Abstract

Inspired by a recently experiment by M. Lettner et al. [Phys. Rev. Lett. 106, 210503 (2011)], we propose a robust scheme to prepare three-dimensional entanglement state between a single atom and a Bose-Einstein condensate (BEC) via stimulated Raman adiabatic passage (STIRAP) technique. The atomic spontaneous radiation, the cavity decay, and the fiber loss are efficiently suppressed by the engineering adiabatic passage. Our strictly numerical simulation shows our proposal is good enough to demonstrate the generation of three-dimensional entanglement with high fidelity and within the current experimental technology.

© 2012 OSA

1. Introduction

Quantum entanglement plays a vital role in many practical quantum information system, such as quantum teleportation [1], quantum dense coding [2], and quantum cryptography [3]. Entangled states of higher-dimensional systems are of great interest owing to the extended possibilities they provide, which including higher information density coding [4], stronger violations of local realism [5,6], and more resilience to error [7] than two dimensional system. Over the past few years, fairish attention has been paid to implement higher-dimensional entanglement with trapped ions [8, 9], photons [1012], and cavity QED [1316].

Moreover, it has been shown that entanglement between two spatially separated subsystems is very useful for distributed quantum computation [17, 18]. Recently, a large number of schemes have been proposed for generating entangled state of atoms, which are individually trapped in distant optical cavities connected by fibers [1925]. The main problems in entangling atoms in these schemes are the decoherence due to leakage of photons from the cavity and fiber modes, and spontaneous radiation of the atoms [26]. By using the stimulated Raman adiabatic passage (STIRAP) [2734], our scheme can overcome these problems. The idea of STIRAP is that the system is initially prepared in a decoherence-free state (dark state), and evolve adiabatically along the dark state to the required state by two delayed but partially overlapping pulses. Many schemes have been proposed to prepare entanglement state via STIRAP [3543].

The Bose-Einstein condensate (BEC) has many advantages over other systems such as long storage times, the high write-read efficiencies, and excellent internal-state preparation [44, 45]. Recently, remote entanglement between a single atom and BEC was experimentally realized [46]. But the efficiency is very low due to the photon loss. In this paper, we takes both the advantages of cavity-fiber system and STIRAP in order to create three-dimensional entanglement state between a single 87Rb atom and a 87Rb BEC at a distance. The atom and BEC are placed inside two high-finesse optical cavities respectively, which connected by an optical fibre. The atom–light interaction is identical for all atoms of the BEC and enhanced greatly because the atoms collectively couple to the same light mode [47, 48]. The entanglement state can be generated with highly fidelity even in the range that the cavity decay and spontaneous radiation of the atoms are comparable with the atom-cavity coupling strength. Our scheme is also robust to the variation of atom number in the BEC. As a result, the highly fidelity three-dimensional entanglement state of the BEC and atom can be realized base on our proposed scheme.

This paper is organized as follows. In Sec. 2, we introduce the basic model of our system. In Sec. 3, the generation of the three-dimensional entanglement state is provided. In Sec. 4, we demonstrate the influences of atomic spontaneous radiation, photon leakage out of the cavities and fiber on the implementation. Finally, in Sec. 5, we discuss experimental feasibility of our scheme and conclude our results.

2. The fundamental model

We consider the situation describe in Fig. 1, where a single 87Rb atom and a 87Rb BEC are trapped in two distant double-mode optical cavities, which are connected by an optical fiber (see Fig. 1). The 87Rb atomic levels and transitions are also depicted in this figure. [46, 49, 50]. The states |gL〉, |g0〉, |gR〉 and |ga〉 correspond to |F = 1, mF = −1〉, |F = 1, mF = 0〉, |F = 1, mF = 1〉 of 5S1/2 and |F = 2, mF = 0〉 of 5S1/2, while |eL〉, |e0〉 and |eR〉 correspond to |F = 1, mF = −1〉, |F = 1, mF = 0〉 and |F = 1, mF = 1〉 of 5P3/2. The atomic transition |ga〉 ↔ |e0〉 of atom in cavity A is driven resonantly by a π-polarized classical field with Rabi frequency ΩA; |e0A ↔ |gLA (|e0A ↔ |gRA) is resonantly coupled to the cavity mode aA,L (aA,R) with coupling constant gA. The atomic transition |gLB ↔ |eLB (|gRB ↔ |eRB) of BEC in cavity B is driven resonantly by a π-polarized classical field with Rabi frequency ΩB; |eRB ↔ |g0B (|eLB ↔ |g0B) is resonantly coupled to the cavity mode aB,L (aB,R) with coupling constant gB. Here we consider BEC for a single excitation, the ground and single excitation states are described by the state vectors |Gf=(1/N)j=1N|gfjk=1,kjN|g0j and |Ef=(1/N)j=1N|efjk=1,kjN|g0j (f = 0, L, R), where |...〉j describe the state of the jth atom in the BEC [46].

