter. For simplicity, we’llconsider ¯ h = 1 throughout this paper. Defining the raising and lowering operators a † i , a i a † i = 12 ( σ xi + iσ yi ) , a i = 12 ( σ xi − iσ yi ) . (2)5ollowing the standard procedure to treat the Hamiltonian (1), we introduce Fermi operators b † i , b i [52] a † i = b † i exp( iπ i − X j =1 b † j b j ) , a i = exp( − iπ i − X j =1 b † j b j ) b i , (3)then applying Fourier transformation we obtain b † i = 1 √ N N/ X p = − N/ e ijφ p c † p , b i = 1 √ N N/ X p = − N/ e − ijφ p c p . (4)where φ p = πpN . Therefore, the Hamiltonian can be written as H = N/ X p =1 ˜ H p , (5)with ˜ H p given by˜ H p = α p ( t )[ c † p c p + c †− p c − p ] + iJ ( t ) δ p [ c † p c †− p + c p c − p ] + 2 h ( t ) , (6)where α p ( t ) = − J ( t ) cos φ p − h ( t ) and δ p = 2 γ sin φ p .As [ ˜ H l , ˜ H m ] = 0 for l, m = 0 , , , . . . , N/
2, the Hamiltonian in the 2 N -dimensionalHilbert space can be decomposed into N/ {| i , c † p c †− p | i , c † p | i , c †− p | i} we ob-tain the matrix representation of ˜ H p ˜ H p = h ( t ) − iJ ( t ) δ p iJ ( t ) δ p − J ( t ) cos φ p − h ( t ) 0 00 0 − J ( t ) cos φ p
00 0 0 − J ( t ) cos φ p . (7)Initially the system is assumed to be in a thermal equilibrium state and therefore itsinitial density matrix is given by ρ p (0) = e − β ˜ H p (0) , (8)where β = 1 /kT , k is Boltzmann constant and T is the temperature.Since the Hamiltonian is decomposable we can find the density matrix at any time t , ρ p ( t ), for the p th subspace by solving Liouville equation given by i ˙ ρ p ( t ) = [ ˜ H p ( t ) , ρ p ( t )] , (9)6hich gives ρ p ( t ) = U p ( t ) ρ p (0) U † p ( t ) . (10)where U p ( t ) is time evolution matrix which can be obtained by solving the equation i ˙ U p ( t ) = U p ( t ) ˜ H p ( t ) . (11)To study the effect of a time-varying coupling parameter J ( t ) we consider the followingforms J exp ( t ) = J + ( J − J ) e − Kt , (12) J cos ( t ) = J − J cos ( Kt ) , (13) J sin ( t ) = J − J sin ( Kt ) , (14) J tanh ( t ) = J + J − J (cid:20) tanh (cid:18) K ( t −
52 ) (cid:19) + 1 (cid:21) . (15)Note that Eq. (11) gives two systems of coupled differential equations with variable co-efficients. Such systems can only be solved numerically which we adopt in this paper. B. An Exact Solution for Proportional J and h In this section we present an exact solution of the system using a general time-dependentcoupling J ( t ) and a magnetic field with the following form: J ( t ) = λ h ( t ) (16)where λ is a constant. Using Eqs. (7), (11) and (16) we obtain i ˙ u ˙ u ˙ u ˙ u = u u u u λ − iδ p iδ p − φ p − λ J ( t ) , (17)and i ˙ u = − φ p J ( t ) u , u = u . (18)Equation (17) can be rewritten as i ˙ u j = J ( t ) H ′ u j . (19)for j = 1 ,
2, where H ′ = λ iδ p − iδ p φ p − λ , u j = u j u j . (20)7ntroducing a unitary rotation matrix S = cos θ e iφ sin θ − e − iφ sin θ cos θ . (21)Using S to diagonalize H ′ we obtain SH ′ S − = λ λ . (22)Where the angles φ and θ were found to be φ = ( n + 1) π, tan 2 θ = δ p φ p + λ , (23)where n = 0 , ± , ± , . . . , thereforesin 2 θ = δ p q δ p + (2 cos φ p + λ ) , cos 2 θ = 2 cos φ p + λ q δ p + (2 cos φ p + λ ) . (24)Finding λ and λ we get λ = s δ p + (2 cos φ p + 2 λ ) − φ p , λ = − s δ p + (2 cos φ p + 2 λ ) − φ p . (25)Now we define v j = Su j and substitute in eq. (19) we get i ˙ v j = (cid:16) SH ′ S − + i ˙ SS − (cid:17) v j . (26)Hence i ˙ v j = λ λ v j . (27)Solving this equation we obtain v = cos θ e − iλ R t J ( t ′ ) dt ′ i sin θ e − iλ R t J ( t ′ ) dt ′ , v = i sin θ e − iλ R t J ( t ′ ) dt ′ cos θ e − iλ R t J ( t ′ ) dt ′ . (28)Finally u is given by u = cos θe − iλ R t J ( t ′ ) dt ′ + sin θe − iλ R t J ( t ′ ) dt ′ , (29) u = − i sin θ cos θ (cid:26) e − iλ R t J ( t ′ ) dt ′ − e − iλ R t J ( t ′ ) dt ′ (cid:27) , (30) u = − u , (31)8 = sin θe − iλ R t J ( t ′ ) dt ′ + cos θe − iλ R t J ( t ′ ) dt ′ , (32) u = u = e i cos φ p R t J ( t ′ ) dt ′ , (33)where sin θ = vuuut q δ p + (2 cos φ p + λ ) − (2 cos φ p + λ )2 q δ p + (2 cos φ p + λ ) , (34)cos θ = vuuut q δ p + (2 cos φ p + λ ) + (2 cos φ p + λ )2 q δ p + (2 cos φ p + λ ) . (35) III. SPIN CORRELATION FUNCTIONS AND ENTANGLEMENT EVALUA-TION
In this section we evaluate different magnetization and the spin-spin correlation functionsof the XY model, then we evaluate the entanglement in the system. The magnetization inthe z -direction is defined as M = 1 N N X j =1 ( S zj ) = 1 N /N X p =1 M p , (36)where M p = c † p c p + c †− p c − p −
1. In terms of the density matrix, it is given by h M z i = T r [ M ρ ( t )] T r [ ρ ( t )] = 1 N /N X p =1 T r [ M p ρ p ( t )] T r [ ρ p ( t )] . (37)The spin correlation functions are defined by S xl,m = h S xl S xm i , S yl,m = h S yl S ym i , S zl,m = h S zl S zm i , (38)which can be written in terms of the fermionic operators as follows [52]: S xl,m = 14 h B l A l +1 B l +1 . . . A m − B m − A m i , (39) S yl,m = ( − l − m h A l B l +1 A l +1 . . . B m − A m − B m i , (40) S zl,m = 14 h A l B l A m B m i , (41)9here A i = b † i + b i , B i = b † i − b i . (42)Using Wick Theorem [53], the expressions (39)-(41) can be evaluated as pfaffians of the form S xl,m = 14 pf F l,l +1 G l,l +1 · · · G l,m − F l,m P l +1 ,l +1 · · · P l +1 ,m − Q l +1 ,m · · · . .P m − ,m − Q m − ,m F m − ,m , (43) S yl,m = ( − l − m pf P l,l +1 Q l,l +1 · · · Q l,m − P l,m F l +1 ,l +1 · · · F l +1 ,m − G l +1 ,m · · · . .F m − ,m − G m − ,m P m − ,m , (44) S zl,m = 14 pf P l,l Q l,m P l,m F l,m G l,m P m,m , (45)where F l,m = h B l A m i , P l,m = h A l B m i , Q l,m = h A l A m i , G l,m = h B l B m i . (46)To evaluate the entanglement between two quantum systems in the chain we use theconcurrence which has been shown to be a measure of entanglement [54]. The concurrence C ( t ) is defined as C ( ρ ) = max(0 , λ a − λ b − λ c − λ d ) , (47)where the λ i ’s are the positive square root of the eigenvalues, in a descending order, of thematrix R defined by R = q √ ρ ˜ ρ √ ρ , (48)and ˜ ρ is the spin-flipped density matrix given by˜ ρ = ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) . (49)10nowing that ρ is symmetrical and real due to the symmetries of the Hamiltonian andparticularly the global phase flip symmetry, there will be only 6 non-zero distinguishedmatrix elements of ρ which takes the form [55] ρ = ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ , . (50)Hence, the roots of the matrix R come out to be λ a = √ ρ , ρ , + | ρ , | , λ b = √ ρ , ρ , + | ρ , | , λ c = (cid:12)(cid:12)(cid:12) √ ρ , ρ , − | ρ , | (cid:12)(cid:12)(cid:12) and λ d = (cid:12)(cid:12)(cid:12) √ ρ , ρ , − | ρ , | (cid:12)(cid:12)(cid:12) .To find the non-zero matrix elements of ρ , one can use the formula of the expectationvalue of an operator in terms of density matrix D ˆ G E = T r ( ρ ˆ G ) / T r ( ρ ) along with themagnetization eq.(37) and the spin correlation functions eq.(39)-(41) which give ρ , = 12 M zl + 12 M zm + S zl,m + 14 , (51) ρ , = 12 M zl − M zm − S zl,m + 14 , (52) ρ , = 12 M zm − M zm − S zl,m + 14 , (53) ρ , = − M zl − M zm + S zl,m + 14 , (54) ρ , = S xl,m + S yl,m , (55) ρ , = S xl,m − S yl,m . (56) IV. RESULTS AND DISCUSSIONA. Constant Magnetic Field
