modeling harmonic generation by a degenerate two-level atom

162 J. Opt. Soc. Am. B/Vol. 13, No. 1 /January 1996 Burlon et al.

Modeling harmonic generationby a degenerate two-level atom

R. Burlon, G. Ferrante, and C. Leone

Dipartimento di Energetica ed Applicazioni di Fisica,Universita di Palermo, Viale delle Scienze, 90128 Palermo, Italy

P. A. Oleinikov and V. T. Platonenko

International Laser Centre, Moscow State University, Vorobyevy Gory, Moscow 119899, Russia

Received April 6, 1995; revised manuscript received July 17, 1995

An analytical theory of the generation of high-order harmonics of laser radiation has been developed on thebasis of a two-level model atom with degenerate levels. Among other parameters, onset, width, and cutoff ofthe plateau in the harmonic spectrum are obtained in simple analytical forms that connect the basic problemparameters and permit a transparent interpretation of the mechanism underlying the spectrum formation forthis specific case. Selected numerical calculations are reported to corroborate the analytical findings and toinvestigate other harmonic-spectrum features.

PACS numbers: 42.50.Hz, 32.80.Rm, 42.65.Ky. 1996 Optical Society of America

1. INTRODUCTION

In recent years the observation of the generation of high-order radiation harmonics in experiments with gaseousjets has been one of the most interesting and intriguingtopics in the realm of laser atomic physics. This topichas prompted many theoretical efforts aimed at elucidat-ing the basic physics underlying the process. Summariesof this research, both experimental and theoretical, alongwith extensive discussions, may be found in several pa-pers and reviews.1 – 11

Progress in the understanding of the physics under-lying high harmonic generation (HHG) was started byKrause and co-workers,12 who numerically integrated theSchrodinger equation and calculated the HHG spectra,showing that the number of the highest harmonics occurswith good approximation when the corresponding maxi-mum harmonics energy is Emax ø Ip 1 3Up. Here Ip isthe ionization energy of the atom, and Up is the pondero-motive energy imparted to the electron by the laser field.This result is then interpreted with the aid of the follow-ing classical model: at some initial time t0, the electronis in the continuum with zero velocity; then it interactswith the laser field, acquiring the average maximum ki-netic energy Ecin 3Up. As a result, when the electronfalls back into the atom, the maximum energy of the emit-ted photon is Ip 1 3Up.

A typical harmonic spectrum, among the many ob-served in the experiments carried out in recent years,consists of three distinct parts, i.e., an initial decreas-ing part, then a plateau, and, finally, a second decreas-ing part. Thus the research of Krause and co-workershas provided the first explanation of the physical mecha-nism underlying the emission of the highest harmonicsin the plateau and the first explanation of the presenceof a cut-off in the harmonic spectrum. But at the same

0740-3224/96/010162-08$06.00

time it has left unanswered the questions of the plateau’sorigin, onset, and width. A modified version of the clas-sical model13 predicts harmonic emission when the freeelectron receives from the field an energy allowing it toreturn to the point at which it was freed. According tothis model, the maximum energy of the emitted harmon-ics is Ip 1 3.17Up, as 3.17Up is the maximum energy thatan electron created in the continuum with zero velocitymay acquire from the field to go back into the initial posi-tion. In this case, as in the previous one, no explanationof the origin and the characteristic features of the har-monic spectrum is given.

At first glance, according to the classical model de-scribed above, one might expect the plateau to start atthe energy Eon ø Ip and to be independent of the laser-field intensity. This expectation seems to be corrobo-rated by the theoretical papers,11 – 13 which contain reportsof HHG spectra calculated with treatments that may beconsidered the quantum-mechanical analog of the classi-cal model. In contrast, the experimental results given inRef. 6 indicate that, although the HHG spectra exhibita plateau starting at the same harmonics for differentvalues of the laser intensities, the energy of the harmon-ics featuring the plateau onset is smaller than Ip. Fur-thermore, the calculations on hydrogen that are given inRef. 14 seem to indicate that the plateau depends on thelaser intensity’s starting at harmonics of lower order, withthe intensity increasing. In any case, the plateau alwaysrises when the harmonic energy is smaller than Ip.