 

Fig. 1 A single 87Rb atom and a 87Rb BEC are trapped in two distant double-mode optical cavities, which are connected by an optical fiber. The states |gL〉, |g0〉, |gR〉 and |ga〉 correspond to |F = 1, mF = −1〉, |F = 1, mF = 0〉, |F = 1, mF = 1〉 of 5S1/2 and |F = 2, mF = 0〉 of 5S1/2, while |eL〉, |e0〉 and |eR〉 correspond to |F = 1, mF = −1〉, |F = 1, mF = 0〉 and |F = 1, mF = 1〉 of 5P3/2. The atomic transition |ga〉 ↔ |e0〉 of atom in cavity A is driven resonantly by a π-polarized classical field with Rabi frequency ΩA; |e0A ↔ |gLA (|e0A ↔ |gRA) is resonantly coupled to the cavity mode aA,L (aA,R) with coupling constant gA. The atomic transition |gLB ↔ |eLB (|gRB ↔ |eRB) of BEC in cavity B is driven resonantly by a π-polarized classical field with Rabi frequency ΩB; |eRB ↔ |g0B (|eLB ↔ |g0B) is resonantly coupled to the cavity mode aB,L (aB,R) with coupling constant gB.

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Initially, if the atom and BEC are prepared in the states |gaA and |G0B respectively, and the cavities and fiber modes are in the vacuum states. In the rotating wave approximation, the interaction Hamiltonian of the atom (BEC)-cavity system can be written as (setting h̄ = 1) [47]

Hac=k=L,R(ΩA(t)|e0Aga|+gA(t)aA,k|e0Agk|+NΩB(t)|EkBGk|)+NgB(t)aB,L|ERBG0|+NgB(t)aB,R|ELBG0|+H.c.,
In the short fibre limit, the coupling between the cavity fields and the fiber modes can be written as the interaction Hamiltonian [20, 23, 24]
Hcf=k=L,Rvk[bk(aA,k++aB,k+)+H.c.].
In the interaction picture the total Hamiltonian now becomes
HI+Hac+Hcf.

3. Generation of the three-dimensional entanglement state

In this section, we begin to investigate the generation of the three-dimensional entangled state in detail. The time evolution of the whole system state is governed by the Schrödinger equation

it|ψ(t)=HI|ψ(t).
The single excitation subspace can be spanned by the following state vectors [51]
|ϕ1=|gaA|G0B|0000c|00f,|ϕ2=|e0A|G0B|0000c|00f,|ϕ3=|gLA|G0B|1000c|00f,|ϕ4=|gRA|G0B|0100c|00f,|ϕ5=|gLA|G0B|0000c|10f,|ϕ6=|gRA|G0B|0000c|01f,|ϕ7=|gLA|G0B|0010c|00f,|ϕ8=|gRA|G0B|0001c|00f,|ϕ9=|gLA|ERB|0000c|00f,|ϕ10=|gRA|ELB|0000c|00f,|ϕ11=|gLA|GRB|0000c|00f,|ϕ12=|gRA|GLB|0000c|00f,
where |nAL nAR nBL nBRc denotes the field state with nAi (i = L, R) photons in the i polarized mode of cavity A, nBi in the i polarized mode of cavity B, and |nL nRf represents ni photons in i polarized mode of the fiber. The Hamiltonian HI has the following dark state:
|D(t)=K{2gA(t)ΩB(t)|ϕ1ΩA(t)ΩB(t)[|ϕ3+|ϕ4|ϕ7|ϕ8]gB(t)ΩA(t)[|ϕ11+|ϕ12]},
which is the eigenstate of the Hamiltonian corresponding to zero eigenvalue. Here and in the following gi, Ωi are real, and K2=gA2ΩB2+4ΩA2ΩB2+2gB2ΩA2. Under the condition
gA(t),gB(t)ΩA(t),ΩB(t),
we have
|D(t)2gA(t)ΩB(t)|ϕ1gB(t)ΩA(t)[|ϕ11+|ϕ12].
Suppose the initial state of the system is |ϕ1〉, if we design pulse shapes such that
limtgB(t)ΩA(t)gA(t)ΩB(t)=0,limt+gA(t)ΩB(t)gB(t)ΩA(t)=12,
we can adiabatically transfer the initial state |ϕ1〉 to a equal superposition of |ϕ1〉, |ϕ11〉 and |ϕ12〉, i.e., 1/3(|gaA|G0B|gLA|GRB|gRA|GLB)|0000c|00f, which is a product state of the three-dimensional atom-BEC entangled state, the cavity mode vacuum state, and the fiber mode vacuum state. The pulse shapes and sequence can be designed by an appropriate choice of the parameters. The coupling rates are chosen such that gA(t) = gB(t) = g, νL = νR = ν = 100g, N = 104, laser Rabi frequencies are chosen as ΩA(t) = Ω0 exp [−(tt0)2/200τ2] and ΩB(t)=Ω0exp[t2/200τ2]+Ω02exp[(tt0)2/200τ2], with t0 = 20τ being the delay between pulses [52]. Figure 2 shows the simulation results of the entanglement generation process, where we choose g = 5Ω0, τ=Ω01. With this choice, conditions (7) and (8) can be well satisfied. The Rabi frequencies of ΩA(t), ΩB(t) are shown in Fig. 2(a). Figure 2(b) and 2(c) shows the time evolution of populations. In Fig. 2(b) P1, P11, and P12 denote the populations of the states |ϕ1〉, |ϕ11〉, and |ϕ12〉. Figure 2(c) show the time evolution of populations of other states {|ϕ2〉, |ϕ3〉, |ϕ4〉, |ϕ5〉, |ϕ6〉, |ϕ7〉, |ϕ8〉, |ϕ9〉, |ϕ10〉}, which are almost zero during the whole dynamics. Finally P1, P11, and P12 arrive at 1/3, which means the successful generation of the 3-dimensional entangled state. Figure 2(d) shows the error probability defined by [53]:
Pe(t)=1|D(t)|φs(t)|2,
here |φs (t)〉 is the state obtained by numerical simulation of Hamiltonian (3) and |D(t)〉 is the dark state defined by Eq. (6). From the Fig. 2(a)–2(d) we conclude that we can prepare the three-dimensional entanglement state between single atom and a BEC with high success probability.

