## Generation of three-dimensional entangled state between a single atom and a Bose-Einstein condensate via adiabatic passage |

Optics Express, Vol. 20, Issue 13, pp. 14547-14555 (2012)

http://dx.doi.org/10.1364/OE.20.014547

Acrobat PDF (830 KB)

### 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

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]

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]

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]

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]

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]

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]

^{87}

*Rb*atom and a

^{87}

*Rb*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

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]

## 2. The fundamental model

^{87}

*Rb*atom and a

^{87}

*Rb*BEC are trapped in two distant double-mode optical cavities, which are connected by an optical fiber (see Fig. 1). The

^{87}

*Rb*atomic levels and transitions are also depicted in this figure. [46

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]

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]

*g*〉, |

_{L}*g*

_{0}〉, |

*g*〉 and |

_{R}*g*〉 correspond to |

_{a}*F*= 1,

*m*= −1〉, |

_{F}*F*= 1,

*m*= 0〉, |

_{F}*F*= 1,

*m*= 1〉 of 5

_{F}*S*

_{1/2}and |

*F*= 2,

*m*= 0〉 of 5

_{F}*S*

_{1/2}, while |

*e*〉, |

_{L}*e*

_{0}〉 and |

*e*〉 correspond to |

_{R}*F*= 1,

*m*= −1〉, |

_{F}*F*= 1,

*m*= 0〉 and |

_{F}*F*= 1,

*m*= 1〉 of 5

_{F}*P*

_{3/2}. The atomic transition |

*g*〉 ↔ |

_{a}*e*

_{0}〉 of atom in cavity

*A*is driven resonantly by a

*π*-polarized classical field with Rabi frequency Ω

*; |*

_{A}*e*

_{0}〉

*↔ |*

_{A}*g*〉

_{L}*(|*

_{A}*e*

_{0}〉

*↔ |*

_{A}*g*〉

_{R}*) is resonantly coupled to the cavity mode*

_{A}*a*(

_{A,L}*a*) with coupling constant

_{A,R}*g*. The atomic transition |

_{A}*g*〉

_{L}*↔ |*

_{B}*e*〉

_{L}*(|*

_{B}*g*〉

_{R}*↔ |*

_{B}*e*〉

_{R}*) of BEC in cavity*

_{B}*B*is driven resonantly by a

*π*-polarized classical field with Rabi frequency Ω

*; |*

_{B}*e*〉

_{R}*↔ |*

_{B}*g*

_{0}〉

*(|*

_{B}*e*〉

_{L}*↔ |*

_{B}*g*

_{0}〉

*) is resonantly coupled to the cavity mode*

_{B}*a*(

_{B,L}*a*) with coupling constant

_{B,R}*g*. Here we consider BEC for a single excitation, the ground and single excitation states are described by the state vectors

_{B}*f*= 0,

*L*,

*R*), where |...〉

*describe the state of the*

_{j}*j*th atom in the BEC [46

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]

*g*〉

_{a}*and |*

_{A}*G*

_{0}〉

*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*

_{B}*h*̄ = 1) [47

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]

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

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]

## 3. Generation of the three-dimensional entanglement state

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]

*n*

_{AL}*n*

_{AR}*n*

_{BL}*n*〉

_{BR}*denotes the field state with*

_{c}*n*(

_{Ai}*i*=

*L*,

*R*) photons in the

*i*polarized mode of cavity

*A*,

*n*in the

_{Bi}*i*polarized mode of cavity

*B*, and |

*n*

_{L}*n*〉

_{R}*represents*

_{f}*n*photons in

_{i}*i*polarized mode of the fiber. The Hamiltonian

*H*has the following dark state: which is the eigenstate of the Hamiltonian corresponding to zero eigenvalue. Here and in the following

_{I}*g*, Ω

_{i}*are real, and*

_{i}*ϕ*

_{1}〉, if we design pulse shapes such that we can adiabatically transfer the initial state |

*ϕ*

_{1}〉 to a equal superposition of |

*ϕ*

_{1}〉, |

*ϕ*

_{11}〉 and |

*ϕ*

_{12}〉, i.e.,

*g*(

_{A}*t*) =

*g*(

_{B}*t*) =

*g*,

*ν*=

_{L}*ν*=

_{R}*ν*= 100

*g*,

*N*= 10

^{4}, laser Rabi frequencies are chosen as Ω

*(*

_{A}*t*) = Ω

_{0}exp [−(

*t*−

*t*

_{0})

