Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency

Commun, Theor. Phys. (Beijing, China) 41 (2004) pp.106- 110@ International Academic PublishersVol. 41, No. 1, January 15, 2004Optical Multi-wave Mixing Process Based on Electromagnetically InducedTransparency*LI Jia-Hua,1 PENG Ju-Cun,2 and CHEN Ai-Xi131 Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China2Department of Physics, Xiaogan Normal University, Xiaogan 432100, China3Wuhan Institute of Physics and Mathematics, the Chinese Academy of Sciences, Wuhan 430071, China(Received April 22, 2003; Revised June 3, 2003)Abstract In this paper, we propose and analyze an optical multi-wave mixing scheme for the generation of coherentlight in a five-level atomic system in the context of electromagnetically induced transparency. A detailed semiclassicalstudy of the propagation of generated mixing and probe fields is demonstrated. The analytical dependence of thegenerated mixing field on the probe field and the respective detuning is predicted. Such a nonlinear optical process canbe used for generating short-wavelength radiation at low pump intensities.PACS numbers: 42.50.Gy, 42.65.Ky, 42.50.HzKey words: optical multi wave mixing, electromagnetically induced transparency, Rabi frequency1 IntroductionKerr nonlinearity. More recently, Wu, et al. analyzedElectromagnetically induced transparencya four-wave-mixing (FWM) scheme in a five-level atomicoptical transparency of a three-level medium at a resonantsystem and hyper-Raman scattering (HRS) in resonanttransition induced by application of a coherent electro-coherent media by the use of EIT, which leads to suppress-ing both two-photon and three-photon absorptions in bothpanying reduction of the group velocity of light by many FWM and HRS schemes and enabling the four-wave mix-orders of magnitude in the EIT transparenrcy windowlB) ing to proceed through real, reonant intermediate stateswithout absorption loss. [12,13] In particular, with the ad-nection with many potential aplications, espeiaily low- vent of Bose Einstein condensation in an atomie gas thereintensity nonlinear optics (i.e, it can reach a level of fewhas been much interest in studying four-wave mixing withmatter wave both experimentally and theoretically in theinformation storage.9] In essence, the transparency mayframework of nonlinear atomic optics.I14- -19]In this context, in the present work, by applying thebe viewed as the result of a combination of the dynamicmethodof Wu et al.l 1 2] we present and analyze an opticalterference between the two dressed states that are cIe-multi-wave mixing scheme for the generation of coherentated by the strong coupling laser. Recently, Schmit andlight in the five-level atomic system by means of electro-magnetically induced transparency, under the conditionwithout considering the wave-mixing process. It turns outthat we apply the weak probe pulsed laser. A detailedthat based on electromagnetically induced transparencymixing and probe fields is demonstrated. The infuencesthe nonlinear susceptibility of the probe field can greatlyof the probe field and the respective detuning on the gen-increase by many orders of magnitude.10] Li, et al. haveerated mixing field are discussed. Such a nonlinear opti-demonstrated a clear experimental observation of an en-cal process can be used for producing the coherent short-hanced nondegenerate four-wave mixing (NDFWM) pro-wavelength radiation. To conclude we give a brief dis-cess, making use of the EIT effect in a threelevel Acussion on the experimental realization of the proposedtype system of Rb atoms with cw diode lasers.21 Laterscheme.on, Deng, et al. have proposed an EIT-based double-Asystem with the inclusion of the generated field, and by 2 Model and Solutions of Atomic Equationsmeans of the time-dependent perturbation treatment theyof Motionhave shown that the possible quantum constructive andLet us considerthe, energy scheme depicted in Fig. 1.destructive interference between different excitation path- In this five-level at中国煤化工e pulsed laserways can result in enhancement and suppression of the (pulse length r) isMHCNMH G2) resonance.