 

Fig. 2 The numerical simulation of Hamiltonian (3) in the entanglement generation process, where we choose g = 5Ω0, τ=Ω01. (a): the Rabi frequency of ΩA(t), ΩB(t). (b): the time evolution of populations of the states |ϕ1〉, |ϕ11〉, and |ϕ12〉 is denoted by P1, P11, and P12 respectively. (c): time evolution of populations of other states {|ϕ2〉, |ϕ3〉, |ϕ4〉, |ϕ5〉, |ϕ6〉, |ϕ7〉, |ϕ8〉, |ϕ9〉, |ϕ10〉}, which are almost zero during the whole dynamics. (d): error probability Pe (t) defined by Eq. (6).

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4. Effects of spontaneous emission and photon leakage

To evaluate the performance of our scheme, we now consider the dissipative processes due to spontaneous decay of the atoms from the excited states and the decay of cavity. We assess the effects through the numerical integration of the master equation for the system in the Lindblad form. The master equation for the density matrix of whole system can be expressed as [25]

dρdt=i[HI,ρ]k=L,R[κfk2(bk+bkρ2bkρbk++ρbk+bk)i=A,Bκik2(aik+aikρ2aik+ρaik+ρaik+aik)]j=a,L,Rγ0jA2(σe0e0Aρ2σgje0Aρσe0gjA+ρσe0e0A)h=1Nk=L,Rj=k,0γkjBh2(σekekBhρ2σgjekBhρσekgjBh+ρσekekBh),
where γ0jA and γkjBh denote the spontaneous radiation rates from state |e0A to |gjA and |ekB to |gjB of the hth atom in the BEC, respectively; κik and κfk denote the photon leakage rates from the cavity fields and fiber modes, respectively; σmni=|min|(m,n=e0,ek,gj) are the usual Pauli matrices. Starting with the initial density matrix |ϕ1〉 〈ϕ1|, by solving numerically Eq. (11) in the subspace spanned by the vectors (5) and |ϕ13〉 = |gLA |G0B |0000〉c |00〉f, |ϕ14〉 = |gRA |G0B |0000〉c |00〉f. Fig. 3 shows the fidelity of the entanglement state as a function of the photon leakage rate κ (κ = κAk = κBk = κfk) and for the atom spontaneous radiation rate γ(γ=j=a,L,Rγ0jA=j=L,0γkjBh=j=R,0γkjBh)=0,0.2g,0.4g,0.6g,0.8g,1.0g (from the top to the bottom). In the calculation, for simplicity we choose γ0aA=γ0kA=γ/3, γk0Bh=γkkBh=γ/2 (k = l,r), the other parameters same as in Fig. 2. From the Fig. 3 we can see that the entanglement state can be generated with highly fidelity even in the range of γ, κg.

 

Fig. 3 Fidelity of the entanglement state (obtained by numerical simulation of master equation (8)) as a function of the photon leakage rate κ and for the atom spontaneous radiation rate γ = 0, 0.2g, 0.4g, 0.6g, 0.8g, 1.0g (from the top to the bottom).

