## Polarization spectroscopy of dressed four-wave mixing in a three-level atomic system

JOSA B, Vol. 26, Issue 9, pp. 1710-1719 (2009)

http://dx.doi.org/10.1364/JOSAB.26.001710

Acrobat PDF (563 KB)

### Abstract

Polarization properties of pure four-wave mixing (FWM) and dressed-FWM processes in a two-level system and a cascade three-level atomic system are theoretically and experimentally investigated. The relative intensities and polarization characteristics of the FWM signals in different laser polarization configurations and different level systems are experimentally investigated and compared. Also, the results are theoretically explained by different transition paths combinations. In the dressed-FWM processes, we study the dependence of dressing effect on the incident field’s polarization. The FWM signal generated by a linearly polarized pumping field is suppressed more by the dressing field than the one generated by a circularly polarized pumping field. However, an opposite effect was observed when the probe field’s polarization is changed. The multidressing mechanisms are used to explain these effects. In addition, the interference and polarization dependence of the coexisting FWM signals in the same atomic system are discussed.

© 2009 Optical Society of America

## 1. INTRODUCTION

9. H. Ma, A. S. L. Gomes, and Cid B. de Araujo, “All-optical power-controlled switching in wave mixing: application to semiconductor-doped glasses,” Opt. Lett. **18**, 414–416 (1993). [CrossRef] [PubMed]

10. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

11. T. B. Bahder and P. A. Lopata, “Fidelity of quantum interferometers,” Phys. Rev. A **74**, 051801(R) (2006). [CrossRef]

## 2. EXPERIMENTAL SETUP

## 3. BASIC THEORY

### 3A. Various Nonlinear Susceptibilities for Different Polarization Schemes

12. S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. R. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A **58**, 3000–3003 (1998). [CrossRef]

*i*

*θ*is the rotated angle of the HWP’s axis from the

*x*axis). Consequently, the polarization has two corresponding components, i.e., horizontal component

*S*direction (the signals in the

*P*direction for the three cases are all generated by

*α*is the rotation angle of the HWP from the

*x*axis and

*δ*is the phase difference between the two polarization [horizontal

### 3B. Nonlinear Susceptibilities for a Zeeman-Degenerate System Interacting with Polarized Fields

*x*axis is the original polarization direction of all the incident fields, and it is also the quantization axis. We then decompose the arbitrary field into two components: parallel to and perpendicular to the

*x*axis, respectively. When this field interacts with atoms, the perpendicular component can be decomposed into equally left-circularly- and right-circularly-polarized components. Different polarization schemes can excite different transition paths in the Zeeman-degenerate atomic systems, and so it is necessary to take into account the Clebsch–Gordan coefficients associated with the various transitions between Zeeman sublevels in all pathways when calculating the FWM intensities. Figure 2 shows the transition schematic configurations for the Zeeman-degenerate two-level and three-level cascade systems interacting with one arbitrarily polarized and two horizontally polarized fields. Table 2, 3 list all the perturbation chains for different cases, respectively. By considering the schematic figures and the tables, we can obtain the expressions of various density matrices corresponding to nonlinear susceptibilities for different polarization schemes.

### 3C. Third-Order Density-Matrix Elements in the Presence of Dressing Fields

13. Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level folding atomic system,” Phys. Rev. A **77**, 063829 (2008). [CrossRef]

## 4. RESULTS AND DISCUSSIONS

*θ*of the HWP. From Table 1 we can see that, for the horizontally polarized component [Fig. 3a], the dependence of the FWM intensity on

*θ*follows

2. L. Museur, C. Olivero, D. Riedel, and M. C. Castex, “Polarization properties of coherent VUV light at **70**, 499–503 (2000). [CrossRef]

4. C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, “Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast **72**, 023811 (2005). [CrossRef]

6. W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A **63**, 063406 (2001). [CrossRef]

*x*axis [the square points in Figs. 3c, 3d]. In Figs. 3c, 3d, the experimental results are well described by the functions

