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harmonics generation and wave - mixing are one of the basic phenomena of nonlinear optics which have been extensively studied both theoretically and experimentally with the advent of lasers @xcite . recent advance in laser technologies has provided ultrahigh intensities for supershort laser pulses that makes achievable non - perturbative regime of harmonic generation , which significantly extends the spectral region accessible by lasers , in particular , for short wavelengths towards vuv / xuv or even x - ray radiation @xcite . such short wavelength radiation is of great interest due to numerous significant applications , e.g. in quantum control , spectroscopy , sensing and imaging etc ..    depending on the laser - atom interaction parameters , harmonic generation may arise from bound - bound @xcite and bound - free - bound transitions via continuum spectrum @xcite . bound - bound mechanism of harmonic generation without ionization is more efficient for generation of moderately high harmonics @xcite . for this mechanism resonant interaction is of importance . besides pure theoretical interest as a simple model , resonant interaction regime exhibits significant enhancement of frequency conversion efficiencies @xcite . however , to access highly excited states of atoms / molecules by optical lasers the multiphoton excitation problem arises . required resonantly - driven multiphoton transition is effective for the systems with the mean dipole moments in the stationary states , or three - level atomic systems with close enough to each other two states and nonzero transition dipole moment between them @xcite . as a candidate , we have studied the hydrogenlike atomic and ionic systems where the atom has a mean dipole moment in the excited stationary states , because of accidental degeneracy for the orbital momentum @xcite . other interesting examples of efficient direct multiphoton excitation are molecules with a permanent dipole moments @xcite , evenly charged molecular ions at large internuclear distances @xcite , and artificial atoms @xcite realized in circuit quantum electrodynamics ( qed ) setups @xcite .    in the work @xcite we have shown that the multiphoton resonant excitation of a three - level atomic / molecular system is efficient by the two bichromatic laser fields . hence , having efficient two - color multiphoton resonant excitation scheme it is of interest to consider multicolor harmonic generation and wave - mixing processes by an atomic or molecular system under the such circumstances when only a bound states are involved in the interaction process , which is the purpose of the current paper . the presence of the second laser provides additional flexibility for the implementation of multiphoton resonance expanding the spectrum of possible combinations . moreover , two - color excitation extends accessible scattering frequencies with sum- and difference components . in the current paper , we employ analytical approach for high - order multiphoton resonant excitation of quantum systems which has been previously developed by us @xcite . expression for the time - dependent mean dipole moment describing coherent part of scattering spectrum is obtained . the results based on this expression are applied to hydrogen atom and evenly charged homonuclear diatomic molecular ion . the main spectral characteristics of the considered process are in good agreement with the results of the performed numerical calculations . estimations show that one can achieve enhanced generation of moderately high harmonics / wave - mixing via multiphoton resonant excitation by appropriate laser pulses . our interest is also motivated by the advent of circuit qed setups @xcite where one can realize artificial atoms of desired configuration . thus , the obtained results may also be is of interest for artificial atoms , and the presented results can be scaled to other systems and diverse domains of the electromagnetic spectrum . the paper is organized as follows . in section ii , we present the analytical model and derive the coherent contribution to the multicolor harmonic spectra . in section iii , we present some results of numerical calculations of the considered issue without a multiphoton resonant approximation and compare the obtained spectra with the analytical results . here , we consider concrete systems , such as hydrogenlike atom and evenly charged molecular ion . finally , conclusions are given in section iv . we consider a three - level quantum system interacting with the two laser fields of frequencies @xmath0 and @xmath1 as shown in fig.