## Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing |

Optics Express, Vol. 20, Issue 22, pp. 24263-24271 (2012)

http://dx.doi.org/10.1364/OE.20.024263

Acrobat PDF (1101 KB)

### Abstract

We propose a scheme to implement the Deutsch’s algorithm through non-degenerate four-wave mixing process. By employing photon topological charges of optical vortices, we demonstrate the ability to realize the necessary four logic gates for all balanced and constant functions. We also analyze the feasibility of the proposed scheme on the single photon level.

© 2012 OSA

## 1. Introduction

## 2. Experimental scheme

20. H. T. Zhou, D. W. Wang, D. Wang, J. X. Zhang, and S. Y. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A **84**(5), 053835 (2011). [CrossRef]

12. W. Jiang, Q. F. Chen, Y. S. Zhang, and G. C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A **74**(4), 043811 (2006). [CrossRef]

21. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**(2), 161–204 (2011). [CrossRef]

21. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**(2), 161–204 (2011). [CrossRef]

22. Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. **21**(14), 1064–1066 (1996). [CrossRef] [PubMed]

## 3. Realization of quantum Deutsch’s algorithm

### 3.1. Coding rule for the logic gates and theoretical analysis

### 3.2. The experimental scheme to realize the four logic gates

## 4. Result testing

4. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

24. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**(25), 257901 (2002). [CrossRef] [PubMed]

^{85}Rb as the energy diagram in Fig. 1(b), the calculated relative transition ratio for signal field is 83.1% [25

25. D. A. Steck, “Rubidium 85 D line data,” http://steck.us/alkalidata.

^{85}Rb chose as the medium to interact with light [20

20. H. T. Zhou, D. W. Wang, D. Wang, J. X. Zhang, and S. Y. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A **84**(5), 053835 (2011). [CrossRef]

^{85}Rb system the frequency difference is around GHz), thus we can filter the noise photons with a multi-pass Fabry-Pérot etalon. It has been demonstrated that the extremely large total suppression of the pump beam can reach 118 dB signal-to-noise ratio [26

26. D. Höckel and O. Benson, “Electromagnetically induced transparency in cesium vapor with probe pulses on the single-photon level,” Phys. Rev. Lett. **105**(15), 153605 (2010). [CrossRef] [PubMed]

5. Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A **78**(5), 053810 (2008). [CrossRef]

## 5. Conclusion

^{85}Rb vapor. Our scheme is an important element in quantum networking with atomic ensembles, and we believe that the proposal can also be applied to quantum computation with higher-dimensional quantum system using OAM of photons.

## Acknowledgments

## References and links

1. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

2. | X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. |

3. | P. Zhang, B. H. Liu, R. F. Liu, H. R. Li, F. L. Li, and G. C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A |

4. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

5. | Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A |

6. | J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. |

7. | S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. |

8. | S. Barreiro, J. W. R. Tabosa, J. P. Torres, Y. Deyanova, and L. Torner, “Four-wave mixing of light beams with engineered orbital angular momentum in cold cesium atoms,” Opt. Lett. |

9. | D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A |

10. | J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A |

11. | H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. |

12. | W. Jiang, Q. F. Chen, Y. S. Zhang, and G. C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A |

13. | D. Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

14. | J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. |

15. | T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and Y. Hirayama, “Measuring the entanglement between double quantum dot charge qubits,” Phys. Rev. B |

16. | D. Jaksch, “Optical lattices, ultracold atoms and quantum information processing,” Contemp. Phys. |

17. | M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett. |

18. | A. N. Oliveira, S. P. Walborn, and C. H. Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B: Quantum Semiclassical Opt. |

19. | P. Zhang, R. F. Liu, Y. F. Huang, H. Gao, and F. L. Li, “Demonstration of Deutsch’s algorithm on a stable linear optical quantum computer,” Phys. Rev. A |

20. | H. T. Zhou, D. W. Wang, D. Wang, J. X. Zhang, and S. Y. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A |

21. | A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. |

22. | Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. |

23. | R. W. Boyd, |

24. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. |

25. | D. A. Steck, “Rubidium 85 D line data,” http://steck.us/alkalidata. |

26. | D. Höckel and O. Benson, “Electromagnetically induced transparency in cesium vapor with probe pulses on the single-photon level,” Phys. Rev. Lett. |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 26, 2012

Revised Manuscript: October 4, 2012

Manuscript Accepted: October 4, 2012

Published: October 9, 2012

**Citation**

Mingtao Cao, Liang Han, Ruifeng Liu, Hao Liu, Dong Wei, Pei Zhang, Yu Zhou, Wenge Guo, Shougang Zhang, Hong Gao, and Fuli Li, "Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing," Opt. Express **20**, 24263-24271 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24263

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### References

- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett.101(23), 233601 (2008). [CrossRef] [PubMed]
- P. Zhang, B. H. Liu, R. F. Liu, H. R. Li, F. L. Li, and G. C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A81(5), 052322 (2010). [CrossRef]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001). [CrossRef] [PubMed]
- Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A78(5), 053810 (2008). [CrossRef]
- J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett.83(24), 4967–4970 (1999). [CrossRef]
- S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett.90(13), 133001 (2003). [CrossRef] [PubMed]
- S. Barreiro, J. W. R. Tabosa, J. P. Torres, Y. Deyanova, and L. Torner, “Four-wave mixing of light beams with engineered orbital angular momentum in cold cesium atoms,” Opt. Lett.29(13), 1515–1517 (2004). [CrossRef] [PubMed]
- D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A67(2), 023803 (2003). [CrossRef]
- J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A56(5), 4193–4196 (1997). [CrossRef]
- H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett.85(2), 286–289 (2000). [CrossRef] [PubMed]
- W. Jiang, Q. F. Chen, Y. S. Zhang, and G. C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A74(4), 043811 (2006). [CrossRef]
- D. Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proc. R. Soc. Lond. A Math. Phys. Sci.400(1818), 97–117 (1985). [CrossRef]
- J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett.74(20), 4091–4094 (1995). [CrossRef] [PubMed]
- T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong, and Y. Hirayama, “Measuring the entanglement between double quantum dot charge qubits,” Phys. Rev. B80, 161309 (2009). [CrossRef] [PubMed]
- D. Jaksch, “Optical lattices, ultracold atoms and quantum information processing,” Contemp. Phys.45(5), 367–381 (2004). [CrossRef]
- M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett.91(18), 187903 (2003). [CrossRef] [PubMed]
- A. N. Oliveira, S. P. Walborn, and C. H. Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B: Quantum Semiclassical Opt.7(9), 288–292 (2005). [CrossRef]
- P. Zhang, R. F. Liu, Y. F. Huang, H. Gao, and F. L. Li, “Demonstration of Deutsch’s algorithm on a stable linear optical quantum computer,” Phys. Rev. A82(6), 064302 (2010). [CrossRef]
- H. T. Zhou, D. W. Wang, D. Wang, J. X. Zhang, and S. Y. Zhu, “Efficient reflection via four-wave mixing in a Doppler-free electromagnetically-induced-transparency gas system,” Phys. Rev. A84(5), 053835 (2011). [CrossRef]
- A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3(2), 161–204 (2011). [CrossRef]
- Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett.21(14), 1064–1066 (1996). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1992).
- J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett.88(25), 257901 (2002). [CrossRef] [PubMed]
- D. A. Steck, “Rubidium 85 D line data,” http://steck.us/alkalidata .
- D. Höckel and O. Benson, “Electromagnetically induced transparency in cesium vapor with probe pulses on the single-photon level,” Phys. Rev. Lett.105(15), 153605 (2010). [CrossRef] [PubMed]

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