Entanglement dynamics of two qubits under the influence of external kicks and Gaussian pulses
EEntanglement dynamics of two qubits under the influence ofexternal kicks and Gaussian pulses
Ferdi Altintas and Resul Eryigit ∗ Department of Physics, Abant Izzet Baysal University, Bolu, 14280-Turkey.
Abstract
We have investigated the dynamics of entanglement between two spin-1/2 qubits that are sub-ject to independent kick and Gaussian pulse type external magnetic fields analytically as well asnumerically. Dyson time ordering effect on the dynamics is found to be important for the sequenceof kicks. We show that ”almost-steady” high entanglement can be created between two initiallyunentangled qubits by using carefully designed kick or pulse sequences.
PACS numbers: 03.65.Ud; 03.67.Mn; 75.10.Jm ∗ email:[email protected] a r X i v : . [ qu a n t - ph ] S e p . INTRODUCTION Control and manipulation of entanglement which is a quantifiable resource for quantuminformation tasks such as quantum computing [1], communication [2] and cryptography [3]have been studied along many directions in the last decade [4–13]. Among these studies, sys-tems that are modelled as 1-D Heisenberg chain with qubits of spin-1/2 particles as the mainunit are one of the prototypical examples [14]. For such systems the control of entanglementbetween the qubits can be manipulated with the help of various type external magneticfields [4–13]. In particular, Heule et al. investigated the feasibility of local operator controlin arrays of interacting qubits modelled as isotropic Heisenberg spin chains [4]. Along similarlines, Caneva et al. explored optimal quantum control by appropriate pulses to affect the re-quired transformations by numerical Krotov algorithm [5]. Wu et al. showed that one qubitgates can be constructed with global magnetic fields and controllable Heisenberg exchangeinteractions [6]. Levy demonstrated a scheme that uses pairs of spin-1/2 particles to formlogic qubits and Heisenberg exchange only to produce all gate operations [7]. Malinovskyand Sola studied phase control of entanglement in two qubit systems and showed that bychanging the relative phase of control pulses, one can control entanglement at will [8]. Sadiek et al. studied the control and manipulation of entanglement evolution for a two qubit systemcoupled through XYZ Heisenberg interaction influenced by a time-varying external field [9].Wang et al. demonstrated that near perfect entanglement can be obtained by applying amagnetic field on a single spin of an isotropic Heisenberg chain of length N [10]. Abliz et al. studied the entanglement dynamics for a two-qubit Heisenberg XXZ model effectedby population relaxation in the presence of various types of magnetic fields and showedthat it is possible to produce, control and modulate high entanglement with the help oftime-dependent external fields despite the existence of dissipation [11].For a general time-dependent external field, time-ordering effects might be importantand the dynamics cannot be found analytically. Most of the aforementioned studies employnumerical methods to investigate the entanglement control [4, 5, 8–10]. Although the nu-merical methods are fast and reliable, analytic solutions provide a more clear picture of thephysics behind the dynamics. For a single qubit, the time evolution of populations and co-herence under the influence of external field in the form of Gaussian pulse or a delta functionkick was investigated by Kaplan et al. and Shakov et al. [15, 16]. The fast pulse or kick is2efined based on the relation between the energy splitting of the qubit ∆ E and the durationof the pulse τ ; if ∆ Eτ << z -direction. We consider one, two, three and four kicks as well as Gaussian pulsesequences and their effect on the dynamics of entanglement between the qubits. We showthat entanglement can be controlled by a careful design of the sequence of kicks.The organization of this paper is as follows: In Sec. II, we introduce the model and basicformalism necessary to solve time evolution exactly. In Sec. III, we discuss time orderingeffect on the dynamics. In Sec. IV, Wootters concurrence as an entanglement measure isbriefly introduced. In Sec. V, the analytic entanglement dynamics of kicked qubits in thepresence of time ordering is discussed by choosing single and multiple (up to four) kicks. InSec. VI, the effect of finite pulse width on the entanglement dynamics is studied numericallyby choosing a Gaussian pulse or pulse sequence as an external field. We conclude as asummary of the important results in Sec. VII. II. THE MODEL AND BASIC FORMULATION