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5. Discussion and conclusion

It is necessary to briefly discuss the experimental feasibility of our scheme. Firstly, trapping 87Rb BEC in cavity QED has also been realized in recently experiment [47]. In this experiment, the atom number can be selected between 2,500 and 200,000 and the relevant cavity QED parameter (g,κ,γ) = 2π × (10.6, 1.3, 3.0) MHz is realizable. So the condition γ, κ < 0.4g can be satisfied with these system parameters for entangling the BEC and atom with fidelity larger than 98%. Secondly, the classical fields Rabi frequency can be selected by changing the laser density in principle. The strong coupling between two cavities by a waveguide has been experimental realized [54]. The coupling strength can be reached as high as 25 GHz, which is much larger than atom-cavity coupling strength and the strength of the classical fields. Finally, atoms in BEC do not fulfill the requirement of identical coupling, but it shows a similar energy spectrum, which can be modeled by the Tavis-Cummings Hamiltonian with an effective collective coupling gBeff=gμ(N), here μ(N)=0.5(10.0017N0.34) is the overlap between BEC spatial atomic mode and cavity mode [47,55]. So the coupling strength gB(t) will decrease with increasing atom number N. We can increase the ΩB(t) accordingly to compensate this. One challenge here is photoassociation driven by the classical laser because it gradually reduces the BEC atom number N [46]. The fidelity as a function of the atom number N of the BEC is plotted in Fig. 4 with the parameters γ = κ = 0.4g, and the other parameters same as in Fig. 2. From the Fig. 2, we can see that our scheme is robust to the variation of atom number in the BEC. Of course if the lost atoms carry away the single excitation, the scheme will be fail.

 

Fig. 4 Fidelity vs the atom number N of the BEC with the parameters γ = κ = 0.4g, and the other parameters same as in Fig. 2.

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In summary, based on the STIRAP technique, we propose a scheme to prepare three-dimensional entanglement state between a BEC and a atom. In this scheme, the atomic spontaneous radiation and photon leakage can be efficiently suppressed, since the populations of the excited states of atoms and cavity (fiber) modes are almost zero in the whole process. We also show that this scheme is highly stable to the variation of atom number in the BEC. Recently, strong atom–field coupling for Bose–Einstein condensates in an optical cavity on a chip [56] and strong coupling between distant photonic nanocavities [54] have been experimentally realized. So our scheme is considered as a promising scheme for realizing entanglement between BEC and atom on a photonic chip.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 60677044, 11005099), the Fundamental Research Funds for the central universities (Grant No. 201013037). L. Chen was also supported in part by the Government of China through CSC (Grant No. 2009633075).

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53. H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70(1), 012305 (2004). [CrossRef]  

54. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6(1), 56–61 (2012). [CrossRef]  

55. S. Leslie, N. Shenvi, K. R. Brown, D. M. Stamper-Kurn, and K. B. Whaley, “Transmission spectrum of an optical cavity containing N atoms,” Phys. Rev. A 69(4), 043805 (2004). [CrossRef]  

56. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom–field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450(7167), 272–276 (2007). [CrossRef]   [PubMed]  