*ϵ*), the polarizations of the FWM signals are also changed. Besides, the excited nonlinear susceptibilities of the P and S polarizations greatly modify the signal’s polarization states, which can be detected by the

*x*axis. Comparing to Fig. 4a, the reduction of the signal intensity is more than 50%. More interestingly, the FWM signals generated by the linearly polarized

14. H. B. Zheng, Y. P. Zhang, Z. Q. Nie, C. B. Li, H. Chang, J. P. Song, and M. Xiao, “Interplay among multidressed four-wave mixing processes,” Appl. Phys. Lett. **93**, 241101 (2008). [CrossRef]

15. Y. P. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. **91**, 061113 (2007). [CrossRef]

16. S. G. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. **98**, 053601 (2007). [CrossRef] [PubMed]

17. S. G. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: Interference and entanglement,” Phys. Rev. A **76**, 013803 (2007). [CrossRef]

*Δ*from 0 to very certain values, the phase difference between the two FWM processes alters from in-phase to out-phase, so the interference can switch back and forth between constructive and destructive values [18

18. Y. P. Zhang and M. Xiao, *Multi-Wave Mixing Processes* (Higher Education Press, Beijing and Springer, Berlin, 2009). [CrossRef]

19. Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual induced transparency,” Phys. Rev. Lett. **99**, 123603 (2007). [CrossRef] [PubMed]

20. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. **102**, 013601 (2009). [CrossRef] [PubMed]

13. Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level folding atomic system,” Phys. Rev. A **77**, 063829 (2008). [CrossRef]

*α*generated in the two-level system when the probe beam

4. C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, “Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast **72**, 023811 (2005). [CrossRef]

## 5. CONCLUSION

## ACKNOWLEDGMENTS

1. | K. Tsukiyama, “Parametric four-wave mixing in Kr,” J. Phys. B |

2. | L. Museur, C. Olivero, D. Riedel, and M. C. Castex, “Polarization properties of coherent VUV light at |

3. | J. Ishii, Y. Ogi, Y. Tanaka, and K. Tsukiyama, “Observation of the two-photon resonant parametric four-wave mixing in the NO |

4. | C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, “Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast |

5. | P. B. Chapple, K. G. H. Baldwin, and H. A. Bachor, “Interference between competing quantum-mechanical pathways for four-wave mixing,” J. Opt. Soc. Am. B |

6. | W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A |

7. | W. R. Garret and Y. Zhu, “Coherent control of multiphoton driven processes: A laser-induced catalyst,” J. Chem. Phys. |

8. | Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A |

9. | H. Ma, A. S. L. Gomes, and Cid B. de Araujo, “All-optical power-controlled switching in wave mixing: application to semiconductor-doped glasses,” Opt. Lett. |

10. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

11. | T. B. Bahder and P. A. Lopata, “Fidelity of quantum interferometers,” Phys. Rev. A |

12. | S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. R. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A |

13. | Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level folding atomic system,” Phys. Rev. A |

14. | H. B. Zheng, Y. P. Zhang, Z. Q. Nie, C. B. Li, H. Chang, J. P. Song, and M. Xiao, “Interplay among multidressed four-wave mixing processes,” Appl. Phys. Lett. |

15. | Y. P. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. |

16. | S. G. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. |

17. | S. G. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: Interference and entanglement,” Phys. Rev. A |

18. | Y. P. Zhang and M. Xiao, |

19. | Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual induced transparency,” Phys. Rev. Lett. |

20. | Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

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

(270.4180) Quantum optics : Multiphoton processes

(300.2570) Spectroscopy : Four-wave mixing

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: June 23, 2009

Manuscript Accepted: July 16, 2009

Published: August 14, 2009

**Virtual Issues**

August 19, 2009 *Spotlight on Optics*

**Citation**

Ruimin Wang, Yigang Du, Yanpeng Zhang, Huaibin Zheng, Zhiqiang Nie, Changbiao Li, Yuanyuan Li, Jianping Song, and Min Xiao, "Polarization spectroscopy of dressed four-wave mixing in a three-level atomic system," J. Opt. Soc. Am. B **26**, 1710-1719 (2009)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-26-9-1710

