Three-Dimensional Quantum State Transferring Between Two Remote Atoms by Adiabatic Passage under Dissipation

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1 Commun. Theor. Phys. (Beijing, China) 54 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 1, July 15, 2010 Three-Dimensional Quantum State Transferring Between Two Remote Atoms by Adiabatic Passage under Dissipation CHEN Ju-Mei (íö), LIANG Lin-Mei ( Ö), LI Cheng-Zu (Ó Ý), and DENG Zhi-Jiao ( ã ) Department of Physics, National University of Defense Technology, Changsha , China (Received October 9, 2009; revised manuscript received December 4, 2009) Abstract Recently, Zhou et al. [Phys. Rev. A 79 (2009) ] proposed a scheme for transferring three-dimensional quantum states between remote atomic qubits confined in cavities connected by fibers through adiabatic passage. In order to avoid the decoherence due to spontaneous emission, Zhou et al. utilized the large detuning atom-field interaction. In the present paper, we discuss the influence of dissipation on the scheme in both the resonant atom-field interaction case and the large detuning case. We numerically analyze the success probability and the transferring fidelity. It is shown that the resonant case is a preferable choice for the technique of the stimulated Raman adiabatic passage (STIRAP) due to the shorter operation time and the smaller probability of dissipation. PACS numbers: Pp, Lx, Pq Key words: quantum state transfer, adiabatic passage, the stimulated Raman adiabatic passage 1 Introduction The transfer of quantum states from one location (A) to another (B) is an important feature in the field of quantum information science for distributing and processing information. [1 2] By combining local quantum information processing with quantum-state transfer between the nodes of the network one could accomplish many quantum information tasks. To implement quantum states transfer, several approaches have been employed. For short distance quantum communications, spin chains, and coupled resonator waveguide are proposed. [3 4] For long distance quantum communications, optical systems such as cavity QED system [5 6] are used to transfer states from one node to another by photons in free space [7] or through a fiber. [8 10] However, it is very difficult to have a perfect optical connection due to excitation loss associated with the optical channel due to scattering and absorption. Zhou et al. [11] proposed a scheme to transfer threedimensional quantum state between distant atomic qubits trapped in spatially separated cavities connected by an optical fiber. Based on the Stimulated Raman adiabatic passage (STIRAP), [12 14] system evolves along dark states with no population of the fiber mode, which can avoid the decoherence due to the imperfection of optical channels. Furthermore, in order to avoid the decoherence due to spontaneous emission, authors utilized the large detuning atom-field interaction to adiabatically eliminate atomic excited states. [15] Since the dark states contain one-photon cavity states, the dark states are not immune to the cavity decay. However, if the laser intensity is much smaller than the atom-cavity coupling strength, the population of the cavity states with non-zero photon can be very small. Nevertheless, with weaker laser intensity, the pulse duration time must be much prolonged in order to fulfill the adiabatic condition, which would enhance the probability of dissipation. Since the effective atom-cavity coupling strength in the large detuning case is much smaller than that in the resonant case, the pulse duration time in the large detuning case will be much longer in the adiabatic limit. It is natural for us to ask how well the transfer works in the resonant case and the large detuning case under the influence of dissipation. In the present paper, we employ the quantum jump approach [16] to analyze numerically the influence of spontaneous emission and the cavity decay. The remainder of this paper is organized in three parts as follows. In Sec. 2, we first describe the model and then consider transferring three-dimensional quantum states in the absence of dissipation. In Sec. 3, we discuss the transferring passage under dissipation by numerical simulation. Finally, we give some discussions and conclusions in Sec State Transfer without Decay First, we introduce the atom-cavity-fiber system. As shown in Fig. 1(a), two identical atoms A and B are separately trapped in two optical cavities A and B and the cavities are connected by an optical fiber. The atomic transition e i i (i = 1, 2) is coupled to the laser field with Rabi frequency Ω(t), while the transition e 1 ( e 2 ) 0 is coupled to the cavity mode a L(R) with coupling strength g L(R) (t). For simplicity, we assume g (i) L = g (i) R = g (i = A, B) in the following. In addition, in the short fiber limit l v/(2πc) 1, where v is the decay rate of the cavity fields into a continuum of fiber modes, only one resonant mode of the fiber interacts with the cavity modes. [8] In the rotating frame, the Hamiltonian of the atom-cavity-fiber system is given by