In this paper, we consider two Heisenberg XXX coupled qubits in a time-dependentexternal magnetic field acting in the z -direction. The typical time-dependent Hamiltonianfor this system may be expressed as [9] (we set ¯ h = 1):ˆ H ( t ) = ˆ H + ˆ H int ( t ) , (1)3here ˆ H = J (cid:88) i = x,y,z ˆ σ i ˆ σ i , ˆ H int ( t ) = − (cid:88) i =1 B iz ( t )ˆ σ iz , (2)where ˆ σ , i ( i = x, y, z ) are the usual Pauli spin matrices, J is the qubit-qubit interactionstrength and B z ( t ) and B z ( t ) are the time-dependent magnetic fields acting on qubit 1 and2, respectively. It should be noted that the qubit-qubit interaction term in Eq. (1) is givenby ˆ H which is constant in time and the time-dependent part of the total Hamiltonian iscalled ˆ H int ( t ) which describes the qubit-magnetic field interaction and assumed to be a singlereal function of t .The most general form of an initial pure state of the two-qubit system is | Ψ(0) (cid:105) = a (0) | (cid:105) + a (0) | (cid:105) + a (0) | (cid:105) + a (0) | (cid:105) , where a i (0) ( i = 1 , , ,
4) are complex numberswith (cid:88) i =1 | a i (0) | = 1, then the probability amplitudes evolve in time under Hamiltonian (1)according to Schr¨odinger equation as: i ddt a ( t ) a ( t ) a ( t ) a ( t ) = J − B T ( t ) 0 0 00 − J + ∆ B ( t ) 2 J
00 2 J − J − ∆ B ( t ) 00 0 0 J + B T ( t ) a ( t ) a ( t ) a ( t ) a ( t ) , (3)where ∆ B ( t ) = B z ( t ) − B z ( t ) and B T ( t ) = B z ( t ) + B z ( t ). The formal solution of Eq. (3)may be written in terms of the time evolution matrix ˆ U ( t ) as a ( t ) a ( t ) a ( t ) a ( t ) = ˆ U ( t ) a (0) a (0) a (0) a (0) . (4)The evolution operator for the general time-dependent Hamiltonian of two qubits is not easyto obtain analytically; a number of systematic procedures are obtained in Refs [23] and [24]based on dynamical groups of the system when time-ordering is not important. Here the4ime evolution operator ˆ U ( t ) may be expressed as:ˆ U ( t ) = ˆ T e − i (cid:82) t ˆ H ( t (cid:48) ) dt (cid:48) = ˆ T e − i (cid:82) t ( ˆ H + ˆ H int ( t (cid:48) ) ) dt (cid:48) = ˆ T ∞ (cid:88) n =0 ( − i ) n n ! (cid:90) t ˆ H ( t n ) dt n ... (cid:90) t ˆ H ( t ) dt (cid:90) t ˆ H ( t ) dt . (5)The only non-trivial time dependence in ˆ U ( t ) arises from time-dependent ˆ H ( t ) and timeordering ˆ T . The Dyson time ordering operator ˆ T specifies that ˆ H ( t i ) ˆ H ( t j ) is properlyordered [15, 16, 25]:ˆ T ˆ H ( t i ) ˆ H ( t j ) = ˆ H ( t i ) ˆ H ( t j ) + θ ( t j − t i ) (cid:104) ˆ H ( t j ) , ˆ H ( t i ) (cid:105) , (6)where θ ( t j − t i ) is the Heaviside step function whose value is zero if ( t j − t i ) is negativeand one if ( t j − t i ) is positive. It should be noted that time ordering imposes a connectionbetween the effects of ˆ H ( t i ) and ˆ H ( t j ) and gives rise to observable, non-local, time orderingeffects when (cid:104) ˆ H ( t j ) , ˆ H ( t i ) (cid:105) (cid:54) = 0 [26, 27]. III. TIME ORDERING
If one takes ˆ T = 1 in Eq. (5) to obtain ˆ U ( t ), the time evolution is said to contain notime ordering. So the difference between the result obtained by an exact treatment of ˆ T in Eq. (5) and ˆ T → T → A. Limit of no time ordering
Replacing ˆ T with 1 in Eq. (5), in the Schr¨odinger picture we have,ˆ U ( t ) = ˆ T e − i (cid:82) t ˆ H ( t (cid:48) ) dt (cid:48) → ∞ (cid:88) n =0 ( − i ) n n ! (cid:20)(cid:90) t ˆ H ( t (cid:48) ) dt (cid:48) (cid:21) n = ∞ (cid:88) n =0 ( − i ) n n ! (cid:20) ˆ H t + (cid:90) t ˆ H int ( t (cid:48) ) dt (cid:48) (cid:21) n = ∞ (cid:88) n =0 ( − i ) n n ! (cid:104)(cid:16) ˆ H + ˆ¯ H int (cid:17) t (cid:105) n = e − i ˆ¯ Ht = ˆ U ( t ) , (7)where ˆ¯ H int t = (cid:82) t ˆ H int ( t (cid:48) ) dt (cid:48) is the averaged interaction field, and ˆ¯ H = ˆ H + ˆ¯ H int and[ ˆ H , ˆ¯ H int ] terms are non-zero. By expanding in powers of [ ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) )], it is straightforward5o show that to leading order in ˆ H int ( t ) and ˆ H , the time ordering effect is given byˆ U − ˆ U (cid:39) − (cid:90) t dt (cid:48)(cid:48) (cid:90) t (cid:48)(cid:48) dt (cid:48) (cid:104) ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) ) (cid:105) = − (cid:104) ˆ H , ˆ H int (cid:105) (cid:90) t dt (cid:48) ( t − t (cid:48) ) f ( t (cid:48) ) , (8)where ˆ H int ( t (cid:48) ) = ˆ H int f ( t (cid:48) ). This leading term disappears if the pulse centroid T k = t/ f ( t (cid:48) ) is symmetric about T k . Furthermore, ˆ U − ˆ U vanishes identically