Generalized dressed and doubly-dressed multi-wave mixing

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 OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (270.4180) Multiphoton processes; (300.2570) Four-wave mixing; (320.7110) Ultrafast nonlinear optics; (030.1670) Coherent optical effects. 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. 20, 982-984 (1995). 2. 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). 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. 23, 804-806 (1998). 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. 82, 5229-5232 (1999). 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. 93, 183601 (2004). 6. Y. P. Zhang, A. W. Brown, and M. Xiao, "Observation of interference between four-wave mixing and sixwave mixing," Opt. Lett. 32, 1120-1122 (2007). 7. Y. P. Zhang and M. Xiao, "Enhancement of six-wave mixing by atomic coherence in a four-level invertedY system," Appl. Phys. Lett. 90, 111104 (2007). 8. H. Kang, G. Hernandez, and Y. F. Zhu, "Slow-light six-wave mixing at low light intensities," Phys. Rev. Lett. 93, 073601(2004). 9. Z. C. Zuo, J. Sun, X. Liu, Q. Jiang, G. S. Fu, L. A. Wu, and P. M. Fu, "Generalized n-photon resonant 2nwave mixing in an (n +1)-level system with phase-conjugate geometry," Phys. Rev. Lett. 97, 193904 (2006). 10. H. Ma and C. B. de Araujo, "Interference between thirdand fifth-order polarization in semiconductor doped glasses," Phys. Rev. Lett. 71, 3649-3652 (1993). 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. 108, 3897-3902 (1998). 12. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1992). 13. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50 (7), 36 (1997). 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). 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). 16. M. Yan, E. G. Rickey, and Y. F. Zhu, "Observation of doubly dressed states in cold atoms," Phys. Rev. A 64, 013412 (2001). #79873 $15.00 USD Received 7 Feb 2007; revised 13 May 2007; accepted 14 May 2007; published 29 May 2007 (C) 2007 OSA 11 June 2007 / Vol. 15, No. 12 / OPTICS EXPRESS 7182 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). 18. Y Wu and L Deng, "Achieving multi-frequency mode entanglement with ultra-slow multi-wave mixing," Opt. Lett. 29, 1144-1146 (2004). 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).


Introduction
High-order multi-wave mixing processes have been the subject of intense research activities for the past few decades.Efficient four-wave mixing (FWM) [1][2][3][4][5][6][7] and six-wave mixing (SWM) [6][7][8][9][10][11] have been experimentally observed in multi-level atomic systems.In general, as the order of the nonlinearity increases, more complex beam geometries are usually required to satisfy the phase-matching conditions.Also, the nonlinear signal decreases by several orders of magnitude with an increase in each order of nonlinearity of the interaction [12].Since higher order nonlinear optical processes are usually much smaller in amplitude than lower order ones, the interplay between nonlinear optical processes of different orders, if it exists, is usually very difficult to observe.In recent years, many schemes have been developed to enhance higher-order nonlinear wave-mixing processes.More importantly, with induced atomic coherence and interference, the higher-order processes can become comparable or even greater in amplitude than the lower order wave-mixing processes.
The destructive interference in three-level or four-level atomic systems generates electromagnetically induced transparency (EIT) [13,14] which reduces linear absorption and enhances FWM processes [1][2][3][4][5].The doubly-dressed four-level system with a metastable excited state shows sharp dark resonance due to destructive interference between the secondarily-dressed states [15].This is in contrast to a four-level system for the observed triple-peak absorption spectrum in which the doubly-dressed system exhibits constructive interference due to decoherence of the Raman coherence [16].However, we show that constructive interference occurs between two FWM excitation paths of doubly-dressed states in a five-level system.These high-order multi-photon interferences and light-induced atomic coherence are very important in nonlinear wave-mixing processes and might be used to open and optimize multi-channel nonlinear optical processes in multi-level atomic systems that are otherwise closed due to high absorption [6,7,17,18].
In this paper we report, for the first time, a generalized scheme for resonantly dressed (2n-2) wave mixing ((2n-2)WM) and doubly-dressed (2n-4) wave mixing ((2n-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 (2n-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.

Generalized dressed (2n-2)WM and doubly dressed (2n-4)WM
For a closed-cycle (n+1)-level cascade system (Fig. 1), where states |i-1> to |i> are coupled by laser field The Rabi frequencies are defined as , where ij μ are the transition dipole moments between level i and level j.To quantitatively understand such phenomenon of interplay between coexisting 2nWM, dressed (2n-2)WM and doubly-dressed (2n-4)WM processes, we need to use perturbation chain expressions involving all the ( ) 00 2,0 10 ( ) Γ is the transverse relaxation rate between states |i> and |0>.Similarly, we can easily obtain (2 3) (2 3) 00 1,0 10 ) 00 ,0 10 , respectively.The non-dressed generalized 2nWM with phase-conjugate geometry has also been considered in an (n+1)-level system [9].When both fields 1 n E − and n E are turned on, there exist three physical mechanisms of interest.First, the (2n-4)WM process will be dressed by the two strong fields 1 n E − and n E and a perturbative approach for such interaction can be described by the following coupled equations: In the steady state, Eqs. ( 1)-( 3) can be solved together with perturbation chain (C n-2 ) to give the doubly-dressed (2n-4)WM . Under the ) This expansion shows that, the doubly-dressed (2n-4)WM process converts to a coherent superposition of signals from (2n-4)WM, (2n-2)WM and 2nWM (2 5 ) ( 2 3 ) ( 2 1 ) 10 10 10 ( ) , or dressed (2n-4)WM and 2nWM  4) results from the (2n-2)WM process dressed by the strong field n E and a perturbative approach for such interactions can be described by the following coupled equations: In the steady state, Eq. ( 5) can be solved together with perturbation chain (C n-1 ) to give . Under the condition 2 ( ′ can also be expanded to be (2 5) (2 5) (2 3) 10 10 10 , and the dressed (2n-4)WM process converts to a coherent superposition of signals from (2n-4)WM and (2n-2)WM.

