Fast generation of N-atom Greenberger-Horne-Zeilinger state in separate coupled cavities via transitionless quantum driving
aa r X i v : . [ qu a n t - ph ] M a y Fast generation of N -atom Greenberger-Horne-Zeilinger state inseparate coupled cavities via transitionless quantum driving Wu-Jiang Shan , Ye-Hong Chen , Yan Xia , ∗ , and Jie Song , † Department of Physics, Fuzhou University, Fuzhou 350002, China Department of Physics, Harbin Institute of Technology, Harbin 150001, China
By jointly using quantum Zeno dynamics and the approach of “transitionless quan-tum driving (TQD)” proposed by Berry to construct shortcuts to adiabatic passage(STAP), we propose an efficient scheme to fast generate multiatom Greenberger-Horne-Zeilinger (GHZ) states in separate cavities connected by opitical fibers onlyby one-step manipulation. We first detail the generation of the three-atom GHZstates via TQD, then, we compare the proposed TQD scheme with the traditionalones with adiabatic passage. At last, the influence of various decoherence factors,such as spontaneous emission, cavity decay and fiber photon leakage, is discussed bynumerical simulations. All the results show that the present TQD scheme is fast andinsensitive to atomic spontaneous emission and fiber photon leakage. Furthermore,the scheme can be directly generalized to realize N -atom GHZ states generation bythe same principle in theory. PACS numbers: 03.67. Pp, 03.67. Mn, 03.67. HKKeywords: Quantum Zeno dynamics; Transitionless quantum driving; Greenberger-Horne-Zeilinger state; Cavity quantum electrodynamics.
I. INTRODUCTION
Quantum entanglement is not only one of the most important features in quantum me-chanics [1], but also a key resource for testing quantum mechanics against local hidden theory[2]. Recently, the entangled states have been applied in many fields in quantum informa-tion processing (QIP), such as quantum computing [3], quantum cryptography [4], quantum ∗ E-mail: [email protected] † E-mail: [email protected] teleportation [5, 6], quantum secret sharing [7], and so on. These promising applicationshave greatly motivated the researches in the generation of entangled states.It is worth noting that a typical entangled state so-called Greenberger-Horne-Zeilinger(GHZ) state | GHZ i = √ ( | i + | i ) , first proposed and named by Daniel M. Green-berger, Machael Horne and Anton Zeilinger [8], has raised much interest. Contrary to otherentangled states, the GHZ state exhibits some special features, such as it is the maximallyentangled state and can maximally violate the Bell inequalities [9]. In 2001, Zheng has pro-posed a scheme to test quantum mechanics against local hidden theory without the Bell’sinequalities by use of multiatom GHZ state [10]. Therefore, great interest has arisen regard-ing the significant role of GHZ state in the foundations of quantum mechanics measurementtheory and quantum communication. At present, the first and main problem we face is howto generate GHZ state by using current technologies. To our knowledge, in some experimen-tal systems, such as trapped ions systems [11], photons systems [12, 13], and atoms systems[14], scientists have realized the generation of such GHZ state. Recently, a promising ex-perimental instrument named cavity quantum electrodynamics (C-QED), which concernsthe interaction between the atom and the quantized field within cavity [15], has arousedmuch attention. Based on C-QED, many theoretical schemes for generating GHZ state havebeen proposed. For example, Li et al. have proposed a scheme to generate multiatom GHZstate under the resonant condition by Zeno dynamics [16], but the scheme is sensitive tothe atomic spontaneous emission and fiber photonic leakage. Hao et al. have proposed anefficient scheme to generate mulitiatom GHZ state under the resonant condition via adia-batic passage [17], but it takes too long time. Chen et al. have proposed a smart scheme toovercome the above drawbacks, but the scheme needs to trap three atoms in one cavity [18],such design is difficult to manipulate each atom in experiment and to construct a large-scalequantum network.On the other hand, in modern quantum