References

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  1. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
    [Crossref] [PubMed]
  2. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
    [Crossref] [PubMed]
  3. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
    [Crossref] [PubMed]
  4. T. Durt, D. Kaszlikowski, J. -L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69(3), 032313 (2004).
    [Crossref]
  5. D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418–4421 (2000).
    [Crossref] [PubMed]
  6. D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88(4), 040404 (2002).
    [Crossref] [PubMed]
  7. M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90(16), 167906 (2003).
    [Crossref] [PubMed]
  8. A. B. Klimov, R. Guzmán, J. C. Retamal, and C. Saavedra, “Qutrit quantum computer with trapped ions,” Phys. Rev. A 67(6), 062313 (2003).
    [Crossref]
  9. I. E. Linington and N. V. Vitanov, “Robust creation of arbitrary-sized Dicke states of trapped ions by global addressing,” Phys. Rev. A 77(1), 010302(R) (2008).
    [Crossref]
  10. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
    [Crossref] [PubMed]
  11. B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100(6), 060504 (2008).
    [Crossref] [PubMed]
  12. A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
    [Crossref]
  13. X. B. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
    [Crossref]
  14. G. W. Lin, M. Y. Ye, L. B. Chen, Q. H. Du, and X. M. Lin, “Generation of the singlet state for three atoms in cavity QED,” Phys. Rev. A 76(1), 014308 (2007).
    [Crossref]
  15. S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
    [Crossref]
  16. L. B. Chen, P. Shi, Y. J. Gu, L. Xie, and L. Z. Ma, “Generation of atomic entangled states in a bi-mode cavity via adiabatic passage,” Opt. Commun. 284(20), 5020–5023 (2011).
    [Crossref]
  17. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000).
    [Crossref] [PubMed]
  18. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
    [Crossref] [PubMed]
  19. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
    [Crossref]
  20. T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79(26), 5242–5245 (1997).
    [Crossref]
  21. S. J. van Enk, H. J. Kimble, J. I. Cirac, and P. Zoller, “Quantum communication with dark photons,” Phys. Rev. A 59(4), 2659–2664 (1999).
    [Crossref]
  22. S. Clark, A. Peng, M. Gu, and S. Parkins, “Unconditional preparation of entanglement between atoms in cascaded optical cavities,” Phys. Rev. Lett. 91(17), 177901 (2003).
    [Crossref] [PubMed]
  23. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96(1), 010503 (2006).
    [Crossref] [PubMed]
  24. Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75(1), 012324 (2007).
    [Crossref]
  25. X. Y. Lü, J. B. Liu, C. L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78(3), 032305 (2008).
    [Crossref]
  26. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
    [Crossref] [PubMed]
  27. J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29(2), 690–697 (1984).
    [Crossref]
  28. U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Külz, and K. Bergmann, “Population switching between vibrational levels in molecular beams,” Chem. Phys. Lett. 149(5–6), 463–468 (1988).
    [Crossref]
  29. U. Gaubatz, P. Rudecki, S. Sciemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results,” J. Chem. Phys. 92(9), 5363–5376 (1990).
    [Crossref]
  30. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70(3), 1003–1025 (1998).
    [Crossref]
  31. J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, “Adiabatic population transfer in a three-level system driven by delayed laser pulses,” Phys. Rev. A 40(11), 6741–6744 (1989).
    [Crossref] [PubMed]
  32. R. G. Unanyan, M. Fleischhauer, B. W. Shore, and K. Bergmann, “Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states,” Opt. Commun. 155(1–3), 144–154 (1998)
    [Crossref]
  33. H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express 4(2), 77–83 (1999).
    [Crossref] [PubMed]
  34. X. L. Song, L. Wang, R. Z. Lin, Z. H. Kang, X. Li, Y. Jiang, and J. Y. Gao, “Observation of CARS signal via maximal atomic coherence prepared by F-STIRAP in a three-level atomic system,” Opt. Express 15(12), 7499–7505 (2007).
    [Crossref] [PubMed]
  35. R. G. Unanyan, N. V. Vitanov, and K. Bergmann, “Preparation of entangled states by adiabatic passage,” Phys. Rev. Lett. 87(13), 137902 (2001).
    [Crossref] [PubMed]
  36. R. G. Unanyan, M. Fleischhauer, N. V. Vitanov, and Klaas Bergmann, “Entanglement generation by adiabatic navigation in the space of symmetric multiparticle states,” Phys. Rev. A 66(4), 042101 (2002).
    [Crossref]
  37. M. Amniat-Talab, S. Guérin, N. Sangouard, and H. R. Jauslin, “Atom-photon, atom-atom, and photon-photon entanglement preparation by fractional adiabatic passage,” Phys. Rev. A 71(2), 023805 (2005).
    [Crossref]
  38. M. Amniat-Talab, S. Guérin, and H. R. Jauslin, “Decoherence-free creation of atom-atom entanglement in a cavity via fractional adiabatic passage,” Phys. Rev. A 72(1), 012339 (2005).
    [Crossref]
  39. N. V. Vitanov, K. A. Suominen, and B. W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32(18), 4535–4546 (1999).
    [Crossref]
  40. Z. Kis and E. Paspalakis, “Arbitrary rotation and entanglement of flux SQUID qubits,” Phys. Rev. B 69(2), 024510 (2004).
    [Crossref]
  41. J. Song, Y. Xia, and H. S. Song, “Entangled state generation via adiabatic passage in two distant cavities,” J. Phys. B 40(23), 4503–4512 (2007).
    [Crossref]
  42. J. Klein, F. Beil, and T. Halfmann, “Robust population transfer by stimulated raman adiabatic passage in a Pr3+ : Y2SiO5 crystal,” Phys. Rev. Lett. 99(11), 113003 (2007).
    [Crossref] [PubMed]
  43. L. B. Chen, M. Y. Ye, G. W. Lin, Q. H. Du, and X. M. Lin, “Generation of entanglement via adiabatic passage,” Phys. Rev. A 76(6), 062304 (2007).
    [Crossref]
  44. Y. Yoshikawa, K. Nakayama, Y. Torii, and T. Kuga, “Long storage time of collective coherence in an optically trapped Bose-Einstein condensate,” Phys. Rev. A 79(2), 025601 (2009).
    [Crossref]
  45. S. Riedl, M. Lettner, C. Vo, S. Baur, G. Rempe, and S. Dürr, “A Bose-Einstein condensate as a quantum memory for a photonic polarization qubit,” Phys. Rev. A 85(2), 022318 (2012).
    [Crossref]
  46. M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, and G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106(21), 210503 (2011).
    [Crossref] [PubMed]
  47. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450(7167), 268–271 (2007).
    [Crossref] [PubMed]
  48. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010).
    [Crossref] [PubMed]
  49. T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317(5837), 488–490 (2007).
    [Crossref] [PubMed]
  50. B. Weber, H. P. Specht, T. Mueller, J. Bochmann, M. Muecke, D. L. Moehring, and G. Rempe, “Photon-photon entanglement with a single trapped Atom,” Phys. Rev. Lett. 102(3), 030501 (2009).
    [Crossref] [PubMed]
  51. S. B. Zheng, “Multi-atom entanglement engineering and phase-covariant cloning via adiabatic passage,” J. Opt. B: Quantum Semiclass. Opt. 7(5), 139–141 (2005).
    [Crossref]
  52. P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79(1), 53–77 (2007).
    [Crossref]
  53. H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70(1), 012305 (2004).
    [Crossref]
  54. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6(1), 56–61 (2012).
    [Crossref]
  55. S. Leslie, N. Shenvi, K. R. Brown, D. M. Stamper-Kurn, and K. B. Whaley, “Transmission spectrum of an optical cavity containing N atoms,” Phys. Rev. A 69(4), 043805 (2004).
    [Crossref]
  56. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom–field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450(7167), 272–276 (2007).
    [Crossref] [PubMed]