in the specialcases of H int ( t (cid:48) ) = 0 , H int ( t (cid:48) ) = ¯ H int [15, 16]. Also the commutator (cid:104) ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) ) (cid:105) (i.e.,the time ordering effect) vanishes for B z ( t ) = B z ( t ) or J = 0 because by using the totalHamiltonian (1) we have[ ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) )] = 2 iJ (cid:0)(cid:0) B z ( t (cid:48) ) − B z ( t (cid:48) ) (cid:1) − (cid:0) B z ( t (cid:48)(cid:48) ) − B z ( t (cid:48)(cid:48) ) (cid:1)(cid:1) (ˆ σ y ˆ σ x − ˆ σ x ˆ σ y ) , (9)which vanishes when either the time-dependent magnetic fields on qubits 1 and 2 are equalor the qubit-qubit interaction is neglected.In general there is no simple analytic form for the exact result ˆ U ( t ), except for specialcases [9, 15, 16]. For the result without time ordering with averaged magnetic fields ¯ B z t = (cid:82) t B z ( t (cid:48) ) dt (cid:48) = α and ¯ B z t = (cid:82) t B z ( t (cid:48) ) dt (cid:48) = β , the time evolution matrix ˆ U ( t ) in Eq. (7) canbe easily calculated as:ˆ U ( t ) = e − i ( ˆ H t + ˆ¯ H int t ) = y y ∗ y ( u + iv ) y ( − w + iz ) 00 y ( w + iz ) y ( u − iv ) 00 0 0 y ∗ y ∗ , (10)where y = e iJt ,y = e i ( α + β ) ,u = cos (Γ) ,v = ( α − β )Γ sin (Γ) ,w = 0 ,z = − J t
Γ sin (Γ) , (11)6here Γ = (cid:112) J t + ( α − β ) and α and β are called the integrated magnetic strengthsassociated with the magnetic fields acting on qubit 1 and 2, respectively.Similarly, the time evolution matrix without time ordering in interaction picture canbe studied [15, 16]. However in Refs. [15] and [16], it was shown that the occupationprobabilities for a kicked qubit for the dynamics in the limit ˆ T → B z ( t ) = B z ( t ) or J = 0, thetime ordering effect defined as ˆ U K ( t ) − ˆ U ( t ) vanishes. IV. MEASURE OF ENTANGLEMENT
For a pair of qubits, Wootters concurrence can be used as a measure of entanglement [29].The concurrence function varies from C = 0 for a separable state to C = 1 for a maximallyentangled state. To calculate the concurrence function one needs to evaluate the matrixˆ R = ˆ ρ ( t )(ˆ σ y ⊗ ˆ σ y ) ˆ ρ ∗ ( t )(ˆ σ y ⊗ ˆ σ y ) , (12)where ˆ ρ ( t ) is the density matrix of the system and ˆ ρ ∗ ( t ) is its complex conjugate. Then theconcurrence is defined as C ( ˆ ρ ) = max { , λ − λ − λ − λ } , (13)where λ , λ , λ and λ are the positive square roots of the eigenvalues of ˆ R in descendingorder.It should be noted that due to the discrete symmetry (conservation of parity underflipping of the ˆ σ ji , i = x, y, z and j = 1 ,
2, i.e. when ˆ σ ji → − ˆ σ ji ) of the total Hamiltonian (1),the states | Φ (cid:105) = a | (cid:105) + a | (cid:105) and | Ψ (cid:105) = a | (cid:105) + a | (cid:105) can never get mixed in timedue to that symmetry [9], as can be seen from Eq. (3). Thus we consider the time evolutionof the concurrence of these states individually. The concurrence function for a pure state | Φ( t ) (cid:105) = a ( t ) | (cid:105) + a ( t ) | (cid:105) with density matrix ˆ ρ ( t ) = | Φ( t ) (cid:105) (cid:104) Φ( t ) | is given by C ( ˆ ρ ) = max { , | a ( t ) a ( t ) |} . (14)7imilarly, for the pure state | Ψ( t ) (cid:105) = a ( t ) | (cid:105) + a ( t ) | (cid:105) with density matrix ˆ ρ ( t ) = | Ψ( t ) (cid:105) (cid:104) Ψ( t ) | , the concurrence function reads as C ( ˆ ρ ) = max { , | a ( t ) a ( t ) |} , (15)where according to Eq. (4), the time-dependent coefficients read a i ( t ) = (cid:88) j =1 U ij ( t ) a j (0) , (16)where U ij ( t ) ( i, j = 1 , , ,
4) are the matrix elements of ˆ U ( t ). V. ENTANGLEMENT DYNAMICS OF KICKED QUBITS
In this part, we will examine the entanglement dynamics of kicked qubits by taking intoaccount the time ordering effects for the initially pure separable | Ψ(0) (cid:105) = | (cid:105) and maximallyentangled | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) Bell states. We will work in the Schr¨odinger picture andpresent analytic expressions for the time evolution operator of the two-qubit system for asingle kick as well as a positive followed by a negative kick and a sequence of two, three andfour equally distanced kicks. For all kick sequences we consider two integrated magneticstrength regimes: α = 2 β and α = 3 β and for convenience we shall set J = 1 and β = 1.From those results we will use Eqs. (14) and (16) to analyze and discuss the time evolutionof the entanglement between the qubits.We have also considered the other Bell states √ ( | (cid:105) ± | (cid:105) ) and √ ( | (cid:105) − | (cid:105) ) aswell as separable states of