2 108 CHEN Ju-Mei, LIANG Lin-Mei, LI Cheng-Zu, and DENG Zhi-Jiao Vol. 54 H I = i=a,b [(Ω (i) (t) e (i) 1 1(i) + g e (i) 1 0(i) a (i) + ( e (i) 1 e(i) 1 + e(i) 2 e(i) 2 )] + L + Ω(i) (t) e (i) 2 2(i) + g e (i) [νb k (a(a) k k=l,r 2 0(i) a (i) R + H.c.) + a (B) k ) + H.c.], (1) where b k is the resonant mode of the fiber with polarization k (k = L, R) and ν is the coupling strength with the cavity modes. Fig. 1 Representation of the scheme. Two identical atoms are confined in two spatially separated doublemode cavities A and B, respectively, and the cavities are connected via an optical fiber. (a) The schematic setup. (b) The level configuration for the atoms. We note that state D 0 = (2) is a stationary state of the Hamiltonian and decoupled from the dynamics. In addition, the Hamiltonian has the following dark states: D 1 gω (B) Ω (A) Ω (B) 00 ( L00 0L0 ) gω (A) , (3) in the subspace { , e , 00 L00, 00 00L, 00 0L0, 0e 1 000, }; and D 2 gω (B) Ω (A) Ω (B) 00 ( R00 0R0 ) gω (A) , (4) in the subspace { , e , 00 R00, 00 00R, 00 0R0, 0e 2 000, }, respectively. Here, in the denotation s A s B n A n B n f, the index s i (i = A, B) denotes the state of the i-th atom, n k (k = A, B) and n f represent the state of the k-th cavity mode and the fiber mode, respectively. We note that if the system evolves along the dark states adiabatically, the atoms are always in ground states and the fiber mode is in the vacuum state. In the dark states D 1 and D 2, the fiber modes are not populated, which means that the fiber links two spatially separate nodes without photons in the fiber mode. It arises from quantum destructive interference. The two states 00 i00 (i = L, R) and 00 0i0 are intermediated by the state 00 00i through the cavity-fiber coupling. Since the two transition paths 00 i i and 00 0i i interfere destructively, thus the fiber mode remains in the vacuum state. In addition, we assume that the condition g Ω (i) (i = A, B) is always satisfied throughout the whole process. Therefore, the population of the cavity modes in excited states is very small and can be neglected. In the following, we implement the three-dimensional quantum state transfer: Φ i = (α 00 + β 10 + γ 20 ) 000 (α 00 + β 01 + γ 02 ) 000, (5) where α, β, and γ satisfy the normalized condition α 2 + β 2 + γ 2 = 1. States D 0,1,2 in Eqs. (2) (4) are to be utilized for the procedure. If the system is initially in the superposition of states D 0,1,2, it will evolve along the initial dark state in the adiabatic limit. We start with the initial state Φ i = (α 00 + β 10 + γ 20 ) 000. (6) Here the initial state is the superposition of states D 0,1,2 : Φ i = α D 0 + β D 1 + γ D 2, (7) under the initial condition Ω (B) Ω (A). In order to implement the state transfer, we perform one STIRAP [12] and the system evolves along dark states during the adiabatic passage. In the adiabatic procedure, the laser Ω (B) (t) precedes the laser Ω (A) (t) with an overlap area; and the population of the component state ( ) will be coherently transferred to ( ) along the dark state D 1 ( D 2 ). However, the population of the component in Φ i remains unchanged because of the steady state D 0. At the end of this passage, the quantum state transfer is accomplished and the state of expression (6) evolves to Φ i = (α 00 β 10 γ 20 ) 000. (8) The three-dimensional quantum state transfer can be implemented by adding an appropriate quantum operation on atom B. 3 State Transfer in Presence of Decay Then we introduce the spontaneous emission and the photon decay and numerically analyze their effects by the quantum jump approach [16] in both the resonant case and the large detuning case. As long as there is no photon leakage from the atomic excited state or the cavity mode or

