## Controllable entanglement preparations between atoms in spatially-separated cavities via quantum Zeno dynamics |

Optics Express, Vol. 20, Issue 12, pp. 13440-13450 (2012)

http://dx.doi.org/10.1364/OE.20.013440

Acrobat PDF (3112 KB)

### Abstract

By using quantum Zeno dynamics, we propose a controllable approach to deterministically generate tripartite GHZ states for three atoms trapped in spatially separated cavities. The nearest-neighbored cavities are connected via optical fibers and the atoms trapped in two ends are tunably driven. The generation of the GHZ state can be implemented by only one step manipulation, and the EPR entanglement between the atoms in two ends can be further realized deterministically by Von Neumann measurement on the middle atom. Note that the duration of the quantum Zeno dynamics is controllable by switching on/off the applied external classical drivings and the desirable tripartite GHZ state will no longer evolve once it is generated. The robustness of the proposal is numerically demonstrated by considering various decoherence factors, including atomic spontaneous emissions, cavity decays and fiber photon leakages, etc. Our proposal can be directly generalized to generate multipartite entanglement by still driving the atoms in two ends.

© 2012 OSA

## 1. Introduction

7. X. B. Zou, K. Pahlke, and W. Mathis, “Conditional generation of the Greenberger-Horne-Zeilinger state of four distant atoms via cavity decay,” Phys. Rev. A **68**, 024302 (2003). [CrossRef]

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13. M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature (London) **467**, 570–573 (2010). [CrossRef]

10. L. F. Wei, Y. X. Liu, and F. Nori, “Generation and control of Greenberger-Horne-Zeilinger entanglement in superconducting circuits,” Phys. Rev. Lett. **96**, 246803 (2006). [CrossRef] [PubMed]

13. M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature (London) **467**, 570–573 (2010). [CrossRef]

12. D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature (London) **438**, 639–642 (2005). [CrossRef]

14. J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. **73**, 565–582 (2001). [CrossRef]

15. J. I. Cirac and P. Zoller, “Preparation of macroscopic superpositions in many-atom systems,” Phys. Rev. A **50**, R2799–R2802 (1994). [CrossRef] [PubMed]

16. J. Hong and H.-W. Lee, “Quasideterministic generation of entangled atoms in a cavity,” Phys. Rev. Lett. **89**, 237901 (2002). [CrossRef] [PubMed]

17. 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**, 3221–3224 (1997). [CrossRef]

21. A. S. Parkins and H. J. Kimble, “Position-momentum Einstein-Podolsky-Rosen state of distantly separated trapped atoms,” Phys. Rev. A **61**, 052104 (2000). [CrossRef]

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

*connected by an optical fiber*. Based on this proposal, various schemes [23

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

32. P.-B. Li and F.-L. Li, “Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system,” Opt. Express **19**, 1207–1216 (2011) [CrossRef] [PubMed]

28. X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A **79**, 052330 (2009). [CrossRef]

29. S. B. Zheng, “Generation of Greenberger-Horne-Zeilinger states for multiple atoms trapped in separated cavities,” Eur. Phys. J. D **54**, 719–722 (2009). [CrossRef]

30. A. Zheng and J. Liu, “Generation of an N-qubit Greenberger-Horne-Zeilinger state with distant atoms in bimodal cavities,” J. Phys. B: At. Mol. Opt. Phys. **44**, 165501 (2011). [CrossRef]

33. P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A **275**, 12–19 (2000). [CrossRef]

35. P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. **196**, 012017 (2009). [CrossRef]

33. P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A **275**, 12–19 (2000). [CrossRef]

36. P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, “Zeno dynamics yields ordinary constraints,” Phys. Rev. A **65**, 012108 (2002). [CrossRef]

35. P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. **196**, 012017 (2009). [CrossRef]

*H*=

_{K}*H*+

*KH*, with

_{a}*H*is for the quantum system investigated and the

*H*describing the additional interaction with the detector, and

_{a}*K*the coupling constant. In the limit

*K*→ ∞, the subsystem of interest is dominated by the evolution operator

*U*(

*t*) = lim

_{K}_{→∞}exp(

*iKH*)

_{a}t*U*(

_{K}*t*) which takes the form

*U*(

*t*) = exp(−

*it*∑

_{n}*P*) [35

_{n}HP_{n}35. P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. **196**, 012017 (2009). [CrossRef]

