Control and manipulation of entanglement between two coupled qubits by fast pulses
CControl and manipulation of entanglement betweentwo coupled qubits by fast pulses
Ferdi Altintas and Resul Eryigit
Department of Physics, Abant Izzet Baysal University, Bolu, 14280-Turkey.E-mail: altintas [email protected], [email protected]
Abstract.
We have investigated the analytical and numerical dynamics ofentanglement for two qubits that interact with each other via Heisenberg XXX-typeinteraction and subject to local time-specific external kick and Gaussian pulse-typemagnetic fields in x - y plane. The qubits have been assumed to be initially prepared indifferent pure separable and maximally entangled states and the effect of the strengthand the direction of external fast pulses on concurrence has been investigated. Thecarefully designed kick or pulse sequences are found to enable one to obtain constantlong-lasting entanglement with desired magnitude. Moreover, the time ordering effectsare found to be important in the creation and manipulation of entanglement by externalfields.PACS numbers: 03.65.Ud; 03.67.Mn; 75.10.Jm a r X i v : . [ qu a n t - ph ] N ov ntanglement Dynamics of Kicked Qubits
1. Introduction
Entanglement exhibited by composite correlated quantum systems has become one ofthe most widely investigated physics subject of recent years [1]. It provides promisingresource for some quantum information tasks, such as quantum teleportation [2],cryptography [3] and computing [4]. For those applications, the generation,manipulation, detection and control of entanglement between the quantum systemsshould be precise and effectively done during carrying out of the required operations. Aspin 1 / et al. showed that one qubit gates in spin-based quantumcomputers can be constructed with a global magnetic field and controllable Heisenbergexchange interactions [9]. Imamoglu et. al. demonstrated that 1-D Heisenberg chainof spin 1/2 particles (qubits) can be used as a prototypical system to study the role ofentanglement in quantum computational tasks [10].Numerous studies have been devoted to the control, production and manipulationof entanglement in Heisenberg spin chains with the help of external magnetic fields [11,12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Among them, Sadiek et al. investigated thetime evolution of entanglement between two Heisenberg XYZ coupled qubits under anexternal sinusoidal time-dependent nonuniform magnetic field and they demonstratedthat the entanglement of the system can be controlled and tuned by varying thetime-dependent magnetic field and the Heisenberg exchange coupling parameters [12].Abliz et al. studied the entanglement dynamics of two-qubit Heisenberg XYZ modelaffected by population relaxation and subject to various types of external magneticfields, such as an inhomogeneous static field, homogeneous exponentially varying andperiodically varying magnetic fields, and they demonstrated that high entanglementcan be produced, controlled and modulated with the help of external time-varyingfields despite the existence of dissipation [14]. Huang and Kais demonstrated thatthe entanglement in one-dimensional spin system, modeled by the XY Hamiltoniancan be localized between nearest-neighbor qubits for certain values of the step functionexternal magnetic fields [17]. In Ref. [11], the possibility of creating and controllingentangled states by changing the relative phase of control pulses was investigated.Huang et al. demonstrated that the dynamics of the nearest-neighbor entanglementfor one-dimensional spin system under sinusoidal magnetic fields displays a periodicstructure with a period related to that of the magnetic field [18]. Blaauboer and DiVincenzo presented a scheme for detecting entanglement between electron spins in adouble-quantum-dot nanostructure and they demonstrated separation and coherentlyrotating of entangled spins in quantum dots by using a time-dependent gate voltageand magnetic field [19].The form of the external magnetic field used in control of entanglement is animportant parameter in manipulation of the entanglement between the qubits. As ntanglement Dynamics of Kicked Qubits et al. and Shakov et al. have investigated the population and coherencedynamics of a single qubit under the influence of kick or Gaussian pulse sequencemagnetic fields [22, 23]. It was shown that the instantaneous pulses (kicks) providea full population transfer in a qubit from one state to the other which has a great dealof interest in quantum computing and control theory [24]. In our recent paper [20],we have extended the schema introduced in Refs. [22, 23] to two coupled qubits andhave investigated the dynamics of entanglement between two Heisenberg qubits underthe influence of strong delta function (kick) and Gaussian pulse type magnetic fieldsdirected along the z -axes. The qubits have been assumed to be initially prepared inthe separable | (cid:105) and the maximally entangled √ ( | (cid:105) + | (cid:105) ) states. We have shownthat the fast pulses provide an efficient way of controlling entanglement between thecoupled qubits and also by this control it is possible to manipulate the transition fromdisentangled to the entangled states of the system. Longitudinal control fields werefound to be ineffective in creating entanglement for the | (cid:105) and | (cid:105) type initial statesor manipulating the entanglement of √ ( | (cid:105) ± | (cid:105) ) type maximally entangled initialstates.In the present study, we extend the work in Ref. [20] to consider the effect of thetransverse field and analyze the dynamics of entanglement between two qubits thatinteract with each other via Heisenberg XXX type interaction that are subject to site-specific time-dependent magnetic control fields. Our aim is to widen the possibility ofcreating and manipulating entanglement with the tailored external kick and Gaussianpulse sequence type fields for arbitrary pure initial states. One important finding of thepresent work is the transverse external fields can control the dynamics of entanglementfor two qubits for initial states that could not be manipulated with the longitudinalfields. We have also showed that the opposite scenario can take place for transverseexternal fast pulses.The paper is organized as follows. In Sec. 2, the model and basic formulation forthe solution of time evolution are briefly discussed. In Sec. 3, Wootters concurrence asa measure of entanglement is introduced. Entanglement dynamics of two qubits undermultiple kicks analytically and multiple Gaussian pulses numerically are investigated inSecs. 4 and 5, respectively. In Sec. 6, we conclude with a summary of important results.