3 No. 1 Three-Dimensional Quantum State Transferring Between Two Remote Atoms by Adiabatic Passage under Dissipation 109 the fiber mode during the transferring process, the Hamiltonian in Eq. (1) becomes H cond =H I i κ (a (i) L 2 a(i) L + a (i) R a(i) R ) i=a,b i κ 0 2 (b L b L + b R b R) i Γ ( e (i) 1 2 e(i) 1 + e(i) i=1,2 2 e(i) 2 ), (9) where κ and κ 0 are the photon decay rates of the cavity modes and fiber mode, respectively, and Γ is the atomic spontaneous emission rate. Note that H cond is non-hermitian, the norm of a state vector evolving under the corresponding Schröinger equation decreases with time in general. For an arbitrary initial state ψ(t i ), P sus (t) = ψ(t i ) U cond (t, t i)u cond (t, t i ) ψ(t i ) (10) defines the probability that no photon has been emitted at time t, where U cond (t, t i ) is the time evolution operator for H cond and the corresponding normalized state vector is ψ(t) = U cond (t, t i ) ψ(t i ) / P sus (t). (11) The transferring fidelity is F = ψ( ) ψ id 2, (12) where ψ id is the ideal final state. We show the numerical results with the initial state ) in Figs According to the conditions of the adiabatic passage [12] the effective Rabi frequency Ω eff and the duration of the adiabatic passage T should satisfy Ω eff T > 10. In our numerical simulation, the duration T is chosen to satisfy Ω eff,max T = 50 for both the resonant case (Ω eff,max = Ω max ) and the large detuning case (Ω eff,max = gω max / [8] ). In addition, the pulses are assumed to have the Gaussian shape exp[ (t T/2 ± t 0 ) 2 /(τ 2 )] with the maximal value Ω max = 0.25g (in experiment, g more than four times larger than Ω max has been achieved [16] ). The corresponding constants for the pulses are T = 200/g, t 0 = 24/g, and τ = 40/g in the resonant case ( = 0) and T = 2000/g, t 0 = 0, and τ = 400/g in the large detuning case with the choice = 10g. Fig. 2 Time evolution of the initial state (/2) 00 + (1/2) 10 + (1/2) 20 ) 000. (a) the Rabi frequencies; (b) the populations of the states; (c) the probabilities of all the excited stats of the atoms and the cavity modes: P exc denotes the probability of all the excited atomic states; P pht represents the probability of all the excited states of the cavity modes; (d) the population of the fiber mode. the left column shows the resonant case, while the right column exhibits the large detuning case. Figure 2 shows the time evolution for the initial state ) with κ = 0.02g, κ 0 = 0.02g, and Γ = 0.02g. In the resonant case (the large detuning case), the populations for states , , are , , ( , , ) respectively; and the transferring fidelity is 99.99% (98.59%) while the suc-

4 110 CHEN Ju-Mei, LIANG Lin-Mei, LI Cheng-Zu, and DENG Zhi-Jiao Vol. 54 cess probability is 97.98% (88.61%) by our numerical calculation. Although the component did not participate in the evolution, its amplitude is increased compared with that in the initial state. The reason is that the other components dissipated in the evolution to enlarge its relative weight in the normalized final state. The probabilities of the excited counterpart are much smaller in the large detuning case than that in the resonant case. However, the probability of dissipation is larger due to the much prolonged duration time to fulfill the adiabatic condition in the large detuning case. which results from the destructive interference of the two transition paths from the two cavity modes to fiber mode. Furthermore, it is shown that, under the same dissipation conditions, the probability and the transferring fidelity are smaller in the large detuning case than that in the resonant case by comparing Fig. 4 and Fig. 3. The probability of the excited states is smaller, but the longer operation time to satisfy the adiabatic condition enhances the probability of dissipation in the large detuning case. Fig. 3 Dependence of the success probability P succ (upper subplot) and fidelity F (lower subplot) on κ and Γ for the initial state (/2) 00 +(1/2) 10 +(1/2) 20 ) 000 in the resonant case. Figures 3 and 4 show the influences of κ and Γ on the success probability P succ and the transfer fidelity F. We see that P succ and F are affected by κ and Γ similarly in the resonant case in Fig. 3. The reason is that the probability P exc (of all the states containing an excited atomic state) is close to the probability P pht (of all the states containing one photon in the cavity modes), which is shown in Fig. 2(c). In the large detuning case, P succ and F are almost unaffected by Γ in that the excited atomic states are adiabatically eliminated [see Fig. 2(g)]. We also did the simulation of the effect of fiber decay rate κ 0 on the success probability and the transferring fidelity, which shows that the fiber decay rate has little influence on them. It is because of the negligible population of the fiber mode, Fig. 4 Dependence of the success probability P succ (upper subplot) and fidelity F (lower subplot) on κ and Γ for the initial state (/2) 00 +(1/2) 10 +(1/2) 20 ) 000 in the large detuning case with the choice = 10g. 4 Discussions and Conclusions We now briefly discuss the experimental feasibility. As a specific example of realization of our scheme proposed here, we consider hyperfine levels of 87 Rb. [17] According to the selective rule of photon absorption and emission, the levels of atoms and polarizations of cavity fields and classical fields can be chosen as shown in Fig. 1(b). In addition, it has been experimentally reported that g = 34 2π MHz, κ = 4.1 2π MHz, and Γ = 2.6 2π MHz. [18] If we adopt the above numbers, i.e. κ/g 0.12, Γ/g 0.077, the operation time is of the order 10 6 s (10 5 s) in the resonant (the large detuning) case, as can be seen from Fig. 2. Under these conditions, for the initial state ) ,