application field, an important method to ma-nipulate the states of a quantum system is adiabatic passage, included “rapid” adiabaticpassage (RAP), stimulated Raman adiabatic passage (STIRAP), and their variants [19].The adiabatic passage covers the shortage with respect to errors or fluctuations of the pa-rameters compared with the resonant pulses, but its evolution speed is very slow, so it maybe useless in some cases. In recent years, shortcuts to adiabatic passage (STAP), whichaccelerates a slow adiabatic quantum process via a non-adiabatic route, has aroused a greatdeal of attention. Many theoretical proposals have been presented to realize QIP, such as fastpopulation transfer [20–23], fast entanglement generation [22, 24], fast implementation ofquantum phase gates [25, 26], and so on. To our knowledge, the main methods to constructeffective shortcuts has two forms: one is invariant-based inverse engineering based Lewis-Riesenfeld invariant (IBLR) [27] and the other is transitionless quantum driving (TQD) [28],which is pointed out by Berry. The two methods are strongly related [29], but also havetheir own characteristics. For example, the former does not need to modify the originalHamiltonian H ( t ), but the algorithm is suitable for some special physical models. The lat-ter needs to modify the original Hamiltonian H ( t ) to the “counter-diabatic driving” (CDD)Hamiltonian H ( t ) to speed up the quantum process. The fixed Hamiltonian H ( t ) can beobtained in theory, but it does not usually exist in real experiment.In addition, the quantum Zeno effect (QZE), first understood by Neumann [30] andnamed by Misra and Sudarshan [31], exhibits a especially experimental phenomenon thattransitions between quantum states can be hindered by frequent measurement. The systemwill evolve away from its initial state and remain in the so-called “Zeno subspace” defined bythe measure due to frequently projecting onto a multi-dimensional subspace [32, 33]. Thisis so-called quantum Zeno dynamics (QZD). Without making using of projection operatorsand non-unitary, “a continuous coupling” can obtain the same quantum Zeno effect insteadof discontinuous measurements [34, 35]. Now, we give a brief introduction of the quantumZeno dynamics in the form of continuous coupling [35]. Suppose that the system and itscontinuously coupling external system are governed by the total Hamiltonian H tot = H s + KH e , where H s is the Hamiltonian of the quantum system to be investigated, H e is anadditional Hamiltonian caused by the interaction with the external system, K is the couplingconstant. In the limit K → ∞ , the evolution operator of system can be expressed as U ( t ) = exp [ − it P n ( Kη n P n + P n H s P n )], where P n is the eigenprojection of H e correspondingto the eigenvalue η n , i.e., H e P n = η n P n [36].Inspired by the above useful works, we make use of Zeno dynamics and TQD to constructSTAP to generate N -atom GHZ state in C-QED. Our scheme has the following advantages:(1) The atoms are trapped in different cavities so that the single qubit manipulation is moreavailable in experiment. (2) The fast quantum entangled state generation for multiparticlein spatially separated atoms can be achieved in one step. (3) Numerical results show that ourscheme is not only fast, but also robust against variations in the experimental parametersand decoherence caused by atomic spontaneous emission and fiber photon leakage. In fact,further research shows that, the total operation time for the scheme is irrelevant to thenumber N of qubits.The paper is organized as follows. In section II, we give a brief introduction to theapproach of TQD proposed by Berry. In section III, we introduce the physical modal andthe systematic approximation by QZD. In section IV, we propose the schemes to generatethe three-atom GHZ state via TQD and adiabatic passage, respectively. The decoherencecaused by various factors is discussed by the numerical simulation. In section V, we directlygeneralize the scheme in section IV to generate N -atom GHZ states in one step. At last, wediscuss the experimental feasibility and make a conclusion about the scheme in section VI. II. TRANSITIONLESS QUANTUM DRIVING