2. The model and basic formulation
In the present study, we consider two Heisenberg XXX-coupled qubits under time-dependent external magnetic fields in x - y plane. The time-dependent Hamiltonian forthis model can be represented as [14, 25] (¯ h = 1):ˆ H ( t ) = ˆ H + ˆ H int ( t ) , (1) ntanglement Dynamics of Kicked Qubits H = J (cid:88) i = x,y,z ˆ σ i ˆ σ i , ˆ H int ( t ) = 12 (cid:88) i =1 B i ( t )(cos( θ )ˆ σ xi + sin( θ )ˆ σ yi ) , (2)where ˆ σ i , ( i = x, y, z ) are the usual Pauli spin operators, B ( t ) and B ( t ) are theexternal time-dependent magnetic fields acting on the qubits 1 and 2, respectively, J isthe qubit-qubit interaction strength and θ is the angle between the magnetic fields andthe x -axes. For simplicity, we will assume 0 ≤ θ ≤ π . Here, we take J to be constantin time and assume all the time dependence in the systems’ Hamiltonian ˆ H ( t ) comesonly from ˆ H int ( t ). It is also possible to control qubit-qubit entanglement evolution bytime dependent coupling strength instead of magnetic fields [26]. It was shown that suchcontrols can be implemented physically, for example by using interacting flux qubits [26].The most general form of an initial pure state of the two-qubit system may begiven by the state vector | 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 numbers with (cid:88) i =1 | a i (0) | = 1. Then, underHamiltonian (1) the probability amplitudes evolve in time according to Schr¨odingerequation as: i ddt a ( t ) a ( t ) a ( t ) a ( t ) = J (cid:101) B ( t ) (cid:101) B ( t ) 0 (cid:101) B ( t ) ∗ − J J (cid:101) B ( t ) (cid:101) B ( t ) ∗ J − J (cid:101) B ( t )0 (cid:101) B ( t ) ∗ (cid:101) B ( t ) ∗ J a ( t ) a ( t ) a ( t ) a ( t ) , (3)where (cid:101) B k ( t ) = e − iθ B k ( t ) ( k = 1 ,
2) and (cid:101) B k ( t ) ∗ is its complex conjugate.Mathematically and conceptually, it is convenient to write the formal solution of Eq. (3)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)Here all the time-dependence of the system is contained in the time evolution operatorˆ U ( t ), while the initial conditions are specified in | Ψ(0) (cid:105) . The time evolution operatorˆ U ( t ) may be expressed as:ˆ U ( t ) = ← T e − i (cid:82) t ˆ H ( t (cid:48) ) dt (cid:48) = ← T e − i (cid:82) t ( ˆ H + ˆ H int ( t (cid:48) ) ) dt (cid:48) = ← 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)Here ← T is the so-called the Dyson time ordering operator which arranges the operatorsin order of increasing of time [27], for example, ← T ˆ H ( t i ) ˆ H ( t j ) = ˆ H ( t j ) ˆ H ( t i ) if t j > t i and ˆ H ( t i ) ˆ H ( t j ) otherwise. It gives rise to observable, nonlocal, time ordering effects if ntanglement Dynamics of Kicked Qubits (cid:104) ˆ H ( t j ) , ˆ H ( t i ) (cid:105) (cid:54) = 0 [22, 23]. Note that the non-trivial time dependence inˆ U ( t ) arises from time dependent ˆ H ( t ) and time ordering ← T .
3. Measure of entanglement
For two-qubit systems, as an entanglement measure, Wootters concurrence is a well-defined quantity [28]. Its value ranges from 0 for a separable state to 1 for a maximallyentangled (Bell) state. The concurrence function is defined as: C ( ˆ ρ ) = max { , (cid:113) λ − (cid:113) λ − (cid:113) λ − (cid:113) λ } , (6)where λ i ( i = 1 , , ,
4) are the eigenvalues of the matrix ˆ ρ ( t )(ˆ σ y ⊗ ˆ σ y ) ˆ ρ ( t ) ∗ (ˆ σ y ⊗ ˆ σ y )in descending order. Here ˆ ρ ( t ) is the density matrix of the qubits and ˆ ρ ( t ) ∗ is itscomplex conjugate. According to Schr¨odinger equation (3), it should be noted that theprobability amplitudes a i ( t ) ( i = 1 , , ,
4) evolve in time interdependently. From thispoint, it is convenient to consider a general time-dependent two-qubit pure state in theform: | Ψ( t ) (cid:105) = a ( t ) | (cid:105) + a ( t ) | (cid:105) + a ( t ) | (cid:105) + a ( t ) | (cid:105) . Then, it is straightforwardto show that the concurrence function (6) for the general pure state with density matrixˆ ρ ( t ) = | Ψ( t ) (cid:105) (cid:104) Ψ( t ) | is given by the simple equation as: C ( ˆ ρ ) = 2 | a ( t ) a ( t ) − a ( t ) a ( t ) | , (7)where the time-dependent coefficients may be given by Eq. (4) as: a i ( t ) = (cid:88) j =1 U ij ( t ) a j (0) , (8)where U ij ( t ) ( i, j = 1 , , ,
4) are the matrix elements of ˆ U ( t ). According to Eqs. (7)and (8), to study entanglement dynamics of two qubits, one has to present the initialpreparation of the qubits, i.e., a i (0), and the matrix elements of the time evolutionmatrix ˆ U ( t ). In this work, we will consider pure states as initial states. An initiallypure state remains pure at all times under the dynamics given by Eq. (3). It is wellknown that for pure states entanglement can quantify all quantum correlations betweentwo two-level systems, while such an identification is complicated for mixed states andentanglement signifies only a particular type of quantum correlation [29].