(1a ) . it is assumed that the system is in a @xmath2 configuration in which a pair of upper levels @xmath3 and @xmath4 with permanent dipole moments are coupled to a lower level @xmath5 . another possible three - level scheme is @xmath6 configuration shown in fig.(1b ) . in this case the lower level @xmath5 is coupled to an upper level @xmath3 which has a strong dipole coupling to an adjacent level @xmath7 . if the separation of the energy levels of the excited states is smaller than laser - atom interaction energy then by a unitary transformation @xcite the problem can be reduced to the @xmath2 configuration fig.(1a ) . as an illustrative example may serve hydrogenlike atom considered in parabolic @xcite and more conventional spherical coordinates . in parabolic coordinates , the atom has a mean dipole moment in the excited states , while in the second case because of the random degeneracy of the orbital moment there is a dipole coupling between the degenerate states , but the mean dipole moment is zero for the stationary states . the inverse with respect to the @xmath2 configuration is the polar @xmath8 configuration , which can be realized for artificial atoms @xcite . hence , as a general model we will consider the scheme of the @xmath2 configuration . type with mean dipole moments in the excited states and ( b ) @xmath6 configuration with the coupling transition between the excited states . the considered configurations are unitary equivalent to each other . ] the hamiltonian for the system within semiclassical dipole approximation is given in form @xmath9@xmath10where , @xmath11 , @xmath12 and @xmath13 are the energies of the stationary states @xmath14 , @xmath15 and @xmath4 respectively , and@xmath16is the interaction part of the hamiltonian with a real matrix element of the electric dipole moment projection @xmath17 @xmath18 , and @xmath19 are slowly varying amplitudes of linearly polarized laser fields , with unit polarization vector @xmath20 and constant relative phase @xmath21 . the diagonal terms in ( interact ) describe the interaction due to the mean dipole moments and are crucial for effective multiphoton coupling . we consider schrdinger equation @xmath22with hamiltonian ( [ ham ] ) at the resonance @xmath23 for efficient multiphoton coupling . the resonance detunings are given by relations @xmath24for @xmath25 pair of photon numbers . here and below , unless otherwise stated , atomic units ( @xmath26 ) are employed . our method of solving the schrdinger equation with hamiltonian ( ham ) has been described in detail in @xcite , and will be excluded here . the time - dependent wave function can be expressed as @xmath27@xmath28@xmath29where @xmath30 are the time - averaged probability amplitudes and @xmath31 are rapidly changing functions on the scale of waves periods . depending on the ratio of frequencies @xmath32 the resonant condition ( [ detuning ] ) can hold for a single pair of photon numbers -normal resonance or can also be satisfied by diverse pairs of photons numbers ( in principle infinity ) -degenerate resonance . let us first consider the case of normal resonance . hence , if resonant condition holds for a pair ( @xmath33 ) then assuming the smooth turn - on of the pump waves , the relation between rapidly oscillating and slow oscillating parts of the probability amplitudes can be written@xmath34@xmath35@xmath36@xmath37where @xmath38@xmath39and@xmath40@xmath41 in deriving these equations we have applied well - known jacobi anger expansion via bessel functions:@xmath42    now let us proceed to the case of degenerate resonance . particularly if @xmath43 , where @xmath44 is an integer number , then there are many channels of resonance transitions and one should take into account all possible transitions and the relation between rapidly oscillating and slow oscillating parts of the probability amplitudes can be written    @xmath45    @xmath46 \frac{s_{2}+n}{s_{2}}e^{is_{1}\varphi } e^{is_{2}\omega _ { 2}t } ,   \label{srpd1}\ ] ]    @xmath47    @xmath48    where @xmath49 is given by resonance condition@xmath50 in the schrdinger picture the coherent part of the dipole spectrum is expressed as follows @xcite:@xmath51where@xmath52is the time - dependent expectation value of dipole operator . with the help of wave function ( [ wf ] ) the expectation value of the dipole operator ( dev1 ) can be written as    @xmath53    @xmath54    @xmath55    combining the solution for slow oscillating parts of the probability amplitudes with ( [ z1 ] ) , ( [ z2 ] ) and ( [ dev2 ] ) one can calculate analytically the expectation value of the dipole operator for an arbitrary initial atomic state . the fourier transform of @xmath56 gives the coherent part of the dipole spectrum . the solution for slow oscillating parts of the probability amplitudes analytically is very complicated and in order to reveal the physics of multiphoton resonant excitation process let us consider systems with inversion symmetry : @xmath57 and @xmath58 . as is seen from ( [ dev2 ] ) , for effective harmonic generation in certain resonance conditions one should provide considerable population transfer between the atomic states @xmath14 and upper levels @xmath3 and @xmath4 . dynamic stark shifts can take the states off  resonance , so appropriate detunings for compensation are chosen . let us first consider the case of normal resonance . for the system initially situated in the ground state , the solution for slow oscillating parts of the probability amplitudes is @xcite : @xmath59@xmath60@xmath61where @xmath62 describes dynamic stark shifts@xmath63@xmath64and @xmath65 expresses the frequency of rabi oscillations @xmath66  replacing the probability amplitudes in ( [ dev2 ] ) by the corresponding expressions ( [ avg1])-([avg3 ] ) and taking into account the relations ( [ rpd1])-([rpd3 ] ) one can derive the final analytical expression for @xmath67 . as is seen from eq . ( [ stark ] ) , the dynamic stark shift is proportional to the ratio @xmath68 , while the multiphoton coupling is proportional to @xmath69 . since large dynamic stark shifts are detrimental for maintenance of considerable population transfer , here we consider systems with @xmath70 . taking into account the smallness of the parameter @xmath71 , from ( [ dev2 ] ) at the first approximation we derive the following compact analytic formula:@xmath72where@xmath73@xmath74    as we can see from ( [ main ] ) , ( [ amp ] ) , the spectrum consists of doublets @xmath75 . at that only harmonics with odd sum of harmonic numbers @xmath76 are existed , as it was expected because of inversion symmetry of the considered problem . now let us consider the degenerate resonance . particularly if @xmath77 , where @xmath44 is odd , the dipole expectation value can be obtained from eqs . ( [ srpd1])-([srpd3 ] ) , ( [ dev2 ] ) and is given as : @xmath78@xmath79where rabi frequenc is given as:@xmath80    the spectrum is noticeably different for even @xmath44 . in particular , for bichromatic field with frequencies @xmath81 and @xmath82 dipole spectrum contains also even harmonics of the main frequency , tunable low - frequency ( much smaller than @xmath81 ) part and six hyper - raman lines per harmonic . dipole moment expectation value is expressed by very bulky formula and will not be given here . in this section we present numerical calculations for hydrogen atom and homonuclear diatomic molecular ion @xmath83 @xcite with the specific parameters of available laboratory lasers . the numerical results will be compared with exact results of the dipole spectrum to estimate accuracy and applicability of generalized rotating wave approximation . the time - dependent schrdinger equation for three - level model with hamiltonian ( [ ham ] ) are considered . the solution for the probability amplitudes has been obtained using runge - kutta algorithm and scattering spectrum is estimated by applying the fast fourier transform method @xcite . for smooth turn - on of the laser fields , we consider envelopes with hyperbolic tangent @xmath84 temporal shape , where @xmath85 characterizes the turn - on time and chosen to be @xmath86 . it is assumed that quantum systems are initially in the ground state @xmath5 . for both systems the main transitions fall in the vacuum ultraviolet range . thus , @xmath87 @xmath88 . and @xmath89 @xmath90 . for hydrogen atom and ion @xmath83 , respectively . this is a spectral domain where strong coherent radiation is difficult to generate and two or higher photon multicolor resonant excitation is of interest .     