## Heralded generation of multipartite entanglement for one photon by using a single two-dimensional nonlinear photonic crystal |

Optics Express, Vol. 21, Issue 7, pp. 7875-7881 (2013)

http://dx.doi.org/10.1364/OE.21.007875

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### Abstract

We propose a compact scheme for the heralded generation of single-photon multipartite entanglement by using a single two-dimensional nonlinear photonic crystal. Studies have shown that by appropriate structure design, the single-photon entanglement shared among three spatially distinct optical modes can be generated through three concurrent spontaneous parametric down-conversion processes by using the other photon in an identical spatial mode as a trigger. Furthermore, we analyze the entanglement of such heralded single-photon tripartite W-type state theoretically. This method can be expanded for the heralded single-photon N-partite entanglement generation. This compact and stable quantum light source may act as a key ingredient in quantum information science.

© 2013 OSA

## 1. Introduction

1. S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A **72**(6), 064306 (2005). [CrossRef]

7. I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics **6**(4), 234–237 (2012). [CrossRef]

1. S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A **72**(6), 064306 (2005). [CrossRef]

2. S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. **66**(3), 252–255 (1991). [CrossRef] [PubMed]

3. L. Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett. **73**(17), 2279–2283 (1994). [CrossRef] [PubMed]

4. D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett. **104**(18), 180504 (2010). [CrossRef] [PubMed]

5. E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett. **88**(7), 070402 (2002). [CrossRef] [PubMed]

6. G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A **85**(2), 022316 (2012). [CrossRef]

7. I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics **6**(4), 234–237 (2012). [CrossRef]

7. I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics **6**(4), 234–237 (2012). [CrossRef]

13. S. Y. Lan, A. G. Radnaev, O. A. Collins, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “A multiplexed quantum memory,” Opt. Express **17**(16), 13639–13645 (2009). [CrossRef] [PubMed]

14. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. **84**(2), 777–838 (2012). [CrossRef]

## 2. Theoretic analysis

^{(2)}can be written as a Fourier serieswhere

20. Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ^{(2)} nonlinear photonic crystal,” Phys. Rev. A **86**(2), 023835 (2012). [CrossRef]

*j*= 1,2,3), and we have

*L*is the length of NPC along z direction and

*j*. For the noncollinear type-0 nondegenerate QPM SPDC, the joint spectrum density can be written as

17. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A **62**(6), 062314 (2000). [CrossRef]

24. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A **61**(5), 052306 (2000). [CrossRef]

*C*

_{a,b}_{(}

_{c}_{)}means the concurrence between

*a*and

*b*(

*c*),

*C*means the concurrence between

_{a,bc}*a*and

*bc*. We can obtain

15. S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science **324**(5928), 764–768 (2009). [CrossRef] [PubMed]

16. P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys. **11**(6), 063029 (2009). [CrossRef]

*j*= 1,2,3) can be easily decomposed in terms of unitary operations that can be implemented with beamsplitters and phaseshifters (see Fig. 3) [23

23. M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A **55**(4), 2564–2579 (1997). [CrossRef]

*W*> as nonlocal observables

_{j}*M*= |

_{j}*W*><

_{j}*W*|. For any state

_{j}*ρ*The single-photon tripartite entanglement is three-body pure state for the relative phase between any two modes is fixed in a single crystal wafer. So for maximal three-mode entanglement,

## 3. Conclusion

4. D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett. **104**(18), 180504 (2010). [CrossRef] [PubMed]

25. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. **76**(5), 722–725 (1996). [CrossRef] [PubMed]

29. S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A **86**(3), 034302 (2012). [CrossRef]

## Acknowledgments

## References and links

1. | S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A |

2. | S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. |

3. | L. Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett. |

4. | D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett. |

5. | E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett. |

6. | G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A |

7. | I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics |

8. | L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature |

9. | C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science |

10. | J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett. |

11. | K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature |

12. | C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett. |

13. | S. Y. Lan, A. G. Radnaev, O. A. Collins, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “A multiplexed quantum memory,” Opt. Express |

14. | J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. |

15. | S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science |

16. | P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys. |

17. | W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A |

18. | V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. |

19. | N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. |

20. | Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ |

21. | J. P. Torres, A. Alexandrescu, S. Carrasco, and L. Torner, “Quasi-phase-matching engineering for spatial control of entangled two-photon states,” Opt. Lett. |