^{2}/200

*τ*

^{2}] and

*t*

_{0}= 20

*τ*being the delay between pulses [52

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

*g*= 5Ω

_{0},

*(*

_{A}*t*), Ω

*(*

_{B}*t*) are shown in Fig. 2(a). Figure 2(b) and 2(c) shows the time evolution of populations. In Fig. 2(b)

*P*

_{1},

*P*

_{11}, and

*P*

_{12}denote the populations of the states |

*ϕ*

_{1}〉, |

*ϕ*

_{11}〉, and |

*ϕ*

_{12}〉. Figure 2(c) show the time evolution of populations of other states {|

*ϕ*

_{2}〉, |

*ϕ*

_{3}〉, |

*ϕ*

_{4}〉, |

*ϕ*

_{5}〉, |

*ϕ*

_{6}〉, |

*ϕ*

_{7}〉, |

*ϕ*

_{8}〉, |

*ϕ*

_{9}〉, |

*ϕ*

_{10}〉}, which are almost zero during the whole dynamics. Finally

*P*

_{1},

*P*

_{11}, and

*P*

_{12}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

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]

*φ*(

_{s}*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.

## 4. Effects of spontaneous emission and photon leakage

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]

*e*

_{0}〉

*to |*

_{A}*g*〉

_{j}*and |*

_{A}*e*〉

_{k}*to |*

_{B}*g*〉

_{j}*of the*

_{B}*h*th atom in the BEC, respectively;

*κ*and

_{ik}*κ*denote the photon leakage rates from the cavity fields and fiber modes, respectively;

_{fk}*ϕ*

_{1}〉 〈

*ϕ*

_{1}|, by solving numerically Eq. (11) in the subspace spanned by the vectors (5) and |

*ϕ*

_{13}〉 = |

*g*〉

_{L}*|*

_{A}*G*

_{0}〉

*|0000〉*

_{B}*|00〉*

_{c}*, |*

_{f}*ϕ*

_{14}〉 = |

*g*〉

_{R}*|*

_{A}*G*

_{0}〉

*|0000〉*

_{B}*|00〉*

_{c}*. Fig. 3 shows the fidelity of the entanglement state as a function of the photon leakage rate*

_{f}*κ*(

*κ*=

*κ*=

_{Ak}*κ*=

_{Bk}*κ*) and for the atom spontaneous radiation rate

_{fk}*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*,

*κ*∼

*g*.

## 5. Discussion and conclusion

^{87}

*Rb*BEC in cavity QED has also been realized in recently experiment [47

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]

*g*,

*κ*,

*γ*) = 2

*π*× (10.6, 1.3, 3.0) MHz is realizable. So the condition

*γ*,

*κ*< 0.4

*g*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

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]

**450**(7167), 268–271 (2007). [CrossRef] [PubMed]

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]

*g*(

_{B}*t*) will decrease with increasing atom number

*N*. We can increase the Ω

*(*

_{B}*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

**106**(21), 210503 (2011). [CrossRef] [PubMed]

*N*of the BEC is plotted in Fig. 4 with the parameters

*γ*=

*κ*= 0.4

*g*, 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.

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]

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]

## Acknowledgments

## References and links

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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 |

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 |

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. |

47. | F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature |

48. | J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature |

49. | T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science |

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. |

51. | S. B. Zheng, “Multi-atom entanglement engineering and phase-covariant cloning via adiabatic passage,” J. Opt. B: Quantum Semiclass. Opt. |

52. | P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. |

53. | H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A |

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. |

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 |

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 |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 2, 2012

Revised Manuscript: April 16, 2012

Manuscript Accepted: May 7, 2012

Published: June 14, 2012

**Citation**

Li-Bo Chen, Peng Shi, Chun-Hong Zheng, and Yong-Jian Gu, "Generation of three-dimensional entangled state between a single atom and a Bose-Einstein condensate via adiabatic passage," Opt. Express **20**, 14547-14555 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14547

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