This short discussion shows that, in spite of the con-siderable success in reproducing the observed data, thereis still no complete understanding of the physical mecha-nism and conditions controlling the origin and the shapeof the harmonic spectrum or of the precise way thatthe basic system (atom plus field) parameters enter theprocess. This circumstance has prompted the writing of

1996 Optical Society of America

Burlon et al. Vol. 13, No. 1 /January 1996 /J. Opt. Soc. Am. B 163

a large number of theoretical treatments of the single-atom HHG response, based on simple models, that arecapable of yielding analytical results and of showing howthe system parameters are controlling the harmonic spec-trum or of somehow providing a physical insight into theprocess. In spite of their simplicity, such model treat-ments appear to display a significant potential in pre-dicting features such as those observed in the experi-ments or that could become of experimental interest intheir own right. (See Ref. 11 for a recent review of mostof such model treatments.) According to many authorsand models, the schematic physical picture accountingfor the HHG spectrum formation is that of an electronembedded in a continuum energy band of width DE sIp 1 3Upd 2 Ip and undergoing transitions between thestates of such a continuum and a deep bound state sep-arated from the continuum by several photons. The en-ergy space between the deep bound state and the con-tinuum is void. The extension of the continuum banddepends on the laser intensity through Up and broadensupward as the field intensity is increased.

It is our aim in this paper to calculate the HHG spec-trum with a mathematical model that bears some simi-larity to the physical picture just described and that, withsome restrictions, is exactly solvable. The model is thatof a two-level atom whose upper level is twofold degen-erate (see Fig. 1). Such a model may mimic a part of amany-level, fully structured real atomic system, especiallywhen few levels are important and when groups of lev-els are energetically very close (quasi-degeneracy). Therestrictions are as follows: (a) the lower level is unaf-fected by the external field, and (b) the dipole moment ofthe transitions w0

0 ! w10, w2

0 is much smaller than thedipole moment of the transition between the two uppersublevels w1

0 and w20. These restrictions, which make

the model exactly solvable in analytical form, correspondto some extent to the physical situation in which an exter-nal field significantly alters the upper part of a spectrum,leaving the lower one unaffected. Note that the assump-tion that the external field does not significantly affectthe ground state is common to practically all the mod-els and theoretical treatments of HHG. However, theproposed degenerate two-level atom model departs in atleast two ways from the physical picture outlined above toexplain the HHG spectrum formation: (a) the two-levelatom model accounts only for bound–bound transitions(which of course are also of interest in their own right),whereas the observed HHG spectra are expected to begenerated by bound–free–bound transitions; and (b) theponderomotive energy Up increases linearly with the fieldintensity I, enlarging the continuum 3Up band upward.In the degenerate two-level atom model, the region of thecontinuum band is mimicked by the linear Stark shift(in a first-order treatment), which increases in propor-tion to the square root of the intensity I 1/2 and movesthe split sublevels symmetrically upward and downwardwith respect to the original position of the unsplit, degen-erate level. These points have to be kept in mind if onewishes to relate the predictions of the degenerate two-level atom to the observed characteristic features of theHHG spectra. As is shown below, the present exactlysolvable model yields in a transparent and neat way thephysical conditions controlling the plateau onset, cutoff,

and width; explicitly links the basic parameters of theatom–laser system; and provides an explanation of theappearance of a plateau in the harmonic spectrum.