two qubits as | (cid:105) , | (cid:105) , | (cid:105) . Under the influence of ˆ H ( t ) ofEq. (1), the states of the type | (cid:105) and | (cid:105) remain separable, while the entanglement of √ ( | (cid:105) ± | (cid:105) ) does not change with time. These may be checked using Eq. (15) and thesolution of the expansion coefficients a ( t ) and a ( t ) in Eq. (3) for the considered initialstates. The dynamics of √ ( | (cid:105) − | (cid:105) ) is same as that of √ ( | (cid:105) + | (cid:105) ) and that of | (cid:105) issame as | (cid:105) that are noted after specifying the propagators for kicked qubits and by usingEq. (14) and Eq. (16). So, we consider only | Ψ(0) (cid:105) = | (cid:105) and | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) asinitial states.One point we want to emphasize that before the field is active, the propagator is equalto e − i ˆ H t and given by Eq. (23). Based on Eqs. (14), (16) and (23), the concurrence of theinitial state | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) is equal to 1, while the concurrence of | Ψ(0) (cid:105) = | (cid:105) | sin(4 J t ) | at time t before the kick. Note that the entanglement dynamics for the initialstate | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) is unperturbed by the qubit-qubit interaction in the absenceof external field, while qubit-qubit interaction creates a high degree of entanglement thatoscillates between 0 and 1 if the qubits are initially prepared in | Ψ(0) (cid:105) = | (cid:105) state. A. Single kick
FIG. 1: Concurrence as a function of dimensionless time,
J t , for an ideal positive kick applied at T = 5 for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a) and | Ψ(0) (cid:105) = | (cid:105) (b). The dashedlines correspond to α = 2 β and the solid lines to α = 3 β . Here we consider two qubits whose states coupled by an interaction field which can beexpressed as a sudden ”kick” at t = T , namely B z ( t ) = αδ ( t − T ) , B z ( t ) = βδ ( t − T ).For such a kick the integration over time is trivial and the time evolution matrix in Eq. (5)becomes [15] ˆ U K ( t ) = e − i ˆ H ( t − T ) e − i (cid:82) T (cid:15)T − (cid:15) ˆ H int ( t (cid:48) ) dt (cid:48) e − i ˆ H T , (17)in the same form as Eq. (10) with elements y = e iJt ,y = e i ( α + β ) ,u = cos (2 J t ) cos( α − β ) ,v = cos (2 J ( t − T )) sin( α − β ) ,w = sin (2 J ( t − T )) sin( α − β ) ,z = − sin (2 J t ) cos( α − β ) , (18)9or t > T . The propagator without time ordering is given in Eq. (10) and as explainedbefore when α = β or J = 0, the time ordering effect, ˆ U K ( t ) − ˆ U ( t ), vanishes after the fieldis active.By inserting Eq. (18) into Eqs. (14) and (16) for the initial states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) )and | Ψ(0) (cid:105) = | (cid:105) , one can obtain the analytic expressions of the concurrence functions afterthe kick ( t > T ). For the maximally entangled state, the concurrence is given as: C ( ˆ ρ ) = max { , (cid:12)(cid:12) cos (∆) + e iJ ( t − T ) sin (∆) (cid:12)(cid:12) } , (19)while for | Ψ(0) (cid:105) = | (cid:105) the concurrence after t = T can be obtained as C ( ˆ ρ ) = 2 max { , | Λ |} , (20)where Λ = (cos(2 J t ) cos(∆) − i cos( ζ ) sin(∆)) ( i cos(∆) sin(2 J t ) + sin( ζ ) sin(∆)), ∆ = α − β and ζ = 2 J ( t − T ).The dynamics of concurrence for the initial Bell state and the separable state for thesingle kick which are given by Eqs. (19) and (20) for t > T are displayed in Fig. 1(a)and (b), respectively. The effect of the kick on the entanglement is pronounced for bothinitial states; the concurrence of the Bell state starts oscillating with an amplitude thatdepends on the ratio of the integrated magnetic strength of the external fields on qubit 1and 2 (i.e., α and β , respectively). The effect of the kick on the system initially in separablestate, | (cid:105) , is similar with the exception that the concurrence variation amplitudes get lowerafter the kick. One should also note that C ( ˆ ρ ) of the initial Bell state is independent of J before the kick, while the frequency of its time-dependence after the kick is proportionalto the qubit-qubit interaction strength J , as can be seen from Fig. 1(a) and the analyticexpression Eq. (18). B. A positive followed by a negative kick
