## Generalized dressed and doubly-dressed multi-wave mixing

Optics Express, Vol. 15, Issue 12, pp. 7182-7189 (2007)

http://dx.doi.org/10.1364/OE.15.007182

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### Abstract

We present a theoretical treatment for generalized dressed and doubly-dressed multi-wave mixing processes. Co-existing four-wave mixing (FWM), six-wave mixing (SWM) and eight-wave mixing processes have been considered in a closed-cycle five-level system. Due to constructive interference of the secondarily-dressed and primarily-dressed excitation pathways, the FWM and SWM signals are greatly enhanced. The dually enhanced FWM channels are opened simultaneously. The dressing fields provide the energy for such large enhancement.

© 2007 Optical Society of America

## 1. Introduction

1. 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]

6. Y. P. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. **32**, 1120–1122 (2007). [CrossRef] [PubMed]

14. 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]

1. 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]

15. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A **60**, 3225–3228 (1999). [CrossRef]

16. M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A **64**, 013412 (2001). [CrossRef]

6. Y. P. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. **32**, 1120–1122 (2007). [CrossRef] [PubMed]

7. Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted-Y system,” Appl. Phys. Lett. **90**, 111104 (2007). [CrossRef]

17. L. Deng and M. G. Payne, “Inhibiting the onset of the three-photon destructive interference in ultraslow propagation-enhanced four-wave mixing with dual induced transparency,” Phys. Rev. Lett. **91**, 243902 (2003). [CrossRef] [PubMed]

18. Y Wu and L Deng, “Achieving multi-frequency mode entanglement with ultra-slow multi-wave mixing”, Opt. Lett. **29**, 1144–1146 (2004). [CrossRef] [PubMed]

*n*-2) wave mixing ((2

*n*-2)WM) and doubly-dressed (2

*n*-4) wave mixing ((2

*n*-4)WM) processes in (

*n*+1)-level atomic systems. Co-existing FWM, SWM and eight-wave mixing (EWM) processes have been considered in a closed-cycle five-level folded system as one example (

*n*=4) of the generalized doubly-dressed (2

*n*-4)WM systems. Such co-existing different order multi-wave mixing processes and the interplay between them have not been reported in multilevel atomic systems, to the best of our knowledge, in the literature. Investigations of such intermixing and interplay between different types of nonlinear wave-mixing processes will help us to understand and optimize the generated high-order multi-channel nonlinear optical signals.

## 2. Generalized dressed (2*n*-2)WM and doubly dressed (2*n*-4)WM

*n*+1)-level cascade system (Fig. 1), where states ∣

*i*-1> to ∣

*i*> are coupled by laser field

*E*[

_{i}*E′*) (

_{i}*ω*,

_{i}**k**

_{i}(

**k′**

_{i}), and Rabi frequency

*G*(

_{i}*G′*)]. The Rabi frequencies are defined as

_{i}*G*=

_{i}*ε*

_{i}*μ*/

_{ij}*ħ*,

*G′*=

_{i}*ε′*

_{i}*μ*, where

_{ij}/ħ*μ*are the transition dipole moments between level

_{ij}*i*and level

*j*. Fields

*E*

_{n-2}and

*E′*

_{n-2}with the same frequency propagate along beams 2 and 3 with a small angle [Fig. 1(a)]. Fields

*E*

_{2},

*E*

_{3}to

*E*

_{n-3}propagate along the direction of beam 2, while a weak probe field

*E*

_{1}(beam 1) propagates along the opposite direction of beam 2. The simultaneous interactions of atoms with fields

*E*

_{1},

*E*

_{2}to

*E*

_{n−2}will induce atomic coherence between ∣0> and ∣

*n*-2> through resonant (

*n*-2)-photon transitions. This (

*n*-2)-photon coherence is then probed by fields

*E′*

_{n-2}and

*E*

_{n-3}to

*E*

_{2}and as a result a (2

*n*-4)WM (ρ

^{(2n-5)}

_{10}) signal of frequency

*ω*

_{1}in beams 4 is generated almost opposite to the direction of beams 3, satisfying the phase-matching condition

**k**

_{2n-4}=

**k**

_{1}+

**k**

_{n-2}-

**k′**

_{n-2}. When two strong dressing fields

*E*

_{n-1}and

*E*are used to drive the transitions ∣

_{n}*n*-2> to ∣

*n*-1> and ∣

*n*-1> to ∣

*n*>, respectively, as shown in Figs. 1(b), there exist one doubly-dressed (2

*n*-4)WM (ρ

_{10}

^{″(2n−5)}), one singly-dressed (2

*n*-2)WM (ρ

_{10}

^{′(2n−3)}) and one 2

*n*WM (ρ

^{(2n-1)}

_{10}) processes, satisfying the same

**k**

_{2n-4}.

