One-Step Generation of Scalable Multiparticle Entanglement for Hot Ions Driven by a Standing-Wave Laser

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1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 2, August 15, 2011 One-Step Generation of Scalable Multiparticle Entanglement for Hot ons Driven by a Standing-Wave Laser YANG Wen-Xing ( ), 1, and CHEN Ai-Xi (íçí) 2 1 Department of Physics, Southeast University, Nanjing , China 2 Department of Applied Physics, School of Basic Science, East China Jiaotong University, Nanchang , China (Received November 22, 2010) Abstract An alternative scheme is proposed for one-step generation of multiparticle cluster state with trapped ions in thermal motion. n this scheme, the ions are simultaneously illuminated by a standing-wave laser tuned to the carrier. During the operations, the vibrational mode is virtually excited, thus the quantum operations are insensitive to the heating. t is shown that the high fidelity multiparticle entanglement could be generated in just one step even including the small fluctuations of parameters. n addition, the ion does not need to be exactly positioned at the node of the standing wave, which is also important from the viewpoint of experiment. PACS numbers: Lx, Dv, Bz Key words: cluster states, trapped ions, standing-wave laser Quantum entanglement is a universal resource in the realm of quantum information. t not only provides a tool to disprove the local hidden variable theory, 1] but also has practical applications in quantum computation and quantum-information processing (QP). 2 4] Quantum entanglement involving a large number qubits is now attracting increasing interest. t has been shown that there are two inequivalent classes of tripartite entangled states, the Greenberger Horne Zeilinger (GHZ) class 5] and W class, 6] under stochastic local operation and classical communication. Briegel and Raussendorf have introduced another class of multiqubit entangled states, i.e., the socalled cluster states. 7] t has a high persistence of entanglement and serves as a resource for GHZ state. More importantly, it acts as a universal resource for the oneway quantum computation, which is performed by local measurement and feedback of their outcomes. 8] n this regards, we should note that the cluster state has been experimentally generated. 9] Furthermore, the experimental demonstration of the cluster-state violation of Bell s inequality has also been reported. 10] So far, a number of systems have been proposed as potentially viable qubit models ] Among a variety of qubits implemented, the ion-trap system 15] is a promising candidate for generation of the cluster state. Trapped ions provide a relatively clean system, because they can be confined for long durations while experiencing only small perturbations from the environment, and can be coherently manipulated. Different theoretical schemes have been proposed for quantum gate operations with the aid of the ion s collective motions and many of them have been experimentally demonstrated with small numbers of qubits. 15] Furthermore, six-ion GHZ state, 16] and the eight-ion W state have been experimentally reported. 17] However, multi-qubit cluster states have not been demonstrated in such a system yet. n this letter, we propose an alternative scheme for one-step preparing the multi-qubit cluster state for the thermal motion of multiple hot ions using a standing-wave field resonant with the ionic carrier frequency. The distinguish feature of the scheme is that the time required to generate the desired state is limited only by the available vibrational frequency, not by inherent mechanisms such as off-resonant excitations. Therefore, our scheme would may the generation of larger cluster states in a resonant time with a single standing-wave laser pulse, which is important for studying the decoherence processes for larger systems. Moreover, the heating effect of the vibrational mode is negligible because the evolution operator of the system is independent of the creating operator and the annihilation operator of the vibrational mode. Recently, we have noticed that a scheme for one-step preparation of cluster state in the ion-trap system was proposed. 