In concluding this section we note that, in the context ofHHG, other authors have also considered two-level atommodels with different physical contents and heuristic po-tential. These models are reviewed in Ref. 11. In addi-tion, simple classical models have also proved useful ininvestigating HHG.15

2. THEORETICAL MODELWe begin by taking as a basis the states wj

0s j 0, 1, 2d,for which the diagonal-matrix elements of the dipole mo-ment djj

0 are zero. Then in the basis, where

w0 w00 ,

w1,2 sw10 6 w2

0dys2d1/2,

we have

d11 2d22 d120 ,

d12 0 ,

d01 sd020 1 d01

0dys2d1/2,

d02 sd020 2 d01

0dys2d1/2. (1)

The electric field is assumed to be E sin vt. Takingthe system wave function in the form

C X

aj exps2iEj ty"dwj

for the amplitudes aj, we obtain the equations

Ùaj iVjj aj 1 iVj0 expsiv10tda0 j fi 0 , (2)

Ùa0 X

iV0j exps2iv10tdaj , (3)

where Vij Vij sin vt and Vij dij Ey", with v10 sE1 2 E0dy" being the transition frequency. The energylevels E1 6 "Vjj are shown in Fig. 1.

Below we look for a solution of the system of Eqs. (2)and (3) in the first order of perturbation theory (as per-turbation is taken, the nondiagonal part of the interactionenergy). For instance, at a0s2`d 1 we obtain

Fig. 1. Schematic of a two-level atom with a twofold degeneracyof the upper level: (a) without the external radiation field,(b) with the external radiation field. "V11 is the linearStark shift.


a0 1 ,

aj Z t

2`

iVj0stdexpsiv10tddt exp

"iZ t

t

Vjj st0ddt0

#.

(4)

The integration in expressions like Eqs. (4) may be car-ried out in closed form, provided that the external fieldamplitude changes sufficiently slowly, yieldingZ t

2`

Vjj st0ddt0 ø 2V

vcos vt

and, for arbitrary natural l,Z t

2`

V expfisv10 6 lvdt0gdt0 øV expfisv10 6 lvdtg

isv10 6 lvd.

Then, using the expansion

expsiz cos wd 1X

ikJkszdexpsikwd

and carrying out the integration over t, we express theaverage dipole moment

d Xj ,k

ajpakdjk exps2ivjktd

as

d X

n2`

dn exps2invtd , (5)

where

dn 21/2 sidnSnanE , (6)

Sn Xl1

sv102 2 v2ds21d1

sv102 2 l2v2d

sJl21 1 Jl11dsJl2n 1 Jl1nd

Xl1

slsnd , (7)

with Jk Jksd12Eyhvd being Bessel functions of integerorder and real argument;

a2N11 a 2v01fsd01

0d2 1 sd020d2g

"sv012 2 v2d

, (8a)

the transition linear polarizability; and

a2N 4vd01

0d020

"sv012 2 v2d

. (8b)

The structure of our results clearly shows that thetransition from the lower state to the upper ones takesplace, in general, by means of multiphoton absorption.When E 0, all the quantities Sn with n fi 1 are zero,whereas S61 1. For small-field amplitudes, dn ø En.In systems having a center of inversion, the productd01

0d020d12

0 0, accordingly d2N 0, i.e., only odd har-monics are generated. Below we consider only such sys-tems and use

d010 0 .

[For instance, we can assume that w00, w1

0, and w20 are

the states of a hydrogenic atom with quantum numbers (1,0, 0), (2, 1, 0), and (2, 0, 0), respectively. The states withm 61 are omitted, as we consider linearly polarizedlight.]

In such systems, for arbitrary initial amplitudes, wehave

d2N11 2i2

s21dN S2N11aE

"ja0s2`dj2

2ja1

0s2`dj2 1 ja20s2`dj2

2

#

d2N s21dN

2S2NaEfa1

0s2`da20ps2`d 1 c.c.g ,

where aj0 is the amplitude of the state wj

0.In the incoherent states the average value a01aa

0p iszero, and accordingly the average value d2N is zero too.The condition of validity of the perturbation treatment de-scribed by Eqs. (2) and (3) may be considered an indica-tion of the smallness of the right-hand side of the secondequation displayed in Eqs. (4) as compared with unity.This condition may also be expressed asÉ X

l2`

lv

v10 1 lvJl expsilvtd

É 2

,,

√d12

0

d020

!2

.