*n*WM, dressed (2

*n*-2)WM and doubly-dressed (2

*n*-4)WM processes, we need to use perturbation chain expressions involving all the ρ

^{(2n-1)}

_{10}, ρ

^{(2n-3)}

_{10}and ρ

^{(2n-5)}

_{10}nonlinear wave-mixing processes for arbitrary field strengths of

*E*. The simple (2

_{i}*n*-4)WM via Liouville pathway (C

_{n-2})

*G*=

_{a}*G*

_{1}∣

*G*

_{2}∣

^{2}⋯∣

*G*

_{n-3}∣

^{2},

*d*= Γ

_{i}_{i0}+

*i*(Δ

_{1}+Δ

_{2}+⋯+Δ

_{i}) with Δ

_{i}=Ω

_{i}-

*ω*. Γ

_{i}_{i0}is the transverse relaxation rate between states ∣

*i*> and ∣0>. Similarly, we can easily obtain ρ

^{(2n-3)}

_{10}=-ρ

^{(2n-5)}

_{10}∣

*G*

_{n-1}∣

^{2}/

*d*

_{n-2}

*d*

_{n-1}and ρ

^{(2n-1)}

_{10}=ρ

^{(2n-5)}

_{10}∣

*G*

_{n-1}∣

^{2}∣

*G*∣

_{n}^{2}/

*d*

_{n-2}

*d*

^{2}

_{n-1}

*d*via perturbation chains (C

_{n}_{n-1})

_{n})

*n*WM with phase-conjugate geometry has also been considered in an (n+1)-level system [9

9. Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. S. Fu, L. A. Wu, and P. M. Fu, “Generalized *n*-photon resonant 2*n*-wave mixing in an (*n* +1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. **97**, 193904 (2006). [CrossRef] [PubMed]

*E*

_{n-1}and

*E*are turned on, there exist three physical mechanisms of interest. First, the (2

_{n}*n*-4)WM process will be dressed by the two strong fields

*E*

_{n-1}and

*E*and a perturbative approach for such interaction can be described by the following coupled equations:

_{n}_{n-2}) to give the doubly-dressed (2

_{n-4})WM

*G*∣

_{n}^{2}<<Γ

_{n0}Γ(

_{n-1)0}and ∣

*G*

_{n-1}∣

^{2}<<Γ

_{(n-1)0}Γ(

_{n-2)0}, ρ′′

^{(2n-5)}

_{10}can be expanded to be

*n*-4)WM process converts to a coherent superposition of signals from (2

*n*-4)WM, (2

*n*-2)WM and 2

*n*WM (ρ

^{(2n-5)}

_{10}+ ρ

^{(2n-3)}

_{10}+ ρ

^{(2n-1)}

_{10}), or dressed (2

*n*-4)WM and 2

*n*WM (ρ′

^{(2n-5)}

_{10}+ ρ

^{(2n-1)}

_{10}), or (2

*n*-4)WM and dressed (2

*n*-2)WM (ρ

^{(2n-5)}

_{10}+ρ′

^{(2n-3)}

_{10}) in the weak dressing field limit.

^{(2n-3)}

_{10}term in Eq. (4) results from the (2

*n*-2)WM process dressed by the strong field

*E*and a perturbative approach for such interactions can be described by the following coupled equations:

_{n}_{n-1}) to give

*G*∣

_{n}^{2}<< Γ

_{n0}Γ

_{(n-1)0}, ρ′

^{(2n-3)}

_{10}can be expanded further to be ρ′

^{(2n-3)}

_{10}≈ ρ

^{(2n-3)}

_{10}+ ρ

^{(2n)}

_{10}, and the dressed (2

*n*-2)WM process converts to a coherent superposition of signals from (2

*n*-2)WM and 2

*n*WM. Third, the ρ′

^{(2n-5)}

_{10}term in of Eq. (4) results from (2

*n*-4)WM process dressed by the strong field

*E*

_{n-1}Similarly, we can obtain

*E*

_{n-1}coupled equations and the perturbation chain (C

_{n-2}). Under ∣

*G*

_{n-1}∣

^{2}<< Γ

_{(n-1)0}Γ

_{(n-2)0}, ρ′

^{(2n-5)}

_{10}can also be expanded to be ρ′

^{(2n-5)}

_{10}≈ ρ

^{(2n-5)}

_{10}+ ρ

^{(2n-3)}

_{10}and the dressed (2

*n*-4)WM process converts to a coherent superposition of signals from (2

*n*-4)WM and (2

*n*-2)WM.

