Coherent Resonance of Saturated Absorption in Spectroscopy of Counterpropagating Waves

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1 Journal of Modern Phyic,,, 8-5 doi:.6/jmp..5b Publihed Online May ( Coherent Reonance of Saturated Aborption in Spectrocopy of Counterpropagating Wave A. A. Chernenko, E. G. Saprykin, A. M. Shalagin, Intitute of Semiconductor Phyic, Siberian Branch, Ruian Academy of Science, Novoibirk, Ruia Intitute of Automation and Electrometry, Siberian Branch, Ruian Academy of Science, Novoibirk, Ruia Novoibirk State Univerity, Novoibirk, Ruia Received ABSTRACT Reult of theoretical reearche of the aturated aborption reonance hape in a method of the probing field on V-type of tranition are repreented. It i hown that in cae of oppoite circulary polarized optical field the reonance i hown in the form of cro, and it form trongly depend on relaxation contant of level and it can be repreented a in the form of a dip, and aborption peak. Thu, the peak form ha excluively coherent character. Atomic tranition are offered, on which obervation of the given effect i poible. Keyword: Nonlinear Spectrocopy; Atomic Tranition; Saturation Aborption; Cro Reonance; Light Wave. Introduction It i known that action of the trong light field on a reonant medium lead to nonlinear effect which are hown in the form of change of the denity population of reonant level (effect of aturation, plitting of level and an interference of level o-called nonlinear interferential effect (NIEF a reult of imultaneou aborption ome photon []. The effect of aturation ha excluively incoherent character, and two other are caued by coherent procee. The method of a probing field [,] allow invetigating the pecified nonlinear effect. The reonance regitered in thi method in the aborption pectrum of the probing field by the reonant ga medium i hown in the form of a dip on the Doppler line contour and carrie generally incoherent (populational character. The coherent procee are hown in a form of the field broadening of the reonance pectrum (effect of level plitting and a an additive in the dip amplitude (contribution of NIEF. And, the value of the NIEF contribution depend on the direction of propagation of the trong and probing light wave; effect i maximum at unidirectional wave, and in cae of oppoite directed wave the contribution of effect to reonance amplitude i trongly uppreed (in relation of the uniform width to the Doppler line width of atomic tranition. So for the two-level medium under optimum condition the hare of NIEF in amplitude of a reonance doen t exceed ome percent []. In pite of the fact that reearche on nonlinear pectrocopy of multilevel quantum ytem are conducted already for a long time, a ituation when the nonlinear reonance in oppoite directed wave would be defined by only coherent procee, a far a we know, in t found until now. There are cae, a in paper [] where to the oberved pectrum feature of the nonlinear reonance, having the incoherent nature, NIEF manifetation wa mitakenly attributed. In thi work we conider quantum ytem in which in cae of the oppoite directed light wave the nonlinear reonance having excluively coherent nature (without a contribution of populated effect with characteritic for coherent procee by an interferential form of a pectrum can be oberved. Such ituation i realized in the V-type of tranition the upper tate of which ha to break up only in the bottom tate, and light wave have to be circular polarized with the counter direction of rotation of polarization vector. A number of atomic tranition on which probably experimental obervation of a reonance of coherent type i offered.. Theoretical Model Let u conider the problem about the aborption pectrum of a probing field in a medium with V-type of tranition in the preence of a trong field with the ame frequency and oppoite propagation direction. The cheme of tranition i preented in Figure. The trong wave i aumed to be plane (with the frequency, wave vector k, and electric field trength Е and reonant to the atomic tranition m n (with the tranition frequency mn. Polarization of the trong wave i either linear or circular (eparate cae. The probe wave i alo mono- Copyright SciRe.