*P*is the eigenprojection of

_{n}*H*= ∑

_{a}

_{n}*λ*corresponding to the eigenvalue

_{n}P_{n}*λ*. As a consequence, the system-detector can be described by the evolution operator

_{n}*U*(

_{K}*t*) ∼ exp(−

*iKH*)

_{a}t*U*(

*t*) = exp[−

*i*∑

*(*

_{n}*Kλ*+

_{n}P_{n}*P*)

_{n}HP_{n}*t*]. This result is of great importance in view of practical applications of the quantum Zeno dynamics, such as to prepare various quantum states [37

37. A. Luis, “Quantum-state preparation and control via the Zeno effect,” Phys. Rev. A **63**, 052112 (2001). [CrossRef]

38. X. B. Wang, J. Q. You, and F. Nori, “Quantum entanglement via two-qubit quantum Zeno dynamics,” Phys. Rev. A **77**, 062339 (2008). [CrossRef]

39. A. Beige, D. Braun, B. Tregenna, and P. L. Knight, “Quantum computing using dissipation to remain in a decoherence-free subspace,” Phys. Rev. Lett. **85**, 1762–1765 (2000). [CrossRef] [PubMed]

42. J. D. Franson, T. B. Pittman, and B. C. Jacobs, “Zeno logic gates using microcavities,” J. Opt. Soc. Am. B **24**, 209–213 (2007). [CrossRef]

28. X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A **79**, 052330 (2009). [CrossRef]

30. A. Zheng and J. Liu, “Generation of an N-qubit Greenberger-Horne-Zeilinger state with distant atoms in bimodal cavities,” J. Phys. B: At. Mol. Opt. Phys. **44**, 165501 (2011). [CrossRef]

*N*atoms trapped individually in

*N*cavity is provided in Sec. 3. Finally, in Sec. 4 we discuss the feasibility of our proposal and give our conclusions.

## 2. Generation of GHZ state of atoms trapped in different cavities by quantum Zeno dynamics

*e*〉 and two dipole-transition forbidden ground states |0〉, |1〉. The first and third atomic transitions |1〉 ↔ |

*e*〉 are driven resonantly by classical lasers with the couplings coefficient Ω

_{1}and Ω

_{3}, respectively. The other atomic transition is resonantly coupled to the corresponding cavity mode with coupling constant

*g*

_{i,r(l)}(

*i*= 1, 2, 3). The subscript

*r*(

*l*) denotes the right (left) circularly polarization. In the short fiber limit, (2

*Lv*̄)/(2

*πc*) ≪ 1, where

*L*is the length of the fiber and

*v*̄ is the decay rate of the cavity fields into a continuum of fiber modes [23

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

*a*and

*b*are the annihilation operators associated with the modes of cavity and fiber respectively and

*v*(

_{i}*i*= 1, 2) is the corresponding cavity-fiber coupling constant. We assume that

*g*

_{1,r}=

*g*

_{2,r(l)}=

*g*

_{3,l}=

*g*and

*v*

_{1}=

*v*

_{2}=

*v*for convenience. If the initial state of the whole system is |1,0,0〉

*|0〉*

_{a}_{c1}|0〉

_{f1}|0,0〉

_{c2}|0〉

_{f2}|0〉

_{c3}, then the evolution of system will be restricted in the subspace spanned by

*i, j,k*〉

*(*

_{a}*i*,

*j*,

*k*= 0, 1,

*e*) denotes the state of atoms in each cavity,

*n*in |

*n*〉

*(*

_{s}*s*=

*c*

_{1},

*f*

_{1},

*c*

_{2},

*f*

_{2},

*c*

_{3}) denotes the photon number in cavities or fibers.