Suppose a system is dominated by a time-dependent Hamiltonian H o ( t ) with instanta-neous eigenvectors | φ n ( t ) i and eigenvalues E n ( t ), H o ( t ) | φ n ( t ) i = E n ( t ) | φ n ( t ) i . (1)When a slow change satisfying the adiabatic condition does , the system governed by H o ( t )can be expressed at time t | ψ ( t ) i = e iξ n ( t ) | φ n ( t ) i ,ξ n ( t ) = − h Z t dt ′ E n ( t ′ ) + i Z t dt ′ h φ n ( t ′ ) | ∂ t ′ φ n ( t ′ ) i , (2)where ∂ t ′ = ∂∂ t ′ . Because the instantaneous eigenstates | φ n ( t ) i do not meet the Schr¨ o dingerequation i ¯ h∂ t | φ n ( t ) i = H | φ n ( t ) i , a finite probability that the system is in the state | φ m = n ( t ) i will occur during the whole evolution process even under the adiabatic condition.To construct the Hamiltonian H ( t ) that drives the instantaneous eigenvector | φ n ( t ) i ex-actly, i.e., there are no transitions between different eigenvectors during the whole evolutionprocess, we define the unitary operator U = X n e iξ n ( t ) | φ n ( t ) ih φ n (0) | , (3)we can formally solve the Schr¨odinger equation H ( t ) = i ¯ h ( ∂ t U ) U † . (4)Substituting eq. (3) into eq.(4), the Hamiltonian H ( t ) can be expressed H ( t ) = i ¯ h X n ( | ∂ t φ n ih φ n | − ¯ h X n | φ n i ˙ ξ n h φ n | ) , (5)the simplest choice is E n = 0, for which the bare states | φ n ( t ) i , with no phase factors, aredriven by H ( t ) = i ¯ h X n | ∂ t φ n ih φ n | . (6) III. PHYSICAL MODAL AND SYSTEMATIC APPROXIMATION BY QZD
For the sake of the clearness, let us first consider the physical modal that three identicalatoms a , a and a are trapped in three linearly arranged optical cavities C , C and C ,respectively. As shown in FIG. 1, each atom possesses one excited level | e i and three groundstates | g l i , | g o i and | g r i . The cavities C and C are single-mode, the cavity C are bi-mode. C , C and C are connected by the optical fibers f , f , respectively. Assuming that thetransition | e i a ↔ | g o i a is resonantly driven by a external classical field with the time-dependent Rabi frequencie Ω ( t ), while the transition | e i ↔ | g l i ( | e i ↔ | g r i )is resonantly coupled to the left-circularly(right-circularly) polarized cavity mode with thecoupling constant g l ( r ) , respectively.In the short-fiber limit, i.e., (2 L ¯ ν ) / (2 πc ) ≪ L is the length of the fibers, ¯ ν is the decayrate of the cavity fields into a continuum of fiber modes and c is the speed of light), onlyone resonant mode of the fiber interacts with the cavity mode [37]. In the rotating frame,the Hamiltonian of the whole system can be written as (¯ h = 1) H tot = H l + H c ,H l = X o =1 , Ω o ( t ) | e i a o h g | + H.c.,H c = g l a l | e i a h g l | + g l a l | e i a h g l | + g r a r | e i a h g r | + g r a r | e i a h g r | + v b † ( a l + a l ) + v b † ( a r + a r ) + H.c., (7)where a † kl ( a † kr ) and a kl ( a kr ) denote the creation and annihilation operators for the left-circularly (right-circularly) polarized mode of cavities C k ( k = 1 , , b † j and b j denote the creation and annihilation operators associated with the resonant mode of fiber f j ( j = 1 , g l = g l = g r = g r = g and v = v = v . If the initial state of the whole system is | g o g l g r i| i C | i C | i C | i f | i f (here | g o g l g r i = | g o g l g r i a a a ), the whole system evolves in the following subspaces | φ i = | g o g l g r i| i C | i C | i C | i f | i f , | φ i = | eg l g r i| i C | i C | i C | i f | i f , | φ i = | g l g l g r i| i C | i C | i C | i f | i f , | φ i = | g l g l g r i| i C | i C | i C | i f | i f , | φ i = | g l g l g r i| i C | i C | i C | i f | i f , | φ i = | g l eg r i| i C | i C | i C | i f | i f , | φ i = | g l g r g r i| i C | i C | i C | i f | i f , | φ i = | g l g r g r i| i C | i C | i C | i f | i f , | φ i = | g l g r g r i| i C | i C | i C | i f | i f , | φ i = | g l g r e i| i C | i C | i C | i f | i f , | φ i = | g l g r g o i| i C | i C | i C | i f | i f , (8)where | ijk i ( i, j, k ∈ [ e, g l , g o , g r ]) denotes the state of the atoms in every cavity, | n i s ( s = C , C , f , f ) means that