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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 12 — Jun. 11, 2007
  • pp: 7182–7189
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Generalized dressed and doubly-dressed multi-wave mixing

Yanpeng Zhang and Min Xiao  »View Author Affiliations


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

High-order multi-wave mixing processes have been the subject of intense research activities for the past few decades. Efficient four-wave mixing (FWM) [1-7

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]

] and six-wave mixing (SWM) [6-11

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]

] 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

12. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1992).

]. 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

13. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50 (7), 36 (1997).

,14

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]

] which reduces linear absorption and enhances FWM processes [1-5

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]

]. 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

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]

]. 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

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]

]. 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

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

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

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

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]

].

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.

2. 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 Ei [E′i) (ωi, k i (k′ i), and Rabi frequency Gi (G′i)]. The Rabi frequencies are defined as Gi = εi μij/ħ, G′i = ε′i μij, where μij are the transition dipole moments between level 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 (2n-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 En are used to drive the transitions ∣n-2> to ∣n-1> and ∣n-1> to ∣n>, respectively, as shown in Figs. 1(b), there exist one doubly-dressed (2n-4)WM (ρ10″(2n−5)), one singly-dressed (2n-2)WM (ρ10′(2n−3)) and one 2nWM (ρ(2n-1) 10) processes, satisfying the same k 2n-4.

Fig. 1. (a). Schematic diagram of phase-conjugate doubly-dressed (2n-4)WM. (b) Energy-level diagram for doubly-dressed (2n-4)WM in a closed-cycle (n+1)-level cascade system.

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 ρ(2n-1) 10, ρ(2n-3) 10 and ρ(2n-5) 10 nonlinear wave-mixing processes for arbitrary field strengths of Ei. The simple (2n-4)WM via Liouville pathway (Cn-2) ρ00(0)ω1ωn2ρn2,0(n2)ωn2ω2ρ10(2n5) gives ρ10(2n5)=i(1)n+1GaGn2(Gn2)*eik2n4rd12d22dn32dn2, where Ga =G 1G 22⋯∣G n-32, di = Γi0+i12+⋯+Δi) with Δii-ωi. Γi0 is the transverse relaxation rate between states ∣i> and ∣0>. Similarly, we can easily obtain ρ(2n-3) 10 =-ρ(2n-5) 10G n-12/d n-2 d n-1 and ρ(2n-1) 10(2n-5) 10G n-12Gn2/d n-2 d 2 n-1 dn via perturbation chains (Cn-1) ρ00(0)ω1ωn1ρn1,0(n1)ωn1ω2ρ10(2n3) and (Cn) ρ00(0)ω1ωnρn,0(n)ωnω2ρ10(2n1), respectively. The non-dressed generalized 2nWM 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 2n-wave mixing in an (n +1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006). [CrossRef] [PubMed]

].

When both fields E n-1 and En are turned on, there exist three physical mechanisms of interest. First, the (2n-4)WM process will be dressed by the two strong fields E n-1 and En and a perturbative approach for such interaction can be described by the following coupled equations:

ρn0t=dnρn0+iGneiknrρ(n1)0
(1)
ρ(n1)0t=dn1ρ(n1)0+iGn1eikn1rρ(n2)0+iGn*eiknrρn0
(2)
ρ(n2)0t=dn2ρ(n2)0+iGn2eikn2rρ(n3)0+iGn1*eikn1rρ(n1)0
(3)

In the steady state, Eqs. (1)-(3) can be solved together with perturbation chain (Cn-2) to give the doubly-dressed (2n-4)WM ρ10(2n5)=i(1)n+1GaGn2(Gn2)*eik2n4r(dn1dn+Gn2)d12d22dn32[dn2(dn1dn+Gn2)+dnGn12]. Under the conditions ∣Gn2<<Γn0Γ(n-1)0 and ∣G n-12 <<Γ(n-1)0Γ(n-2)0, ρ′′(2n-5) 10 can be expanded to be

ρ10(2n5)=dn2ρ10(2n5)(1dn2Gn12dn22dn1+Gn12Gn2dn22dn12dn)
(i.e.,ρ10(2n5)=ρ10(2n5)+ρ10(2n3)+ρ10(2n1)=ρ10(2n5)+ρ10(2n1)=ρ10(2n5)+ρ10(2n3))
(4)

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 (ρ(2n-5) 10+ ρ(2n-3) 10 + ρ(2n-1) 10), or dressed (2n-4)WM and 2nWM (ρ′(2n-5) 10 + ρ(2n-1) 10), or (2n-4)WM and dressed (2n-2)WM (ρ(2n-5) 10 +ρ′(2n-3) 10) in the weak dressing field limit.