2 A. A. CHERNENKO ET AL. 9 chromatic (with the frequency =, the wave vector k = k, and the electric field trength Е with circular polarization. When olving the problem, the aturation of a medium by the probing wave i taking into account under the aumption of it weakne a compared to the trong wave. The ga i aumed to be rarefied enough to neglect colliion. The medium i aumed to be optically thin. We conider the problem in a coordinate ytem with the quantization exi along the direction of wave vector k of the trong field. In thi coordinate ytem light field induce tranition between magnetic ublevel (Figure with variation М = ± (at linear polarization, or tranition with М = + or М = (at circular polarization depending on a direction of it rotation. In olving the problem we tart from kinetic equation for a denity matrix of the atomic ytem. According to [], the dynamic of diagonal element of the denity matrix ( i = ii in the model of relaxation contant i decribed by the ytem of equation: i ii Qi kik Re( ivij ji t k j ( Re( i V j For off-diagonal element of the denity matrix the next ytem i valid: ik ( ik iik ik iv [, ] ik iv [, ] ik ( t Here Г i and Г ik are the width of the level and tranition, Q i i the number of exitation act of the i-th level per unit time, the ummand in ytem of ij ji ki k equation ( determine the pontaneou decay of the upper tate m to the lower tate n and i abent in the equation for the population of upper level, V and V are operator of the interaction between the atom and the trong and probing field. The operator are defined a V = Gexp(i(kr t +H.с. and V = G exp(i(k r t +H.с., where the operator G = de, and G = de, and d i the operator of the reduced dipole moment of atomic tranition. Let' mark, that the given tatement of the problem and the obtained olution are fair both for tranition between exited tate, and for a cae, when the lower tate n i the ground tate of atom. In thi cae the width of the lower tate i determined by mean value of the interaction time of particle with the light field. Following [], we eek olution of the ytem of equation for the denity matrix in the following form (the firt harmonic approximation by the difference of wave frequencie = : the diagonal element are found a i = i + + i exp(it + - i exp( it; off-diagonal element on the allowed and forbidden tranition ac- k cordingly are found a ik = R ik exp( it + R ik exp( i t + R ik exp( i t and ik = r ik + r + ik exp(it + r - ik exp( it, where =. The ubtantiation of the given kind of olution and their accuracie for the V-, Λ- and J = -J = type of tranition are given by u in work [,5]. In the approximation of rotating optical field the ytem of equation for the denity matrix in the conidered tranition cheme in the tationary cae i reduced to a ytem of algebraic equation for i, i, R ik, R ik, R ik, r ik, and r ik. Taking into account the hermiticity of thee coefficient, we write only independent equation for a cae of the linear polarized trong field. The equation for a cae of the circular polarized trong field are eaily received from reduced below, equating to zero one of the circular trength component of trong field. In a cae of the V- cheme (Figure population of the lower ( and the upper ( and level are decribed by a following ytem of equation: n Q Akk R igkrk k, k, Re( ig R ( ( e( n i Ak k i GR GR k, ig ( R GR GR m Q Re( ig R (a (б (в ( m i i( GR GR (г m Q Re( igr Re( ig R (д ( m i i( GR GR GR (е The ytem of equation for polarization at the allowed ( and and at the forbidden tranition ha the next form: m n Figure. Scheme of interaction of light field with V-type of atomic tranition, - hift of level. Copyright SciRe.