We start with studying the dynamics of the nearest neighbor concurrence C ( i, i + 1) forthe completely anisotropic system, γ = 1, when the coupling parameter is J exp as well as J tanh and the magnetic field is a constant using the numerical solution. In Figure 1 we studythe dynamics of the concurrence with the parameters J = 0 . , J = 2 , h = 1 and differentvalues of the transition constant K = 0 . K in addition to the coupling parameter and magnetic field. The11arger the transition constant is, the lower is the asymptotic value of the entanglement andthe more rapid decay is. This result demonstrates the non-ergodic behavior of the system,where the asymptotic value of the entanglement is different from the one obtained underconstant coupling J . In Fig. 2 we study the effect of the system size N on the dynamics t (J −1 ) C ( i , i + ) (a) (a) t (J −1 ) C ( i , i + ) (b) (b) t (J −1 ) C ( i , i + ) (c) (c) t (J −1 ) C ( i , i + ) (d) (d) FIG. 1: C ( i, i + 1) as a function of t with J = 0 . , J = 2 , h = 1 , N = 1000 at kT = 0 and (a) J = J exp , K = 0 . J = J exp , K = 10 ; (c) J = J tanh , K = 0 . J = J tanh , K = 10. of the concurrence. We select the parameters J = 0 . , J = 2 , h = 1 and K = 1000. Wenote that for all values of N the concurrence reaches an approximately constant value butthen starts oscillating after some critical time t c , that increases as N increases, which meansthat the oscillation will disappear as we approach an infinite one-dimensional system. Such12scillations are caused by the spin-wave packet propagation [48]. We next study the dynamics FIG. 2: C ( i, i + 1) as a function of t (units of J − ) with J = J exp , J = 0 . , J = 2 , h = 1 , K = 1000 at kT = 0 and N varies from 100 to 300. of the nearest neighbor concurrence when the coupling parameter is J cos with different valuesof K , i.e. different frequencies, which is shown in Fig. 3. We first note that C ( i, i + 1) showsa periodic behavior with the same period of J ( t ). It has been shown in a previous work[51] that for the considered system at zero temperature the concurrence depends only onthe ratio J/h . When J ≈ h the concurrence has a maximum value. While when J >> h or J << h the concurrence vanishes. In Fig. 3, one can see that when J = J max , C ( i, i + 1)decreases because large values of J destroy the entanglement, while C ( i, i + 1) reaches amaximum value when J = J = 0 .
5. As J ( t ) vanishes, C ( i, i + 1) decreases because of themagnetic field domination. In Fig. 4 we study the dynamics of nearest neighbor concurrencewhen J = J sin . As can be seen, C ( i, i + 1) shows a periodic behavior with the same periodas J ( t ). We note that we get larger values of C ( i, i + 1) compared to the previous case J = J cos . This indicates the importance of an initial concurrence to maintain and yieldhigh concurrence as time evolves. Comparing our results with the previous results of timedependent magnetic field [48], we note that the behavior of C ( i, i + 1) when J = J cos issimilar to its behavior when h = h sin , where h sin = h (1 − sin ( Kt )), and vice versa.13
50 100 15000.050.10.1500.51 t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (a) J(t)h(t) (a) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (b) J(t)h(t) (b) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (c) J(t)h(t) (c)
FIG. 3:
Dynamics of nearest neighbor concurrence with γ = 1 for J cos where J = 0 . , h = 1 at kT = 0and (a) K = 0 . K = 0 . K = 1.
