Atomic coherence and interference phenomena in resonant nonlinear optical interactions

a rXiv:q uant-ph/05114v126M ay2SPIE Vol.279815th International Conference on Nonlinear Optics ICONO’9527June -1July 1995,St.Petersburg,Russia Coherent Phenomena and Amplification without Inversion Atomic coherence and interference phenomena in resonant nonlinear optical interactions 1A.K.Popov Institute for Physics Russian Academy of Sciences,Krasnoyarsk University and Krasnoyarsk Technical University 660036,Krasnoyarsk,Russia.Fax:(3912)438923,E-mail:popov@ksc.krasn.ru and S.G.Rautian Institute of Automation and Electrometry Russian Academy of Sciences 630090,Novosibirsk,Russia 1Abstract Interference effects in quantum transitions,giving rise to amplification without inversion,optical trans-parency and to enhancements in nonlinear optical frequency conversions are considered.Review of the relevant early theoretical and experimental results is given.The role of relaxation processes,spon-taneous cascade of polarizations,local field effects,Doppler-broadening,as well as specific features of the interference in the spectral continuum are discussed.Keywords:atomic coherence and interference,resonant nonlinear interactions,bound-free transi-tions,amplification without inversion,relaxation-induced processes,local field effects,inhomogeneous broadening,frequency-conversion,V UV generation 2Introduction There has been considerable interest recently in the study of laser-induced quantum coherence and interference,which leads to fundamental effects in high resolution nonlinear spectroscopy,to amplifica-tion of radiation without the requirement of population inversion (AW I )and to resonantly enhancedrefraction at vanishing (without)absorption (ERW A ),to coherent population trapping and construc-tive contributions in resonantly enhanced nonlinear-optical frequency conversions and,at the same time,to distractive contributions in absorption of the fundamental and generated radiations 1,2.Wide range of applications are expected 3.Resonant nonlinear optical interference effects have been subject of the extensive both theoretical and experimental studies since the discovery of masers and lasers (see for example 2and ref.therein).In this paper we briefly review some early and recent results of Russian research groups on this topic.3Resonant nonlinear optical interference processes3.1Destructive and constructive interference in classical and quantum opticalphysicsInterference is one of the fundamental physical phenomena.Two oscillations at one and the same,or close,frequencies may interfere both in constructive and destructive ways.One can manipulateby the resulting oscillations with variation of the relative phase and the amplitudes of the interfering oscillators in order to enhance or,on the contrary,to eliminate the oscillations of any nature.In-terference is widely used in optical physics,including quantum optics.The concept of interference is more general,then the notions of elementary quantum-optical processes,such as one-photon,multistep and multiphoton transitions.These notions were introduced and classify at their frequency-correlation properties in the framework of the perturbation theory.Indeed,in resonant interactions,these proper-ties may be drastically changed with growth of the intensity of the coupled fields 4;2b,c .The latter may give rise to such effects in nonlinear spectroscopy of Doppler broadened transition,as compensation of the residual inhomogeneous broadening in