The propagator for a sequence of either identical or non-identical kicks can be easilyobtained by multiplication of several matrices of the form of Eq. (17) [15, 16]. For example,one may consider a sequence of two kicks of opposite sign at times t = T and t = T ,namely, B z ( t ) = α ( δ ( t − T ) − δ ( t − T )), B z ( t ) = β ( δ ( t − T ) − δ ( t − T )). Following the10rocedure given in Eq. (17), one obtains the time evolution matrix for t > T as [15]:ˆ U K ( t ) = e − i ˆ H ( t − T ) e − i (cid:82) T (cid:15)T − (cid:15) ˆ H int ( t (cid:48) ) dt (cid:48) e − i ˆ H ( T − T ) e − i (cid:82) T (cid:15)T − (cid:15) ˆ H int ( t (cid:48) ) dt (cid:48) e − i ˆ H T , (21)where the elements of Eq. (21) is the same form as Eq. (10) with parameters y = e iJt ,y = 1 ,u = cos (2 J t ) cos(∆) + cos (2 J ( t − T s )) sin(∆) ,v = (cos( ζ ) − cos( ζ )) sin(∆) cos(∆) ,w = (sin( ζ ) − sin( ζ )) sin(∆) cos(∆) ,z = − sin (2 J t ) cos(∆) − sin (2 J ( t − T s )) sin(∆) , (22)where ζ i = 2 J ( t − T i ) , ∆ = ( α − β ) and T s = T − T . For this case, the time evolutionmatrix without time ordering is given byˆ U ( t ) = e − i ˆ H t = e − iJt e iJt cos(2 J t ) − ie iJt sin(2 J t ) 00 − ie iJt sin(2 J t ) e iJt cos(2 J t ) 00 0 0 e − iJt , (23)because for a positive kick followed by a negative kick the averaged interaction Hamiltonian,ˆ¯ H int t = 0. For the cases J = 0, or T s = 0, or α = β , the time ordering effect defined asˆ U K ( t ) − ˆ U ( t ) goes to zero, as expected.The entanglement dynamics under the positive-negative kick sequence at times t > T isobtained by using the expression Eq. (22) in Eqs. (14) and (16) and are displayed in Fig. 2(a)and (b) for the initial Bell and separable states, respectively. The effect of the negativekick at T is found to be opposite for the | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) and | Ψ(0) (cid:105) = | (cid:105) initialstates; for the | (cid:105) state, the dependence of concurrence on the integrated magnetic strengthvanishes while for the initial Bell state, amplitude of concurrence oscillations change with α and β . One peculiar result from Fig. 2(a) is the observation that the negative kick has noinfluence on the dynamics of concurrence for α = 3 β magnetic fields (solid line in Fig. 2(a)).11 IG. 2: Concurrence as a function of
J t for an ideal positive kick applied at T = 5 followedby an ideal negative kick at T = 10 for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a) and | Ψ(0) (cid:105) = | (cid:105) (b). The dashed lines correspond to α = 2 β and the solid lines to α = 3 β . The positive-negative kick sequence also demonstrates the strong effect of time ordering onthe entanglement dynamics. As mentioned before, based on the propagator for ˆ T → t > T . On theother hand, as can be seen from Figs. 2(a) and 2(b), the time-ordered propagator leads todifferent results in the concurrence for both initial states. C. Two positive kicks
To show the difference between positive and negative kicks applied after the first positivekick on the entanglement dynamics of two qubits, one may consider a sequence of two positivekicks applied at times t = T and t = T , namely, B z ( t ) = α ( δ ( t − T ) + δ ( t − T )) , B z ( t ) = β ( δ ( t − T ) + δ ( t − T )). Following the procedure given in Eq. (17), one obtains the timeevolution matrix Eq. (21) for t > T in the form as Eq. (10) with parameters y = e iJt ,y = e i ( α + β ) ,u = cos (2 J t ) cos(∆) − cos (2 J ( t − T s )) sin(∆) ,v = (cos( ζ ) + cos( ζ )) sin(∆) cos(∆) ,w = (sin( ζ ) + sin( ζ )) sin(∆) cos(∆) ,z = − sin (2 J t ) cos(∆) + sin (2 J ( t − T s )) sin(∆) , (24)12here ζ i = 2 J ( t − T i ) , ∆ = ( α − β ) and T s = T − T . Here the propagator without timeordering can be calculated by replacing ¯ B z t → α and ¯ B z t → β in Eq. (10) and note for α = β or J = 0, ˆ U K ( t ) − ˆ U ( t ) vanishes, as expected according to Eqs. (8) and (9). FIG. 3: Concurrence as a function of
J t for a sequence of two ideal positive kicks applied at T = 5and T = 10 for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a) and | Ψ(0) (cid:105) = | (cid:105) (b). Thedashed lines correspond to α = 2 β and the solid lines to α = 3 β . The effect of two consecutive positive kicks on the dynamics of concurrence for two qubitsis displayed in Fig. 3(a) and 3(b) for the initial Bell state, √ ( | (cid:105) + | (cid:105) ) and separable state, | (cid:105) , respectively. Comparing the analytic expressions of the time-evolution operators forpositive-negative and positive-positive kick sequences of Eq. (22) and Eq. (24), respectively,along with the Fig. 2 and Fig. 3, the effect of the sign of the kicks in the sequence is tochange the amplitude of the concurrence oscillations. The oscillation amplitude of C ( ˆ ρ ) forthe Bell state as well as separable state increases for the positive-positive sequence comparedto that of positive-negative sequence of kicks. Also, α/β dependence of the amplitude isdifferent as can be seen from a comparison of Fig. 2 and 3. D. Three and four positive kicks