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### References

- K. Tsukiyama, “Parametric four-wave mixing in Kr,” J. Phys. B 29, L345-L351 (1996). [CrossRef]
- L. Museur, C. Olivero, D. Riedel, and M. C. Castex, “Polarization properties of coherent VUV light at 125 nm generated by sum-frequency four-wave mixing in mercury,” Appl. Phys. B 70, 499-503 (2000). [CrossRef]
- J. Ishii, Y. Ogi, Y. Tanaka, and K. Tsukiyama, “Observation of the two-photon resonant parametric four-wave mixing in the NO C2Π(v=0) state,” Opt. Commun. 132, 316-320 (1996). [CrossRef]
- C. J. Zhu, A. A. Senin, Z. H. Lu, J. Gao, Y. Xiao, and J. G. Eden, “Polarization of signal wave radiation generated by parametric four-wave mixing in rubidium vapor: Ultrafast (~150-fs) and nanosecond time scale excitation,” Phys. Rev. A 72, 023811 (2005). [CrossRef]
- P. B. Chapple, K. G. H. Baldwin, and H. A. Bachor, “Interference between competing quantum-mechanical pathways for four-wave mixing,” J. Opt. Soc. Am. B 6, 180-183 (1998). [CrossRef]
- W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, “Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes,” Phys. Rev. A 63, 063406 (2001). [CrossRef]
- W. R. Garret and Y. Zhu, “Coherent control of multiphoton driven processes: A laser-induced catalyst,” J. Chem. Phys. 106, 2045-2048 (1997). [CrossRef]
- Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003). [CrossRef]
- H. Ma, A. S. L. Gomes, and Cid B. de Araujo, “All-optical power-controlled switching in wave mixing: application to semiconductor-doped glasses,” Opt. Lett. 18, 414-416 (1993). [CrossRef] [PubMed]
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002). [CrossRef]
- T. B. Bahder and P. A. Lopata, “Fidelity of quantum interferometers,” Phys. Rev. A 74, 051801(R) (2006). [CrossRef]
- S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. R. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000-3003 (1998). [CrossRef]
- Z. Q. Nie, H. B. Zheng, P. Z. Li, Y. M. Yang, Y. P. Zhang, and M. Xiao, “Interacting multi-wave mixing in a five-level folding atomic system,” Phys. Rev. A 77, 063829 (2008). [CrossRef]
- H. B. Zheng, Y. P. Zhang, Z. Q. Nie, C. B. Li, H. Chang, J. P. Song, and M. Xiao, “Interplay among multidressed four-wave mixing processes,” Appl. Phys. Lett. 93, 241101 (2008). [CrossRef]
- Y. P. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two four-wave mixing channels via atomic coherence,” Appl. Phys. Lett. 91, 061113 (2007). [CrossRef]
- S. G. Du, J. M. Wen, M. H. Rubin, and G. Y. Yin, “Four-wave mixing and biphoton generation in a two-level system,” Phys. Rev. Lett. 98, 053601 (2007). [CrossRef] [PubMed]
- S. G. Du, E. Oh, J. M. Wen, and M. H. Rubin, “Four-wave mixing in three-level systems: Interference and entanglement,” Phys. Rev. A 76, 013803 (2007). [CrossRef]
- Y. P. Zhang and M. Xiao, Multi-Wave Mixing Processes (Higher Education Press, Beijing and Springer, Berlin, 2009). [CrossRef]
- Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual induced transparency,” Phys. Rev. Lett. 99, 123603 (2007). [CrossRef] [PubMed]
- Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009). [CrossRef] [PubMed]

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