Assuming d020 to be sufficiently small, the inequality is

fulfilled for values of the shift

V11 d11E

"

d120E

"(9)

that are as large as, for instance, the transition frequencyv10.

3. ANALYTICAL ESTIMATESFormulas (5)–(7) are the basic results of our model, fromwhich we can proceed to obtain information on the har-monic generation, either by direct numerical evaluationor by considering their limiting behavior. The lattermethod is explored here; in Section 4 we discuss the re-sults of the numerical analysis.

First, we note that the wave functions of the degeneratestates in the presence of the field, completed by the time-dependent parts, are

w1 1

p2

sw10 1 w2

0dexp

√2

i"

E1t

! Xs

s21ds

3 Js

√V11

v

!exps2isvtd , (10a)

w2 1

p2

sw10 1 w2

0dexp

√2

i"

E1t

! Xs

s21ds

3 Js

√2

V11

v

!exps2isvtd , (10b)

where the Bessel functions Js are a measure of the in-tensity of the photon replica E1 1 s"v, i.e., a measure ofthe possibility that the electron will populate the virtualexcited state with energy E1 1 s"v. If V11yv , 1, theBessel function does not exhibit oscillating behavior ver-sus s, and, as a consequence, no plateau may be expectedbecause the virtual states in which the upper energy sub-levels are replicated have a vanishingly small probabilityof being populated. V11yv . 1 is thus a first condition


to be satisfied in obtaining oscillating Bessel functions,yielding intense replicas with comparable intensity andeventually being responsible for the plateau formation.Even more important in understanding the features ex-hibited by a plateau is the relation between the excitationfrequency v10 sE1 2 E0dy" and the ac Stark shift of theexcited level. To discuss this relation we first rewrite thelth term of Sn [Eq. (7)] in a slightly different form, namely:

slsnd ´sldRsl, nd , (11)

with

´sld 2sv10

2 2 v2dsv10

2 2 l2v2d

√lv

V11

!Jl

√V11

v

!, (11a)

Rsl, nd Jl2n

√V11

v

!1 Jl1n

√V11

v

!. (11b)

´sld is seen as the measure of the possibility of excitingthe upper level when the atom absorbs l laser photons,whereas Rsl, nd is seen as the measure of the possibilityof radiating the nth harmonics if the atom has absorbedl photons to get into the upper level. The two terms en-tering Rsl, nd indicate that a given harmonic, in general,may be radiated from two different electron virtual states(from two replicas of the states w1 and w2) and that thepossibility of its being irradiated is proportional to the val-ues of the Bessel functions, which give the weights of thecorresponding replicas. Thus the structure of Eq. (11)emphasizes the two-step nature of the process of har-monic generation: first the electron reaches an excitedstate, which may be replicated many times with differ-ent weights by the exciting field, and then the electronradiates the harmonic while being in one of these virtualstates. It must be noted that in the r space the motionof the excited electron is confined because of the pres-ence of the nucleus, and it is the nucleus, playing the roleof a third body, which may make real what otherwise isonly virtual. Furthermore, the structure of ´sld explic-itly shows the importance of resonant and quasi-resonanttransitions in this kind of process; this feature is exploitedbelow and proves to be crucial for the plateau appearance.For a similar observation, see Ref. 11.

To proceed, we assume that the behavior of Sn [Eq. (7)]is dominated by a single, resonant term for which v10 øl0v. Accordingly, from Eq. (7) we retain only such aterm, namely:

S2N11 X

l

sls2N 1 1d ø sl0 s2N 1 1d

´sl0dRsl0, 2N 1 1d , (12)

where we have replaced n with 2N 1 1.Furthermore, we use known properties of the Bessel

functions appearing in relation (12). Let us consider thecase in which v10 . V11, which, thanks to our resonanceapproximation, also implies that l0 . V11yv. ´sl0d playsthe role of a constant factor in the following considera-tions:

(a) For small N, we expect that the values of the indicesof the Bessel functions are dominated by l0, which islarger than the argument V11yv; accordingly, for the first

small values of N, Jl0 2 s2N 1 1d and Jl0 1 s2N 1 1d willbe rapidly decreasing functions of N. (This decreasingbehavior is expected to account for the initial part of theharmonic spectrum.)