One may consider a sequence of n -positive kicks applied at times t = T , t = T , ..., t = T n ,namely B z ( t ) = n (cid:88) i =1 αδ ( t − T i ) , B z ( t ) = n (cid:88) i =1 βδ ( t − T i ). For example, following the proceduregiven in Eq. (17), one obtains the time evolution matrix for three positive kicks at times13 IG. 4: Concurrence as a function of dimensionless time,
J t , for 4-successive ideal positive kicksfor the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a) and | Ψ(0) (cid:105) = | (cid:105) (b). Here the dashed linescorrespond to α = 2 β and the solid lines to α = 3 β and we take T = 5 , T = 10 , T = 15 and T = 20.FIG. 5: (Colour online) The contour plot of concurrence versus J t and the ratio, α/β , for the initialpure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a) and | Ψ(0) (cid:105) = | (cid:105) (b). Here the contour plots include fourideal positive kicks applied at T = 5 , T = 10 , T = 15 and T = 20. (There are ten equidistantcontours of concurrence in the plots between 0 (black) and 1 (white).) t > T as: ˆ U K ( t ) = e − i ˆ H ( t − T ) e − i (cid:82) T (cid:15)T − (cid:15) ˆ H int ( t (cid:48) ) dt (cid:48) e − i ˆ H ( T − T ) e − i (cid:82) T (cid:15)T − (cid:15) ˆ H int ( t (cid:48) ) dt (cid:48) × e − i ˆ H ( T − T ) e − i (cid:82) T (cid:15)T − (cid:15) ˆ H int ( t (cid:48) ) dt (cid:48) e − i ˆ H T , (25)14n the same form as Eq. (10) with parameters y = e iJt ,y = e i ( α + β ) ,u = cos (2 J t ) cos(∆) − (cid:88) i,j =1 i 75. The α/β periodicity of the maximum of C ( ˆ ρ ) increasesafter each kick for the initial Bell state. The dependence of C ( ˆ ρ ) on α/β for the initial sep-arable state | (cid:105) is more complicated compared to the case of initial Bell state. The almostperiodic structures exist also in Fig. 5(b); their periodicity changes after each kick, but it isnot easy to obtain an expression for that change. Most importantly, the high entanglementregions, which are indicated in white in the contour plots have long life times for each pos-itive kicks for the initial Bell state, while for the separable state they are distributed in anarrower area compared to the initial Bell state case and have long lifetimes only after 1 st ,2 nd and 4 th kicks. VI. ENTANGLEMENT DYNAMICS OF QUBITS PERTURBED BY A SE-QUENCE OF GAUSSIAN PULSES Depending on the physical implementation of the qubit, it might be difficult to ob-tain an external field that can be considered as a kick. Instead a Gaussian pulse withfinite width can be applied (for example, half-cycle electromagnetic pulses with width near τ = 1 ps may be experimentally achievable [30, 31]). Thus in this part, we will discuss theentanglement dynamics of two qubits under the influence of Gaussian pulses of the form16 iz ( t ) = α i √ πτ e − ( t − Tk )2 τ ( α , = α, β ) centered at T k with width τ . The dynamics of entangle-ment in the presence of time ordering for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) )and | Ψ(0) (cid:105) = | (cid:105) are evaluated by numerically integrating the corresponding equations inEq. (3) and using Eq. (14). Here we will investigate how the entanglement depends on thepulse width τ by choosing a single pulse, a positive pulse followed by a negative pulse, andmultiple positive pulses up to four centered at times T = 5 , T = 10 , T = 15 and T = 20.For all pulse sequences we will consider two integrated magnetic strength regimes: α = 2 β and α = 3 β and for convenience we shall set J = 1 and β = 1. One should note that in thelimit τ → 0, the results of entanglement dynamics of kicked qubits in the presence of timeordering should be the same that are analyzed in the previous section. A. Single Pulse In Fig. 6, we show the results of a calculation of the concurrence for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) and | Ψ(0) (cid:105) = | (cid:105) when strongly perturbed by a single Gaussianpulse centered at t = T with width τ . According to Schr¨odinger equation (3) of the systemconsidered here, the expansion coefficients a ( t ) and a ( t ) evolve independently and a ( t )and a ( t ) obey a first-order coupled differential equation set, for example for a single pulse,as: i ˙ a ( t ) = (cid:18) − J − ( α − β ) √ πτ e − ( t − T τ (cid:19) a ( t ) + 2 J a ( t ) ,i ˙ a ( t ) = (cid:18) − J + ( α − β ) √ πτ e − ( t − T τ (cid:19) a ( t ) + 2 J a ( t ) , (29)which are solved numerically by using a fourth order Runge-Kutta algorithm. The mostimportant observation from Fig. 6 is the existence of almost constant high concurrence forthe initially separable state at α = 2 β integrated magnetic strength and high width Gaussianpulse, while the entanglement continues to have high amplitude oscillations for α = 3 β ; itsvalue for α = 2 β is almost constant at around 1 for J τ ; the dimensionless pulse widthgreater than 0.15. On the contrary for the initial Bell state, the oscillation amplitude of C ( ˆ ρ ) increases with the J τ of the pulse for each magnetic ratio ( α/β = 2 and α/β = 3).17 IG. 6: Concurrence as a function of J t for a single Gaussian pulse with width τ for the initialpure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a), (c), (e) and (g) and | Ψ(0) (cid:105) = | (cid:105) (b), (d), (f) and (h).The dashed lines correspond to α = 2 β and the solid lines to α = 3 β . Here we assume fourdimensionless pulse width as: J τ = 0 . , . , . , . B. Positive-negative and positive-positive pulse sequence In Fig. 7 and 8, we show the results of a calculation for the concurrence for the initialpure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) and | Ψ(0) (cid:105) = | (cid:105) when strongly perturbed by a single18 IG. 7: Concurrence as a function of J t for a positive followed by a negative Gaussian pulseshaving the same width τ for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a), (c), (e) and (g)and | Ψ(0) (cid:105) = | (cid:105) (b), (d), (f) and (h). The dashed lines correspond to α = 2 β and the solid linesto α = 3 β . Here we assume four dimensionless pulse width as: J τ = 0 . , . , . , . Gaussian pulse centered at t = T followed by a negative or positive Gaussian pulse centeredat t = T with the same width τ . For the double pulse sequence a ( t ) and a ( t ) obey the19oupled equations: i ˙ a ( t ) = (cid:18) − J − ( α − β ) √ πτ ( e − ( t − T τ ± e − ( t − T τ ) (cid:19) a ( t ) + 2 J a ( t ) ,i ˙ a ( t ) = (cid:18) − J + ( α − β ) √ πτ ( e − ( t − T τ ± e − ( t − T τ ) (cid:19) a ( t ) + 2 J a ( t ) , (30)where + sign in the ± on the right-hand side is for positive-positive pulse sequence, while − sign is for positive-negative pulse sequence.The concurrence dynamics for positive-negative and positive-positive pulse sequencesobtained from the numerical solutions of Eq. (30) are displayed in Figs. 7 and 8, respec-tively for the initial Bell and the separable states at different dimensionless pulse widthvalues ( J τ = 0 . , . , . , . C ( ˆ ρ )oscillation amplitudes for α = 3 β at times t > T . The almost constant high entanglementcan be obtained for the | (cid:105) initial state for positive-negative pulse sequence as can be seenfrom Fig. 7(h) for α = 3 β and t > T . One peculiarity of this figure is that high entan-glement is obtained for α = 2 β after the first pulse, while it is obtained for α = 3 β afterthe second pulse. For the positive-positive Gaussian pulse sequence, the difference frompositive-negative sequence becomes small as the width of the pulse gets larger as can bededuced from a comparison of Figs. 7 and 8. On the other hand, for a small pulse width,the difference is significant. For example for J τ = 0 . 05 and α = 3 β , the initial Bell statehas nearly constant entanglement around 1 (see Fig. 7(a)) for positive-negative pulse se-quence after negative pulse, while it oscillates between 1 and 0.25 for positive-positive pulsesequence (see Fig. 8(a)). C. A sequence of four positive pulses The effect of integrated magnetic strength and the pulse width on the dynamics of con-currence for two qubits perturbed by a sequence of four positive Gaussian pulses is dis-played in Fig. 9(a)-(h) for initial Bell state, | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ), and separable state, | Ψ(0) (cid:105) = | (cid:105) . For this four positive pulse sequence the concurrence may be calculated by20 IG. 8: Concurrence as a function of J t for a sequence of two positive Gaussian pulses havingthe same witdh τ for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a), (c), (e) and (g) and | Ψ(0) (cid:105) = | (cid:105) (b), (d), (f) and (h). The dashed lines correspond to α = 2 β and the solid lines to α = 3 β . Here we assume four dimensionless pulse width as: J τ = 0 . , . , . , . i ˙ a ( t ) = (cid:32) − J − ( α − β ) √ πτ (cid:88) i =1 e − ( t − Ti )2 τ (cid:33) a ( t ) + 2 J a ( t ) ,i ˙ a ( t ) = (cid:32) − J + ( α − β ) √ πτ (cid:88) i =1 e − ( t − Ti )2 τ (cid:33) a ( t ) + 2 J a ( t ) . (31)These figures can be compared with Figs. 5(a) and 5(b) to discern the effect of the pulsewidth. As the pulse gets wider, the α/β dependent oscillatory structures in the figure coa-lesce to produce non-periodic structures, especially after third and fourth pulses. The highentanglement regions, which are shown in white, still can have long lifetimes, as indicated bywhite straight perpendicular sections in the contour plots of Fig. 9. In the case of ideal kick,the maximally entangled state is found to be unaffected by the external field for α/β = 1 and α/β ∼ = 2 . , . , . , . , . 