Raman-like and cascade configurations 4;2b,c ;5.Quantum interference may occur when coherent superposition of real states is involved in a process 6.Alternatively,interfering frequency-degenerate intraatomic oscillations may originate from different correlated quantum pathways,contributing in one and the same frequency.For example,in the weak-field approximation,these can be one-and two-photon contributions to an optical process,associated with the radiation at a given frequency.Such process may be thought as that started from the coherent superposition of closely spaced real energy-level and quasi-level (virtual state),created by the auxiliary strong field 2b,c ;4.Such a coherent superposition can be produced even more easily than in the case of real doublet state.In general,even in the cases,when many elementary processes contribute to an optical process and their classification is troublesome,one can explain and predict experimental results with the aid of the notion of interfering frequency-degenerated components of nonlinear polarization.The amplitudes of the components can be varied with the intensities and phases –with the frequency-detunings of the driving fields.3.2Effect of energy levels population and relaxation,density matrix approach In general case of open energy-level configuration with all the levels being populated and various relaxation processes involved,density-matrix method is the most convenient for the analysis of a resonant nonlinear-optical response.Explicit formulae,describing spectral properties of a weak probe field in the presence of an auxiliary strong one,in cascade,V and Λconfigurations can be easily derived in the similar way 2b,c .We shall show that on the example of the energy-level schematic,givenE 1at frequency ω1≈ωgl and E 3at frequency ω3≈ωmn are els configuration.Fields E 2at frequency ω2and E 4at frequency ω4are probeWe shall derive the conditions to achieve AW I at the transitionwell as at transition ml ,so that both V and Λconfigurations Frequency of the probe field may be both higher and compared to the driving field.energy-level configuration,shown in the Fig.1.Density equations in the interaction representation,relevant to theunder consideration,can be written in the form:lg r 1·exp (i Ω1t ),ρnm =r 3·exp (i Ω3t ),ρng =r 2·exp (i Ω1t )+˜r 2·exp [i (Ω1+Ω3−Ω4)t ],ρlm =r 4·exp (i Ω4t )+˜r 4·exp [i (Ω1−Ω2+Ω3)t ],ρln =r 12·exp [i (Ω1−Ω2)t ]+r 43·exp [i (Ω4−Ω3)t ],ρii =r i ,P 2r 2=iG 2∆r 2−iG 3r ∗32+ir ∗12G 1,d 2˜r 2=−iG 3r ∗41+ir ∗43G 1,P 4r 4=i [G 4∆r 4−G 1r 41+r 43G 3],d 4˜r 4=−iG 1r 32+ir 12G 3P 41r 41=−iG ∗1r 4+ir ∗1G 4,P 43r 43=−iG 4r ∗3+ir 4G ∗3,P 32r 32=−iG ∗2r 3+ir ∗2G 3,P 12r 12=−iG 1r ∗2+ir 1G ∗2,Γm r m =−2Re {iG ∗3r 3}+q m ,Γn r n =−2Re {iG ∗3r 3}+γgn r g +γmn r m +q n ,Γg r g =−2Re {iG ∗1r 1}+q g ,Γl r l =−2Re {iG ∗1r 1}+γgl r g +γml r m +q l ,∆r1=r l−r g,∆r2=r n−r g,∆r3=r n−r m,∆r4=r l−r m.WhereΩ1=ω1−ωlg,Ω3=ω3−ωmn,Ω2=ω2−ωgn,Ω4=ω4−ωml,G1=−E1d lg/2¯h,G2=−E2d gn/2¯h,G3=−E3d nm/2¯h,G4=−E4d ml/2¯h,P1=Γlg+iΩ1,P2=Γng+iΩ2,P3=Γnm+iΩ3,P4=Γlm+iΩ4,,P12=Γln+i(Ω1−Ω2), P43=Γln+i(Ω4−Ω3),P32=Γgm+i(Ω3−Ω2),P41=Γgm+i(Ω4−Ω1),d2=Γng+i(Ω1+Ω3−Ω4),d4=Γlm+i(Ω1−Ω2+Ω3).HereΩi are frequency detuning from the resonances,G i—Rabi frequencies,∆r i—power–depending population differences,Γij—homogeneous half linewidths,Γ−1i—lifetimes,γij—relax-ation rates from i to j states,q i—population rate by a incoherent source.Density matrix amplitudes r i determine absorption/gain and refraction indexes,˜r i—determine four–wave mixing driving