3 5 A. A. CHERNENKO ET AL. ( mn i R ig( igr igr (a ( mn i( R ig ( ig r ( mn i R ig ( ig r ig (б r (в ( mn i R ig ( ig ( (г ig r ( mn i( R ig ( ig r (д ( mn i R ig ( ig ( (е ig r ( m i r i( G R G R G R ( m i( r i( G R G (5а R (5б ( m i( r i( GR GR G R (5в In equation ( - (5 Г n and Г m are the width of the lower and upper level, Г mn i the width of the tranition line, i the width of the forbidden tranition between magnetic ublevel of the upper tate; ik = ik and ik = ik are the value of frequency detuning of the trong and probing field from the frequencie ik of tranition between the magnetic ublevel of the m and n tate. The motion of atom i taken into account by the following change in the equation: ik ik kv, ik ik k v, аnd (k kv, where v i the velocity of atom. The ituation of counter propagation of wave with equal frequencie wa then analyzed: =, and k = k. The hape of the aborption line of the probing field (per atom wa determined by the following relation: / = mn Re(i(R /G, where the deignation <...> mean the averaging by the Maxwell velocity ditribution of particle, and = mn d /c mn i the reonant aborption cro ection. In calculation the probabilitie A ki of the decay of magnetic ublevel by each of the pontaneou channel were et to be imilar and equal to A mn /.. Reult of Numerical Solution of the Initial Equation Stationary ytem of equation ( - (5 that were ob- tained above for element of the denity matrix were olved numerically upon varying the value of width of the upper Г m and lower Г n atom level, the parameter of the radiation branching from the upper level а (a = A mn /Γ m and the intenitie of optical field uing the relaxation characteritic of the - p 8 tranition of the neon atom (А mn =.88 7 с -, Г m = с -, Г n = 5 6 с, Г mn = (Г m + Г n /. The value of level width varied from the aforeaid value to Г m = Г n = Г mn, at the ame time, the tranition width Г mn remained contant and the value of branching parameter а varied within the range.. The line width of the forbidden tranition between the ublevel of the upper tate relied to width of the upper level. The Doppler line width i taken to be equal kv T = 5. 9 с -, and the variation range of particle velocitie at integration wa kv T with a tep kv T = ( - - kv T. The aturation parameter of the trong (κ and probing (κ p field were choen in the form and ( de / / ( mnn p ( de / / ( m nn,.. Strong Field of Linear Polarization where E and E are the trength of the circular compo- nent of the trong and probi ng field and d i the re- duced dipole moment of tranition. The value of aturation parameter varied within the limit κ 5 and κ p κ. Below, reult are preented for mall value of the aturation parameter of the probing field κ p = -. Reult of numerical imulation of the aborption pectrum of the probing wave of circular polarization in the preence of a trong aturating field of linear polarization are preented in Figure and. When the upper tate i degenerated the hape of the nonlinear reonance i repreented a a dip in the center of the Doppler contour of the aborption line. It amplitude and width depend on the aturation parameter of optical field, on width of tranition level and on the branching parameter а. With elimination of the degeneracy (plitting of the upper level one can oberve a dip of the aturated aborption at the hifted tranition frequency in the hape of the aborption line of the circular polarized probing wave (named a РR, the ign of it hift i defined by the ign of circular polarization and a cro reonance (CR at the unhifted frequency тn. Amplitude and ign of the cro reonance depend eentially on the relation between the level width Г m and Г n and on the branching parameter а, at the ame time, the effect of thee characteritic on the parameter of the parent (PR reonance i expreed much weaker. It i een from Figure that, depending on the value of the branching parameter a, the cro reonance can manifet itelf not only a a dip, but alo a an aborption peak. The maximal value of the peak amplitude i reached for the cloed tranition (for parameter а = both when the value of the level width are imilar (Г m Г n and when they are coniderably different (Г m >> Г n. Thu it appear, that at a imilar value of the aturation parameter κ of the trong field, the peak amplitude in cae of imilar value of level width i coniderably larger than the peak amplitude in cae of a trong difference between the width (curve and 5. Copyright SciRe.

4 A. A. CHERNENKO ET AL. 5 α α Ω/kv T Figure. Reonance hape at different value а and Γ n : = Γ mn ; =5, p = - ; Γ n = Γ mn ( -,. Γ mn (5-8; a= (, 5,.98 (6,.75 (, 7,.6 (;. (, 8..8 α/α Ω/kv T Figure. Behavior of the reonance hape from intenity of the trong field: = Γ mn ; a = ; p = -, =. (;. ( 5. ( 5 (; Γ n =. Γ mn (continuou line, Γ mn (dahed line. With a decreae in the value of the branching parameter cro reonance i tranformed to a dip from the peak form. Moreover, in the cae of imilar value of level width Г m = Г n the flip i oberved for value of the branching parameter а.6; for the relation between the level width Г m >> Г n the change in the ign of the reonance amplitude occur near the value а.98. Note alo that, in cae of imilar value of the level width the dip amplitude of the cro reonance i coniderably le and the width i coniderably larger than in cae of when Г m >> Г n (compare curve and 8. Taking the approximation of a weak probing field and neglecting the polarization at the combination frequency - one can obtain from the ytem of equation ( - (5 an analytical olution for polarization at the probing field frequency R ( a: G R i( ( mn i G ( ig (( ( i ( i( mn (6 Copyright SciRe.