(b) With a further increase in N, the Bessel functionJl2s2N11d changes behavior when

l0 2 s2N 1 1d ø V11yv , (13)

stopping to decrease and starting to oscillate (withN ), whereas the other Bessel function continues to de-crease rapidly because l0 1 s2N 1 1d .. V11yv. Thencondition (13), or

s2N 1 1dv ø l0v 2 V11 ø v10 2 V11 , (14)

marks the end of the decreasing part of the harmonicspectrum and the beginning of a plateau.

(c) Continuing with a further increase in N, we entera domain in which s2N 1 1d . l0, and there are againvalues of N for which

jl0 2 s2N 1 1dj s2N 1 1d 2 l0 ø V11yv . (15)

For still greater values of N, the Bessel function Jl0 2

s2N 1 1d again decreases rapidly [Jl0 1 s2N 1 1d andmaintains its decreasing behavior for all the values of N ].Then condition (15), or

s2N 1 1dv ø V11 1 l0v ø v10 1 V11 , (16)

marks the end of the plateau (the end of the oscillatingbehavior of the Bessel function) and a new decreasing partof the harmonic spectrum. Using relations (12) and (14)we also obtain the plateau width

D ø v10 1 V11 2 sv10 2 V11d ø 2V11 . (17)

A schematic diagram of the physical picture correspond-ing to the formation of the harmonic spectrum with itsplateau characteristics is given in Fig. 2, providing atransparent interpretation of the physics underlying thespectrum formation. The upper level of our model atomis Stark split by the amount 2"V11. All the transitionsfrom inside the band of width 2"V11 back to the groundstate will form the plateau, with the onset correspond-ing to the transitions from the bottom of the band andthe cutoff corresponding to the transitions from the top.The bottom of the Stark shift is at "sv10 2 V11d, whereasthe top is at "sv10 1 V11d. Transitions from outside theStark shift correspond to the decreasing parts of the har-monic spectrum. The band 2"V11 represents a portionof the energy space in which the electron virtual stateshave intensities of comparable value. According to ouranalysis, the formation of a well-defined plateau is theresult of two conditions that must be fulfilled simultane-ously: (a) only one channel (the l0th), or very few, domi-nates the transition, and (b) the radiation field must orig-inate a sufficiently high number of intense virtual statesaround the upper level. The simultaneous fulfillment ofconditions (a) and (b) puts limitations on the inequalityv10 ø l0v . V11 and provides further insight into the


Fig. 2. Connection between the field-modified (Stark-split) en-ergy spectrum of a two-level atom and the expected shape ofthe harmonic spectrum. The upper level is Stark split by theamount 2"V11 2d12E. All the transitions that originate in-side the band of width 2"V11 back to the ground state will formthe plateau, with the onset corresponding to transitions from thebottom of the band and the cutoff corresponding to transitionsfrom the top. The bottom of the band is located at "sv10 2 V11d,the top, at "sv10 1 V11d. Transitions from outside the bandcorrespond to the decreasing parts of the harmonic spectrum.

physics of the process. In fact, if l0 is much larger thanV11yv, the small quasi-resonant denominator in the fac-tor

´sl0d 2sv10

2 2 v2dsv10

2 2 l02v2d

√l0v

V11

!Jl0

√V11

v

!

is compensated by the smallness of the Bessel function Jl,and the l0 term is no longer the leading one. This resultimplies that in the infinite sum of Eq. (7) other terms withl smaller than l0 become as important as, or even moreimportant than, the l0 term. Thus the transition to theupper level becomes a superposition of several differentchannels, which spread the electron population over low-lying virtual energy states, from which the radiation ofintense high-order harmonics is unlikely. Of course, onecan consider the situation of an exact resonance, whichwill single out only one channel as before. However, asstated at the beginning of this section, the exact resonancewill not be enough to produce a plateau if the field is notintense enough (and if V11 is much smaller than v10):few or no intense electron replicas will have comparableintensity and will be available to give rise to the plateau[in mathematical terms, the Bessel function Jl0 2 s2N 1