*The project supported by National Fundamental Research Program of China under Grant No. 2001CB309310 and National NaturalScience Foundation of China under Grant Nos. 90103026, 10125419, and 10121503No.1Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency107One of the two lower levels is coupled to the upper level (assumingh = 1),(|1) →|2)) by a strongly coherent coupling laser to cre-|4 >m-ate an Autler -Townes doublet and two sign laser fieldsare tuned to the transition |1) →|3)> and |3> →|4), re-w。spectively. To avoid significant group velocity mismatch|3 >and for mathematical simplicity, we will assume that allthe involving fields are continuous single-frequency laser|wm(cw laser) with the exception of the probe field. It shouldwsbe noted, however, that the transition |1> - + |3) is dipole-12 >forbidden in line with the well-known selection rule, yet wecan choose sufficiently strong laser field to stimulate elec-v。|wtric quadrupole transitions, which still leads to observed11 >emission and absorption.In the present analysis the semiclassical Hamiltonian.10>describing the atom -field interaction for the system underconsideration can be written as in the Schrodinger pictureFig. 1 EIT-based five-level atomic diagram for analysis.H= εj|i>(i|+ (S2p ei0|2><0| + h.c.) + (Sc ei0|2><1| +h.c)+ (2o1 ei91|3><1|+ h.c)j=0+ (S2s2ei02|4><3| + h.c.) + (S2mei0m|4><0| + h.c.),(1)where On = kin .r- wnt corresponds to the positive-frequency part of the respective field, Sn(n = C,P, s1, s2,m)are one-half Rabi frequencies for the corresponding transtions, i.e, hp = D2oE(wp)/(2h), Sc = D21 E(wc)/(2h),S2s1 = D31 E(ws1)/(2h), S2s2 = D43E(ws2)/(2h), Sm = D4oE(wm)/(2n), with Dku denoting the dipole moment for thetransition between levels |k^> and |l), and Ej = hwj is the energy of the atomic state |j). For simplicity of analysis, inwhat follows we will take Eo = 0 for the ground state |0>. Turning to the interaction picture, the Hamiltonian can berewritten asHo= (up -wo)1)<1| +wp|2><2| + (wp - we + ws1)|3><3| + (up -wc + wsl + ws2)|4><4|,(2)Hmt = Owe|1><1| + Ouwp|2>(2| + Ow3|3><3| + Ows|4><4| + (SIpeit"12><0|+ Se eikeT|2><1|+ S2s1 eiko1"|3><(1| + S2s2 eike2°"|4>(3| + Sm eikmTl4><0| + h.c.),(3)where Owp = O。= (w2-wo)-wp is the single-photon detuning, Owe =△p-△c = (w1 -wo)- (wp- wc),Ow3=△p-Oc+δ1 = (w3 - wo)- (wp-wc) -WsI, and△w4= Op- Oc+δ1 +δm = (w4-wo)-(wp -wc)-Ws1 -Ws2.are the separate two-photon, three-photon, and four-photon detunings, respectively.Let us assume that the wave function has the form|4>= Ao|0> + A1 el(le,- ke)*[1) + A2elkp"|2> + Ase(kp- ketea)"[3)>) + Aekm"|4>(4)with Aj standing for the probability amplitude of the(0 +2)A2=-i9oA1- iS2pAo,atomic state, respectively.