5 5 A. A. CHERNENKO ET AL. Here, the difference of level population are determined via population of the lower level, branching parameter а and rate of timulated tranition W ik a: ( W m ; ( W m ; W ik n mn ( mn ik ; (7 N, ( a W n ( a W n W W m where N Q n i the population of the lower level in the abence of the trong field. In the velocity (balanced approximation the preence of a cro reonance i aociated with amplitude of the Bennett dip or peak in the population of the common level. The pecificity of the V-cheme i that the common level i the lower level and, in the abence of the pontaneou radiation from the upper level to the third level (branching parameter а =, a dip in the population of the common level i abent (it follow from (7 that N. It i compenated by the arrival of the pontaneouly emitted particle. In thi cae the parent reonance i caued excluively by the non-equilibrium population of the upper level. Under thee condition, it follow from olution (6 and averaging over the velocity ditribution of particle that the populational part in the hape of the CR (the firt ummand in olution (6 i repreented by the Doppler contour and the peak (Figure, curve and 5 are caued only by coherent procee (the econd ummand in (6, leading to an increae in the aborption coefficient. The performed calculation how that, in the given cheme of tranition, the contribution of coherent procee to the CR hape i maximal at equal value of level width, and for cae of Г m >> Г n their contribution i le almot by an order of magnitude. Thi i illutrated in Figure by the dependence for two relation between value of level width both for the CR peak hape (curve and 5 and for the dip (curve and 8. The effect of the aturating field intenity on the CR peak hape i hown in Figure. Here, for all relation between level width and any value of the branching parameter, an increae in the trong field intenity lead to an increae in the CR amplitude and width both for the peak form and for the dip. However, the functional relation of amplitude for the peak and for the dip form of the CR in the range of the ued intenitie of the aturating field appear different. If the CR amplitude in the form of a dip increae by a law cloe to the quare root dependence, the relation of the amplitude of the coherent CR peak under thee condition i of linear character, with the quare root dependence for the PR amplitude. Let u mark the character of influence of the probing m field intenity: an increae of the probing field intenity reult in decreaing in the amplitude of the Doppler lining contour of the aborption line and in the amplitude of the cro (for the dip and peak hape and parent reonance, thu an increae of the width of thee reo- nance i oberved. However, here we did not take into account a capability of appearance of the pectrum feature tipulated by patial modulation of the nonlinear medium uceptibility in a field of tanding wave. Thee problem were conidered earlier (ee, for example, activity [6,7] and reduced link in them. The nonlinear interference effect at the account of aturation by the probing field were calculated alo in activity [8], but only for a cae of unidirectional wave, in abence of the cro reonance... Strong Field of Circular Polarization The cae of circular polarization of the trong and probing wave i of interet. If the ign of rotation of electric field vector of the trong and probing wave in a medium are imilar, only the parent reonance will take place, it hift in the frequency cale will depend on the of plit- direction of rotation of field vector and a value ting of the upper level. In cae of the counter circular polarized vector of field only a cro reonance will be oberved in the aborption line pectrum of the probing wave. The CR i located at the half um of tranition frequencie in the region of the interaction of counter propagating wave with the ame group of moving atom. The graph correponding to thi cae are preented in Figure. Here the aturated aborption line hape, calculated for two value of plitting of the upper level and a number of value of the atomic tranition parameter are hown. It i viible, that for any value of level plitting within the limit of the Doppler line width the reonance i exhibited on the ame light wave frequency. And, in the abence of branching parameter (а = reonance i repreented a the peak of aborption (curve - 5. A it wa marked above, in thi cae the population part of the CR i abent, and it hape i caued excluively by the nonlinear coherent proce (NIEF agree with []. The preented relation (the mot obviou are curve and demontrate the alternating hape which i typical for the NIEF and a decreae in the effect with an increae in the difference between the level lifetime. A decreae in the branching parameter а lead to appearance of the populational contribution and to tranformation of the peak form into a dip (curve 5-8. Thu, in cae of the counter propagating and counter circular polarized wave the oberved reonance of aturated In practice, thee are experiment in which the probing wave i obtained a reult of reflection from the mirror behind the cell with a ga. Copyright SciRe.