## 3. Interplay among coexisting FWM, SWM and EWM

*n*=4) of the generalized doubly-dressed (2

*n*-4)WM system described above can be employed as an example to study the intermixing and interplay between FWM, SWM and EWM processes (Table 1). The laser beams are aligned spatially in the pattern as shown in Fig. 2(a), with seven beams (

*E*

_{1},

*E*

_{2},

*E*′

_{2},

*E*

_{3},

*E*′

_{3},

*E*

_{4},

*E*′

_{4}) propagating through the atomic medium with small angles between them in a square-box pattern [Fig. 2(a)]. For a closed-cycle folded five-level system, Figs. 2(b)-2(f) generally correspond to the cases of blocking beams

*E*′

_{2}and

*E*′

_{3}(EWM) [Fig. 2(b)],

*E*′

_{2}and

*E*′

_{4}(dressed SWM) [Fig. 2(c)], or

*E*′

_{3}and

*E*′

_{4}(doubly dressed FWM) [Fig. 2(e)], respectively. However, the doubly-dressed FWM (ρ′′

^{(3)}

_{10}) process [Figs. 2(e) and 2(f)] converts to a coherent superposition of signals from FWM (ρ

^{(3)}

_{10}), SWM (ρ

^{(5)}

_{10}) and EWM (ρ

^{(7)}

_{10}). The dressed SWM (ρ′

^{(5)}

_{10}) process [Figs. 2(c) and 2(d)] converts to a coherent superposition of signals from FWM and SWM in the weak dressing field limit. Under ∣

*G*

_{3}∣

^{2}<<Γ

_{30}Γ

_{20}and ∣

*G*

_{4}∣

^{2}<<Γ

_{10}Γ

_{40}, Eq. (4) reduces to

^{(3)}

_{10}= -

*G*/

_{a}*d*

^{2}

_{1}

*d*

_{2}, ρ

^{(5)}

_{10}=

*G*∣

_{a}*G*

_{3}∣

^{2}/(

*d*

^{2}

_{1}

*d*

^{2}

_{2}

*d*′

_{3}), ρ

^{(7)}

_{10}= -

*G*∣

_{a}*G*

_{3}∣

^{2}∣

*G*

_{4}∣

^{2}/(

*d*

^{2}

_{1}

*d*

^{2}

_{2}

*d*′

^{2}

_{3}

*d*

_{4})

**k**

_{F}=

**k**

_{1}+

**k**

_{2}-

**k**′

_{2},

*d*′

_{3}= Γ

_{30}+

*i*Δ

^{a}

_{3}, Δ

^{a}

_{3}=Δ

^{a}

_{2}-Δ

_{3}, Δ

^{a}

_{2}=Δ

_{1}+Δ

_{2}and

*G*=

_{a}*i*

*G*

_{1}

*G*

_{2}(

*G*′

_{2})

^{*}e

*i*

**k**

_{F}∙

**r**

^{a}

_{3}/Γ

_{20}or Δ

_{4}/Γ

_{30}. Figure 3(a) shows that as the dressed field

*G*

_{4}is increased a dip appears at the line center first, then the spectrum splits into two separate peaks. This is a typical Autler-Townes (AT) splitting (The left and right peaks of Fig. 3(a) correspond to the ∣+> and ∣-> levels created by the dressed field

*G*

_{4}in Fig. 2(d), respectively). The two peaks are located asymmetrically due to Δ

_{3}≠ 0. Figures 3(b) and 3(c) present the suppression and enhancement of the dressed SWM signal intensity. The SWM signal intensity with no dressing field is normalized to 1. At the exact three-photon resonance Δ

^{a}

_{3}= 0, we see that the SWM signal intensity is suppressed when the frequency of the dressing field is scanned across the resonance (Δ