4. Entanglement dynamics of two coupled qubits under the influence ofthree positive kicks
Depending on the complexity of the time evolution in ˆ H int ( t ), the dynamics mayor may not be solved analytically. Most of the studies employ numerical solutionsof the time evolution in order to control entanglement between two level quantumsystems [11, 12, 13, 14, 15, 17, 18, 20]. However, analytic solutions are more convenientand easy to analyze, if they are available. For one qubit case, Shakov et al. listed someprogressive approximations in which the time evolution may be solved analytically [23].These approximations are the qubit having degenerate basis states, the external field ntanglement Dynamics of Kicked Qubits U ( t ) in Appendices A, Band C, respectively. They will be used in Eqs. (7) and (8) to calculate the dynamics ofconcurrence for various initial states. In Ref. [20], it was found that the entanglementof the qubits initially in | (cid:105) (or | (cid:105) ) and √ ( | (cid:105) ± | (cid:105) ) can be manipulated easilywith an external field in z -direction, while for the initial states of the type | (cid:105) , | (cid:105) orlinear combination of them were immune to such a manipulation. So, we consider thosestates that cannot be affected by longitudinal external fields and analyze their dynamicsunder a transverse field.In the following, we will discuss the effect of transverse field on the dynamics of thesystem which is initially in a separable, i.e., | Ψ(0) (cid:105) = | (cid:105) or in a maximally entangledstate i.e., | Ψ(0) (cid:105) = √ ( | (cid:105) + | (cid:105) ). To see the effect of kicks on the entanglementdynamics for these initial states, one should note that before the kick the qubits evolvein accordance with the time independent Hamiltonian ˆ H and the propagator is givenby ˆ 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 . (9)Based on Eqs. (7), (8) and (9), the initially separable state remains separable, while theconcurrence for the initial Bell state is equal to 1 at any time before the kick. That is,qubit-qubit interaction has no effect in the absence of external fields.Here we consider two qubits whose states are strongly perturbed by three positivekicks applied at times t = T , t = T and t = T . The magnetic fields on qubits can begiven as B ( t ) = α (cid:88) i =1 δ ( t − T i ) and B ( t ) = β (cid:88) i =1 δ ( t − T i ), where α and β are calledintegrated magnetic strengths [20], and δ ( t ) is the dirac delta function. For such a kicksequence, the integration over the time is trivial and the time evolution matrix (5) inthe presence of time ordering can be obtained easily [20, 22, 23]. In the Appendix part,the evolution matrices after each kick are presented and by using these propagatorsin Figs. 1(a) and 1(b), the effects of the sequence of three kicks on C ( ˆ ρ ) for | (cid:105) and √ ( | (cid:105) + | (cid:105) ) initial states are displayed, respectively, with θ = π/ α/β = 2 ntanglement Dynamics of Kicked Qubits Α (cid:144) Β= Α (cid:144) Β= Jt C H Ρ ï L H a L Α (cid:144) Β= Α (cid:144) Β= Jt C H Ρ ï L H b L Figure 1. C (ˆ ρ ) versus Jt for a sequence of three positive kicks applied at times T = 5 , T = 10 and T = 15 for the initially pure states | (cid:105) (a) and √ ( | (cid:105) + | (cid:105) ) (b)with θ = π . Here the dashed lines correspond to α/β = 2 and the solid lines to α/β = 3,and we set β = 1 and J = 1. or α/β = 3. The concurrence for the considered initial states in the time domains5 < t <
10, 10 < t <
15 and t >
15 may be found by using Eqs. (A.2), (B.2) and (C.2),respectively, in Eqs. (7) and (8). Both figures show the pronounced effect of the kick onentanglement dynamics; the concurrence of these initial states starts oscillating just aftera single kick with an increase in their oscillation amplitudes with the increase in the ratio α/β (see the time domain 5 < t < t = 5 inducesentanglement from the initially unentangled qubits (Fig. 1(a)) and yields oscillationsin the concurrence function for the Bell state (Fig. 1(b)). It is worth mentioning herethat for the magnetic fields acting in the z -axes, it was found that the qubits remainunentangled or maximally entangled state at all times for | (cid:105) or √ ( | (cid:105) + | (cid:105) ) initialstates despite the existence of strong external kicks [20]. Comparing one, two and threekick regions, the sole effect of the number of kicks is found to change the amplitude (orminimum) of C ( ˆ ρ ) for initial Bell state. As can be seen from Fig. 1(b), after secondkick, the amplitude of concurrence oscillations increases compared to that of first kick,while it decreases after third kick. On the other hand, for | (cid:105) state