((Making use of the Schrodinger equation in the interac-+ 713 )A3= -i2s1A1 - i02*2e-i8hrA4,tion picture i8/8tl亚= Himt|亚>, the equations of motionfor the probability amplitude of the atomic wave functions+ 4)A4= -i2mAo - iSs2ei6k."A3 ,(5)can be readily obtained aswhere we have int中国煤化工= n1 + iOwe,8GHAo= -i0yA42 - i92mA4,2= 72 + iOwp,MHCNM HG + iOws andphasematchrelationδk=kp+ks1+ks2-kc-km(品+元)4-=-i9;Aa2 - i92;As,γ;(j=1~ 4) is the relaxation rate of the state |j>.108LI Jia-Hua, PENG Ju-Cun, and CHEN Ai-XiIn order to correctly describe the propagation of the varying envelope equations for the Rabi frequencies Rpoptical pulse, the atomic equations of motion must beand Mm are given bysimultaneously solved with the Maxwell's equations forθ18both the probe and the generated mixing fields in a self-(+ ia)9p = iko2A2A0,consistent manner. In the limit of plane waves and slowlyvarying amplitude, the positive frequency parts of these(品+二0 )s2m = ikosAAs,(8)\0zT cOt,fields satisfywhere we have introduced the notations(品+-元)Ep = 4rikpP[+),ko2 = 2πNup|Do212(tc)，ko4 = 2πNwm|Do4l2/(tc) .1818(6Following the standard procedure described before in(品+ ia)Em = 4rikmPlIt),Refs. [12], [13], [20], and [21], the probe pulsed laser iswhere Ep(PP+ ) and Em(Pt) are the slowly varying en- assumed to be sufficiently weak that the ratio (Rp/Lc)2velopes of the probe and generated mixing field ampli-is much less than unity, so that in essence all of thetudes (polarizations). In order to treat the propagationatomic population remains in the ground state |0>, i.e,of the generated MWM field through a resonant medium，Ao ≈1. Again, we consider the coupling field we andthe polarization of the medium must be obtained. In the sign fields Ws1 and Ws2 to be continuous single-frequencyfollowing using the atomic wave function (4), we havelaser，but the probe field Wp and the generated mix-P(r,0)= N(亚D|亚ing field Wm have a time-dependent beam profile. Withthese assumptions, performing Fourier transformations= p(+)(up)ekpr + p(+)(wc)eilkerA;(t)= (1/√2π)j_x a;(w)exp(- iwt)dw,j= 1 ~4 and+ p[t(ws1)eike1r + P(2 )(ws2)ete2r"S;(t) = (1/√2元)S2∞W;(w)exp(-iwt)dw,j = p, m forboth Eqs. (5) and Eqs. (8), we straightforwardly obtain+ p(+)(wm)elkmr + h.c，the following equations .where N and D are the atomic concentration and theOw1a1 = S2*a2 + S2*1as，dipole moment operator of the relevant transition, respectively. Pht)(wn) represents the positive frequencyOw2a2= Sea1+ Wp,part of the polarization at the. corresponding field,△w3a3= S2s1a1 + S02*2e-ik"as,i.e.，Pp+)(wp) = NDo2A2At，PC+(wc) = ND12A2A1,P(t)(ws1) = ND13A3A1， P62(ws2) = ND34A4Ag,Owsa4= Wm + So2ei8kra3(9)pMt)(wm) = NDo4A4Ag. For both the probe at wpwhere we have introduced the notations△w1 =w+ i51,md generated mixing fields, combining P}+(wp) =△的2 = w+ i172, △的3 = w+ i73, and Ow4 = w+ i174.NDo2A2A and Plt(wm)NDo4A4A with Rp =Equations (9) can be solved in terms of Wp and Wm withD2oE(wp)/(2h) and Sm = D4oE(wm)/(2h), the slowly the resulta2=S82*192*2Ow1 O03Ows - O01|S2。2|2 - 00|S31|2Wp,S*92s1S2g28k.r. 。O01O的2O的3 - O3|S2c|2 - 02|9231|204=SWp(Wm.(10)Likewise, combining品- i阅)w。-ika0r, (品- i:)Wm = iu,(11)then inserting Eqs. (10) into Eqs. (11), we have3Wp8Wmr-i("+12)W,-=igeWm， 0H”-i(% +1s- 8k)Wm = igsW,(12)where we have defined the new parametersWm = Wm ei8k=,f= ko 00103004二019.2二045中国煤化工28*192，f4= ko4'0的O的0的3一0s31S2.12 - 02182.CYHCNMHG .92= ko2n04-S= Ow1 O2O的3OG4 -△w1 021S2|2 -△的2O04|S231|2 -△的3O04|S2c|2 + |S2。2|2|212No.1Optical Multi-wave Mixing Process Based on Electromagnetically Induced Transparency109After some algebra but with no further assumptions, the solutions to Eqs. (12) areWp(z,w) = Q2 exp(iλ+z) + R2 exp(iX_z),Wm(z,w) = [Q4 exp(iX+z) + Rs exp(iλ_ z)] exp(i8kz),(13)where the constants Qj and Rj are determined by the boundary condition at z = 0 andλ+=“f2+ f4- δk:土;√(f2- f4 + 8k)2 + 49294.23 DiscussionsFor given Wp(0,w) and the MWM emission process, Wm(0,w) = 0, the generated MWM field Wm(z,w) is thengiven byWm(z,w) = Wp(0,w)94Wl(z,w) =λ+-λ_(14)W[(+→-号15-010明which is the main result of the present study.If we use Gaussian pulse shape, for the probe pulsed laser at the entrance to the medium, Sp(0,t) = S2p(0,0)e-(t/T)2.Carrying