_{4}= 0). The presence of the weak dressing field can either suppress or enhance the SWM signal when Δ

^{a}

_{3}≠ 0 [Fig. 3(b)]. Such suppression and enhancement mainly result from the absorption and dispersion of SWM and EWM signals and their interference. When

*G*

_{4}/Γ

_{30}=50 the SWM signal is strongly enhanced by a factor of 620 in the presence of the dressing field when Δ

^{a}

_{3}/Γ

_{30}=-50 [blue curve in Fig. 3(c)], which is mainly due to the three-photon (∣0>→∣1>→∣2>→∣+1>) resonance. In general, the constructive and destructive interferences between the ∣+> and ∣-> SWM channels (Table 1) result in the enhancement and suppression of SWM signal, respectively. However, such enhancement mainly originates from the dispersion of dressed SWM in the weak dressing field limit [19

19. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. **87**, 073601 (2001). [CrossRef] [PubMed]

^{a}

_{2}/Γ

_{20}or Δ

_{3}/Γ

_{30}[Figs. 2(e) and 2(f)]. A dressing field

*G*

_{3}(

*G*

_{3}>

*G*

_{4}) creates dressed atomic states ∣+> and ∣-> from the unperturbed states ∣2> and ∣3> [black curve in Fig. 4(a)] (the separation between the two peaks is Δ

_{AT1}≈ 2{

*G*

_{3}[

*G*

^{2}

_{3}+2Γ

_{30}(Γ

_{20}+Γ

_{30})]

^{1/2}-Γ

^{2}

_{30}}

^{1/2}). When the dressing field

*G*

_{4}is tuned close to one of the primarily-dressed states ∣+> (or ∣->), basically the dressing field only couples the dressed state ∣+> (or ∣->) to the state ∣4> and leaves the other dressed state ∣-> (or ∣+>) unperturbed [16

16. M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A **64**, 013412 (2001). [CrossRef]

_{4}= Δ

_{ATI}/2 there exist the secondarily-dressed states ∣++> and ∣+-> around the primarily-dressed state ∣+>, the FWM signal amplitude for the two doubly-dressed states will approximately be half of the amplitude for the singly-dressed state ∣-> [blue curve in Fig. 4(a)]; when Δ

_{4}= -Δ

_{ATI}/2 the secondarily-dressed states ∣-+> and ∣--> (around the primarily-dressed state ∣->) can be generated (red curve in [Fig. 4(a)]. That is to say, the two FWM Liouville pathways,

_{30}, the doubly-dressed four-level system also exhibits a constructive interference [16

16. M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A **64**, 013412 (2001). [CrossRef]

15. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A **60**, 3225–3228 (1999). [CrossRef]

*G*

_{3}=

*G*

_{4}=0) is normalized to 1 (i.e., ρ′′

^{(3)}

_{10}/ρ

^{(3)}

_{10}). There are two groups of suppression and enhancement curves due to the primarily-dressed states created by the field

*G*

_{4}(

*G*

_{4}>

*G*

_{3}). When Δ

^{a}

_{2}=Δ

_{4}=0 the splitting Δ

_{AT2}is approximately proportional to Δ

_{AT2}≈ 2{

*G*

_{4}[

*G*

^{2}

_{4}+2Γ

_{40}(Γ

_{30}+Γ

_{40})]

^{1/2}-Γ

_{40}}

^{1/2}. Based on the same secondarily-dressed state ∣+> created by the field

*G*

_{3}, the two photon (∣0>→∣1>→∣+>) resonant FWM signals (corresponding to each group of curves) are enhanced dramatically [Figs. 4(b) and 4(c)]. Specifically, at exact two-photon resonance Δ

^{a}

_{2}= 0 we see that the FWM signal intensity is suppressed when the frequency of the dressing field

*G*

_{3}is scanned across Δ

_{3}=±Δ

_{AT2}/2. The presence of the weak dressing field

*G*

_{3}(

*G*

_{3}<

*G*

_{4}) can either suppress or enhance the FWM signal when Δ

^{a}

_{2}≠0 [Fig. 4(b)]. With two strong dressing fields

*G*

_{3}=

*G*

_{4}=50Γ

_{30}the FWM signal with dual peaks is enhanced by a factor of 440 when Δ

^{a}

_{2}+Δ

_{4}=-50Γ

_{30}. The dual enhanced FWM channels have been opened simultaneously [Fig. 4(c)] by the two strong dressing fields, which provide the energy for such large enhancement.