18] However, the scheme requires two different standing-wave lasers with phase difference π/2; and the scheme only considers contribution of the center-of-mass vibrational mode, which is experimentally problematic. n our scheme, only one standing-wave laser is needed. Furthermore, unlike the scheme, 18] the ions do not need to be exactly positioned at the node of the standing-wave, which is also important from the viewpoint of experiment. Finally, we show in this letter the high fidelity multiparticle entanglement could be generated in Supported by the National Natural Science Foundation of China under Grant Nos , , and National Fundamental Research Program of China under Grant No. 2007CB wenxingyang@seu.edu.cn c 2011 Chinese Physical Society and OP Publishing Ltd

2 264 Communications in Theoretical Physics Vol. 56 just one step even including the small fluctuations of parameters. We consider N identical two-level ions confined in a trap. We drive the ions with a standing-wave laser beam, tuned to the ions transition. The standing wave can be constructed from pure back reflection since the superposition of a traveling wave and the reflected wave is just a standing wave. n the rotating-wave approximation, the Hamiltonian for this system is given by (assuming = 1) 19 20] H = ν p a p a p + ω 0 σ z,j + Ω 2 σ + j sin N ] η jp (a p + a p ) + φ e iω0t + H.c., (1) σ z,j = (1/2)( e j e j g j g j ), σ + j = e j g j, and σ j = g j e j, with e j and g j being the excited and ground states of the j-th ion, a p and a p are the creation and annihilation operators for the p-th vibrational mode with vibrational frequency ν p, ω 0 is the transition frequency for the two-level ion, and Ω and φ are the Rabi frequency and phase of the laser field. η jp is the Lamb-Dicke parameter incorporating the relative displacement of the j-th ion in the p-th mode. We can rewrite H as H = ν p a p a p + ω 0 σ z,j + Ω 2 σ + j sin N η jp (a p + a p )cos φ + cosη(a p + a p ]e )sin φ iω0t + H.c.. (2) Let us consider the behavior of the trapped ions in the Lamb-Dicke regime, η n p + 1 1, with n p being the mean phonon number of the p-th mode. n the case we can retain terms only to the first order in η. n the interaction picture the interaction Hamiltonian is given by N ] H = Ω σ x,j η jp (a p e iνpt + a e iνpt )cosφ + sin φ, (3) σ x,j = (1/2)(σ j + + σ j ). We define the new atomic basis + j = 1 ( g j + e j ), (4) 2 j = 1 2 ( g j e j ). (5) Then we can rewrite the Hamiltonian H as H = Ω cosφ η jp a p e iνpt( S z,j σ j 1 ) 2 σ+ j + a e iνpt( S z,j σ+ j 1 2 σ j )] + sin φs z,j, (6) S + j = + j j, S j = j + j, and S z,j = + j + j j j. The interaction Hamiltonian involves the competition between the creation and annihilation processes of one phonon. Since the internal state transition accompanied by the creation and annihilation of one phonon is not resonant, the corresponding terms oscillate at the frequency ν p. The Hamiltonian is given by the product of the part only involving the internal state with that only concerning the external state. The time evolution of the Hamiltonian (6) is decided by Schrödinger equation, i d ψ(t) dt Perform the unitary transformation with Then we obtain = H ψ(t). (7) ψ(t) = e ih(1) t ψ (t) H (1) i d ψ (t) dt = Ω sinφ S z,j. (8) = H (2) ψ (t), (9) H (2) = Ω cosφ η jp a p e iνpt( S z,j σ j e iω sin φt 1 2 σ+ j eiω sin φt) + a e iνpt( S z,j σ+ j eiω sin φt 1 2 σ j e iωsin φt)] + sin φs z,j. (10) Let us assume that Ω sinφ ν p, we can neglect the terms oscillating fast. Then H (2) reduces to = Ω η jp (a p e iνpt + a e iνpt )cosφ S z,j. (11) H (2) Since H (1), H (2) ] = 0 the evolution operator corresponding to the Hamiltonian (6) can be expressed as U(t) = U 1 (t) U 2 (t), (12) U 1 (t) = exp(iαtj p,x ), with α = Ω sinφ, refers to