the quantum field state of system contains n photons. | n n i C means that the number of left-circularly photon is n and the number of right-circularlyphoton is n in the cavity C .Under the Zeno condition g, v ≫ Ω , Ω , the Hilbert subspace is split into nine invariantZeno subspace Z = {| φ i , | ψ i , | φ i} , Z = {| ψ i} ,Z = {| ψ i} , Z = {| ψ i} , Z = {| ψ i} , Z = {| ψ i} ,Z = {| ψ i} , Z = {| ψ i} , Z = {| ψ i} , (9)where the eigenstates of H c are | ψ i = N ( | φ i − gv | φ i + | φ i − gv | φ i + | φ i ) , | ψ i = N ( −| φ i + ε | φ i − η | φ i − χ | φ i + χ | φ i + η | φ i − ǫ | φ i + | φ i ) , | ψ i = N ( −| φ i − ε | φ i − η | φ i + χ | φ i − χ | φ i + η | φ i + ǫ | φ i + | φ i ) , | ψ i = N ( | φ i − µ | φ i − ζ | φ i + δ | φ i − θ | φ i + δ | φ i − ζ | φ i − µ | φ i + | φ i ) , | ψ i = N ( | φ i + µ | φ i − ζ | φ i − δ | φ i − θ | φ i − δ | φ i − ζ | φ i + µ | φ i + | φ i ) , | ψ i = N ( −| φ i + ε | φ i − η | ψ i + χ | φ i − χ | φ i + η | φ i − ǫ | φ i + | φ i ) , | ψ i = N ( −| φ i − ε | φ i − η | ψ i − χ | φ i + χ | φ i + η | φ i + ǫ | φ i + | φ i ) , | ψ i = N ( | φ i − µ | φ i + ζ | φ i − δ | φ i + θ | φ i − δ | φ i + ζ | φ i − µ | φ i + | φ i ) , | ψ i = N ( | φ i + µ | φ i + ζ | φ i + δ | φ i + θ | φ i + δ | φ i + ζ | φ i + µ | φ i + | φ i ) , (10)with the corresponding eigenvalues λ = 0 , λ = − p ( g + 2 v − A ) / , λ = p ( g + 2 v − A ) / ,λ = − p (3 g + 2 v − A ) / , λ = p (3 g + 2 v − A ) / , λ = − p ( g + 2 v + A ) / ,λ = p ( g + 2 v + A ) / , λ = − p (3 g + 2 v + A ) / , λ = p (3 g + 2 v + A ) / , (11)where the parameters are ǫ = p g + 2 v − A √ g , η = − g + 2 v − A gv , χ = p g + 2 v − A ( g + A )2 √ gv ,µ = p g + 2 v − A √ g , ζ = − g − v + A gv , δ = p g + 2 v − A ( − g + A )2 √ gv , θ = − g + Av ,ǫ = p g + 2 v + A √ g , η = − g + 2 v + A gv , χ = p g + 2 v + A ( − g + A )2 √ gv ,µ = p g + 2 v + A √ g , ζ = g + 2 v + A gv , δ = p g + 2 v + A ( g + A )2 √ gv , θ = g + Av , (12)in addition, A = p g + 4 v and N w is the normalization factor of the eigenstate | ψ w i ( w =1 , , · · · , k th Zeno subspace Z k is P βk = | β ih β | , ( | β i ∈ Z k ) . (13)The Hamiltonian in Eq. (8) can be approximately given by H tot ≃ X k,β,γ λ k P βk + P βk H l P γk = X k =2 λ k | ψ k ih ψ k | + N (Ω | φ ih ψ | + Ω | φ ih ψ | + H.c. ) . (14)If the initial state is | g o g l g r i| i C | i C | i C | i f | i f , it reduces to H eff = N (Ω | φ ih ψ | + Ω | φ ih ψ | + H.c. ) , (15)which can be treated as a simple three-level system with an excited state | ψ i and twoground states | φ i and | φ i . Then we obtain the eigenvectors and eigenvalues of the effectiveHamiltonian H eff as | η o ( t ) i = cos θ ( t )0 − sin θ ( t ) , | η ± ( t ) i = 1 √ sin θ ( t ) ± θ ( t ) , (16)with the corresponding eigenvalues η = 0 and η ± = ± N Ω, and tan θ = Ω Ω and Ω = p Ω + Ω . IV. THE GENERATION OF THE THREE-ATOM GHZ STATE VIATRANSITIONLESS QUANTUM DRIVING AND ADIABATIC PASSAGEA. Adiabatic passage method
For the sake of the clearness, we first briefly present how to generate the three-atom GHZstate via adiabatic passage. When the adiabatic condition |h n | ∂ t n ± i| ≪ | λ ′± | is fulfilled welland the initial state is | ψ (0) i = | φ i , the state evolution will always follow | n ( t ) i closely.To generate the three-atom GHZ states via the adiabatic passage and meet the boundaryconditions of the fractional stimulated Raman adiabatic passage (STIRAP),lim t →−∞ Ω ( t )Ω ( t ) = 0 , lim t → + ∞ Ω ( t )Ω ( t ) = tan α, (17)we need properly to tailor the Rabi frequencies Ω ( t ) and Ω ( t ) in the original Hamiltonian H tot Ω ( t ) = sin α Ω exp [ − ( t − t − t f / t c ] , Ω ( t ) = Ω exp [ − ( t + t − t f / t c ] + cos α Ω exp [ − ( t − t − t f / t c ] , (18)where Ω is the pulse amplitude and t f is the operation time. t c and t are some relatedparameters to be chosen for the best performance of the adiabatic passage process. Inorder to achieve better performance and meet the boundary conditions, we suitably chosethe parameters that tan α = 1, t = 0 . t f and t c = 0 . t f . As shown in Fig. 2, thetime-dependent Ω ( t ) / Ω and Ω ( t ) / Ω versus t/t f are plotted with the fixed values t and t c . With the above parameters, we obtain our wanted three-atom GHZ state | ψ ( t f ) i =( | φ i − | φ i ) / √ B. Transitionless quantum driving method