nonlinear polarizations.The equations and their solution for the cascade atomic configurations can be derived by the simple change of the detunings signs2b.3.3Laser–induced atomic coherence and classification of resonant nonlinear ef-fectsSolution of the coupled density–matrix equations may be represented in the form:r1,3=iG1,3∆r1/P1,r2,4=iG2,4R2,4/P2,4,R2=∆r2(1+g7+v7)−v3(1+v7−g8)∆r3−g3(1+g7−v8)∆r1(1+g4+v4)+[v5+v4(v5−g6)+g5+g4(g5−v6)],(2)∆r1=(1+æ3)∆n1+b1æ3∆n3(1+æ1)(1+æ3)−a1æ1b1æ3,∆r2=∆n2−b2æ3∆r3−a2æ1∆r1,∆r4=∆n4−a3æ1∆r1−b3æ3∆r3;r m=n m+(1−b2)æ3∆r3,r g=n g+(1−a3)æ1∆r1,r n=n n−b2æ3∆r3+a1æ1∆r1,(3) r l=n l−b1æ3∆r3+a3æ1∆r1,∆r i(E1=0,E3=0)=∆n i;g1=|G1|2P∗12P2,g3=|G1|2P41P4,g5=|G1|2P41d∗2,g7=|G1|2P∗12d∗4,v1=|G3|2P∗32P2,v3=|G3|2P43P4,v5=|G3|2P43d∗2,v7=|G3|2P∗32d∗4;æ1=æ01Γ2lgΓlΓgΓlg|G1|2,æ3=æ03Γ2mnΓmΓnΓmn|G3|2;a1=γgn a2Γn(Γg−γgl)=γgnΓlΓl(Γm−γmn)=γml b3Γl(Γm+Γn−γmn).By substituting frequency deviationsΩi for that Doppler-shiftedΩi−k i v(v is atomic velocity)we can take into account the effect of atomic motion.Imaginary part of density-matrix amplitudes r2and r4 represent absorption or gain at the corresponding probe-field frequencies.At G3=0equations(2) and(3)convert in solutions forΛand V schemesr2=i G2,r4=iG4.(4)Following2b,c we can classify resonant nonlinear effects as1)power saturation of the populations (eq.(4));2)strong-field induced splitting of the probe-field resonances,or ac Stark effect(denom-inators in eqs.(4));and3)nonlinear interference effect(NIEF)(second and third terms in the nominators of eqs.(2)).4Difference in absorption and emission spectra due to the nonlinear interference effects,amplification without inversion,resonance–enhanced refraction without absorptionPower of emitted or absorbed radiation,for example at the frequencyω2,which is proportional to Re(−iG∗2r2),can be considered as a difference between pure emission(the term,proportional to r g) and pure absorption(the rest terms in eqs.(2)).The difference in frequency-dependence of these terms, induced by the auxiliary drivingfield,is the origin of AW I2b.Refractive index atω2is determined by Im(−iG∗2r2)and,in general,laser-induced minimum in absorption may coincide with the resonance-enhanced maximum in refraction1,3.Thus,laser-induced resonance splitting and NIEF transform only spectral shape of absorption/gain and refractive indices,give rise to difference in the line shapes of spontaneous(or pure induced)emission and absorption,but do not affect the integral intensity of the spectral lines2b,c:dΩ2Re(−ir2/G2)=∆r2, dΩ4Re(−ir4/G4)=∆r4.(5) Indeed,NIEF give rise to electromagnetically induced transparency(EIT)and to AW I at the transi-tions gn(or ml),when contributions of second and third terms in the nominators of eqs.(2)are equal or dominate over∆r2(or over∆r4),correspondingly.From the above presented density-matrix equa-tions one can see that the coherence at the transitions gm and ln(r32and r12),induced in cooperation of the strong and the probefields,is the source of the EIT and AW I effects.A great number of elementary processes,introduced and defined for the bare states in the frame-work of the perturbation theory,may give contribution to the absorption/gain indexα(Ωi).Consider, for example,α(Ω4)at the frequencyω4>ω1(Fig.1),reduced by it’s maximum valueα0(0)in the absence of the all strongfields,for the case when E3=0.From the eqs.(2)onefinds:α(Ω4)P4·∆r4−g1∆r1α0(0)≈Γ24∆r4iΩ4∆n4}≈Γ24∆r4Γ214+(Ω4−Ω1)2·|G1|2(∆r1−∆r4) (n l−n m)Ω24−ΓgmΓlmΩ24(n l−n m)(7)The last terms in eqs.