3 No. 2 Communications in Theoretical Physics 265 the carrier excitation. The evolution operator U 2 (t) can be written as N U 2 (t) = e ia(t)j2 p,x e ib(t)aj p,x e ic(t)a J p,x, (13) with J p,x = N η jps z,j /η. Here η = η j1 is the Lamb-Dicke parameter of the center-of-mass mode. e ia(t)j2 p,x refers to the two-phonon process, which couples the internal states of two nearest-neighbor ions, while e ib(t)jp,xap e ic(t)jp,xa p refers to the vibrational sideband excitation. Using the Schrödinger equation idu 2 (t)/dt = H (2) U 2 (t), we obtain A(t) = 1 (η cosφ) 2 t + 1 ] (e iνpt 1), (14) ν p iν p B(t) = η cosφ 1 iν p (e iνpt 1), (15) C(t) = η cosφ 1 iν p (e iνpt 1). (16) We should note that the dynamics also consist of higher order excitations. n the Lamb-Dicke limit, higher order excitations can be neglected since the corresponding excitation strength is much smaller than that for the two-phonon process. 21] f the interaction time satisfies t = 2kπ/ν p (k is an integer), it leads the evolution operator to N U 2 (t) = e iβj2 p,x t, (17) with β = (ηω cos φ) 2 /ν p. t is obviously that A(t) is a periodic function and U 2 (t) induces only the internal state evolution, which is independent of the states of the vibrational modes. We can rewrite U 2 (t) as exp( ih e (t)), H e (t) = λ j (t)( + j + j + j j ) + λ m,n (t) λ m,n (t) = λ j (t) = S z,m S z,n, (18) η mp η np β/2, (19) ηjp 2 β. (20) 4 The first and second terms of H e (t) arise from the stark shifts, which are phonon-number independent due to the competition between the creation and annihilation operators in the Hamiltonian of Eq. (6). The other terms arise from the virtual two-phonon transition + m n n p m + n n p mediated by m n n p ±1 and + m + n n p ±1, and the transition + m + n n p m n n p mediated by + m n n p ± 1 and m + n n p ± 1. These transitions are resonant and phonon-number independent due to the destructive interference of different transition paths. 22] Then up to an unimportant global phase factor, we have U(t) = exp itα S z,j λ m,n (t) S z,m S z,n ].(21) Setting the parameters to satisfy the condition αt = 2λ m,n (t) g/2, we have the total evolution operator as U(t) = exp ig 1 ] 4 (1 S z,m)(1 S z,n ). (22) The initial state of each ion is prepared as g j = ( + j + j )/ 2. When the condition gt = (2l + 1)π (l = 0, 1, 2,...) is satisfied, we obtain a cluster state, as ψ N = 1 2 N/2 N ( + N n ( 1) N n n=1 m=n+1 ) S z,m + j, (23) with Sz N+1 1. The above compact notation if easily understood by multiplying out the right-hand side. The operator S z,m acts on the states ± of the qubits m = n + 1,...,N, with n = 1, 2,...,N 1. The cluster state in Eq. (23) is one of the eigenstates of the correlation operator K (a) and can be completely specified by the following eigenvalue equation, K (a) φ κ C = ( 1) κa φ κ C, (24) K (a) S z,m S z,n, (25) n nbgh(n) with κ 0, 1, nbgh denotes the set of all occupied nearest-neighbor sites of the lattice site m. All the states φ κ C are equally suitable for computation. The detailed proof for the application of the cluster states in a one-way quantum computer is given in Refs. 7 8,23]. We should note that though the internal state evolution of the trapped ions is independent of the thermal phonon of the external vibrational modes after a period t decided by ν p t = 2π, the internal state is entangled with the external vibrational mode during the interaction. Therefore, the present scheme is required that the decay of the external mode is negligible during the interaction. According to the experimental report, 24] we can set the frequency of the vibrational mode ν p = 2π 8.8 MHz, thus the interaction time is on the order of 2π/ν p = 10 7 s. The decoherence rate for phonon state of the trapped ions is about 10 3 ν p /2π, thus the phonon decay time is about 10 3 s, much longer than the interaction time. After the interaction, the internal states are disentangled with the external vibrational modes and then the phonon decay will not affect the entanglement preparation. On the other hand, if we choose the S 1/2 ground state and metastable D 5/2 state of the 40 Ca + ion to be the electronic ground and excited state, respectively, the lifetime of the excited state is about 1 s, 15,25] much longer than the required preparation time for the cluster states. Thus the losses