To reduce the evolution time and obtain the same state as the adiabatic passage, weuse the approach of TQD to construct STAP. As introduced in the above, STAP speedsup a slow adiabatic passage via a non-adiabatic passage route to achieve a same outcome,and the TQD method is a important route to construct shortcuts. According to the ideasproposed by Berry [28], the instantaneous states in Eq. (16) do not meet the Schr¨odingerequation, i.e., i∂ t | n k i 6 = H eff | n k i ( k = 0 , ± ), so the situation that the system starts fromthe state | ψ n (0) i and ends up in the state | ψ m = n ( t ) i occurs in a finite probability evenunder the adiabatic condition. To drive the instantaneous states | n k i ( k = 0 , ± ) exactly, welook for a Hamiltonian H ( t ) related to the original Hamiltonian H eff according to Berry’stransitionless tracking algorithm [28]. From section II, we know the simplest Hamiltonian0 H ( t ) possessed the form, H ( t ) = i X , ± | ∂ t n k ( t ) ih n k ( t ) | . (19)Substituting Eq. (16) in Eq. (19), we obtain H ( t ) = i ˙ θ | φ ih φ | + H.c., (20)where ˙ θ = [ ˙Ω ( t )Ω ( t ) − Ω ( t ) ˙Ω ( t )] / Ω . This is our wanted CCD Hamiltonian to constructSTAP, and we will detail how to construct this Hamiltonian in experiment later.For the present system, the CDD Hamiltonian H ( t ) is given in Eq. (18), but it isirrealizable under current experimental condition. Inspired by Refs. [17, 19], we find analternative physically feasible (APF) Hamiltonian whose effect is equivalent to H ( t ). Thedesign is shown in Fig. 3, the atomic transitions is not resonantly coupled to the classicallasers and cavity modes with the detuning ∆. The Hamiltonian of the system reads H ′ tot = H c + H l + H d , where H d = P k =1 ∆ | e i k h e | . Then, similar to the approximation by QZD insection III, we also obtain an effective Hamiltonian for the non-resonant system H ′ eff = N (Ω | φ ih ψ | + Ω | φ ih ψ | + H.c. ) + 3∆ N | ψ ih ψ | . (21)When the large detuning condition 3∆ N ≥ Ω , Ω is satisfied, we can adiabatically elimi-nate the state | ψ i and obtain the final effective Hamiltonian H fe = − Ω | φ ih φ | − Ω | φ ih φ | − Ω Ω
3∆ ( | φ ih φ | + | φ ih φ | ) . (22)For simplicity, we set Ω = Ω = Ω( t ). The front two terms caused by Stark shift can beremoved and the Hamiltonian becomes H ( t ) = Ω x | φ ih φ | + H.c., (23)where Ω x = − Ω . The equation has a similar form with Eq. (20), but the effective couplingsbetween i ˙ θ and Ω x exist 3 π/ → − i Ω . Then, the eigenstates of H eff become | η ′ o ( t ) i = cos θ ( t )0 i sin θ ( t ) , | η ′± ( t ) i = 1 √ sin θ ( t ) ± − i cos θ ( t ) , (24)1and the CDD Hamiltonian H ( t ) becomes H ( t ) = − ˙ θ | φ ih φ | − ˙ θ | φ ih φ | . (25)Compared Eq. (23) with Eq. (25), we can easily get the CDD Hamiltonian when thecondition Ω x = − ˙ θ is satisfied.Ω ( t ) = Ω ( t ) = Ω( t ) = p
3∆ ˙ θ. (26) C. Numerical simulations and analyses
Next we will show that it takes less time to get the target state on the situation governedby the APF Hamiltonian H ′ tot via TQD than by the original Hamiltonian H tot via adiabaticpassage. The time-dependent population for any state | ψ i is defined as P = |h ψ | ρ ( t ) | ψ i| ,where ρ ( t ) is the corresponding time-dependent density operator. We present the fidelityversus the laser pulses amplitude Ω o and the operation time t/t f via adiabatic passage inFig. 4. As shown in Fig. 4, we can know that the bigger the laser pulse amplitude is,the less time that the system evolution to the target state needs. However, we need tosatisfy the Zeno conditions g, v ≫ Ω , Ω , so we set Ω = 0 . g . In Fig. 5, we display thetime-dependent populations of the states | φ i , | ψ target i , and | φ i via adiabatic passage. Asdepicted in Fig. 4 and Figs. 5, the operation time needs t f ≥ /g to achieve an idealresult at least. It is awkward in some case.Next we will detail the evolution governed by the APF Hamiltonian H ′ tot via TQD. Ac-cording to eq. (24) we finally get a GHZ state | ψ ( t f ) = √ ( | φ i + i | φ i ). In Fig. 6, wepresent the relationship between the fidelity of the three-atom GHZ state (governed by theAPF Hamiltonian) and two parameters ∆ and t f when Ω = 0 . g to satisfy the Zeno condi-tion, where the fidelity of the three-atom GHZ state is defined as F = |h GHZ | ρ ( t f ) | GHZ i| ( ρ ( t f ) is the desity operator of the whole system when t = t f ). We find that a wide rangefor parameters ∆ and t f can obtain a high fidelity of the three-atom GHZ state, and thefidelity increases with the increasing of ∆ and the decreasing of t f . In order to satisfy thelarge detuning condition, we set ∆ = 2 . g . The Fig. 6 reveals that the operation time needs t f ≥ /g via TQD at least. In Figs. 7 we plot the operation time for the creation of theGHZ state governed by H ′ tot and by H tot with the parameters that t f = 72 /g , Ω = 0 . g ,∆ = 2 . g and g = v . Numerical results show that the APF Hamiltonian H ′ tot can govern2the evolution to a perfect GHZ state | ψ ( t f ) i from | ψ i in a relatively short interaction timewhile the original Hamiltonian H tot can not.In above analysis, we do not consider the influence of decoherence caused by variousfactors, such as spontaneous emissions, cavity decays and fiber photon leakages. In fact, thedecoherence is unavoidable during the evolution of the whole system in experiment. Themaster equation of the whole system is written as˙ ρ = − i [ H tot , ρ ]+ X k =1 γ k σ − k ρσ + k − σ + k σ − k ρ − ρσ + k σ − k )+ X k =1 κ c k a l,k ρa + l,k − a + l,k a l,k ρ − ρa + l,k a l,k )+ X k =2 κ c k a r,k ρa + r,k − a + r,k a r,k ρ − ρa + r,k a r,k )+ X k =1 κ f k b k ρb + k − b + k b k ρ − ρb + k b k ) , (27)where γ k is the atomic spontaneous emission rate for the k th atom and κ c ( f ) is the decayrate of the k th cavity ( k th fiber), σ − k denotes the atomic transition from the ground states | m i ( m = g , g l , g r ) to the excited state | e i . For the sake of simplicity, we assume that γ = γ = γ = γ , κ c = κ c = κ c = κ c and κ f = κ f = κ f . As shown in Fig. 8, weplot the fidelity governed by the APF Hamiltonian H ′ tot and by the original Hamiltonian H tot and the dimensionless parameters γ/g , κ c /g and κ f /g , respectively. We can draw aconclusion that the fidelities are almost unaffected by the fiber decay both via TQD and viaadiabatic passage. We focus on the main decoherence factors included the cavity decay andthe atomic spontaneous emission. As shown in Figs. 9, we plot the fidelity versus the cavitydecay and the atomic spontaneous emission. We can know the most important decoherencefactor is the cavity decay. This result can be understood from Ref. [18] that if the Zenocondition can not be satisfied very well, the populations of the intermediate states includingthe cavity excited states can not be suppressed ideally.From the above anslysis, we can obviously know that the evolution time from the initialstate to the target state via TQD is t f = 72 /g when Ω = 0 . g , ∆ = 2 . g , t = 0 . t f , t c = 0 . t f and g = v , while the evolution time for the adiabatic passage is t f = 400 /g when3Ω = 0 . g , t = 0 . t f , t c = 0 . t f and g = v . So, the benefit of the TQD method is shownobviously that the speed via TQD method is faster than that via adiabatic passage. It ismore worthy to note that the fidelity of the target state via TQD is almost equal to thatvia adiabatic passage. So our scheme has a huge advantage compared with the proposalsvia adiabatic passage. That means the present scheme via STAP method is not only fastbut also robust.As we all know, it is necessary for a good scheme to tolerate the deviations of the ex-perimental parameters, because it is impossible to avoid the operational imperfection inexperiment. Define that δx = x ′ − x is the deviation of the ideal value x , x ′ is the actualvalue. In Fig. 10, we plot the fidelity of the target state | ψ target i versus the deviations of theexperimental parameters g , v , Ω , and T ( T = t f denotes the operation time). Numericalresults demonstrate that our scheme is robust against the fluctuation of the experimentalparameters. V. THE GENERATION OF THE N -ATOM GHZ STATE VIATRANSITIONLESS QUANTUM DRIVING Next we briefly present the generalization of the scheme in Section IV to generate N -atom GHZ state by the same principle. We consider the physical configuration shown inFig. 