case, the effectof the number of kicks is complicated compared to the Bell state case, because eachkick can change the amplitude, maximum and minimum of C ( ˆ ρ ). It is obvious thatafter each kick the minimum of C ( ˆ ρ ) increases for initially separable state. The mostimportant observation from these figures is the possibility of obtaining almost steadyhigh concurrence around 1 after third kick for α/β = 3 and both initial states. Thisshows that for certain system parameters (here we set α = 3, β = 1, J = 1 and T = 5), it is possible to create highly entangled qubits from an initially separable stateby perturbing the qubits via instantaneous pulses. Also note that after second kick,the minimum of C ( ˆ ρ ) for Bell state can go to 0 for α/β = 2. As mentioned above,the qubit-qubit interaction has no effect on the entanglement of the considered statesbefore the kick, while their oscillation frequencies depend on the qubit-qubit interactionstrength, J , after the kick as can be seen from Figs. 1(a) and 1(b) as well as from thematrix elements in the Appendix part. ntanglement Dynamics of Kicked Qubits Figure 2. (Colour online) (a) shows the contour plot of C (ˆ ρ ) versus Jt and α/β with θ = π . (b) shows the contour plot of C (ˆ ρ ) versus Jt and θ/π with α/β = 3. Herethe contour plots are for | (cid:105) initial state and include three positive kicks applied attimes T = 5 , T = 10 and T = 15, and we set β = 1 and J = 1. In these contourplots we have assumed twenty equidistant contours of concurrence between 0 (black)and 1 (white). To further elucidate the effect of the strength and the direction of the magneticfields on C ( ˆ ρ ), we have displayed the contour plot of C ( ˆ ρ ) versus J t and α/β forthe separable initial state in Fig. 2(a) and Bell state in Fig. 3(a) with θ = π/ C ( ˆ ρ ) versus J t and θ/π have been plotted for | (cid:105) and √ ( | (cid:105) + | (cid:105) ) with α/β = 3, respectively. In these figures, twenty equidistant contoursof concurrence between 0 (black) and 1 (white) are shown. From Fig. 2(b), it can bededuced that the entanglement dynamics for the initially separable state is independentof the direction of external fields as long as the field is transverse, which also can be seenfrom the relevant time evolution matrix elements in Appendices where the concurrenceis θ -independent. Fig. 2(a) shows two pronounced results for α/β -dependence of theconcurrence for | (cid:105) state: i) The separable state is found to remain almost separabledespite the strong external kicks for α/β = 1 and α/β ∼ = 7 . , . , .
5. ii) The longlived high entanglement regions which are indicated in white straight stripes sectionshave long lifetimes only after the second and the third kicks. From the comparison ofthe one, two and three positive kick sections, it seems that each kick widens the longlived high entanglement α/β -area. This shows the necessity of using multiple kicks tocreate almost steady high entanglement. Also note that each kick creates a different α/β oscillatory structure for C ( ˆ ρ ).Contrary to the initially separable state case, the concurrence for the initial Bellstate strongly depends on the direction of the external magnetic fields if θ/π > . θ/π < .
04, the entanglement between two qubits ntanglement Dynamics of Kicked Qubits Figure 3. (Colour online) (a) shows the contour plot of C (ˆ ρ ) versus Jt and α/β with θ = π . (b) shows the contour plot of C (ˆ ρ ) versus Jt and θ/π with α/β = 3. Herethe contour plots are for √ ( | (cid:105) + | (cid:105) ) initial state and include three positive kicksapplied at times T = 5 , T = 10 and T = 15, and we set β = 1 and J = 1. In thesecontour plots we have assumed twenty equidistant contours of concurrence between0 (black) and 1 (white). is almost unperturbed by the external positive kicks. Similar to | (cid:105) state case, theconcurrence for √ ( | (cid:105) + | (cid:105) ) is found to be unaffected from three kicks for α/β = 1and α/β ∼ = 4 . , . , . , . , . , .
75 that can be deduced from Fig. 3(a). Thisfigure also shows that every kick region has a different α/β periodic structure andlong-lived high entanglement can be effectively obtained and controlled only after thirdpositive kick for the initial Bell state after perturbing the entanglement dynamicsof qubits with external kicks, for example in the time domain 15 < t <
20 for α/β ∼ = 2 . , . , . , . , . , .
0. From Fig. 3(b), one can see that it is possibleto get a constant long-lived entanglement with desired magnitude after third kick byadjusting θ . For example, C ( ˆ ρ ) ≈ . C ( ˆ ρ ) ≈ . C ( ˆ ρ ) ≈ .
08 and C ( ˆ ρ ) ≈ .
63 for θ/π = 0 . θ/π = 0 . θ/π = 0 .
25 and θ/π = 0 .
36, respectively.