4 266 Communications in Theoretical Physics Vol. 56 due to the spontaneous emission can be neglected. Considering the qubit on the transition S 1/2 D 5/2 in 40 Ca +, a study showed that magnetic field fluctuations and laser frequency fluctuations are the major decoherence sources. The possible improvements in experimental setup to overcome the influences of decoherence are demonstrated in Ref. 25]. The coherence time of the qubit is about 1 ms, much longer than the preparation time of the cluster states, which ensures that the cluster states can be generated before the decoherence of the qubits occurs. n order to realize the scalable quantum computation, a longer coherence time is needed. n this regards, we should note that some authors have proposed a promising choice (i.e., use the hyperfine ground states S 1/2, F = 3, m = 0 and S 1/2, F = 4, m = 0 of 43 Ca + to act as the qubits and encode the quantum information on them) for much longer coherence time. 15] t is worth noting that we have assuming that Ω sin φ ν p and thus discarded the terms H (2) = Ω cosφ η jp a p e iνpt( 1 2 σ j e iω sin φt 1 2 σ+ j eiωsin φt) +a e iνpt( 1 2 σ+ j eiωsin φt 1 2 σ j e iωsin φt)].(26) These terms induce Stark shifts on the states + j and j. We assume that the Stark shifts for ± j are γ + and γ (γ + = γ = γ), respectively. n our scheme, the ions do not need to be exactly positioned at the node of the standing-wave. Without losing generality, we assume φ = π/4. We can calculate the fidelity of 20-qubit cluster states as shown in Fig. 1. From Fig. 1, one can find that the fidelity is insensitive to the variation of the Stark shift γ induced by the Hamiltonian (26). Even the stark shift γ = 10η 2 Ω, the fidelity of a 20-qubit cluster state can be higher than 97%. n addition, we also calculate the fidelity of N-qubit cluster states as shown in Fig. 2. The fidelity of a 30-qubit cluster state can be as high as 95%. Fig. 1 (Color online) The fidelity versus the Stark shifts γ + = γ = γ induced by the Hamiltonian (26) for generating a 20-qubit cluster state. Fig. 2 (Color online) The fidelity versus the number of cluster state qubits for the Stark shifts induced by the Hamiltonian (26) γ + = γ = γ = 0.1η 2 Ω. n conclusion, we have proposed an efficient scheme for one-step generating cluster states of trapped ions using a single standing-wave field resonant with the ionic carrier frequency. The distinct feature of the scheme is that the required time is only limited by the available frequency of the vibrational mode, not by the inherent mechanisms such as off-resonant excitations. This makes it promising to obtain cluster states in a reasonable time, which is of importance from viewpoint of experiment. t is unnecessary to exactly position the ion at the node of the standing wave. The numerical calculation of the fidelity by using the available physical parameters shows that the present scheme is mostly within the reach of the current experimental techniques in ion trap. References 1] J.S. Bell, Physics (Long sland City, N.Y.) 1 (1964) ] A.K. Ekert, Phys. Rev. Lett. 67 (1991) ] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69 (1992) ] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70 (1995) ] D.M. Greenberger, M.A. Horne, and A. Zeilinger, in Bell s Theorem, Quantum Theory, and Conceptions of the Universe, ed. M. Kafatos Kluwer, Dordrecht (1989). 6] W. Dür, G. Vidal, and J.. Cirac, Phys. Rev. A 62 (2000) ] H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86 (2001) 910.

5 No. 2 Communications in Theoretical Physics 267 8] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86 (2001) ] P. Walther, et al., Nature (London) 434 (2005) ] P. Walther, et al., Phys. Rev. Lett. 95 (2005) ] J.Q. You, X. Wang, T. Tanamoto, and F. Nori, Phys. Rev. A 75 (2005) ] Z.R. Lin, G.P. Guo, T. Tu, F.Y. Zhu, and G.C. Guo, Phys. Rev. Lett. 101 (2008) ] J.Q. Li, G. Chen, and J.Q. Liang, Phys. Rev. A 77 (2008) ] Z.Y. Xue and Z.D. Wang, Phys. Rev. A 75 (2007) ] R. Blatt and D. Wineland, Nature (London) 453 (2008) ] D. Leibfried, et al., Nature (London) 438 (2005) ] H. Häffner, et al., Nature (London) 438 (2005) ] Y.Q. Zhang, S. Zhang, and X.R. Jin, Phys. Scr. 78 (2008) ] Y. Wu and X. Yang, Phys. Rev. Lett. 78 (1997) ] S.B. Zheng, Phys. Rev. Lett. 90 (2003) ] The effective strength for the n-th order excitation is proportional to η n. 22] A. Sørensen and K. Mølmer, Phys. Rev. Lett. 82 (1999) ] R. Raussendorf, et al., Phys. Rev. A 68 (2003) ] C.A. Sackett, et al., Nature (London) 404 (2000) ] F. Schmidt-Kaler, et al., J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 623.

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