## Modulating the multi-wave mixing processes via the polarizable dark states

Optics Express, Vol. 17, Issue 18, pp. 15468-15480 (2009)

http://dx.doi.org/10.1364/OE.17.015468

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### Abstract

We have observed multi-wave mixing (MWM) processes in reversed-Y (RY) type system in ^{87}*Rb* atoms with electromagnetically induced transparency (EIT) windows at different laser polarization configurations. Interesting rules of changing the MWM processes and EIT profiles are obtained. We have found that the degenerate Zeeman sublevels and their dressed-state effects are responsible for these observed phenomena. Polarizable dark states are used to describe the multi-level dressed states. The experimental data are in good agreement with the results from the theoretical calculation that takes into account all the 16 Zeeman sublevels in the RY system.

© 2009 Optical Society of America

## 1. Introduction

## 2. Theoretical model and analysis

*atoms are involved in the experimental schemes used in this work. The laser beams are spatially aligned as shown in Fig. 1(b). In Fig. 1(a), energy levels |0〉(5*

^{87}Rb*S*

_{1/2},

*F*=2), |1〉 (5

*P*

_{3/2},

*F*′=2), |2〉 (5

*D*

_{3/2},

*F*″=1) and |3〉 (5

*S*

_{1/2},

*F*=1) form the RY-type four-level atomic system. Strong coupling laser beam

*E*

_{2}(

*ω*

_{2},

**k**

_{2}, and Rabi frequency

*G*

_{2}) together with

*E*′

_{2}(

*ω*

_{2},

**k**′

_{2}, and Rabi frequency

*G*′

_{2}) with small angle (0.5°) and the same frequency detuning Δ

_{2}(=

*ω*

_{21}-

*ω*

_{2}), connecting the transition between |1〉 to |2〉, propagate in the opposite direction of the weak probe beam

*E*

_{1}(

*ω*

_{1},

**k**

_{1}, and Rabi frequency

*G*

_{1}), which has the frequency detuning Δ1 (=

*ω*

_{10}-

*ω*

_{1}) and connects the transition between |0〉 to |1〉, as shown in the inset of Fig. 1(b). Two additional coupling beams

*E*

_{3}(

*ω*

_{3},

**k**

_{3}, and Rabi frequency

*G*) and

_{3}*E*′

_{3}(

*ω*

_{3},

**k**′

_{3}, and Rabi frequency

*G*′

_{3}), with the same frequency detuning Δ

*(=*

_{3}*ω*

_{13}-

*ω*

_{3}) connecting the transition between |3〉 to |1〉, also propagate in the opposite direction of the weak probe beam

*E*

_{1}[Fig. 1(b)] with small angles. When all five laser beams (

*E*

_{1},

*E*

_{2},

*E*′

_{2},

*E*

_{3}and

*E*′

_{3}) are turned on simultaneously, only one ladder-type EIT subsystem will form due to the two-photon Doppler-free configuration [2

2. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. **51**, 576–584 (1995).
[CrossRef] [PubMed]

*ω*

_{2}laser beams. Meanwhile, there will be co-existing MWM processes that generate signal beams at frequency

*ω*(

_{M}*ω*=

_{M}*ω*) in such multi-level system. First, without the coupling beams

_{1}*E*

_{3}and

*E*′

_{3}, one pure FWM

*E*process will be generated in the ladder system (|0〉→|1〉→|2〉) satisfying the phase-matching condition of

_{F}**k**

_{F}=

**k**

_{1}+

**k**

_{2}-

**k**′

_{2}. The signal emerging from the “P” polarization direction is detected by an avalanche photodiode detector (APD). Second, when all beams (except with

*E*′

_{2}blocked) are turned on, the strong coupling beam

*E*′

_{2}will dress the energy level |1〉 to create the dressed states |+〉 and |-〉. In such case, there exist a dressed FWM process

**k**′

**=**

_{F}**k**

_{1}+

**k**

_{3}-

**k**′

_{3}and a coexisting six-wave mixing (SWM) process, in which two photons from

*E*

_{2}and one photon each from

*E*

_{1},

*E*

_{3}and

*E*′

_{3}participate in the SWM process to generate

*E*with phase-matching condition of

_{S}**k**

_{S}=

**k**

_{1}+

**k**

_{2}-

**k**

_{2}+

**k**

_{3}-

**k**

_{3}′.