5. Entanglement dynamics of two qubits under the influence of Gaussianpulses
The kicked approximation is based on the energy level of the qubit, ∆ E , and the widthof the pulse, τ , and is valid if ∆ Eτ <<
1. Depending on the physical implementation ofthe qubit, it might be difficult to obtain an external field as a delta function kick. Insteadof kicks, Gaussian pulses can be used. As is well known, the kicked approximation isthe limiting case of Gaussian pulses (cid:32) i.e., lim τ → α K √ πτ e − ( t − TK )2 τ = α K δ ( t − T K ) (cid:33) , thusGaussian pulses enable us to consider the effects of using finite-width pulses on ntanglement Dynamics of Kicked Qubits | (cid:105) andmaximally entangled √ ( | (cid:105) + | (cid:105) ) states. The concurrence for the considered initialstates may be calculated by solving numerically the set of first order coupled ordinarydifferential equations in Eq. (3): i ˙ a ( t ) = J a ( t ) + 12 e − iθ B ( t ) a ( t ) + 12 e − iθ B ( t ) a ( t ) ,i ˙ a ( t ) = 12 e iθ B ( t ) a ( t ) − J a ( t ) + 2 J a ( t ) + 12 e − iθ B ( t ) a ( t ) ,i ˙ a ( t ) = 12 e iθ B ( t ) a ( t ) + 2 J a ( t ) − J a ( t ) + 12 e − iθ B ( t ) a ( t ) ,i ˙ a ( t ) = 12 e iθ B ( t ) a ( t ) + 12 e iθ B ( t ) a ( t ) + J a ( t ) , (10)with replacing B ( t ) → α √ πτ (cid:88) i =1 e − ( t − Ti )2 τ and B ( t ) → β √ πτ (cid:88) i =1 e − ( t − Ti )2 τ . Here theGaussian pulses are assumed to have the same width τ and centered at times t = T , T and T . We will determine how the concurrences of the initially separable and Bellstates depend on the pulse width. It should be noted here that if τ were chosen smallenough, the results obtained in the previous section could be reached.In Figs. 4(a)-4(f), we have shown C ( ˆ ρ ) versus J t for a system strongly perturbedby three narrow Gaussian pulses for | (cid:105) and √ ( | (cid:105) + | (cid:105) ) initial states with θ = π and α/β = 2 or α/β = 3. Comparing Figs. 1(a) and 1(b) with the subfigures in Fig. 4,the width of the pulse changes the minimum, maximum and the amplitude of C ( ˆ ρ ) for | (cid:105) state, while for Bell state, this effect is in its minimum (or amplitude). The mostpronounced observation from these figures is the existence of constant high concurrencenearly 1 at times t >
15 for α/β = 3 and
J τ = 0 . α/β ratio, it is possible to obtain a state very closeto Bell state from an initially separable state by using a Gaussian pulse sequence.To further elucidate the interplay between a pulse sequence, relative magneticstrength on qubits, pulse width and entanglement, we display the contour plot of C ( ˆ ρ )as function of J t and α/β at three different pulse widths
J τ = 0 . , . , . < t <
10 in Fig.5(c)). Contrary to Fig. 2(a), α/β periodicity doesnot exist in Figs. 5(a)-5(c); increasing the pulse width coalesces the α/β -dependentoscillatory structure and produce non-periodic structures. Furthermore, | (cid:105) stateremains separable if and only if α/β = 1 under the strong influence of Gaussian pulses. ntanglement Dynamics of Kicked Qubits J Τ= Α (cid:144) Β= Α (cid:144) Β= Jt C H Ρ ï L H a L J Τ= Jt C H Ρ ï L H b L J Τ= Jt C H Ρ ï L H c L J Τ= Jt C H Ρ ï L H d L J Τ= Jt C H Ρ ï L H e L J Τ= Jt C H Ρ ï L H f L Figure 4. C (ˆ ρ ) versus Jt for a sequence of three positive Gaussian pulses centeredat T = 5 , T = 10 and T = 15 for the initially pure states | (cid:105) ((a), (c) and (e)) and √ ( | (cid:105) + | (cid:105) ) ((b), (d) and (f)) with θ = π . Here the dashed lines correspond to α/β = 2 and the solid lines to α/β = 3, and we consider three dimensionless pulsewidths as Jτ = 0 . Jτ = 0 . Jτ = 0 . J = 1 and β = 1. On the other hand, when τ → α/β = 1 as well as for α/β ∼ = 7 . , . , .