Second, the ρ′(2n-3) 10 term in Eq. (4) results from the (2n-2)WM process dressed by the strong field En and a perturbative approach for such interactions can be described by the following coupled equations:

ρn0t=dnρn0+iGneiknrρ(n1)0andρ(n1)0t=dn1ρ(n1)0+iGn1eikn1rρ(n2)0+iGn*eiknrρn0
(5)

In the steady state, Eq. (5) can be solved together with perturbation chain (Cn-1) to give ρ10(2n3)=ρ10(2n5)Gn12dndn2(dndn1+Gn2). Under the condition ∣Gn2 << Γn0Γ(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 (2n-2)WM process converts to a coherent superposition of signals from (2n-2)WM and 2nWM. Third, the ρ′(2n-5) 10 term in of Eq. (4) results from (2n-4)WM process dressed by the strong field E n-1 Similarly, we can obtain ρ10(2n5)=dn2ρ10(2n5)dn1dn1dn2+Gn12 via the E n-1 coupled equations and the perturbation chain (Cn-2). Under ∣G n-12 << Γ(n-1)0Γ(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 (2n-4)WM process converts to a coherent superposition of signals from (2n-4)WM and (2n-2)WM.

3. Interplay among coexisting FWM, SWM and EWM

Fig.2. (a). Three-dimensional beam geometry to achieve required phase-matching conditions. (b) Five-level atomic system for EWM process with blocking beams E2 and E3. (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 E2 and E4. (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 E3 and E4.

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(a), with seven beams (E 1, E 2, E2, E 3, E3, E 4, E4) 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 E2 and E3 (EWM) [Fig. 2(b)], E2 and E4 (dressed SWM) [Fig. 2(c)], or E3 and E4 (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 32 <<Γ30Γ20 and ∣G 42 <<Γ10Γ40, Eq. (4) reduces to

ρ10(3)=Ga(d3d4+G42)d12[d2(d3d4+G42)+G32+d4]Gad12d2+GaG33d12d22d3+GaG32G42d12d22d32d4=ρ10(3)+ρ10(5)+ρ10(7)+ρ10(3)+ρ10(7)=ρ10(3)+ρ10(5)
(6)

where ρ(3) 10 = -Ga/d 2 1 d 2, ρ(5) 10 = GaG 32/(d 2 1 d 2 2 d3), ρ(7) 10 = -GaG 32G 42/(d 2 1 d 2 2 d2 3 d 4)ρ10(3)=Gad12d2(1G32d3d2) and ρ10(5)=GaG32d12d22d3(1G42d3d4) with k F = k 1 + k 2-k2, d3 = Γ30+iΔa 3, Δa 3a 23, Δa 212 and Ga = i G 1 G 2(G2)*ei k Fr

Table 1. Phase-matching conditions and perturbation chains of EWM, dressed SWM and doubly-dressed FWM in a closed-cycle five-level system.

table-icon
View This Table

We investigate the dressed SWM spectrum versus Δa 320 or Δ430. 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 Δ3 ≠ 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 Δa 3 = 0, we see that the SWM signal intensity is suppressed when the frequency of the dressing field is scanned across the resonance (Δ4 = 0). The presence of the weak dressing field can either suppress or enhance the SWM signal when Δa 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 430 =50 the SWM signal is strongly enhanced by a factor of 620 in the presence of the dressing field when Δa 330 =-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]

].