5 No. 1 Three-Dimensional Quantum State Transferring Between Two Remote Atoms by Adiabatic Passage under Dissipation 111 we may obtain P succ 91.10%, F 99.75% in the resonant case, and P succ 77.78%, F 90.18% in the large detuning case. The fidelity and the success probability are high enough in the resonant case, while the fidelity and the success probability are relatively small in the large detuning case. Thus the resonant atom-field interaction is a better choice for the technique of STIRAP due to the higher success probability and fidelity and shorter operation time. Furthermore, in real experiments, it is very challenging to control precisely the atoms trapped in the Lamb Dicke limit. However, the system s adiabatic evolution only depends on the ratios Ω (i) ( r, t)/g( r, t) and Ω (i) ( r, t)/ω (i ) ( r, t) (i, i = 1, 2). [19] All the ratios are independent of the atomic position r if all the applied laser pulses are collinear with their respective cavities axes, and thus share the same spatial mode structure with their respective cavities. Thus our scheme could work robustly beyond Lamb Dicke limit. In conclusion, we have shown three-dimensional quantum states transfer between two remote atomic qubits in a cavity QED quantum network via adiabatic passage, in the presence of dissipation. According to the conditions for adiabatic passage, [12] the multiplication of the effective Rabi frequency and the implementation time should be larger than 10. The smaller effective Rabi frequency in the large detuning case requires much longer operation time, which enhances the probability of dissipation. It is shown that the resonant case is a preferable choice for the technique of STIRAP due to the shorter operation time and the resultant smaller probability of dissipation. References [1] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000). [2] H.J. Kimble, Nature (London) 453 (2008) 1023; N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [3] S. Bose, Phys. Rev. Lett. 91 (2003) [4] P.B. Li, Y.G. Gu, Q.H. Gong, and G.C. Guo, Phys. Rev. A 79 (2009) [5] H.J. Kimble, Phys. Scr., T. 76 (1998) 127. [6] H. Mabuchi and A.C. Doherty, Science 298 (2002) [7] J.I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi, Phys. Rev. Lett. 78 (1997) [8] T. Pellizzari, Phys. Rev. Lett. 79 (1997) [9] A. Serafini, A. Mancini, and S. Bose, Phys. Rev. Lett. 96 (2006) [10] A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kimble, Phys. Rev. Lett. 98 (2007) [11] Y.L. Zhou, Y.M. Wang, L.M. Liang, and C.Z. Li, Phys. Rev. A 79 (2009) [12] K. Bergmann, H. Theuer, and B.W. Shore, Rev. Mod. Phys. 70 (1998) [13] C.E. Carroll and F.T. Hioe, Phys. Rev. A 42 (1990) [14] Z. Kis and F. Renzoni, Phys. Rev. A 65 (2002) ; N. Sangouard, X. Lacour, S. Guérin, and H.R. Jauslin, Eur. Phys. J. D 37 (2006) 451; H. Goto and K. Ichimura, Phys. Rev. A 70 (2004) ; S.B. Zheng, Phys. Rev. Lett. 95 (2005) ; X. Lacour, N. Sangouard, S. Guérin, and H.R. Jauslin, Phys. Rev. A 73 (2006) ; N. Sangouard, X. Lacour, S. Guérin, and H.R. Jauslin, Phys. Rev. A 72 (2005) ; C. Marr, A. Beige, and G. Rempe, Phys. Rev. A 68 (2003) ; Z. Kis and E. Paspalakis, Phys. Rev. B 69 (2004) ; M.A. Talab, S. Guérin, N. Sangouard, and H.R. Jauslin, Phys. Rev. A 71 (2005) ; D. Møller, L.B. Madsen, and K. Mølmer, Phys. Rev. A 75 (2007) [15] X.Y. Lü, J.B. Liu, C.L. Ding, and J.H. Li, Phys. Rev. A 78 (2008) [16] M.B. Plenio and P.L. Knight, Rev. Mod. Phys. 70 (1998) 101; M.O. Scully and M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge (1999); Z.J. Deng, K.L. Gao, and M. Feng, Phys. Rev. A 74 (2006) ; X.L. Zhang, M. Feng, and K.L. Gao, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 3211; C.Y. Chen, X.L. Zhang, Z.J. Deng, K.L. Gao, and M. Feng, Phys. Rev. A 74 (2006) ; Z.J. Deng, K.L. Gao, and M. Feng, J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 351. [17] N.V. Vitanov, M. Fleischhauer, B.W. Shore, and K. Bergmann, Adv. At. Mol. Opt. Phys. 46 (2001) 55; J. McKeever, A. Boca, A.D. Boozer, J.R. Buck, and H.J. Kimble, Nature (London) 425 (2003) 268; R. Miller, T.E. Northup, K.M. Birnbaum, A. Boca, A.D. Boozer, and H.J. Kimble, J. Phys. B 38 (2005) S551. [18] A. Boca, R. Miller, K.M. Birnbaum, J.M. Raimond, and S. Harohce, Phys. Rev. A 66 (2004) [19] L.M. Duan, A. Kuzmich, and H.J. Kimble, Phys. Rev. A 67 (2003)