5, as was the case for the kick sequence. Peculiarly, Fig. 5(c) showsthat there is a sudden transition between long lasting maximally entangled and almostseparable states of the system just after a pulse for α = 13 β and α = 19 . β . Moreover,the subfigures in Fig. 5 demonstrates the possibility of obtaining desired values of steadyentanglement by carefully designed pulse or pulse sequence and system parameters.In Fig. 6, the contour plot of C ( ˆ ρ ) versus J t and α/β are displayed for √ ( | (cid:105) + | (cid:105) )initial state for the system under three Gaussian pulses having dimensionless pulsewidths as J τ = 0 . , . | (cid:105) state case, C ( ˆ ρ ) of Bell state isundisturbed by the highly wider Gaussian pulses if and only if α/β = 1. In Fig. 3(a),we have noted that the long lived high entanglement regions can only be obtained afterthird kick after disturbing the entanglement dynamics for the Bell state, while theseregions also appears after applying first and second Gaussian pulses as can be seen fromFigs. 6(b) and 6(c).Note that we have not displayed the contour plots of C ( ˆ ρ ) versus θ/π and J t for theGaussian pulse magnetic fields, because they provide no extra information compared to ntanglement Dynamics of Kicked Qubits Figure 5. (Colour online) The contour plot of C (ˆ ρ ) versus Jt and α/β with θ = π and Jτ = 0 . Jτ = 0 . Jτ = 0 . | (cid:105) initialstate and include three positive Gaussian pulses centered at times T = 5 , T = 10 and T = 15, and we set J = 1 and β = 1. In these contour plots we have assumed twentyequidistant contours of concurrence between 0 (black) and 1 (white). the ideal kick case; for the separable | (cid:105) initial state, C ( ˆ ρ ) is independent of θ regardlessof the value of the pulse width τ . For the initial Bell state, the overall θ -dependence isvery similar to that of the kick sequence control.We have also considered | (cid:105) separable, | Φ − (cid:105) = √ ( | (cid:105)−| (cid:105) ) and | Ψ ± (cid:105) = √ ( | (cid:105)±| (cid:105) ) Bell states as initial states and obtained some interesting dynamics. The dynamicsof | (cid:105) initial state under control sequences considered in the present study are foundto be exactly same as that of | (cid:105) initial state discussed above. From the propagatorsof Appendices and Eqs. (7) and (8), one can easily show that | Ψ + (cid:105) = √ ( | (cid:105) + | (cid:105) )initial state is immune to the applied transverse field; its concurrence remains 1 for ntanglement Dynamics of Kicked Qubits Figure 6. (Colour online) The contour plot of C (ˆ ρ ) versus Jt and α/β with θ = π and Jτ = 0 . Jτ = 0 . Jτ = 0 . √ ( | (cid:105) + | (cid:105) ) initial state and include three positive Gaussian pulses centered attimes T = 5 , T = 10 and T = 15, and we set J = 1 and β = 1. In these contourplots we have assumed twenty equidistant contours of concurrence between 0 (black)and 1 (white). every possible value of θ and α/β . Note that entanglement of this initial state canbe manipulated by using a longitudinal control field as shown in Ref. [20]. Note thatthe interaction Hamiltonian, ˆ H int ≈ cos θ (ˆ σ x + ˆ σ x ) + sin θ (ˆ σ y + ˆ σ y ) locally transformsBell states among themselves for θ = 0 and θ = π/
2. For example ˆ σ x . | Φ + (cid:105) = | Ψ + (cid:105) and ˆ σ x . | Φ − (cid:105) = | Ψ − (cid:105) as well as − i ˆ σ y . | Φ − (cid:105) = | Ψ + (cid:105) and − i ˆ σ y . | Φ + (cid:105) = | Ψ − (cid:105) . It is wellknown that entanglement is unaffected by local transformations, so one can easily showthat the dynamics of | Φ + (cid:105) at θ = 0 is the same as that of | Φ − (cid:105) at θ = π/ ntanglement Dynamics of Kicked Qubits | Ψ + (cid:105) = √ ( | (cid:105) + | (cid:105) ) which is found to be unaffected by the control sequence asdiscussed above. Also | Φ + (cid:105) at θ = π/ | Φ − (cid:105) at θ = 0 are transformed to | Ψ − (cid:105) and have similar entanglement dynamics; the concurrence for the initial state | Ψ − (cid:105) isindependent of θ . At the intermediate values of θ , the local transformations mix differentBell states and it is not straightforward to unentangle them. All of them can be easilychecked by using the propagators in the Appendices and the Eqs. (7) and (8). The effectof transverse fast pulses on the entanglement dynamics of | (cid:105) or | (cid:105) separable statesis qualitatively similar to the results for longitudinal fast pulse case [20], thus we havenot covered them.It was pointed out that the observable non-local time-ordering effects in time areexpected to be present if and only if the commutator of the total Hamiltonian (1) atdifferent times is nonzero, i.e, [ ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) )] (cid:54) = 0 [22, 23]. By using Eq. (1), it is easy toshow that the commutator [ ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) )] is equal to[ ˆ H ( t (cid:48)(cid:48) ) , ˆ H ( t (cid:48) )] = J D a − a − a ∗ aa ∗ − a − a ∗ a ∗ , (11)where D = (( B ( t (cid:48) ) − B ( t (cid:48) )) − ( B ( t (cid:48)(cid:48) ) − B ( t (cid:48)(cid:48) ))) and a = e − iθ . Eq. (11) showsthat the commutator vanishes for the cases when J = 0 and / or B ( t ) = B ( t ). Byusing the numerical solutions of Eq. (10) for the considered initial states and thepropagators (A.2), (B.2) and (C.2) in Eqs. (7) and (8), in the case of no qubit-qubitinteraction ( J = 0) as well as equal magnitude external fields ( α = β ), it is easy to showthat the concurrence for Bell state is equal to 1, while | (cid:105) state remains separable at alltimes and C ( ˆ ρ ) of both initial states is unaffected from the external kicks and Gaussianpulses. This situation can be also observed from the contour plots for α = β = 1 case.This finding indicates that ability to manipulate the entanglement in the system byusing external fields is closely related to the nonlocal time-ordering effects.