*Rb*vapor at frequency

*ω*. First, for the FWM signal

_{M}*E*(by blocking laser beams

_{F}*E*

_{3}and

*E*′

_{3}), the nonlinear atomic polarization

*P*

^{(3)}(

*ω*

_{1}) along the i (i=x,y) direction, from first-order perturbation theory, is given by [15

15. 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**, 023811-1–023811-7 (2005).
[CrossRef]

*χ*and

_{x}*χ*are the effective susceptibilities. If there is a half-wave plate placed on the path of the probe beam

_{y}*E*

_{1}, then

*θ*,

*ϕ*are the rotated angles of the half-wave plate and quarter-wave plate, respectively, relative to the x-axis.

*E*(when blocking the coupling laser beam

_{S}*E*′

_{2}), the fifth-order nonlinear polarization

*P*

^{(5)}(

*ω*

_{1}) along the i (i=x,y) direction is then given by

*θ*(or

*ϕ*) before going into the

^{87}

*Rb*atomic vapor cell. By blocking different incident beams, the probe beam and the generated FWM signal (or SWM signal) [13

13. Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two Four-wave Mixing channels via atomic coherence,” Appl. Phys. Lett. **91**, 061113-1–061113-3 (2007).
[CrossRef]

*E*

_{1}(

*ω*

_{1}), whereas the APD receives the horizontally polarized component of the generated MWM signal

*E*(

_{M}*ω*).

_{M}## 3. Experimental results

^{87}

*Rb*atoms by three external cavity diode lasers (ECDL) and linewidths of less than or equal to 1 MHz. Each output power is as follow: 0.7 mw of probe field

*E*

_{1}; 65 mw of coupling beams

*E*

_{2},

*E*′

_{2}and 15 mw of coupling laser beams

*E*

_{3},

*E*′

_{3}. The cell whose length is 5cm is heat up to 60°C and the density is 2.5×10

^{11}/cm

^{3}.

^{87}

*Rb*atoms, by blocking laser beam

*E*′

_{2}, for probe field with different polarizations rotated by a half-wave plate. The coupling beams are linearly polarized in the S polarizations direction and the probe beam is linearly polarized in the P polarization direction initially (defined as

*θ*=0°).

16. H. Y. Ling, Y. Q Li, and M. Xiao, “Coherent population trapping and electromagnetically induced transparency in multi-Zeeman-sublevel atoms,” Phys. Rev. A , **53**, 1014–1026 (1996).
[CrossRef] [PubMed]

17. S. Li, B. Wang, X. Yang, Y. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A. **74**, 033821-1–033821-12 (2006).
[CrossRef]

*E*

_{3}and

*E*′

_{3}), which forms an EIT configuration. For the case as given in Fig. 3(a), the density-matrix

*ρ*considering all the Zeeman levels is solved with the following equations:

*ω*

_{10}is the

*F*=2→

*F*′=2 transition frequency and

*ω*

_{21}is the

*F*′=2→

*F*″=1 transition frequency.

*ω*is the frequency of the coupling field and

_{c}*ω*is frequency of the probe laser. Ω

_{p}*is the Rabi frequency for the transition indicated by its subscripts. The expression of {*

_{N}*dρ*/

*dt*} describes all the relaxation processes in the system. Γ

_{1}and Γ

_{2}are the spontaneous decay rates of the 5

*D*

_{3/2},

*F*′=1 and 5

*P*

_{3/2},

*F*′=2 excited states, respectively. Γ

*is the spontaneous emission rate from an excited state |*

_{ji}*j*> to a ground state |

*i*>. Γ′, Γ″ and Γ‴ are the decoherence rates for the relaxation processes other than the spontaneous decay. To obtain linear susceptibility, we need to solve the density-matrix equations (11) under the steady-state condition. Under the weak probe field approximation [2

2. J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. **51**, 576–584 (1995).
[CrossRef] [PubMed]

*are the differences between the laser frequency and the corresponding atomic transition frequency.*

_{ij}18. For detail in transition probabilities of D1 and D2 line in Rb, see http://steck. Us/alkalidata.

19. Y. Chen, C. Lin, and I. A. Yu, “Role of degenerate Zeeman levels in electromagnetically induced transparency,” Phys. Rev. A. **61**, 053805-1–053805-6 (2000).
[CrossRef]

*E*

_{3}and

*E*′

_{3}are open, there are doubly-dressed effects for the EIT curve. The analytical solutions will be changed as following.