(7)describe Raman-like coupling and originate both from the nominator and the denominator in eq.(6).Population inversion between initial andfinal bare states(r m=n m>r g) is required for amplification of the probefield.b.Resonance:Ω1=Ω4=0.Conditions for AW I and EIT are:|G1|2 g1∆r1≥∆r4,orFigure2:Interference enhanced frequency–mixing.a.Triple resonance enhanced frequency-conversion. Levels2and1as well as3and2are coupled by the strong either one-or multiphoton interac-tions.Levels1and2are coupled by the weakfield. b.Enhancement in frequency-mixing due to the autoionizing-like resonance,induced in spectral continuum by the auxiliary strongfields E and E3. c. Enhancement in three-photon resonant four-wave mixing,induced by the auxilliary strongfield E. mentioned Autler-Townes(ac Stark splitting)as well as local-field effects.Consider energy-level scheme,shown in Fig.2a.Strongfields at frequenciesω3andω2couple unpopulated levels3and2(Rabi frequency G3)and2and1(Rabi frequency G2),respectively.Field atω1≈ω10as well as generatedωs=ω1+ω2+ω3are weak,do not change populations of the levels and are accounted for only in the lowest order of the perturbation theory.Absorption and refraction indexes for the probefields atω1andωs are represented by the imaginary and real parts ofχ1(−ω1;ω1)=(χ01/P01)f1,χs(−ωs;ωs)=(χ0s/P03)f s(9) Nonlinear susceptibility is:χNL(−ωs;ω1+ω2+ω3)=(χNL/P01P02D03)f,(10)whereχ01,χ0s andχNL0are resonant values of the susceptibilities at negligibly small G2and G3.Factorsf1,f2and f describe effects of the strongfields.Simple density-matrix calculations,similar to given in2b,e;10a.yield:f1={1+g2/P01P02[1+(g3/P02D03)]}−1,(11)f s={1+g3/P03D02[1+(g2/D02D01)]}−1,(12)f=f1[1+g3/D03P02]−1=[1+(g2/D02D01)+(g3/D03P02)]−1(13) P01=1+ix1,P02=1+ix0,P03=1+ix s;D01=1+iy1,D02=1+iy0,D03=1+iy s;x1=(ω1−ω10)/Γ10=0,x02=(ω1+ω2−ω21)/Γ20=0,x s=(ωs−ω30)/Γ30=0;y1=(ωs−ω3−ω2−ω10)/Γ10=0,y02=(ωs−ω3−ω21)/Γ20=0,y s=(ω1+ω2+ω3−ω30)/Γ30=0;g2=G22/Γ10Γ2,g3=G23/Γ30Γ20,Γij are homogeneous halfwidth of the corresponding transitions.In the case,when E s is not a probe field,but generated radiation,ωs=ω1+ω2+ω3and D0i=P0i.Factors f1,f s and f are different and describe splitting of the corresponding resonances.Frequency-dependence and difference from unity of the factors f1,f s and f is determined by the coherence, induced at the transition02by the two combinations of strong and weakfields(E1,E2and E s,E3).Generated power P∝g2g3|χNL|2,depends not only on imaginary but on real part ofχNL too,and because of that may not deplete in the spectral range of induced transparency and phase-matching. Each resonance increases|χNL|2by the factor of x−2i,which may be on the order ser-induce spectral structures in real parts ofχ1andχs(dispersion caused by the coherence at the02 transition),provide additional means to phase-match frequency-conversion by the small detunings of the fundamental radiations from the resonances.Triple resonance may yield total enhancement in generated power on the order of1018.Due to the induced transparency,number density of the atoms N and consequently P∝N2may be increased by several orders of the magnitude in addition.At high number density of the atoms,localfields may significantly differ from the external electro-magneticfields both in amplitudes and in phases.As it was shown in12,13,that may drastically change spectral properties of the induced transparency as well as of the generating nonlinear polarization. Similar to11,12,making use Lorentz-Lorenz approximation,localfield effects can be