22. | H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun |

23. | M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A |

24. | V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A |

25. | C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. |

26. | J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature |

27. | C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express |

28. | N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A |

29. | S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A |

**OCIS Codes**

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: December 17, 2012

Revised Manuscript: March 14, 2013

Manuscript Accepted: March 17, 2013

Published: February 25, 2013

**Citation**

J. Shi, P. Xu, M. L. Zhong, Y. X. Gong, Y. F. Bai, W. J. Yu, Q. W. Li, H. Jin, and S. N. Zhu, "Heralded generation of multipartite entanglement for one photon by using a single two-dimensional nonlinear photonic crystal," Opt. Express **21**, 7875-7881 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-7875

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### References

- S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A72(6), 064306 (2005). [CrossRef]
- S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett.66(3), 252–255 (1991). [CrossRef] [PubMed]
- L. Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett.73(17), 2279–2283 (1994). [CrossRef] [PubMed]
- D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, “Purification of single-photon entanglement,” Phys. Rev. Lett.104(18), 180504 (2010). [CrossRef] [PubMed]
- E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, “Teleportation of a vacuum--one-photon qubit,” Phys. Rev. Lett.88(7), 070402 (2002). [CrossRef] [PubMed]
- G. Björk, A. Laghaout, and U. L. Andersen, “Deterministic teleportation using single-photon entanglement as a resource,” Phys. Rev. A85(2), 022316 (2012). [CrossRef]
- I. Usmani, C. Clausen, F. Bussières, N. Sangouard, M. Afzelius, and N. Gisin, “Heralded quantum entanglement between two crystals,” Nat. Photonics6(4), 234–237 (2012). [CrossRef]
- L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature414(6862), 413–418 (2001). [CrossRef] [PubMed]
- C. W. Chou, J. Laurat, H. Deng, K. S. Choi, H. de Riedmatten, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316(5829), 1316–1320 (2007). [CrossRef] [PubMed]
- J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, “Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling,” Phys. Rev. Lett.99(18), 180504 (2007). [CrossRef] [PubMed]
- K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature452(7183), 67–71 (2008). [CrossRef] [PubMed]
- C. Simon, H. de Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memories,” Phys. Rev. Lett.98(19), 190503 (2007). [CrossRef] [PubMed]
- S. Y. Lan, A. G. Radnaev, O. A. Collins, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, “A multiplexed quantum memory,” Opt. Express17(16), 13639–13645 (2009). [CrossRef] [PubMed]
- J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys.84(2), 777–838 (2012). [CrossRef]
- S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science324(5928), 764–768 (2009). [CrossRef] [PubMed]
- P. Lougovski, S. J. van Enk, K. S. Choi, S. B. Papp, H. Deng, and H. J. Kimble, “Verifying multi-partite mode entanglement of W states,” New J. Phys.11(6), 063029 (2009). [CrossRef]
- W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A62(6), 062314 (2000). [CrossRef]
- V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett.81(19), 4136–4139 (1998). [CrossRef]
- N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett.84(19), 4345–4348 (2000). [CrossRef] [PubMed]
- Y. X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A86(2), 023835 (2012). [CrossRef]
- J. P. Torres, A. Alexandrescu, S. Carrasco, and L. Torner, “Quasi-phase-matching engineering for spatial control of entangled two-photon states,” Opt. Lett.29(4), 376–378 (2004). [CrossRef] [PubMed]
- H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat Commun2, 429 (2011). [CrossRef] [PubMed]
- M. Z-dotukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle entanglements via multiport beam splitters,” Phys. Rev. A55(4), 2564–2579 (1997). [CrossRef]
- V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A61(5), 052306 (2000). [CrossRef]
- C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett.76(5), 722–725 (1996). [CrossRef] [PubMed]
- J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410(6832), 1067–1070 (2001). [CrossRef] [PubMed]
- C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Express19(25), 25685–25695 (2011). [CrossRef] [PubMed]
- N. Sangouard, C. Simon, T. Coudreau, and N. Gisin, “Purification of single-photon entanglement with linear optics,” Phys. Rev. A78(5), 080301(R) (2008). [CrossRef]
- S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A86(3), 034302 (2012). [CrossRef]

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