2012 (2)

S. Riedl, M. Lettner, C. Vo, S. Baur, G. Rempe, and S. Dürr, “A Bose-Einstein condensate as a quantum memory for a photonic polarization qubit,” Phys. Rev. A 85(2), 022318 (2012).
[Crossref]

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6(1), 56–61 (2012).
[Crossref]

2011 (3)

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, and G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106(21), 210503 (2011).
[Crossref] [PubMed]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

L. B. Chen, P. Shi, Y. J. Gu, L. Xie, and L. Z. Ma, “Generation of atomic entangled states in a bi-mode cavity via adiabatic passage,” Opt. Commun. 284(20), 5020–5023 (2011).
[Crossref]

2010 (1)

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010).
[Crossref] [PubMed]

2009 (2)

B. Weber, H. P. Specht, T. Mueller, J. Bochmann, M. Muecke, D. L. Moehring, and G. Rempe, “Photon-photon entanglement with a single trapped Atom,” Phys. Rev. Lett. 102(3), 030501 (2009).
[Crossref] [PubMed]

Y. Yoshikawa, K. Nakayama, Y. Torii, and T. Kuga, “Long storage time of collective coherence in an optically trapped Bose-Einstein condensate,” Phys. Rev. A 79(2), 025601 (2009).
[Crossref]

2008 (5)

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O’Brien, K. J. Resch, A. Gilchrist, and A. G. White, “Manipulating biphotonic qutrits,” Phys. Rev. Lett. 100(6), 060504 (2008).
[Crossref] [PubMed]

I. E. Linington and N. V. Vitanov, “Robust creation of arbitrary-sized Dicke states of trapped ions by global addressing,” Phys. Rev. A 77(1), 010302(R) (2008).
[Crossref]

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

X. Y. Lü, J. B. Liu, C. L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78(3), 032305 (2008).
[Crossref]

2007 (10)

Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75(1), 012324 (2007).
[Crossref]

X. L. Song, L. Wang, R. Z. Lin, Z. H. Kang, X. Li, Y. Jiang, and J. Y. Gao, “Observation of CARS signal via maximal atomic coherence prepared by F-STIRAP in a three-level atomic system,” Opt. Express 15(12), 7499–7505 (2007).
[Crossref] [PubMed]

G. W. Lin, M. Y. Ye, L. B. Chen, Q. H. Du, and X. M. Lin, “Generation of the singlet state for three atoms in cavity QED,” Phys. Rev. A 76(1), 014308 (2007).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom–field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450(7167), 272–276 (2007).
[Crossref] [PubMed]

T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317(5837), 488–490 (2007).
[Crossref] [PubMed]

P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79(1), 53–77 (2007).
[Crossref]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450(7167), 268–271 (2007).
[Crossref] [PubMed]

J. Song, Y. Xia, and H. S. Song, “Entangled state generation via adiabatic passage in two distant cavities,” J. Phys. B 40(23), 4503–4512 (2007).
[Crossref]

J. Klein, F. Beil, and T. Halfmann, “Robust population transfer by stimulated raman adiabatic passage in a Pr3+ : Y2SiO5 crystal,” Phys. Rev. Lett. 99(11), 113003 (2007).
[Crossref] [PubMed]

L. B. Chen, M. Y. Ye, G. W. Lin, Q. H. Du, and X. M. Lin, “Generation of entanglement via adiabatic passage,” Phys. Rev. A 76(6), 062304 (2007).
[Crossref]

2006 (1)

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96(1), 010503 (2006).
[Crossref] [PubMed]

2005 (3)

M. Amniat-Talab, S. Guérin, N. Sangouard, and H. R. Jauslin, “Atom-photon, atom-atom, and photon-photon entanglement preparation by fractional adiabatic passage,” Phys. Rev. A 71(2), 023805 (2005).
[Crossref]