11, where N atoms a , a , · · · , a N are trapped in N cavities C , C , · · · , C N connected by N − f , f , · · · , f N − , respectively. The level configurations of theatoms between two ends are the same as that of the atom a in the three-atom case, andthe level configurations of a and a N are the same as those of a and a in the three-atomcase, respectively. The Hamiltonian of the present system can be written as in the rotationframework H total = H ′ l + H ′ c ,H ′ l = Ω ′ | e i a h g | + Ω ′ N | e i a N h g | + H.c.,H ′ c = N − X i =1 g i,l a i,l | e i a i h g l | + N X j =2 g j,r a j,r | e i a j h g r | N − X k =2 [ v k − b † k − ( a k − ,l + a k,l ) + v k b † k ( a k,r + a k +1 ,r )] + H.c.. (28)Let us consider the situation where N is an odd number, i.e., N = 2 l + 1 , ( l = 1 , , , · · · ).Suppose that the initial state of the atoms is | g g l g r g l g r · · · g r i while all the cavities andfibers are vacuum, then the system can be expended in the following subspace | φ ′ i = | g g l g r · · · g r i| i all , | φ ′ i = | eg l g r · · · g r i| i all , | φ ′ i = | g l g l g r · · · g r i| i c , | φ ′ i = | g l g l g r · · · g r i| i f , | φ ′ i = | g l g l g r · · · g r i| i c , | φ ′ i = | g l eg r · · · g r i| i all , | φ ′ i = | g l g r g r · · · g r i| i c , | φ ′ i = | g l g r g r · · · g r i| i f , | φ ′ i = | g l g r g r · · · g r i| i c , | φ ′ i = | g l g r e · · · g r i| i all , · · ·| φ ′ N − i = | g l g r · · · g r g i| i all , (29)where | i all means that there is none photon in all boson modes, | n n i s i ( s = C, f. i =1 , , · · · , N ) means that there are n left-circularly photon and n right-circularly photonin the corresponding cavity C i or fiber f i .Similar to the above procedure from Eq. (10) to Eq. (16), we get an effective Hamiltonian H eff ( N ) = N ′ (Ω ′ | φ ′ ih ψ ′ | + Ω ′ N | φ ′ N − ih ψ ′ | + H.c. ) , (30)where | ψ ′ i = N ′ ( N X i =1 | φ ′ i − i − N − X i =1 gv | φ ′ i i ) . (31)In addition, the eigenstates and eigenvalues of the Hamiltonian in Eq. (30) can be writtenas | χ o ( t ) i = cos θ ′ ( t )0 − sin θ ′ ( t ) , | χ ± ( t ) i = 1 √ sin θ ′ ( t ) ± θ ′ ( t ) , (32)5with the corresponding eigenvalues χ ′ = 0 and χ ′± = ± N ′ Ω ′ , where tan θ ′ = Ω ′ Ω ′ N andΩ ′ = p Ω ′ + Ω ′ N . Substituting Eq. (32) in Eq. (19), we obtain H ′ ( t ) = i ˙ θ ′ | φ ′ ih φ ′ N − | + H.c., (33)where ˙ θ ′ = [ ˙Ω ′ ( t )Ω ′ N ( t ) − Ω ′ ( t ) ˙Ω ′ N ( t )] / Ω ′ .Inspired by the above idea in section IV, we make the system into a non-resonant systemto construct the CDD Hamiltonian in Eq. (33). Therefore, the Hamiltonian of the presentsystem reads H ′ total = H ′ l + H ′ c + H ′ d , where H ′ d = P Ni =1 ∆ | e ih e | . Similar to the aboveprocedure from Eq. (22) to Eq. (23) in Section IV, we obtain the final effective Hamiltonian H ′ fe ( N ) = − Ω ′ | φ ′ ih φ ′ | − Ω ′ N | φ ′ N − ih φ ′ N − | − Ω ′ Ω ′ N
3∆ ( | φ ′ ih φ ′ N − | + | φ ′ N − |ih φ ′ | ) . (34)For simplicity, we set Ω ′ = Ω ′ N = Ω ′ , the front two terms caused by Stark shift can beomitted and the Hamiltonian becomes H N = Ω ′ x ( t ) | φ ′ ih φ ′ N − | + H.c., (35)where Ω ′ x ( t ) = − Ω ′ . To guarantee their consistency, we put a change that Ω N → − i Ω N .Then the eigenstates of H eff ( N ) become | χ ′ o ( t ) i = cos θ ′ ( t )0 i sin θ ′ ( t ) , | χ ′± ( t ) i = 1 √ sin θ ′ ( t ) ± − i cos θ ′ ( t ) , (36)and the CDD Hamiltonian H ( t ) becomes H ′ ( t ) = − ˙ θ ′ | φ ′ ih φ ′ N − | − ˙ θ ′ | φ ′ N − ih φ ′ | . (37)Compared Eq. (35) with Eq. (37), we can easily get the CDD Hamiltonian when thecondition Ω ′ x = − ˙ θ ′ is satisfied.Ω ′ ( t ) = Ω ′ N ( t ) = Ω ′ ( t ) = p
3∆ ˙ θ ′ . (38) VI. EXPERIMENTAL FEASIBILITY AND CONCLUSIONS
Now experimental feasibility needs to be discussed. The configuration of Rb can besuitable for our proposals. Under current experimental condition a set of CQED parameters6 g = 2 π × M Hz , γ = 2 π × . M Hz , and κ c = 2 π × . M Hz are available with thewavelength in the region 630 − .