*g,v*≫ Ω

_{1}, Ω

_{3}, the above Hilbert subspace is split into nine invariant Zeno subspaces [35

**196**, 012017 (2009). [CrossRef]

*λ*

_{1}= 0,

*N*(

_{i}*i*= 1, 2, 3,..., 9) being the normalization factor for the eigenstate |

*ψ*〉. Under above condition, the system can be effectively described by the following Hamiltonian It reduces to if the initial state is |

_{i}*ϕ*

_{1}〉. The effective Hamiltonian

*H*

_{eff}implies that the evolution of system is restricted in the subspace, wherein the cavity modes are kept in the vacuum. Consequently, after the evolution time

*t*the state of the system becomes

*can be generated, i.e., where*

_{a}*e*〉 in the first and third atom, would provoke a slow evolution of the state away from GHZ for

*t*>

*τ*. Therefore, once the GHZ state has been generated, one must switch the lasers off. Since

*H*in Eq. (2) becomes zero and

_{l}*H*|

_{a–c–f}*ψ*(

*τ*)〉 = 0, the GHZ state is preserved. Furthermore, if we perform a single-qubit rotation

*R*(

_{x}*π*/4)(= exp(

*iσ*·

_{x}*π*/4)) on the middle atom, then the tripartite GHZ state reduces to the state: [(|1,0〉

_{1,3}+

*i*|0,1〉

_{1,3})|0〉

_{2}+ (

*i*|1,0〉

_{1,3}+ |0,1〉

_{1,3})|1〉

_{2}]/2. Consequently, the EPR entanglement between two distant atoms can be deterministically obtained by projective measurement on the middle atom (no matter it is found at which state) [43

43. L. F. Wei, Yu-xi Liu, and F. Nori, “Testing Bell’s inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B **72**, 104516 (2005). [CrossRef]

*g,v*≫ Ω

_{1}, Ω

_{3}is satisfied robustly. Now, we discuss how the ratio Ω

_{3}/

*g*influences the fidelity

*F*= |〈Ψ(

*τ*)|

*φ*(

*τ*)〉|

^{2}, with |

*φ*(

*τ*)〉 being the relevant final state evolved by the original

*H*

_{total}defined in Eq. (1). On the other hand, the ratio

*v*/

*g*is another important factor affecting the fidelity [23

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

25. 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**, 012324 (2007). [CrossRef]

28. X.-Y. Lv, L.-G. Si, X.-Y. Hao, and X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A **79**, 052330 (2009). [CrossRef]

_{3}/

*g*corresponds to the higher fidelity. We also note that, even at the relatively-low ratio

*v*/

*g*= 0.5, the fidelity is still high, i.e., above 97%. This is very important for the practical application of our scheme, as the large cavity-fiber coupling is not easy to be satisfied in the realistic experiments. Furthermore, in order to obtain high fidelity in moderate time, we can choose typically the ratios: Ω

_{3}/

*g*= 0.04 and

*v*/

*g*= 1.

*t*reads |

*φ*〉 = ∑

_{total}

_{i}*c*(

_{i}*t*)|

*ϕ*〉 within the subspace spanned by the basic state vectors in Eq. (4). The occupation probability for each state vector |

_{i}*ϕ*〉 during the evolution is

_{i}*P*(

_{i}*t*) = |

*c*(

_{i}*t*)|

^{2}, and satisfies the condition ∑

_{i}*P*(

_{i}*t*) = 1. By solving the Schrödinger equation, the variation of the occupation probability

*P*(

_{c}*t*) (≡ ∑

_{i=3,5,7,9}

*P*(

_{i}*t*)) is portrayed in Fig. 4 (red line). As shown in Fig. 4, the occupation probability of the cavity photonic state is less than 0.04. Therefore, our claim that the scheme is immune to the cavity decay seems justifiable. Meanwhile, Fig. 4 also describes the occupation probabilities

*P*(

_{f}*t*)(≡ ∑

_{i=4,8}

*P*(

_{i}*t*), green line) and

*P*(

_{e}*t*)(≡ ∑

_{i=2,6,10}

*P*(

_{i}*t*), black line) for those states, wherein the photon is in the fiber and one of the atoms in its excited state, respectively. Since

*P*>

_{e}*P*in Fig. 4, our protocol is more sensitive to atomic spontaneous emission than the fiber loss. To check this, let us investigate the evolution of the system governed by the following non-Hermitian Hamiltonian where

_{f}*γ*is the spontaneous emission rate for atoms and

*κ*

_{c(f)}denotes the decay rate of the cavity modes (fiber modes). It is seen from Fig. 5 that the atomic spontaneous emission is the dominant factor of degrading the fidelity of the generated GHZ state. This can also be seen directly in the explicit form of the state |

*ψ*

_{1}〉, where the population probability of the atomic excited state is larger than that of the photon in fiber.