*S*

_{1/2},

*F*=2→5

*P*

_{3/2},

*F*′=2 one-photon transition. Besides its direct dependence on the polarization of the probe beam, the SWM signal spectrum is also modulated by the EIT effect since it transmits through the medium in the EIT window. Figures 5(b1) and 5(b2) show the corresponding differences of the experiment data and theoretically calculated curves which represent different coupling paths of the total SWM processes via different polarization configurations. The left peak is always in the positive part of the dispersion-like EIT curve which gets dramatically modulated effect due to EIT. This means that the fifth-order susceptibility

*χ*

^{(5)}

*must be taken into account due to dressed effect, which affects the evolution of the peak together with the polarization dependence of cos*

_{ijklmn}^{2}2

*θ*. The right peak is in the negative region of the dispersion-like EIT shape, where it is dominated by the polarization property due to the increase in absorption. We can conclude that the evolution of the SWM spectrum is modulated by the modified EIT spectrum.

*E*

_{3}and

*E*′

_{3}while other beams are turned on. The theoretical results can be obtained from the same procedure as above by eliminating the absent items, that is, LHC subsystems in the RY system. The EIT spectrum also shows the same profile, including the dispersion-like curve. However, the positive peak changes from a single peak into two peaks. Although the way of polarization with a quarter-wave plate is different from with a half-wave plate, its rules of evolution can be explained by the same method as for the half-wave plate. Schemes (d), (e) and (f) in Fig. 3 show the ways of coupling in different polarization configurations. Differ from using a half-wave plate whose period is 90 degrees, the right-hand elliptically polarized beam is present during 0~45 and 45~90 degrees [Fig. 3(e)], and a pure RHC-polarized beam at 45 degrees [Fig. 3(f)]. However, within 0~45 degrees, the RHC component increases gradually while the linear component decreases. In the region of 45~90 degrees the opposite process is true. Taking the case of 0~45 degrees as an example, we can take the right-hand elliptically-polarized beam being composed of a vertical, linearly-polarized beam and a RHC-polarized beam. Therefore, in our experiment, the original symmetric EIT configurations [Fig. 3(d)] are replaced by two linear EIT and three RHC EIT subsystems [Fig. 3(e)] that are asymmetric due to the difference in the dipole moments among different Zeeman sublevels. The destruction of this symmetry results in different polarizable dark states leading to the modified EIT spectrum. It is different from the case with half-wave plate, which does not destroy the symmetry in the EIT spectral shape. The situation in the rest of the period (45~90 degree) can be discussed in the same way. So, the spectrum by rotating a quarter-wave plate is different from the case of using a half-wave plate.

*χ*

^{(3)}

*(*

_{ijkl}*ω*;

_{F}*ω*

_{1},-

*ω*

_{2},

*ω*

_{2}and the polarization effect of the probe field in the form of

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today. |

2. | J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A. |

3. | A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, “Transporting and time reversing light via atomic coherence,” Phys. Rev. Lett. |

4. | P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, “Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium,” Opt. Lett. |

5. | B. Lu, W. H. Burkett, and M. Xiao, “Nondegenerate Four-Wave Mixing in a double-Λ system under the influence of coherent population trapping,” Opt. Lett. |

6. | K. J Jiang, L. Deng, and M. G. Payne, “Observation of quantum destructive interference in inelastic Two-wave Mixing,” Phys. Rev. Lett. |

7. | H. Kang, G. Hernandez, J. Zhang, and Y Zhu, “Phase-contrlled light switching at low light levels,” Phys. Rev. A. |

8. | V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from Four-Wave Mixing,” Science. |

9. | S. W. 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. |

10. | V. B. Danielle, A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. |

11. | C. Cohen-Tannoudji and S. Reynaud, “Dreesed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B. |

12. | H. Zheng, Y. Zhang, Z. Nie, C. Li, H. Chong, J. Song, and M. Xiao, “Interplay among multi-dressed Four-wave Mixing processes,” Appl. Phys. Lett. |

13. | Y. Zhang, B. Anderson, A. W. Brown, and M. Xiao, “Competition between two Four-wave Mixing channels via atomic coherence,” Appl. Phys. Lett. |