1dsV11yvd will exhibit oscillating behavior for few or novalues of N ]. The conclusions of this analysis, which isrestricted to the case in which v10 . V11, are that, toyield a well-structured plateau, the transition frequencyand the ac Stark shift must be comparable and that theexternal field must be strong enough to produce intensereplicas inside the Stark shift.

If v10 is instead much larger than V11, our analysispredicts either no plateau or a plateau with a smallerwidth, with a cutoff occurring at a harmonics of lower or-der. We anticipate that the exact numerical evaluationof Eqs. (5)–(7) is in complete agreement with the presentestimates. The analysis of the opposite situation as well,

in which V11 . v10, provides instructive insight into theprocess. However, insofar as we are strictly concernedwith the peculiarities of bound–bound transitions in adegenerate two-level model atom, we are faced with theunusual situation in which the upper level broadens tosuch an extent that its bottom is lower than the initialstate. In this case our assumption on the weak depopu-lation of the initial state requires both additional discus-sion and verification. If, instead, we wish to exploit theinformation yielded by the present model to gain someinsight into the physics of HHG in bound–free transi-tions, then we must keep in mind the differences noted inSection 1, especially the fact that the continuum energyband moves upward only when the field intensity is in-creased and the fact that the bottom of the band (Ip) isthought to remain fixed. If we keep these differences inmind, we can see that no qualitative new information isexpected from the analysis of the case in which V11 . v10

(except that a very large and regular plateau like those re-ported in Ref. 16 may appear). Finally, we observe thatS2N11 is, in general, a sign-changing function of the laserelectric field, through V11, because of the presence of theBessel functions. This result is immediately evident ifwe consider the cases in which the resonant approxima-tion holds well. We point out this behavior of S2N11 be-cause it may be important when summing fields (see thereview given in Ref. 6 and conferences cited therein). Inparticular, the sign-changing character of S2N11 is quali-tatively in agreement with the experimental results re-ported in Ref. 17, in which the author, to explain suchbehavior, assumes that the phases of the dipole momentsd2N11 undergo very rapid changes when the intensity ofthe exciting field is smoothly varied.

4. CALCULATIONSThe numerical estimates, formulas (14), (16), and (17),give complete information on the characteristics and thedynamics of the plateau of the harmonic spectrum asfunctions of the basic problem parameters: the transi-tion frequency sv10d, or the number of laser photons sl0drequired for connecting the ground state to the upper de-generate level; the dipole moment sd12

0d that connects thetwo sublevels of the degenerate level; and the externalelectric field (E), or the Stark shift of the degenerate levelsV11 d12

0Ey"d.From our general results it is also possible to extract

analytically further information on other aspects of theharmonic spectrum, such as the height of the differentharmonics at a given intensity and the dependence ofa given harmonic on the intensity. However, for theseissues, we prefer to take advantage of the exact numericalevaluation of our basic formulas.

The numerical calculations reported below are in-tended, first, to confirm the results of the analysis yieldingformulas (14), (16), and (17); and, second, to complementthe information on the harmonic spectrum concerningthe harmonics intensities and the role of the detuningd v10 2 l0v.

Figures 3 and 4 present calculated results of the exactexpression of Sn

2 versus the harmonic order n 2N 1 1for two values of V11 (25v and 15v) and for four valuesof v10 (two quasi-resonance values 25.1v and 35.1v, and


Fig. 3. Sn2 versus the harmonic order n 2N 1 1. The har-

monic spectra in the quasi-resonant and the off-resonant casesat V11 15 v are compared. Solid curve, v10 25.1v; dottedcurve, v10 25.5v.