accounted for by the substituting one-photon resonances on that red-shifted(by substituting x1and x s for x1+C1and x s+C s,C1=N|d10|2/3ǫ0Γ10;C s=N|d30|2/3ǫ0Γ30,ǫ0-is permittivity of free space).Due to the fact that this does not influence transition frequencies between the excited states and that of the multiphoton transitions,the introduced shifts may drastically change effects of strong electromagnetic fields atω2andω3on both dressed linear and nonlinear responses.Equations,given above,can be easily generalized on the cases of the higher order processes.For example,when1–0and/or3–2,2–1are multiphoton transitions,generalization can be done simply by substituting one-photon Rabi frequencies and detunings for the corresponding multiphoton ones. Manipulations by the nonlinear susceptibility,absorption and refractive indexes for the generating radiation with the auxilliary strongfields,coupled to the adjacent transitions(both bound and con-tinuum states,Figs.2b,c.),were considered in ref.2f,10.6Nonlinear interference effects at bound-free transitions,Laser-induced autoionizing-like resonances(laser induced continuum structure)Nonlinear interference phenomena,similar to those at bound-bound transitions,including AW I and EIT,can occur at the transitions to ionization continuum.Appropriate theory was developed in ref.2f,10a,14.Similar case,relevant to the zone bands in crystals,was considered in ser induced autoionizing like resonances–laser induced continuum structure(LICS)was observed in the experiments ref.16,and since the end of80′s studies of the resonant interference processes in the context of LICS,AW I and EIT,first at bound–free and then at bound–bound transitions,have involved a number of research groups17,18.Potential feasibilities to manipulate both by LICS and by the splitting of the discrete resonances in order to enhance short-wavelengths frequency-mixing output and to decrease resonant absorption of the both fundamental and generated radiations can be shown with the example of Fig.2b.,generalized for the case,whenω1is close toω10,and radiations atω2,ω3andωare strong.The example combines opportunities to manipulate by two LICS and by depletion of absorption at the discrete transitions. Contribution of strong off-resonant k levels are taken into account too.By that,the detunings |ω1-ωgm|,|ω1+ω2−ωgn|and|ω−ω3−ωnl|are assumed being much less than all the rest.Density -matrix calculations give the expressions for nonlinear susceptibilityχ(3)(ωµ=ω1+ω2+ω3),which determines generated power at the frequencyωµ,as well as for absorption indexesα(ω1)andα(ωµ) for probe radiations at corresponding frequences as follows19:χ(3)(ωµ=ω1+ω2+ω3)/χ(3)0µ=K/(D gm X),(14)α(ω1)/α01=Re{[1−g mn/(D gm X)]/D gm},(15)α(ωµ)/α0µ=1−k3βl+k3βl(y l+q gl)2/(1+y l2)−−Re{k4g nn A2(1−iq gn)2/Y}(16) whereχ(3)0µ,α01andα0µ-are corresponding resonant values at the intencities of all thefields beeing negligibly weak.The rest parameters are as follows:K=1−k1βl[(1−iq nl)(1−iq lg)]/[(1−iq ng)(1+ix l)],(17)A=1−k1βl[(1−iq ln)(1−iq gl)]/[(1−iq gn)(1+iy l)],(18) X=(1+g nn)[1+ix n+g mn/D gm(1+g nn)−k2βlβn(1−iq nl)2/(1+ix l)],(19)Y=(1+g nn)[1+iy n+g mn/p gm(1+g nn−k2βlβn(1−iq nl)2/(1+iy l)],(20)D gm=1+i(ω1−ωgm)/Γgm,p gm=1+i(ωµ−ω3−ω2−ωgm)/Γgm,(21)x l=(ω1+ω2+ω3−ω−ωgl−δll)/(Γgl+γll),x n=(ω1+ω2−ωgn−δnn)/(Γgn+γnn),(22) y l=(ωµ−ω−ωgl−δll)/(Γgl+γll),y n=(ωµ−ω3−ωgn−δnn)/(Γgn+γnn),(23) k1=(γglγln)/(γgnγnn),k2=(γnlγln)/(γllγnn),k3=(γglγlg)/(γggγll),k4=(γgnγng)/(γggγnn),(24)g mn=|G mn|2/ΓgmΓgn,βl=g ll/(1+g ll),βn=g nn/(1+g nn),(25)g ii=γii/Γgi,q ij=δij/γij,γij=π¯h G iǫGǫj|ǫ=¯hωµ+Re{ k G ik G kj/p gk},(26)δij=¯h P dǫG iǫGǫj/(¯hωµ−ǫ)+Im{ k G ik G kj/p gk}(27)Factors0≥k i≥1,depending on whether continuum states are not degenerate or degenerate(unity).Comparing eqs.