at four - photon two - color resonance ( @xmath91 ) of hydrogen atom with @xmath92 @xmath90@xmath93 @xmath90 , and @xmath94 @xmath95 the solid ( red ) line corresponds to numerical calculations ; the dashed ( green ) line corresponds to the approximate solution ( for better visibility the latter has been slightly shifted to the right ) . ]     at five - photon two - color resonance ( @xmath96 ) of hydrogen atom with @xmath92 @xmath90@xmath97 @xmath90 , and @xmath98 @xmath95 the solid ( red ) line corresponds to numerical calculations ; the dashed ( green ) line corresponds to the approximate solution . ] figure 2 displays the multicolor harmonic and wave - mixing emission rate ( coherent part ) as a function of the ratio @xmath99 ( we assume @xmath100 and @xmath101 ) at the four - photon two - color resonant excitation of hydrogen atom with xef excimer ( 351 nm , @xmath102 ) and ti : sapphire ( 780 nm , @xmath103 ) laser systems with @xmath92 @xmath104 .. for the hydrogen atom @xmath105 @xmath90 . and @xmath106 @xmath88 . here and below , for the chosen parameters the dynamic stark shift is compensated . the latter provides almost complete population transfer . the solid ( red ) line corresponds to numerical calculations , while the dashed ( green ) line corresponds to the approximate expression ( [ main ] ) . note that the numerical and analytical calculations coincide with high accuracy , so for visual convenience to distinguish these curves the spectrum corresponding to analytical calculations ( [ main ] ) has been slightly shifted to the right for figs . 2 - 5 . figure 3 displays the multicolor harmonic and wave - mixing emission rate at the five - photon two - color resonant excitation of hydrogen atom with ar+ ( 488 nm , @xmath102 ) and ti : sapphire ( 724 nm , @xmath107 ) laser systems with @xmath108 @xmath90 .     at seven - photon two - color resonance ( @xmath109 ) of n@xmath110 molecular ion with @xmath111 @xmath90@xmath112 @xmath90 , and @xmath113 @xmath114 the solid ( red ) line corresponds to numerical calculations ; the dashed ( green ) line corresponds to the approximate solution . ] @xmath115@xmath116 , and @xmath117 @xmath118    in fig . 4 we plot harmonic / wave - mixing emission rate at the seven - photon two - color resonant excitation of n@xmath119 molecular ion with xef excimer ( 351 nm , @xmath120 ) and ti : sapphire ( 820 nm , @xmath107 ) laser systems with @xmath121 @xmath90 . for this system we take @xmath122 @xmath88 . and @xmath106 @xmath90 ..    for a degenerate case of resonance in fig . 5 we plot harmonic / wave - mixing emission rate at @xmath123 with @xmath124 @xmath88 . ( 849 nm ) at waves electric fields @xmath92 @xmath104 .. for the case @xmath125 the radiation spectrum is similar to non - degenerate case . however , there are a few differences . scattering spectrum is richer in satellites of multicolor harmonics , in addition there is low - frequency radiation on rabi frequencies . in fig . 6 we plot low - frequency part of radiation spectrum . here , the presented triplet which can be tuned by laser parameters lies in thz / ir region . @xmath90@xmath126 , and @xmath127 @xmath128 ]    as is seen from these figures in the coherent spectrum there are as harmonics of the individual waves as well as frequencies with sum / difference components and its harmonics , in accordance with the analytical ansatz ( main_d ) . from eq . ( [ amp ] ) it is clear that for effective harmonic generation one should provide large dipole interaction energy @xmath129 , since the bessel function exponentially decreases with increasing index at the given argument . the bessel function @xmath130 at large argument values reaches its maximum at @xmath131 . thus , the cutoff frequency depends linearly on the amplitudes of the laser fields . let us make some estimations for reasonable interaction parameters . the average number of photons at the frequency @xmath81 emitted at each lasers shots of duration @xmath85 , rayleigh length @xmath132 on the atomic / molecular ensemble of density @xmath133 can be estimated as @xcite : @xmath134the incident pulse duration is assumed to be @xmath135  @xmath136 , the rayleigh length is taken to be @xmath137 , and for emitters density we assume @xmath138@xmath139 . for the setup of figs . 