Interplay among coexisting FWM, SWM and EWM
(3) (5) & & One important example (n=4) of the generalized doubly-dressed (2n-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 where (3)    | | ( 1) Table 1.Phase-matching conditions and perturbation chains of EWM, dressed SWM doubly-dressed FWM in a closed-cycle five-level system.

Blocked beams
Doubly dressed FWM (3)   10 ρ′ ′ Dressed SWM (5)   10 ρ′ EWM (7)   10 We investigate the dressed SWM spectrum versus 3  G 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 → + ) 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].
, interfere constructively, leading to an enhanced FWM signal.Due to the decoherence of the Raman coherence 30 ρ , the doubly-dressed four-level system also exhibits a constructive interference [16].By contrast, the doubly-dressed system with a metastable excited state shows sharp dark resonance due to destructive interference between the secondarily-dressed states [15].The coexistence of these three nonlinear wave-mixing processes in this five-level system can be used to evaluate the high-order nonlinear susceptibility (7)  χ by beating the FWM and EWM, or SWM and EWM signals.Since (3)    ρ ρ >> 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 (7)  χ can be measured by homodyne detection with the FWM (or SWM) signal as the strong local oscillator.
Multi-wave mixing possesses the features of excellent spatial signal resolution, free choice of interaction volume and simple optical alignment.Moreover, phase matching can be achieved for a very wide frequency range from many hundreds to thousands of 1 cm − .Specifically, in doubly-dressed (2n-4)WM, the coherence length is given by , with θ being the angle between beams 2 and 3 [Fig.1(a)], where 0 n is the refractive index.For a typical experiment, θ is very small ( < 0.5 ) so that c L is larger than the interaction length L, as has been demonstrated in Refs.[6,7].Thus the phase mismatch due to such small angles between laser beams can be neglected.Moreover, the angle θ can be adjusted for individual experiments to optimize the tradeoff between better phase matching and larger interaction volume or better spatial resolution in Figs.1(a) and 2(a).

Conclusion
We presented a generalized treatment for high-order (up to 2n) 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.
wave-mixing processes for arbitrary field strengths of i E .The simple (2n-4)WM via Liouville pathway (

Fig. 2 . 2 E′ and 3 E′ 2 E′ and 4 E′
Fig.2.(a).Three-dimensional beam geometry to achieve required phase-matching conditions.(b) Five-level atomic system for EWM process with blocking beams 2 E′ and 3 E′ . (c) Five- level atomic system for dressed SWM process and (d) the corresponding dressed-state picture, there exist co-existing SWM and EWM processes with blocking beams 2 E′ and 4 E′ . (e) Five- level atomic system for doubly-dressed FWM process and (f) the corresponding dressed-state picture which shows the primarily-dressed state |-> and the secondarily-dressed states |++> and |+-> (the primarily-dressed state |+> channel is not shown here for simplicity), there exist coexisting FWM, SWM and EWM processes with blocking beams 3 E′ and 4 E′ .

4 E′ 2 E′ and 3 E′ 2 E′ and 4 E′ 3 E′ and 4 E′ρ′G
) 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 (EWM) [Fig.2(b)], (dressed SWM) [Fig.2(c)], or (doubly dressed FWM) [Fig.2(e)], respectively.However, the doubly-dressed FWM ( (3) 10 ρ′′ ) process [Figs.2(e) and 2(f)] converts to a coherent superposition of signals from FWM ( ) process [Figs.2(c) and 2(d)]converts to a coherent superposition of signals from FWM and SWM in the weak dressing field limit<< Γ Γ , Eq. (4) reduces to (a) correspond to the |+> and |-> levels created by the dressed field 4 G in Fig. 2(d), respectively).The two peaks are located asymmetrically due to 3 0 Δ ≠ .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 3 0 a Δ = , we see that the SWM signal intensity is suppressed when the frequency of the dressing field is scanned across the resonance ( 4 0 Δ = ).The presence of the weak dressing field can either suppress or enhance the SWM signal when 3 0 a Δ ≠ [Fig.3(b)].Such suppression and enhancement mainly result from the absorption and dispersion of SWM and EWM signals and their interference.When 4 the SWM signal is strongly enhanced by a factor of 620 in the presence of the dressing field when 3 3 0 / 50 a Δ Γ =− [blue curve in Fig. 3(c)], which is mainly due to the three-photon ( 0 1 2 → →