## 3. Generalization to N-atom entanglement

*N*-atom GHZ states. Let us consider the configuration shown in Fig. 6, where

*N*atoms are individually trapped in

*N*cavities connected by

*N*− 1 short fibers. The level configuration of the atoms between two ends are chosen the same as that of the middle atom in the above 3-atom case. The Hamiltonian of the present system reads where

*N*is odd number (

*N*= 2

*k*+ 1,

*k*= 1,2,3,...), and

*g*

_{i,r(l)}=

*g*,

*v*=

_{i}*v*. If the initial state of the whole system is prepared at the state |1,0,0,...,0〉

*|0〉*

_{a}*, then the system will evolve within the subspace Γ*

_{all}_{full}spanned by the vectors: {|

*ϕ*′

_{1}〉, |

*ϕ*′

_{2}〉, |

*ϕ*′

_{3}〉,..., |

*ϕ*′

_{8k+3}〉}:

*means that all boson modes are in the vacuum state,*

_{all}*n*(

*n*

_{1},

*n*

_{2}) in |

*n*〉

*or |*

_{s}*n*

_{1},

*n*

_{2}〉

*(*

_{s}*s*=

*c*,

_{i}*f*) denotes the photon number in the corresponding resonator while other cavities or fibers are in the vacuum state. Similar to the above procedure, we get the effective Hamiltonian where Set

_{i}*N*-atomic GHZ state can be generated. In order to test the effectiveness of our proposal, we consider specifically, for example, the case of five atoms. In Fig. 7, we plot the time-evolution behaviors of the occupation probability in the initial state |

*ϕ*′

_{1}〉 governed by total Hamiltonian (Eq. (13)) and by the effective Hamiltonian (Eq. (17)). It is shown that the numerical results under these two Hamiltonians agree with each other reasonably well. Therefore, our effective model is valid.

## 4. Discussions and conclusions

^{87}Rb atom, whose relevant atomic levels are shown in Fig. 8. Where the first (third) atom is coupled resonantly to an external

*π*-polarized classical field and a

*π*

_{+}(

*π*

_{−}) polarized photon modes of the cavity, and the middle atom is coupled resonantly to a

*π*

_{+}and a

*π*

_{−}polarized modes. Based on the above discussions, in order to obtain fidelity larger than 90%, one should keep the decay rate

*γ*< 0.0055

*g*. This condition can be satisfied in recent experiments [44

44. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A **71**, 013817 (2005). [CrossRef]

45. J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A **67**, 033806 (2003). [CrossRef]

*g*/2

*π*= 750 MHz,

*γ*/2

*π*= 2.62 MHz,

*κ*/2

_{c}*π*= 3.5 MHz. Also, a near perfect fiber-cavity coupling with an efficiency larger than 99.9% can be realized using fiber-taper coupling to high-Q silica microspheres [46

46. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. **91**, 043902 (2003). [CrossRef] [PubMed]

47. K. J. Gordon, V. Fernandez, P. D. Townsend, and G. S. Buller, “A short wavelength GigaHertz clocked fiber-optic quantum key distribution system,” IEEE J. Quantum Electron. **40**, 900–908 (2004). [CrossRef]

^{5}Hz. With these parameters, it seems that the present entanglement-generation scheme with a high fidelity larger than 93% could be feasible with the present experimental technique. The duration of the Zeno pulses Ω

_{1}and Ω

_{3}utilized to generate the desirable three-atom GHZ state can be estimated as

## Acknowledgments

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44. | S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A |

45. | J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A |

46. | S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. |

47. | K. J. Gordon, V. Fernandez, P. D. Townsend, and G. S. Buller, “A short wavelength GigaHertz clocked fiber-optic quantum key distribution system,” IEEE J. Quantum Electron. |

**OCIS Codes**

(270.5580) Quantum optics : Quantum electrodynamics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 26, 2012

Revised Manuscript: May 12, 2012

Manuscript Accepted: May 15, 2012

Published: May 31, 2012

**Citation**

Wen-An Li and Lian-Fu Wei, "Controllable entanglement preparations between atoms in spatially-separated cavities via quantum Zeno dynamics," Opt. Express **20**, 13440-13450 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13440

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