3 and 4 with the chosen parameters , the average number of radiated photons at frequencies up to @xmath140 per shot is @xmath141 , which is two orders of magnitude larger than what one expects to achieve with tunneling harmonics generation huillier we have presented a theoretical treatment of the multicolor harmonics generation and wave - mixing in a three - level atomic - molecular system under two - color multiphoton resonant excitation . the coherent part of the dipole spectrum was investigated . with the help of an approximate analytical expression for the dynamic wave function of a three - level atom driven by intense laser fields , we obtained an analytical expression for the time - dependent expectation value of the dipole operator . then the results obtained were applied to the hydrogen atom and homonuclear diatomic molecular ion . the spectrum shows harmonics of the individual waves as well as frequencies with sum / difference components and its harmonics . these peaks have quite large amplitudes . the latter is the result of multiphoton resonant interaction of the system with the driving bichromatic laser radiation due to the mean dipole moment in the stationary states . the cutoff frequency depends linearly on the amplitudes of laser fields . the presence of the second laser field can make easier the implementation of efficient population transfer and harmonic generation as well as allows generation of new frequencies . analytical calculations in the generalized rotating wave approximation @xcite allow an explanation of the obtained spectrum . the numerical simulations are in good agreement with the analytical results . the considered scheme may serve as a promising method for efficient production of multicolor high harmonics . it should be noted that the obtained results can be applied to other systems in diverse domains of the electromagnetic spectrum . a. mcpherson , g. gibson , h. jara , u. johann , t. s. luk , i. a. mcintyre , k. boyer , and c. k. rhodes , j. opt . b * 4 * , 595 ( 1987 ) ; . m. ferray , a. lhuillier , x. f. li , l. a. lompre , g. mainfray and c. manus , j. phys . b * *  21 * * , l31 ( 1988 ) . m. protopapas , c. h. keitel , and p. l. knight , rep . * 60 * , 389 ( 1997 ) ; p. salires a. lhuillier , p. antoine , and m. lewenstein , adv . at . , opt . phys . * 41 * , 83 ( 1999 ) ; t. brabec and f. krausz , rev . phys . * 72 * , 545 ( 2000 ) ; p. agostini and l. f. dimauro , rep . . phys . * 67 * , 813 ( 2004 ) ; m. c. kohler , t. pfeifer , k. z. hatsagortsyan , and c. h. keitel , adv . at . * 61 * , 159 ( 2012 ) .    z. h. kafafi , j. r. lindle , r. g. s. pong , f. j. bartoli , l. j. lingg , and j. milliken , chem . . lett . * 188 * , 492 ( 1992 ) ; p. haljan , t. fortier , p. hawrylak , p. b. corkum , and m. yu . ivanov , laser phys . * 13 * , 452 ( 2003 ) ; g. p. zhang , phys . lett . * 95 * , 047401 ( 2005 ) ; d. golde , t. meier , and s. w. koch , phys . b * 77 * , 075330 ( 2008 ) . p. b. corkum , phys . lett . * 71 * , 1994 ( 1993 ) ; k. j. schafer , b. yang , l. f. dimauro , and k. c. kulander , phys . lett . * 70 * , 1599 ( 1993 ) ; m. lewenstein , ph . balcou , m. yu . ivanov , anne lhuillier , and p. b. corkum , phys . rev . a * 49 * , 2117 ( 1994 ) ; v. averbukh , o. e. alon , and n. moiseyev , phys . a * 64 * , 033411 ( 2001 ) ; c. figueira de morisson faria and i. rotter , phys . rev . a * 66 * , 013402 ( 2002 ) .
we consider harmonics generation and wave - mixing by two - color multiphoton resonant excitation of three - level atoms / molecules in strong laser fields . 
 the coherent part of the spectra corresponding to multicolor harmonics generation is investigated . 
 the obtained analytical results on the basis of generalized rotating wave approximation are in a good agreement with numerical calculations . 
 the results applied to the hydrogen atom and homonuclear diatomic molecular ion show that one can achieve efficient generation of moderately high multicolor harmonics via multiphoton resonant excitation by appropriate laser pulses .