75 (see Fig. 5(a)). Comparing Fig. 5(a) with Figs. 9(a), (c), (e)and (g), the initial Bell state is found to be unperturbed by the highly wider Gaussian pulsesif and only if α/β = 1; especially seen obviously for the dimensionless pulse width greaterthan 0.15. One should note that this is one of the conditions in which time ordering effectsvanishes.The last point we want to emphasize that what happens the entanglement dynamicsbetween two qubits if the time ordering effect vanishes. As mentioned before the timeordering effect vanishes for the special cases either α = β or J = 0. From correspondingequations it can be noted that for the case α = β , the concurrence function for the initialstate | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) is equal to 1, while for the initial state | Ψ(0) (cid:105) = | (cid:105) is equalto | sin (4 J t ) | and unaffected by the sequence of the kicks or Gaussian pulses. For the othercase, J = 0, under the influence of kick or Gaussian pulse sequences the concurrence for theinitial Bell state is always equal to 1, while the separable state remains separable at any time.These show an important fact that since the time ordering provides a connection betweeninteractions at different times, it is responsible from the nonlocal correlations between thequbits in time. VII. CONCLUSION We have investigated the dynamics of entanglement for two qubits that interact witheach other via Heisenberg XXX-type interaction under a time-dependent external magnetic22eld. Initial state of the system is considered to be pure Bell or separable states. The mainaim of the study was to investigate the controllability of the entanglement with a sequenceof pulse or kick type external fields.The effect of time ordering in the dynamics of concurrence is found to be important;concurrence calculated when the time ordering is neglected is found to be completely differentthan when it is taken into account. Time-dependent concurrence obtained after one, two,three and four kicks at different magnetic field strengths indicate that one can employcarefully chosen kick or kick sequences to produce high entanglement between two initiallynon-entangled qubits.We have also considered the effect of the pulse width of the external field on the en-tanglement dynamics by modelling the external field as a Gaussian pulse or a sequence ofGaussian pulses. Increasing the width of the pulse is found to enhance the control of highand steady entanglement.One should note that the external control field considered in the present study acts onboth of the qubits at the same time. It might be possible to use pulse sequences acting anindividual qubits at different times to obtain a better control of entanglement. References [1] Nielson M A and Chuang I L 2000 Quantum Computation and Quantum Information, (Cam-bridge: Cambridge University Press)[2] Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K 1996 Phys. Rev. A Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. 9] Sadiek G, Lashin E I and Abdalla M S 2009 Physica B Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Rev. Lett. Phys.Rev. A J. Phys. B: At. Mol.Opt. Phys. J. Magn. Reson. Nature Phys. Rev. Lett. J. Chem. Phys. Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. A J. Phys. B Phys. Rev. Lett. Phys. Rev. A Commun. Pure Appl. Math. Phys. Rev. Lett. Phys. Rev. Lett. Appl. Phys. Lett. IG. 9: (Colour online) The contour plot of concurrence versus J t and α/β , for a sequence of fourpositive Gaussian pulses of width τ for the initial pure states | Φ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ) (a), (c), (e)and (g) and | Ψ(0) (cid:105) = | (cid:105) (b), (d), (f) and (h). Here we assume four dimensionless pulse width as: J τ = 0 . , . , . , . 2. (There are ten equidistant contours of concurrence in the plots between 0(black) and 1 (white).)2. (There are ten equidistant contours of concurrence in the plots between 0(black) and 1 (white).)