6. Conclusion
We have investigated the possibility of creation and control of entanglement betweentwo coupled qubits by using time-dependent external magnetic fields in x - y plane inthe form of delta function kicks and Gaussian pulses for initially pure separable andmaximally entangled states. Analytical (for kick sequence) and numerical (for Gaussianpulses) results presented and discussed in the paper indicate a number of interestingphenomena.Transverse fast pulses can be employed to create long lasting steady entanglementbetween two coupled qubits initially in a separable state | (cid:105) (or | (cid:105) ). Furthermore,entanglement of such a state can be finely-tuned by changing the integrated magneticstrength of the external field at qubit positions, while the direction of transverse pulsesare found not to effect the entanglement of this particular initial state. Similarly, ntanglement Dynamics of Kicked Qubits H ( t ) , ˆ H ( t )] is found to beimportant in the manipulation and control of entanglement by fast pulses. It is foundthat in the case of no time ordering (i.e., [ ˆ H ( t ) , ˆ H ( t )] = 0 which holds for the cases J = 0 and / or α = β ), the transverse fast pulses are found to be ineffective in creationand manipulation of entanglement between qubits initially in any type of pure states.We have also compared the effect of the external field being longitudinal ortransverse and found that with a transverse control one can both create and manipulateentanglement, while the longitudinal fast pulses can only manipulate the entanglementof a state which is already nonzero [20]. Moreover, the transverse fast pulses enable oneto manipulate, create and control entanglement for a number of initial states, while themanipulation of entanglement with the longitudinal fast pulses was found to be doneonly with a limited class of initial states [20].Longitudinal and transverse control fields in the form of kick and Gaussian pulsesequences can be used in a proper sequence to create, control and destroy entanglementfrom arbitrary initial pure states of two coupled qubits. The same formalism can beapplied to initially mixed states by using the time evolution matrix elements providedin Appendices to evolve the density matrix of the system as ˆ ρ ( t ) = ˆ U ( t ) ˆ ρ (0) ˆ U † ( t ) whichmight be interesting in analyzing the time evolution of more-general-than entanglementtype quantum correlations in mixed states, such as quantum discord [36]. Appendix A. Single positive kick
Here, we consider two qubits whose states are strongly perturbed by external fieldswhich may be expressed as a sudden kick at t = T . The time dependent magneticfields on qubits 1 and 2 may be expressed as B ( t ) = αδ ( t − T ) and B ( t ) = βδ ( t − T ),respectively, where α and β are called integrated magnetic strengths. For such a kickthe integration over the time is trivial and the time evolution matrix in Eq. (5) becomesas [20, 23]: 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 , (A.1)with matrix elements U = e − iJt cos (cid:18) α (cid:19) cos (cid:32) β (cid:33) = U ,U = 12 e − iJt (cid:18) e iJt cos (cid:18) ∆2 (cid:19) + cos (cid:18) Ω2 (cid:19)(cid:19) = U ,U = − e − iJt (cid:18) e iJt cos (cid:18) ∆2 (cid:19) − cos (cid:18) Ω2 (cid:19)(cid:19) = U , ntanglement Dynamics of Kicked Qubits U = 12 ie − iJt e − iθ (cid:18) e iJT sin (cid:18) ∆2 (cid:19) − sin (cid:18) Ω2 (cid:19)(cid:19) = e − iθ U ,U = − ie − iJt e − iθ (cid:18) e iJT sin (cid:18) ∆2 (cid:19) + sin (cid:18) Ω2 (cid:19)(cid:19) = e − iθ U ,U = − e − iJt e − iθ sin (cid:18) α (cid:19) sin (cid:32) β (cid:33) = e − iθ U ,U = − e iJ ( t − T ) e iθ (cid:32) cos (cid:32) β (cid:33) sin (cid:18) α (cid:19) sin( ξ ) + i cos (cid:18) α (cid:19) sin (cid:32) β (cid:33) cos( ξ ) (cid:33) = e iθ U ,U = − ie − iJt e − iθ (cid:18) e iξ sin (cid:18) ∆2 (cid:19) + sin (cid:18) Ω2 (cid:19)(cid:19) = e − iθ U , (A.2)where ∆ = ( α − β ) , Ω = ( α + β ) and ξ = 2 J ( t − T ). It should be noted that thepropagator given by Eq. (A.1) is valid only at times t > T . Appendix B. Two positive kicks