50 100 15000.050.10.1500.51 t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (a) J(t)h(t) (a) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (b) J(t)h(t) (b) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (c) J(t)h(t) (c)
FIG. 4:
Dynamics of nearest neighbor concurrence with γ = 1 for J sin with J = 0 . , h = 1 at kT = 0 and(a) K = 0 . K = 0 . K = 1. . A Time-Dependent Magnetic Field In this section we use the exact solution to study the concurrence for four forms ofcoupling parameter J exp , J tanh , J cos and J sin when J ( t ) = λh ( t ) where λ is a constant. Wehave compared the exact solution results with the numerical ones and they have showncoincidence. The dynamics of C ( i, i + 1) for h ( t ) = 1 and J = J exp , J = 0 . , J = 1 with t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (a) J(t)h(t) (a) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (b) J(t)h(t) (b) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (c) J(t)h(t) (c)
FIG. 5:
Dynamics of nearest neighbor concurrence with γ = 1 at kT = 0, J = 0 . , J = 1 , K = 0 . h ( t ) = 1 ; (b) h ( t ) = J ( t ) = J exp ; (c) h ( t ) = J ( t ) = J tanh . K = 0 . C ( i, i + 1) for h ( t ) = J ( t ) = J exp , J = 0 . , J = 1 and K = 0 .
1, as one can see the time-dependent magnetic field caused the asymptotic value of C ( i, i + 1) to decrease. A similarbehavior occurs when h ( t ) = J ( t ) = J tanh , J = 0 . , J = 1 with K = 0 . C ( i, i + 1) when h ( t ) = J ( t ) = J cos and h ( t ) = J ( t ) = J sin respectively, where J = 0 . K = 1. As can be noticed theconcurrence in this case does not show a periodic behavior as it did when h ( t ) = 1 in Figs. 3and 4.In Fig. 7(a) we study the behavior of the asymptotic value of C ( i, i + 1) as a functionof λ at different values of the parameters J , J and K where J ( t ) = λh ( t ). Interestingly,the asymptotic value of C ( i, i + 1) depends only on the initial conditions not on the formor behavior of J ( t ) at t >
0. This result demonstrates the sensitivity of the concurrenceevolution to its initial value. Testing the concurrence at non-zero temperatures demonstratesthat it maintains the same profile but with reduced value with increasing temperature as canbe concluded from Fig. 7(b). Also the critical value of λ at which the concurrence vanishesdecreases with increasing temperature as can be observed, which is expected as thermalfluctuations destroy the entanglement. t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (a) J(t)h(t) (a) t (J −1 ) C ( i , i + ) J ( t ) , h ( t ) (b) J(t)h(t) (b)
FIG. 6:
Dynamics of nearest neighbor concurrence with γ = 1 at kT = 0 with J = h = 0 . , K = 1 for (a) J cos and h cos ; (b) J sin and h sin . Finally, in Fig. 8 we study the partially anisotropic system, γ = 0 .
5, and the isotropicsystem γ = 0 with J (0) = 1. We note that the behavior of C ( i, i + 1) in this case issimilar to the case of constant coupling parameter studied previously [51]. We also notethat the behavior depends only on the initial coupling J (0) and not on the form of J ( t )where different forms have been tested. 17 λ C ( i , i + ) (a) J =0.5J =1J =2 0 0.5 1 1.5 2 2.500.050.10.150.20.250.3 λ C ( i , i + ) (b) J(0)=0.5, kT=0.5J(0)=0.5, kT=1J(0)=1, kT=0.5J(0)=1, kT=1
FIG. 7:
The behavior asymptotic value of C ( i, i + 1) as a function of λ with γ = 1 at (a) kT = 0 ; (b) kT = 0 . , λ C ( i , i + ) (a) λ C ( i , i + ) (b) FIG. 8:
The behavior asymptotic value of C ( i, i + 1) as a function of λ at kT = 0 with (a) γ = 0 . γ = 0. V. CONCLUSIONS AND FUTURE DIRECTIONS
We have studied the dynamics of entanglement in a one-dimensional XY spin chaincoupled through a time-dependent nearest neighbor coupling and in the presence of a time-dependent magnetic field at zero and finite temperatures. We presented a numerical solutionfor the system for general J ( t ) and h ( t ) and an exact solution for proportional J ( t ) and h ( t ).For an exponentially increasing J ( t ) we found that the asymptotic value of the concurrencedepends on the exponent transition constant value, which confirms the non-ergodic behavior18f the system. For a periodic J ( t ) we found that the concurrence shows a periodic behaviorwith the same period as J ( t ). On the other hand for both periodic coupling and magneticfield with same period, the concurrence loses its periodic behavior. When J ( t ) = λh ( t ) where λ is a constant we found that the asymptotic value of the concurrence depends only on theinitial conditions regardless of the form of the coupling parameter or the magnetic field. Infuture, we would like to study the effect of an impurity spin on the entanglement along thedriven one-dimensional spin chain. It will be also interesting to study the decoherence ofa spin pair (quantum gate) as a result of coupling to a driven one-dimensional spin chainacting as its environment. Acknowledgments
This work was supported in part by the deanship of scientific research, King Saud Uni-versity. 19
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