6 A. A. CHERNENKO ET AL. 5 α α Ω kv T Figure. Shape of reonance at orthogonally circular polarization of field: κ = 5; κ p = - ; = (,, = Г mn (-8; a=(-,.9(5,.75 (6,.6 (7,. (8; Г n = Г mn (, 8,. Г mn (,. aborption i a cro reonance actually. It i located in the center of the unhifted tranition line at any value of plitting of the upper level within the Doppler width of the line (in particular, in abence of plitting and ha under certain condition extremely coherent nature. According to the reult of [], interference phenomena in the cae of counter propagating wave that are abent in the approximation of firt nonlinear correction can appear in the next order with repect to aturation. Thi fact explain the aforementioned linear character of the increae of the coherent CR amplitude in Figure with the quare rooted dependence of the PR amplitude. In concluion we hall indicate on a capability of experimental obervation of the coherent type aturated aborption reonance. The given reonance will be realized only in the V-type of tranition in atom between tate with the full moment J = and J =, thu the upper tate of tranition hould break up mainly on the lower tate to upply the value of branching parameter a ~. Tranition S, P from the ground tate of the alkaline-earth metal atom, and of the Hg, Cd, Zn and Yb atom atify to thee requirement. The wave length of thee tranition are located in pectral area of generation of exiting power tunable laer (in the viible range, or their econd harmonic (in the UV range.. Acknowledgement Thi work wa upported by the NSh project and the Fundamental Optical Spectrocopy and It Application Program (project 9.5 of the Branch of Phyical Science, Ruian Academy of Science. REFERENCES [] S. G. Rautian, G. I. Smirnov and A. M. Shalagin, Nonlinear Reonance in Spectra of Atom and Molecule, Nauka, Novoibirk, 979, [in Ruian]. [] V. S. Letokhov and V. P. Chebotayev, Nonlinear Laer Spectrocopy of Ultrahigh Reolution, Nauka, Mocow, 99. [in Ruian]. [] Yu. V. Bogdanova, Nonlinear Reonance in the Signal of Magnetic and Frequency Scanning of Three-level Λ- and V- Sytem, Optic of Atmophere and Ocean, Vol. 6, No. 7,, pp [] E. G. Saprykin, A. A. Chernenko and A. M. Shalagin, Effect of the Atomic Tranition Parameter on Cro Reonance Shape in Spectrocopy of Probing Field, Izvetiya Vuzov. Fizika, No. /,, pp [5] E. G. Saprykin, A. A. Chernenko and A. M. Shalagin, On the Shape of Cro Reonance in Counterpropagating Wave Spectrocopy, Optic and Spectrocopy, Vol., No. 5,, pp doi:./sx85 [6] M. G. Stepanov, Autler-Towne Doublet Probed by Strong Field, Journal of Phyyc B: Atomic, Molecular and Optical Phyic, Vol., 999, pp [7] S. A. Babin, E. V. Podivilov, V. V. Potapov, D. V. Churkin, and D. A. Shapiro, Nonlinear Reonance, Induced by Maximum Spatial Harmonic of Coherence, Journal of Experimental and Theoretical Phyic, Vol., No.,, pp [8] A. K. Popov, S. A. Mylivet and Toma, F. Geoge, Nonlinear Interference Effect and All-Optical Switch- Brodened Me- ing to Optically Dene Inhomogeneouly dia Phyical Review A, Vol. 7, 5. Copyright SciRe.