The next example is the positive-positive kick sequence applied at times t = T and t = T , namely, B ( t ) = α ( δ ( t − T ) + δ ( t − T )) and B ( t ) = β ( δ ( t − T ) + δ ( t − T )).Following the procedure given in Eq. (A.1), one obtains the time evolution matrix attimes t > T as [20, 23]: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 , (B.1)with matrix elements U = 14 e − iJt (cid:16) e iJT (cos(∆) −
1) + cos(∆) + 2 cos(Ω) (cid:17) = U ,U = 14 e − iJt (cid:16) e iJt + e iJ ( t − T ) (cos(∆) −
1) + e iJt cos(∆) + 2 cos(Ω) (cid:17) = U ,U = − e − iJt (cid:16) e iJt + e iJ ( t − T ) (cos(∆) −
1) + e iJt cos(∆) − (cid:17) = U ,U = 12 ie − iJt e − iθ (cid:16) e iJT cos(2 J T ) sin(∆) − sin(Ω) (cid:17) = e − iθ U ,U = − ie − iJt e − iθ (cid:16) e iJT cos(2 J T ) sin(∆) + sin(Ω) (cid:17) = e − iθ U ,U = 14 e − iJt e − iθ (cid:16) e iJT − − (1 + e iJT ) cos(∆) + 2 cos(Ω) (cid:17) = e − iθ U ,U = 14 ie − iJt e iθ (cid:16) e iJ ( t − T ) (1 + e iJT ) sin(∆) − (cid:17) = e iθ U ,U = − ie − iJt e − iθ (cid:16) e iJ ( t − T ) (1 + e iJT ) sin(∆) + 2 sin(Ω) (cid:17) = e − iθ U , (B.2)where ∆ = ( α − β ) and Ω = ( α + β ). Here, we have assumed equally distanced kicksapplied at times T = T and T = 2 T . ntanglement Dynamics of Kicked Qubits Appendix C. Three positive kicks
The final example is the sequence of three positive kicks applied at times t = T , t = T ,and t = T namely, B ( t ) = (cid:88) i =1 αδ ( t − T i ) and B ( t ) = (cid:88) i =1 βδ ( t − T i ). Following theprocedure given in Eq. (A.1), one can obtain the time evolution matrix at times t > T as [20]: 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 , (C.1)with matrix elements for T = T, T = 2 T and T = 3 T , U = 18 e − iJt (cid:18) (3 − e iJT − e iJT ) cos (cid:18) ∆2 (cid:19) + (1 + e iJT ) cos (cid:18) (cid:19) + 4 cos (cid:18) (cid:19)(cid:19) = U ,U = 12 e − iJt (cid:18) cos (cid:18) (cid:19) + e iJ ( t − T ) cos (cid:18) ∆2 (cid:19) ((1 + cos(4 J T )) cos(∆) + i sin(4 J T ) − (cid:19) = U ,U = 12 e − iJt (cid:18) cos (cid:18) (cid:19) − e iJ ( t − T ) cos (cid:18) ∆2 (cid:19) ((1 + cos(4 J T )) cos(∆) + i sin(4 J T ) − (cid:19) = U ,U = 12 ie − iJt e − iθ (cid:18) e iJT (cos(4 J T ) + 2 cos(2
J T ) cos(∆)) sin (cid:18) ∆2 (cid:19) − sin (cid:18) (cid:19)(cid:19) = e − iθ U ,U = − ie − iJt e − iθ (cid:18) e iJT (cos(4 J T ) + 2 cos(2
J T ) cos(∆)) sin (cid:18) ∆2 (cid:19) + sin (cid:18) (cid:19)(cid:19) = e − iθ U ,U = 18 e − iJt e − iθ (cid:18) (2 e iJT + e iJT −
3) cos (cid:18) ∆2 (cid:19) − (1 + e iJT ) cos (cid:18) (cid:19) + 4 cos (cid:18) (cid:19)(cid:19) = e − iθ U ,U = 12 ie − iJt e iθ (cid:18) e iJ ( t − T ) (cos(4 J T ) + 2 cos(2
J T ) cos(∆)) sin (cid:18) ∆2 (cid:19) − sin (cid:18) (cid:19)(cid:19) = e iθ U ,U = − ie − iJt e − iθ (cid:18) e iJ ( t − T ) (cos(4 J T ) + 2 cos(2
J T ) cos(∆)) sin (cid:18) ∆2 (cid:19) + sin (cid:18) (cid:19)(cid:19) = e − iθ U , (C.2)where ∆ = ( α − β ) and Ω = ( α + β ). References [1] Nielson M and Chuang I 2000 Quantum Computation and Quantum Communication (CambridgeUniversity Press, Cambridge, England)[2] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A, and Wootters W K 1993
Phys. Rev. Lett. Phys. Rev. Lett. ntanglement Dynamics of Kicked Qubits [4] Gruska J 1999 Quantum Computing (McGraw-Hill)[5] Loss D and Di Vicenzo D P 1998 Phys. Rev. A Nature (London)
Phys. Rev. A Nature (London)
Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Physica B
Phys. Rev. A Phys. Rev. A Phys. Rev. A ArXiv :quant-ph/1008.2784[17] Huang Z and Kais S 2005
Int. J. Quantum Inf. Phys. Rev. A Phys. Rev. Lett. J. Phys. A: Math. Theor. Phys. Rev. A Phys. Rev.A J. Phys. B: At. Mol. Opt.Phys. Phys. Rev. Lett.
Phys. Lett. A
Phys. Lett. A
J. Phys. B: At. Mol. Opt. Phys. Phys. Rev. Lett. J. Phys. A: Math. Gen. J. Magn. Reson.
Nature
Phys. Rev. Lett. J. Chem. Phys. Phys. Rev. Lett. Phys. Rev. Lett.88