Fig. 4. Same as Fig. 3, but with V11 25 v. Solid curve,v10 35.1v; dotted curve, v10 35.5v.

two off-resonance values 25.5v and 35v). Besides con-firming the analytical estimates given above concerningonset, width, and cutoff of the plateau, calculations showthat in the quasi-resonance cases the intensities of theharmonics are, on average, larger by approximately anorder of magnitude in comparison with the correspondingquantities of the off-resonance case. An exception is theinitial decreasing part of the harmonic spectrum, wherethe values of Sn

2 for the two cases are almost indistin-guishable. As far as the oscillatory behavior for the twocases inside the plateau is concerned, no particular re-mark seems in order. What is remarkable is the symme-try of the plateau with respect to its center, which occursat nv ø v10sn 25 in Fig. 3 and n 35 in Fig. 4).

Figure 5 presents calculated results showing the be-havior of Sn

2 of three selected harmonics sn 9, 35, 63dversus V11yv, which amounts to a study of how the in-tensities of the harmonics depend on the electric field.The three harmonics are chosen to be representative ofthe three characteristic parts of the harmonic spectrum.n 35 is inside the plateau, whereas n 9 and n 63are outside but near the plateau. As we observed thatincreasing the electric field enlarges the plateau in thedirections of both smaller and larger n, increasing V11yv,

the harmonics with n 9 and n 63 are expected to fallinside the plateau. Figure 5 shows that (a) increasingthe laser electric field increases the probabilities of all theharmonics (inside and outside the plateau) by several or-ders of magnitude until V11 becomes equal to v10; (b) forV11 $ v10 a regime of saturation and oscillating behavioris entered; (c) Sn

2 for n inside the plateau is an oscillat-ing function, whereas Sn

2 values for n outside the plateauare monotonically growing functions until they enter theplateau. Once inside the plateau, they too become os-cillating. Note that, for the calculated results presentedin Fig. 5, for completeness we have included the domainV11 . v10 as well, whereas for all the other calculationswe have restricted ourselves to V11 , v10. An extendeddiscussion about the domain V11 . v10 is given at theend of Section 3.

In Fig. 6 we report calculated results of the exact ex-pression of Sn

2 versus the harmonic order n 2N 1 1 forV11 5v and for two values of v10 s15.5v and 35.5v).As discussed above, calculations show that, when theStark shift V11 is small and not comparable with the dis-tance between the two energy levels sv10d, no plateau ap-

Fig. 5. Behavior of three selected harmonics sn 9, 35, 63)versus V11yv, a quantity dependent linearly on the laser electricfield E. v10 35.5v. The arrow marks the point at whichv10 ø V11.

Fig. 6. Sn2 versus the harmonic order n 2N 1 1, with

V11 5v. Solid curve, v10 15.5v; dotted curve, v10 35.5v.The arrow marks the point at which n sv10 1 V11dyv.


Fig. 7. Sn2 versus the harmonic order n 2N 1 1, with

V11 15v. Upper solid curve, v10 25.5v; dotted curve,v10 35.5v; lower solid curve, v10 45.5v. The arrows markthe point at which n sv10 1 V11dyv.

pears (dashed line in Fig. 2). The absence of a plateau,in this case, corresponds to v10yV11 ø 7. The calcula-tions used to obtain the results shown in Fig. 7 considerin more detail the effect of the different values of the ra-tio v10yV11; moreover, a larger value of V11 is considered.In particular, V11yv 15 and v10yv 25.5, 35.5, 45.5.We obtain the ratios v10yV11 1.7, 2.37, 3.0. With thesedata we expect to cover the intermediate regime, goingfrom a well-structured plateau (where v10yV11 1.7) tosituations exhibiting plateaus of lesser and lesser extent.The arrows on the curves mark the order of the harmon-ics having energy E v10 1 V11. In the cases in whichv10yV11 2.37, 3.0, the plateaus have a smaller widthand are cut off at values of n smaller than sv10 1 V11dyv;nevertheless, the harmonic spectrum displays the signa-ture that the value sv10 1 V11dyv marks the separationof two different regimes.