9% [39]. The opticalfiber decay at a 852nm wavelength is about 2.2dB/km [40], which means the fiber decayrate is about κ f = 1 . × Hz . With the above parameters, we obtain a relatively highfidelity about 97 . N -atom GHZ state in separate coupled cavities via transitionless quantum driving (TQD)only by one-step manipulation. We apply a promising method to construct STAP by jointutilization of the Zeno dynamics and the approach of TQD in the cavities QED system.The method features are that we do not need to control the time exactly and the evolutionprocess is fast. Because the atoms are trapped in separate coupled cavity, the single qubitmanipulation can be realized easily. When considering dissipation, we can see that themethod is robust against the decoherences caused by the atomic spontaneous emission andfiber decay. The results show that the scheme has a high fidelity and may be possible toimplement with the current experimental technology. So, the scheme is fast, robust andeffective. We hope the scheme can be used to generate multi-atom GHZ state in the future. ACKNOWLEDGEMENT
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Quantum Electron. , 900-908(2004). e(cid:13) g(cid:13) L(cid:13) g(cid:13)
R(cid:13) g(cid:13) e(cid:13) g(cid:13)
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W(cid:13) l(cid:13) g(cid:13) l(cid:13) g(cid:13) r(cid:13) g(cid:13) r(cid:13) g(cid:13) a(cid:13) a(cid:13) a(cid:13)
C(cid:13)
C(cid:13)
C(cid:13) f(cid:13) f(cid:13)
FIG. 1: The structure of the experimental setup and atoms. Three identical atoms a , a and a are trapped in three separated cavities C , C and C , which are linked by two fibers f , f . t/t f W / W and W / W W / W W / W FIG. 2: The laser pulses Ω / Ω and Ω / Ω versus t/t f . e(cid:13) g(cid:13) L(cid:13) g(cid:13)
R(cid:13) g(cid:13) e(cid:13) g(cid:13)
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R(cid:13) g(cid:13) l(cid:13) g(cid:13) l(cid:13) g(cid:13) r(cid:13) g(cid:13) r(cid:13) g(cid:13) a(cid:13) a(cid:13) a(cid:13)
C(cid:13)
C(cid:13)
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D(cid:13)D(cid:13)D(cid:13) f(cid:13) f(cid:13)
W(cid:13)
W(cid:13)
FIG. 3: The structure of the experimental setup and atoms for the APF Hamiltonian. W /g g T FIG. 4: The fidelity versus the laser pulses amplitude Ω and the operation time t/t f . gt P opu l a t i on | f æ | y target æ (a) gt P opu l a t i on | f æ | f æ (b) FIG. 5: (a) The population P target of the target state | ψ target i and the population P of theinitial state | ψ (0) i governed by the original Hamiltonian H tot via the adiabatic passage. (b) Thepopulation P ( t ) of the states | φ i and the population P ( t ) of the states | φ i governed by theoriginal Hamiltonian H tot via adiabatic passage. The parameters are collectively with the fixedvalues Ω = 0 . g , g = v , t = 0 . t f , t c = 0 . t f and t f = 400 /g . D /ggt f F FIG. 6: The fidelity F of the target state | ψ ( t f ) i governed by H ′ tot versus the interaction time gt f and the detuning ∆ /g . gt P opu l a t i on | f æ | f æ (a) gt P opu l a t i on | f æ | f æ (b) FIG. 7: The population P ( t ) of the state | φ i and the population P ( t ) of the state | φ i governedby (a) the APF Hamiltonian H ′ tot with ∆ = 2 . g . (b) The original Hamiltonian H tot collectivelywith the fixed values Ω = 0 . g , g = v , t = 0 . t f , t c = 0 . t f , and t f = 72 /g . F g /g k c /g k f /g (a) F g /g k c /g k f /g (b) FIG. 8: The fidelity of the target state | ψ ( t f ) i governed by (a) the APF Hamiltonian H ′ tot with∆ = 2 . g , t f = 72 /g and Ω = 0 . g . (b) the original Hamiltonian H tot with t f = 153 /g , andΩ = 0 . g collectively with the fixed values t = 0 . t f , and t c = 0 . t f versus the dimensionlessparameters γ/g , κ c /g , and κ f /g , respectively. g /g k c /g F (a) g /g k c /g F (b) FIG. 9: The fidelity of the target state | ψ ( t f ) i governed by (a) the APF Hamiltonian H ′ tot with∆ = 2 . g , t f = 72 /g , and Ω = 0 . g . (b) the original Hamiltonian H tot with t f = 153 /g , andΩ = 0 . g collectively with the fixed values t = 0 . t f , and t c = 0 . t f versus the dimensionlessparameters γ/g and κ c /g . −0.1 −0.05 0 0.05 0.1−0.1−0.0500.050.1 d g/g d v / v (a) −0.1 −0.05 0 0.05 0.1−0.1−0.0500.050.1 dW / W d T / T (b) FIG. 10: The fidelity of the target state | ψ target i versus the deviations of (a) g and v , (b) T andΩ . C(cid:13)
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C(cid:13) -(cid:13)
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N(cid:13) f(cid:13) -(cid:13) f(cid:13) a(cid:13) a(cid:13)
N(cid:13) a(cid:13) -(cid:13)
N(cid:13) a(cid:13)
FIG. 11: The set-up diagram for the generation of N -atom GHZ states. The N -atoms are respec-tively trapped in N -cavities which are linked by N −−