M. Amniat-Talab, S. Guérin, and H. R. Jauslin, “Decoherence-free creation of atom-atom entanglement in a cavity via fractional adiabatic passage,” Phys. Rev. A 72(1), 012339 (2005).
[Crossref]

S. B. Zheng, “Multi-atom entanglement engineering and phase-covariant cloning via adiabatic passage,” J. Opt. B: Quantum Semiclass. Opt. 7(5), 139–141 (2005).
[Crossref]

2004 (4)

H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70(1), 012305 (2004).
[Crossref]

S. Leslie, N. Shenvi, K. R. Brown, D. M. Stamper-Kurn, and K. B. Whaley, “Transmission spectrum of an optical cavity containing N atoms,” Phys. Rev. A 69(4), 043805 (2004).
[Crossref]

Z. Kis and E. Paspalakis, “Arbitrary rotation and entanglement of flux SQUID qubits,” Phys. Rev. B 69(2), 024510 (2004).
[Crossref]

T. Durt, D. Kaszlikowski, J. -L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69(3), 032313 (2004).
[Crossref]

2003 (4)

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90(16), 167906 (2003).
[Crossref] [PubMed]

A. B. Klimov, R. Guzmán, J. C. Retamal, and C. Saavedra, “Qutrit quantum computer with trapped ions,” Phys. Rev. A 67(6), 062313 (2003).
[Crossref]

X. B. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

S. Clark, A. Peng, M. Gu, and S. Parkins, “Unconditional preparation of entanglement between atoms in cascaded optical cavities,” Phys. Rev. Lett. 91(17), 177901 (2003).
[Crossref] [PubMed]

2002 (4)

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298(5597), 1372–1377 (2002).
[Crossref] [PubMed]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88(4), 040404 (2002).
[Crossref] [PubMed]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref] [PubMed]

R. G. Unanyan, M. Fleischhauer, N. V. Vitanov, and Klaas Bergmann, “Entanglement generation by adiabatic navigation in the space of symmetric multiparticle states,” Phys. Rev. A 66(4), 042101 (2002).
[Crossref]

2001 (1)

R. G. Unanyan, N. V. Vitanov, and K. Bergmann, “Preparation of entangled states by adiabatic passage,” Phys. Rev. Lett. 87(13), 137902 (2001).
[Crossref] [PubMed]

2000 (2)

D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418–4421 (2000).
[Crossref] [PubMed]

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature 404(6775), 247–255 (2000).
[Crossref] [PubMed]

1999 (3)

H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express 4(2), 77–83 (1999).
[Crossref] [PubMed]

S. J. van Enk, H. J. Kimble, J. I. Cirac, and P. Zoller, “Quantum communication with dark photons,” Phys. Rev. A 59(4), 2659–2664 (1999).
[Crossref]

N. V. Vitanov, K. A. Suominen, and B. W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32(18), 4535–4546 (1999).
[Crossref]

1998 (2)

R. G. Unanyan, M. Fleischhauer, B. W. Shore, and K. Bergmann, “Robust creation and phase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states,” Opt. Commun. 155(1–3), 144–154 (1998)
[Crossref]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70(3), 1003–1025 (1998).
[Crossref]

1997 (2)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
[Crossref]

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79(26), 5242–5245 (1997).
[Crossref]

1993 (1)

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref] [PubMed]

1992 (1)

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
[Crossref] [PubMed]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref] [PubMed]

1990 (1)

U. Gaubatz, P. Rudecki, S. Sciemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results,” J. Chem. Phys. 92(9), 5363–5376 (1990).
[Crossref]

1989 (1)

J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, “Adiabatic population transfer in a three-level system driven by delayed laser pulses,” Phys. Rev. A 40(11), 6741–6744 (1989).
[Crossref] [PubMed]

1988 (1)

U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Külz, and K. Bergmann, “Population switching between vibrational levels in molecular beams,” Chem. Phys. Lett. 149(5–6), 463–468 (1988).
[Crossref]

1984 (1)

J. Oreg, F. T. Hioe, and J. H. Eberly, “Adiabatic following in multilevel systems,” Phys. Rev. A 29(2), 690–697 (1984).
[Crossref]

Amniat-Talab, M.