*χ*

^{(7)}by beating the FWM and EWM, or SWM and EWM signals. Since ∣ρ

^{(3)}

_{10}∣ ≫∣ρ

^{(7)}

_{10}∣ and ∣ρ

^{(5)}

_{10}∣ ≫∣ρ

^{(7)}

_{10}∣ is generally true and the FWM, SWM and EWM signals are diffracted in the same direction with same frequency, the real and imaginary parts of χ

^{(7)}can be measured by homodyne detection with the FWM (or SWM) signal as the strong local oscillator.

^{-1}. Specifically, in doubly-dressed (2

*n*-4)WM, the coherence length is given by

*L*=2

_{c}*c*/[

*n*

_{0}(

*ω*

_{n-2}/

*ω*

_{1}) ∣

*ω*

_{n-2}-

*ω*

_{1}∣θ

^{2}], with θ being the angle between beams 2 and 3 [Fig. 1(a)], where

*n*

_{0}is the refractive index. For a typical experiment, θ is very small (<0.5°) so that

*L*is larger than the interaction length

_{c}*L*, as has been demonstrated in Refs. [6

6. Y. P. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. **32**, 1120–1122 (2007). [CrossRef] [PubMed]

7. Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted-Y system,” Appl. Phys. Lett. **90**, 111104 (2007). [CrossRef]

## 4. Conclusion

*n*) nonlinear wave-mixing processes in closed-cycle (

*n*+1)-level atomic systems. Dressed and doubly-dressed laser beams can enhance the high-order nonlinear wave-mixing processes and produce co-existing wave-mixing processes. An example of a five-level folded atomic system is used to illustrate the co-existing FWM, SWM, and EWM processes, and the great enhancement, as well as suppression of the FWM and SWM signals at different parametric conditions. Understanding the higher-order multi-channel nonlinear optical processes can help in optimizing these nonlinear optical processes, which have potential applications in achieving better nonlinear optical materials and opto-electronic devices.

## Acknowledgments

## References and links

1. | 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. |

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

3. | B. Lu, W.H. Burkett, and M. Xiao, “Nondegenerate four-wave mixing in a double-L system under the influence of coherent population trapping,” Opt. Lett. |

4. | V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. |

5. | D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. |

6. | Y. P. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. |

7. | Y. P. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted-Y system,” Appl. Phys. Lett. |

8. | H. Kang, G. Hernandez, and Y. F. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. |

9. | Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. S. Fu, L. A. Wu, and P. M. Fu, “Generalized |

10. | H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarization in semiconductor doped glasses,” Phys. Rev. Lett. |

11. | D. J. Ulness, J. C. Kirkwood, and A. C. Albrecht, “Competitive events in fifth order time resolved coherent Raman scattering:Direct versus sequential processes,” J. Chem. Phys. |

12. | R. W. Boyd, |

13. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

14. | 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 |

15. | M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A |

16. | M. Yan, E. G. Rickey, and Y. F. Zhu, “Observation of doubly dressed states in cold atoms,” Phys. Rev. A |

17. | L. Deng and M. G. Payne, “Inhibiting the onset of the three-photon destructive interference in ultraslow propagation-enhanced four-wave mixing with dual induced transparency,” Phys. Rev. Lett. |

18. | Y Wu and L Deng, “Achieving multi-frequency mode entanglement with ultra-slow multi-wave mixing”, Opt. Lett. |

19. | H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” 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:**

Nonlinear Optics

**History**

Original Manuscript: February 7, 2007

Revised Manuscript: May 13, 2007

Manuscript Accepted: May 14, 2007

Published: May 29, 2007

**Citation**

Yanpeng Zhang and Min Xiao, "Generalized dressed and doubly-dressed multi-wave mixing," Opt. Express **15**, 7182-7189 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7182

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### References

- 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]
- Y. Li and M. Xiao, "Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms," Opt. Lett. 21, 1064-1066 (1996). [CrossRef] [PubMed]
- B. Lu, W. H. Burkett, and M. Xiao, "Nondegenerate four-wave mixing in a double-L system under the influence of coherent population trapping," Opt. Lett. 23, 804-806 (1998). [CrossRef]
- V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, "Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229-5232 (1999). [CrossRef]
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