(14)and(16)with corresponding equations from ref.10a,2f,one can see additional interference LICS structures in generating nonlinear polarization,absorption and refraction indexes, produced in cooperation by the E3and Efields(terms,proportional toβn and g n),which provide with the supplementary means in absorption spectroscopy and for enhancements of generated short-wavelength radiation.7Relaxation-induced coherence processesAs it was outlined above,relaxation may influence interference processes both in negative and positive ways.Consider examples,when role of relaxation is positive.7.1AW I due to interference in spontaneous cascade of polarizationsThe features in absorption and emission spectra,discussed above,are caused by interference of con-tributions of probefield and combination of probe and auxiliary strongfield in atomic polarization. As it was outlined above,there may be other sources of interfering intraatomic oscillations.One of the means to obtain AW I without making use of auxiliary strongfields has been suggested recently in ref.20.The origin is interference through the correlations in spontaneous decay.Consider four-level atomic configuration shown in Fig.3a.All four transitions are allowed.Suppose,that the transition frequencyωmn is close toωm1n1,andωm1mis close toωn1n,that is difference∆∆=ωm1n1−ωmn=ωm1m−ωn1n(28)is small.In this case interference between considered four radiating channels is possible.It is caused by the coherence transfer due to interaction with the vacuum oscillations,besides the populationsFigure3:Energy-level schematics for relaxation-induced coherent processes. a.AW I through spon-taneous cascade of polarizations. b.Relaxation-induced F W M.decay and spontaneous emissions of photons.For the absorption index in the frequency range around ωmn calculations give:α(Ω)=λ2Γ2+Ω2+N n1m1A m1n1[Γ1ΓΓ1f(Ω)]},(29)C= 2J m+1(Γ−iΩ)[Γ1−i(Ω−∆)]=ΓΓ1[ΓΓ1−Ω(Ω−∆)]4πΩ2{N nm A mnΓ+N n1m1A m1n1(Γ1−KC)},(33)According to eq.33,absorption index may occur negative(AW I),if the requirementsK>O,(KC/Γ1−1)N n1m1A m1n1Γ1>N nm A mnΓ(34)are met.When K≤0,∆=0,the condition(|K|C/Γ−1)N n1m1A m1n1Γ>N nm A mnΓ1(35)means appearance of AW I in the line center(Ω=0).Similar phenomena may occur in the spectralrange of the doubletωm1m ,ωn1n.Thus,in the considered atomic configuration AW I may be providedby the correlations in the spontaneous decay without any external action.7.2Collision-induced four-wave mixingConsider example,when collisions and spontaneous relaxation,as well as external magneticfield,break destructive interference8a.This remove elimination of for-wave mixing process and provides with the test,selectively sensitive to the specific modes of relaxation.The experiment was carried out with He−Ne laser,λ=1.52µm,which is resonant to2s2–2p4transition of Ne.The upper level consist of three Zeeman’s sublevel(J1=1),the lower one is singlet(J0=0).Fundamental beam consisted of two linear and orthogonal polarized components E1and E2,frequency-shift∆=ω2−ω1being much less than natural transition linewidth.Intensity of the radiation atω1was much greater then that at ω2.Collision and magneticfield sensitive four-wave mixing output E s atωs=2ω1−ω2=ω2−2∆and with the same polarization as E2was detected.Growth of the F W