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Figures (4)

Fig. 1
Fig. 1

A single 87Rb atom and a 87Rb BEC are trapped in two distant double-mode optical cavities, which are connected by an optical fiber. The states |gL〉, |g0〉, |gR〉 and |ga〉 correspond to |F = 1, mF = −1〉, |F = 1, mF = 0〉, |F = 1, mF = 1〉 of 5S1/2 and |F = 2, mF = 0〉 of 5S1/2, while |eL〉, |e0〉 and |eR〉 correspond to |F = 1, mF = −1〉, |F = 1, mF = 0〉 and |F = 1, mF = 1〉 of 5P3/2. The atomic transition |ga〉 ↔ |e0〉 of atom in cavity A is driven resonantly by a π-polarized classical field with Rabi frequency ΩA; |e0A ↔ |gLA (|e0A ↔ |gRA) is resonantly coupled to the cavity mode aA,L (aA,R) with coupling constant gA. The atomic transition |gLB ↔ |eLB (|gRB ↔ |eRB) of BEC in cavity B is driven resonantly by a π-polarized classical field with Rabi frequency ΩB; |eRB ↔ |g0B (|eLB ↔ |g0B) is resonantly coupled to the cavity mode aB,L (aB,R) with coupling constant gB.

Fig. 2
Fig. 2

The numerical simulation of Hamiltonian (3) in the entanglement generation process, where we choose g = 5Ω0, τ = Ω 0 1. (a): the Rabi frequency of ΩA(t), ΩB(t). (b): the time evolution of populations of the states |ϕ1〉, |ϕ11〉, and |ϕ12〉 is denoted by P1, P11, and P12 respectively. (c): time evolution of populations of other states {|ϕ2〉, |ϕ3〉, |ϕ4〉, |ϕ5〉, |ϕ6〉, |ϕ7〉, |ϕ8〉, |ϕ9〉, |ϕ10〉}, which are almost zero during the whole dynamics. (d): error probability Pe (t) defined by Eq. (6).

Fig. 3
Fig. 3

Fidelity of the entanglement state (obtained by numerical simulation of master equation (8)) as a function of the photon leakage rate κ and for the atom spontaneous radiation rate γ = 0, 0.2g, 0.4g, 0.6g, 0.8g, 1.0g (from the top to the bottom).

Fig. 4
Fig. 4

Fidelity vs the atom number N of the BEC with the parameters γ = κ = 0.4g, and the other parameters same as in Fig. 2.

Equations (11)

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H a c = k = L , R ( Ω A ( t ) | e 0 A g a | + g A ( t ) a A , k | e 0 A g k | + N Ω B ( t ) | E k B G k | ) + N g B ( t ) a B , L | E R B G 0 | + N g B ( t ) a B , R | E L B G 0 | + H . c . ,
H c f = k = L , R v k [ b k ( a A , k + + a B , k + ) + H . c . ] .
H I + H a c + H c f .
i t | ψ ( t ) = H I | ψ ( t ) .
| ϕ 1 = | g a A | G 0 B | 0000 c | 00 f , | ϕ 2 = | e 0 A | G 0 B | 0000 c | 00 f , | ϕ 3 = | g L A | G 0 B | 1000 c | 00 f , | ϕ 4 = | g R A | G 0 B | 0100 c | 00 f , | ϕ 5 = | g L A | G 0 B | 0000 c | 10 f , | ϕ 6 = | g R A | G 0 B | 0000 c | 01 f , | ϕ 7 = | g L A | G 0 B | 0010 c | 00 f , | ϕ 8 = | g R A | G 0 B | 0001 c | 00 f , | ϕ 9 = | g L A | E R B | 0000 c | 00 f , | ϕ 10 = | g R A | E L B | 0000 c | 00 f , | ϕ 11 = | g L A | G R B | 0000 c | 00 f , | ϕ 12 = | g R A | G L B | 0000 c | 00 f ,
| D ( t ) = K { 2 g A ( t ) Ω B ( t ) | ϕ 1 Ω A ( t ) Ω B ( t ) [ | ϕ 3 + | ϕ 4 | ϕ 7 | ϕ 8 ] g B ( t ) Ω A ( t ) [ | ϕ 11 + | ϕ 12 ] } ,
g A ( t ) , g B ( t ) Ω A ( t ) , Ω B ( t ) ,
| D ( t ) 2 g A ( t ) Ω B ( t ) | ϕ 1 g B ( t ) Ω A ( t ) [ | ϕ 11 + | ϕ 12 ] .
lim t g B ( t ) Ω A ( t ) g A ( t ) Ω B ( t ) = 0 , lim t + g A ( t ) Ω B ( t ) g B ( t ) Ω A ( t ) = 1 2 ,
P e ( t ) = 1 | D ( t ) | φ s ( t ) | 2 ,
d ρ d t = i [ H I , ρ ] k = L , R [ κ f k 2 ( b k + b k ρ 2 b k ρ b k + + ρ b k + b k ) i = A , B κ i k 2 ( a i k + a i k ρ 2 a i k + ρ a i k + ρ a i k + a i k ) ] j = a , L , R γ 0 j A 2 ( σ e 0 e 0 A ρ 2 σ g j e 0 A ρ σ e 0 g j A + ρ σ e 0 e 0 A ) h = 1 N k = L , R j = k , 0 γ k j B h 2 ( σ e k e k B h ρ 2 σ g j e k B h ρ σ e k g j B h + ρ σ e k e k B h ) ,

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