Fig. 3. (a). Dressed SWM signal intensity versus Δa 320 (The maximum is normalized to 1) (Γ3020 = 1, Γ1020 = 0.5, Δ420=6, G 420=0 (black curve), G 420=2 (blue curve), G 420=10 (red curve) , G 420 = 20 (magenta curve)). (b) Dressed SWM signal intensity (normalized by no dressing field (G 4 =0) case, i.e., ρ′(5)10(5) 10 versus Δ4302030 =1, Γ1030 =0.5, G 430=0.5, Δ3a30=0 (black curve), Δ3a30=-0.5 (blue curve), Δa 330=-2 (red curve), Δ3a30=-6 (magenta curve)). (c) Dressed SWM signal intensity (normalized by no dressing field case) versus Δ4302030 = 1, Γ1030 = 0.5, G 430 = 50, Δa 330 = -30 (black curve), Δa 3/ Γ30 = -50 (blue curve), Δa 330 = -70 (red curve), Δa 330 = -100 (magenta curve)).

Next, we consider the doubly-dressed FWM spectrum versus Δa 220 or Δ330 [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 ΔAT1 ≈ 2{G 3[G 2 3+2Γ302030)]1/22 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]

]. Thus, when Δ4 = ΔATI/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 Δ4 = -ΔATI/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, ρ00(0)ω1ρ10(1)ω2ρ(++)0(2)ω2ρ10(3) and ρ00(0)ω1ρ10(1)ω2ρ(+)0(2)ω2ρ10(3), interfere constructively, leading to an enhanced FWM signal. Due to the decoherence of the Raman coherence ρ30, 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]

]. 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

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]

].

Fig.4. (a). Doubly-dressed FWM signal intensity versus Δa 22030 = Γ2010, G 320 =13, G 420 = 5, Δ320=0, Δ420=200 (black curve), Δ420=10.5 (blue curve), Δ420 = -10.5 (red curve)). (b) Doubly-dressed FWM signal intensity (normalized by the two dressing field G 3=G 4=0 case, i.e., ρ′′(3) 10) versus Δ3302030 = 1, Γ1030 = 0.5, G 330 = 0.5, G 4Γ30 = 50, Δ1 = Δ4 = 0, Δ230 = 0 (black curve), Δ230 = -0.5 (blue curve), Δ230=-2 (red curve), Δ230=-6 (magenta curve)). (c) Doubly-dressed FWM signal intensity (normalized by no dressing field case) versus Δ3302030 = 1, Γ1030=0.5, G 330 = G 430 = 50, Δ14=0, Δ230=-30 (black curve), Δ230=-50 (blue curve), Δ2Γ30 = -70 (red curve), Δ230 = -100 (magenta curve)).

In Figs. 4(b) and 4(c) the doubly-dressed FWM signal intensity with no dressing field (G 3 =G 4 =0) is normalized to 1 (i.e., ρ′′(3) 10(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 Δa 24=0 the splitting ΔAT2 is approximately proportional to ΔAT2 ≈ 2{G 4[G 2 4 +2Γ403040)]1/240}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 Δa 2 = 0 we see that the FWM signal intensity is suppressed when the frequency of the dressing field G 3 is scanned across Δ3=±ΔAT2/2. The presence of the weak dressing field G 3 (G 3<G 4) can either suppress or enhance the FWM signal when Δa 2 ≠0 [Fig. 4(b)]. With two strong dressing fields G 3=G 4=50Γ30 the FWM signal with dual peaks is enhanced by a factor of 440 when Δa 24=-50Γ30. 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.

The coexistence of these three nonlinear wave-mixing processes in this five-level system can be used to evaluate the high-order nonlinear susceptibility 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.

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 cm-1. Specifically, in doubly-dressed (2n-4)WM, the coherence length is given by Lc =2c/[n 0(ω n-2/ω 1) ∣ω n-2-ω 1∣θ2], 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 Lc is larger than the interaction length 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

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]

]. 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).

4. 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.

Acknowledgments

We acknowledge the funding support from the National Science Foundation.

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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). [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]

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]

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]

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

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  12. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1992).
  13. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 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). [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]
  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]
  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]

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