14. | Y. Zhang and M. Xiao, “Generalized dressed and doubly-dressed multi-wave mixing,” Opt. Express. |

15. | 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. |

16. | H. Y. Ling, Y. Q Li, and M. Xiao, “Coherent population trapping and electromagnetically induced transparency in multi-Zeeman-sublevel atoms,” Phys. Rev. A , |

17. | S. Li, B. Wang, X. Yang, Y. Han, H. Wang, M. Xiao, and K. C. Peng, “Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms,” Phys. Rev. A. |

18. | For detail in transition probabilities of D1 and D2 line in Rb, see http://steck. Us/alkalidata. |

19. | Y. Chen, C. Lin, and I. A. Yu, “Role of degenerate Zeeman levels in electromagnetically induced transparency,” Phys. Rev. A. |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(190.4180) Nonlinear optics : Multiphoton processes

(270.1670) Quantum optics : Coherent optical effects

(190.4223) Nonlinear optics : Nonlinear wave mixing

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: June 22, 2009

Revised Manuscript: July 19, 2009

Manuscript Accepted: July 20, 2009

Published: August 17, 2009

**Citation**

Huaibin Zheng, Yanpeng Zhang, Utsab Khadka, Ruimin Wang, Changbiao Li, Zhiqiang Nie, and Min Xiao, "Modulating the multi-wave mixing processes via the polarizable dark states," Opt. Express **17**, 15468-15480 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15468

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### References

- S. E. Harris, "Electromagnetically Induced Transparency," Phys. Today. 50, 36-42 (1997). [CrossRef]
- J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, "Electromagnetically Induced Transparency in ladder-type inhomogeneously broadened media: Theory and experiment," Phys. Rev. A. 51, 576- 584 (1995). [CrossRef] [PubMed]
- A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, "Transporting and time reversing light via atomic coherence," Phys. Rev. Lett. 88, 103601 (2002). [CrossRef]
- P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, "Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium," Opt. Lett. 20, 982-984 (1995). [CrossRef] [PubMed]
- B. Lu, W. H. Burkett, and M. Xiao, "Nondegenerate Four-Wave Mixing in a double-? system under the influence of coherent population trapping," Opt. Lett. 23, 804-806 (1998). [CrossRef]
- K. J Jiang, L. Deng, and M. G. Payne, "Observation of quantum destructive interference in inelastic Two-wave Mixing," Phys. Rev. Lett. 98, 083604 (2007). [CrossRef]
- H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, "Phase-contrlled light switching at low light levels," Phys. Rev. A. 73, 011802 (2006). [CrossRef]
- V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, "Entangled images from Four-Wave Mixing," Science. 321, 544-547 (2008). [CrossRef] [PubMed]
- S. W. 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, 0536014 (2007). [CrossRef]
- V. B. Danielle, A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, "Generation of paired photons with controllable waveforms," Phys. Rev. Lett. 94, 183601 (2005).
- C. Cohen-Tannoudji and S. Reynaud, "Dreesed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam," J. Phys. B. 10, 345-363 (1977). [CrossRef]
- H. Zheng, Y. Zhang, Z. Nie, C. Li, H. Chong, J. Song, and M. Xiao, "Interplay among multi-dressed Four-wave Mixing processes," Appl. Phys. Lett. 93, 241101 (2008). [CrossRef]
- Y. 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]
- Y. Zhang and M. Xiao, "Generalized dressed and doubly-dressed multi-wave mixing," Opt. Express. 15. 7182-7189 (2007). [CrossRef] [PubMed]
- 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]
- H. Y. Ling, Y. Q Li, and M. Xiao, "Coherent population trapping and electromagnetically induced transparency in multi-Zeeman-sublevel atoms," Phys. Rev. A, 53, 1014-1026 (1996). [CrossRef] [PubMed]
- S. Li, B. Wang, X. Yang, Y. Han, H. Wang, M. Xiao, and K. C. Peng, "Controlled polarization rotation of an optical field in multi-Zeeman-sublevel atoms," Phys. Rev. A. 74, 033821 (2006). [CrossRef]
- For detail in transition probabilities of D1 and D2 line in Rb, see http:// steck. Us/ alkalidata.
- Y. Chen, C. Lin, and I. A. Yu, "Role of degenerate Zeeman levels in electromagnetically induced transparency," Phys. Rev. A. 61, 053805 (2000). [CrossRef]

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