5. CONCLUSIONSWe have given the analytical and numerical treatment ofthe generation of high-order harmomics of laser radiation,using the mathematical model of a two-level atom with adegenerate upper level. Excitation of the upper level oc-curs by multiphoton absorption. Onset, width, and cutoffof the plateau in the harmonic spectrum are obtained insimple analytic forms, connecting the basic problem pa-rameters and permitting a clear physical picture of themechanism underlying the plateau formation. The up-per level is Stark split by an amount 2"V11 2d12

0E.All the transitions from inside the band of width 2"V11

back to the ground state will form the plateau, with theonset corresponding to transitions from the bottom of theband and the cutoff corresponding to transitions from thetop. Thus the plateau is an image of the ac Stark split-ting. Such a picture corresponds to the case in whichthe excitation of the upper level occurs through a sin-gle, quasi-resonant transition and in which the field isstrong enough to give an ac Stark shift smaller than butcomparable with the transition energy. When the aboveconditions are not met, the plateau loses its well-definedstructure, and the cutoff occurs at harmonics of smallerorder. The plateau itself may not even appear.

The peculiarity of the present model is that in the ex-cited state the electron populates a band of width 2"V11,from which it may radiate harmonics when decaying backto a deep ground state. Furthermore, the fact that theelectron is confined implies indirectly a motion aroundthe atomic nucleus, which plays the necessary role of thethird body required for emission of radiation. Thusthe model basically contains all the physical ingredi-ents present either in bound–free quantum-mechanicalmodels or in the simple classical models that considerthe motion of a classical electron under the action oftwo forces. Qualitatively, the contacting points are evenmore significant if a few distinctions are made: (1) Inour model the maximum energy that the field may impartto the electron in the excited state is "V11 (proportional toI 1/2), whereas in the bound–free models the same quan-tity is cUp (proportional to Iyv2d, with c being a constantwith a value of approximately 3. (2) In our model theenergy quantity connecting the ground state to a band ofwidth 2"V11 is the excitation energy "v10, whereas in thebound–free model it is the ionization energy Ip. (3) Inour model, increasing the field splits the excited level E1

into sublevels that move upward and downward in the en-ergy space with respect to E1, whereas in the bound–freepicture the bottom of the band of width cUp is expectedto remain fixed at Ip. If the above distinctions are keptin mind and the necessary identifications between thebound–bound and the bound–free cases are made, for awell-structured plateau our cutoff condition "sv10 1 V11dchanges to Ip 1 cUp; the plateau onset should be local-ized around Ip and should be independent of the intensity;and accordingly the plateau width should be øcUp. Thedifferent behavior of E1 and Ip versus the intensity con-stitutes another significant difference in the form of theplateau: in the bound–bound case the original level E1 islocalized in the middle of the band 2"V11; thus the repli-cas start from that point, going symmetrically to higherand lower values. This effect gives a true, horizontalplateau, as noted in our discussion of Figs. 3 and 4. Inthe bound–free case the original level Ip is localized at thebottom of the band cUp, and the transitions responsiblefor the plateau formation involve replicas correspondingonly to higher-energy values as compared with Ip. Thiseffect is expected to give a descending plateau.

In conclusion, in the authors’ opinion, the merit of theanalytical model discussed in this paper is that it providesdetailed insight into many aspects of the mechanism ofHHG from atoms, which, with necessary distinctions andidentifications, may prove useful in understanding theexperimental observations as well.

ACKNOWLEDGMENTSThis research has been carried out in the framework ofthe General Agreement on Scientific Cooperation betweenLomonosov Moscow State University (Russia) and theUniversity of Palermo (Italy). Additional support hasbeen provided by the University of Palermo ComputationCentre, the Italian Ministry of University and ScientificResearches, the National Group of Structure of Matter ofthe Italian National Research Council, and the SicilianRegional Committee for Nuclear and Structure of MatterResearches.


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modeling harmonic generation by a degenerate two-level atom

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