out Fourier transformation, then we find Wp(0,η) = Sp(0, 0)(τ/√2)e-n/4, where we have introduced thedimensionless variable η = wT. Under perfect phase match conditions δk = 0, we then arrive atWm(z,n)= i2(0,0)r en2/49(1) i()z/e/ sim[[A(m)z]√2A(m)S2p(0,0)τ e-n2/4 (ip()>/er{;f4(n)一f2(m)[A(n)z 1Wp(z,n) =2A(n)sin[F eT+ cos“(0)=]},(15)where; D1D3D4 - Dr|τS2s2|2 - D4|TS2s.1|2; (T82c)(r82*)(5S2*2)f2(m) = ko2cT292(m) = ko2cT2S()S(n)fa(m) = koscr2; D1D2D3 - D3|τS2[2 - D2|τS31|g4() = ko4cT2(5S2*)(T82s1)(TL32)D(n)=η+f2(m)+ f4(n)A(n)= [(2()- 5s(四)2 + 9()<()])S()= D1D2D3D4- D、D2|TS9s2|2 - D2D4|rS2s1|2 - D3D4|r9cI2 + |TS232P|τSo|°，D1=η-(Op- Oc)τ + inT,D2=η- OpT + in2T,D3=η-(Op- Oc +δl)τ + ingT,D4=η-(Ap-△c+δ1 + δm)T + in4T.In order to show that the MWM scheme is capableof ' |0), 5S1/2，F=2 as |1>, 5P1/2 as |2)， 5D3/2 as |3), andgenerating the MWM field at low pump intensities, we nP3/2 as |4) (n > 10). The associated transitions arechoose the Rabi frequencies of the probe Sp and one sign |0) →|2> and |1>→|2) at 795 nm (72 ≈5.6 MHz),field S2。2 well below the saturation levels, yet another sign|1>→|3) at 762 nm(73 = 0.76 MHz) and |3)→|4>field Ms1 well above the saturation levels, and present nu- at 1.3 ~ 1.5 pum (74≈0.09 MHz)，all accessible withmerical calculations, as shown in Fig. 2. The results show diode lasers. For the concerning parameters, we canthat Wm is maximized at δ1T = 0 and δmT = 0 under thechoose atomic density N ~ 1012 cm-3, length of Rb cellusual EIT conditions, demonstrating the enhancement of L ~ 6 mm, the probe pulse length τ ~ 10-11 s, Rabithe MWM via the resonant intermediate state manifested frequencies |S2c|T = 100, |Sp|T = 10, |S2s1|r = 100, andby the EIT- induced suppression of the three photon and|Qs2|T = 5. The wavelength of the generated coherentfour-photon absorptions.light is of the ordenfr 1nelectromag-Now we give a brief discussion on the experimentalnetically induced tYH中国煤化工ing flde isrealization of our scheme. An experimental candidate forapplied to the .C N M H Gereby stingthe proposed system can be found in 87Rb cold atoms with up a quantum interference and allowing the weak probe-the designated states chosen as follows: 5S1/2, F= 1 as pulsed |0〉 →|2〉 field to propagate in a refractively thick110LI Jia-Hua, PENG Ju-Cun, and CHEN Ai-XiRb atomic medium. Then the |0> →|2〉field is increased atoms to evolve smoothly into a population- trapped Su-slowly as compared to the Rabi frequency of the|1〉→|2>perposition state with approximately equal and oppositelyfield, which contributes significantly to the ground-state. phased probability amplitudes between states |0) and |1>.0.0.3直0.210-2三0.10.02345Fig.2 Plot of the generated MWM field Wm as a function η for a five-level atomic system. The parameters are|9p|T = 10， |Sc|τ = 100, |S2s1|τ = 100, |S。2|T = 5, γ1T = 0.01, 72τ = 500, η3T = 5, γ4τ = 0.1, ko2cT2 = 100,ko4cT2= 10, z/cT=2. (a) OpT= Ocτ= δ1τ= δmτ=0; (b) OpT=△cT=δ1τ=0, 8mτ= 20; (c)△pτ= OcT= 0,δ1τ≈δmτ= 20; and (d)△pT= 10,△cT=5, δ1τ =δmT= 20. .4 ConclusionsIn summary, in the present work we present and analyze an optical multi-wave mixing scheme for the generation ofcoherent light in the five-level atomic system by means of electromagnetically induced transparency, under the conditionthat we apply the weak probe pulsed laser. A detailed semiclassical study of the propagation of the generated mixingand probe fields is demonstrated. The influences of the probe field and the respective detuning on the generatedmixing field are discussed. We further show that such a nonlinear optical process can be used for producing theshort-wavelength radiation. 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