M signal with the increase of collision rate and strength of magneticfield was observed,that can be explained as follows.Eachfield and emitting nonlinear polarization P NL(ωs)may be represented as combination of twocircular polarized components P NL+(ωs)and P NL−(ωs).Formulae for each of these components of non-linear polarization consist of two terms.One of them describes F W M of the radiations with one and the same polarizations in two-level subsystem,another one–F W M of the waves with opposite polar-izations in three-level Zeeman’s subsystem(Fig.3b).In the schematic under consideration,it turned out,that the two contributions interfere in the distractive way and completely eliminate each other, provided by the relaxation rates of population and quadruple moment(alignment)in the upper level are equal.It is obvious that trapping of the spontaneous radiation from the upper level,anisotropic collisions,as well as external magneticfield break the counterbalance and,therefore,induce F W M output.Such dependence was observed in the experiments.External magneticfield turns the second channel into fully resonant double-V schematics.8Review of early theory and experiments on NIEF,AW I and related phenomenaCoherence phenomena in three-level systems were studied since discovery of masers.Feasibility to attain AW I in these systems was discussed in some of publications of that period both for microwave21 and optical transitions22.AW I in optical two-level systems was predicted in ref.23andfirst was observed in radio-frequency transitions24,2d.In optical range AW I and corresponding features in refractive index were observed in ref.2e,25.Studies of coherence and interference phenomena in quantum transitions is growing research area,since they are embedded in many optical processes of basic and practical importance.9Concluding remarksAs it was outline,interference is basic and very general phenomenon of optical physics,which may play a crucial role in many experimental schematics of resonant nonlinear optics.Some of such schematics are shown in the Fig.4.Fig.4a.shows upconversion of weak infrared radiation at the frequencyω2.Fields E1and E3 are strong.Destructive interference of oscillations at the frequencyωs−ω3=ωng=ω1+ω2was shown to be one of the main process,limiting the conversion efficiency26.Fig.4b.–interference of multiphoton transition and one-photon,induced by the generating radiation eliminates population of the upper level.Fig.4c.–off-resonant7th-order seventh-harmonic generation interfere with resonant 9th-order seventh-harmonic generation,that was used for detection of the processes26.Figs.4d,e.–interference of contributions of the doublet sublevels in two-photon and off-resonant one-photon transitions.Figs.4f,g.–interference of doublet sublevels in F W M.Figure4:Interfering channels,embedded in resonant nonlinear optical processes.Pressure-induced resonance wasfirst proposed and experimentally proved in8a and later in27.The entire analogy between the schemes3b and4f is seen from the formula for the driving coherence(scheme4f)ρ(2)n′n∝V n′gρ(1)gn+ρ(1)n′g V gn∝[1Ω1−iΓng]1(Ω2+iΓn′g)(Ω1−iΓng)[1−iΓnn′−Γn′g−Γng11References.1.For a survey of recent research on AWI and ERWA see:a.M.Fleischhauer,C.H.Keitel,M.O.Scully,Chang Su,B.T.Ulrich,Shi-Yao Zhu,Phys.Rev.,Vol. A46,1468,1992;b.Papers from”Atomic Coherence and Interference”,Springer-Verlag,1977;b.A.K.Popov,Introduction in Nonlinear Spectroscopy,North-Holland, 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