H-atom ionization by elliptically polarized microwave fields: The overlap criterion

PHYSICAL REVIEW A JULY 1997VOLUME 56, NUMBER 1

H-atom ionization by elliptically polarized microwave fields: The overlap criterion

Krzysztof Sacha and Jakub ZakrzewskiInstytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagiellon´ski, ulica Reymonta 4, 30-059 Krako´w, Poland

~Received 11 February 1997!

The threshold for H-atom ionization by elliptically polarized microwave fields is discussed within theclassical-mechanics framework using the Chirikov overlap criterion. It is shown that the trends observed in therecent experiment@M. R. W. Bellermannet al.Phys. Rev. Lett.76, 892~1996!# are qualitatively reproduced bythe theory; the origin of the remaining discrepancy is discussed. Increased stability of some orbits with respectto the perturbation due to the elliptically polarized microwaves has been related to vanishing widths of thecorresponding resonance islands. Analytic Chirikov overlap prediction is compared with results of numericalsimulations.@S1050-2947~97!04207-8#

PACS number~s!: 32.80.Rm, 32.80.Wr, 32.80.Fb, 05.451b

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I. INTRODUCTION

The ionization of hydrogen atoms by linearly polariz~LP! microwaves has been intensively studied for more th20 years~for a recent review see@1#!. Similarly, a consider-able understanding of the effects induced by circularly poized ~CP! radiation is now available@2#. By far much less isknown for the general case of elliptically polarized~EP! mi-crowaves. Until very recently the studies considered onlydependence of the ionization threshold on the microwpolarization in the regime of low frequencies~i.e., when themicrowave frequencyv!vK , wherevK is the Kepler fre-quency corresponding to the initial atomic state!. Here ex-perimental results for alkali-metal atoms@3,4# have been re-produced by classical simulations@5#. The regime of highfrequencies has been partially discussed within the frawork of quantum localization theory using the so-called Kpler map@6#. This approach, however, has been questio~at least for the limiting case of CP microwaves! by Nauen-berg@7#. Similarly, quantum-numerical simulations@2# haveshown the limitation of the Kepler map approach appliedCP microwaves.

In view of the very few results available, the recent stuof Bellermanet al.group@8# may be a cornerstone triggerinthe investigation of the EP case, a situation somewhat mcomplicated than the limiting cases of both the LP andCP cases. In the former case the conservation of the angmomentum projection onto the polarization axisLz makesthe dynamics effectively two dimensional. In the latter siation, while Lz is not conserved, the transformation to tframe rotating with the microwave frequency removesexplicit oscillatory time dependence@2#. Both these simpli-fications are no longer possible in the general EP microwfield and the problem becomes truly multidimensional, pviding new challenges to the theory.

Despite these basic differences between the various pization cases, the experimental results@8# reveal a similaritybetween the ionization threshold behavior as a functionthe scaled frequencyv05v/vK in the vicinity of the pri-mary resonance between the driving field and the Kepmotion, i.e., [email protected],1.4#, provided the microwave fieldamplitude is appropriately rescaled. The experimental dare quite faithfully reproduced by classical fully thre

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dimensional~3D! simulations; moreover, as briefly describeby Bellermannet al. @8#, the polarization independence othe threshold may be understood by a classical analysithe pendulum Hamiltonian, which is valid in the vicinity othe primary resonance.

Bellermannet al. point out, however, that at microwavamplitudes higher than the 10% ionization threshold, evalues that lead to 50% ionization, the polarization of tmicrowave field becomes more important. Specifically,ionization yieldP(F) as a function of the microwave maxmal amplitudeF increases slower for CP than for LP radition @8#. The ionization threshold~i.e., F value leading to agiven, typically chosen to be 10%, fraction of atoms beieffectively ionized @9#! is determined by the behavior othose initial states of a given principal quantum numbern0that are most vulnerable to microwave perturbation~the ini-tial sample of atoms having a well definedn0 is a mixture ofdifferent angular quantum numbers in the experiment@8#!.To understand the atomic response for higher microwamplitudes one should consider the atomic states more rtant to perturbation, which ionize at higherF values anddetermine the behavior ofP(F).

The aim of this paper is to provide an analysis of tionization threshold dependence for the state-specific expment, i.e., assuming that the atoms are prepared in a wdefined initial state (n0 ,l 0 ,m0). Such complete informationis sufficient to understand theP(F) behavior for an arbitraryinitial mixture of states. Since in the studied regime of frquencies the classical simulations reproduce experimedata quite well@8# we limit ourselves here to a classicdynamics picture, determining the thresholds using the Cikov overlap criterion@10#. We shall in fact limit the discus-sion here to the simplified 2D model of an atom, restrictithe motion of the electron to the polarization plane. Suchapproach proved to be quite useful in both classical@11–13#and quantum@2# studies of the ionization in the presenceCP microwaves. Clearly, such a restricted model~assumingthe quantization axis to be perpendicular to the polarizatplane we can treat the initial states ofl 05m0 only! mayprovide qualitative information only. Still, it may be sufficient to understand the sensitivity of the ionization signalthe microwave polarization.

The results obtained below, while providing some und

719 © 1997 The American Physical Society

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720 56KRZYSZTOF SACHA AND JAKUB ZAKRZEWSKI

standing of the existing data@8#, should be of relevance fofuture experiments. In particular, they are applicable to psible experiments with initial states of well-defined quantunumbers. Present experimental techniques@14# allow for thepreparation of both elongated and circular or elliptic statThus it should be possible to prepare quantum stateswell-defined quantum numbers before the atoms entermicrowave cavity.

In Sec. II we shall present the classical dynamics analof the microwave excitation of H atoms. Applying the Chikov overlap criterion, we derive the analytic estimate for tonset of unbounded diffusion in action space for atoms iminated by the microwave field of arbitrary elliptical polaization @Eq. ~2.21!#. Provided the microwave field is turneon for sufficiently long times~atoms cross the cavity withsufficiently low velocity! this estimate coincides with thionization threshold. For shorter interaction times, the emate provides a lower bound for the microwave amplituthat may result in effective excitation and/or ionization. Uing the derived estimate, we discuss in Secs. III and IVprediction for fields of different polarization~the CP resultshas been presented in@13#!. The analytic prediction is compared with the results of numerical simulations. Section Vdevoted to a comparison of our results with the experimof Bellermannet al. @8#. We conclude in Sec. VI.

II. RESONANCE OVERLAP ANALYSIS

Consider a hydrogen atom perturbed by an elliptically plarized microwave field in the dipole approximation. Rstricting the motion of the electron to the polarization playields the Hamiltonian~in atomic units! of the 2D system:

H5H01FH1 , ~2.1!

where

H05px21py

2

22

1

Ax21y2~2.2!

and

H15xcosvt1aysinvt. ~2.3!

F (v) denotes the amplitude~frequency! of the microwaveradiation, whilea determines the degree of ellipticity. Iparticular,a50 corresponds to a linear polarization, whia51 corresponds to a circular polarization of the microwafield. The EP field may be decomposed into two CP warotating in the opposite sense:

H1511a

2~xcosvt1ysinvt !1

12a

2~xcosvt2ysinvt !,

~2.4!

which allows us to express the Hamiltonian in the actioangle variables by directly transposing the expressions vfor a CP case@11,13#,

H~u,J,f,L,t !5H0~J!1FH1, ~2.5!

where

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H0~J!521

2J2, ~2.6!

H1~u,J,f,L,t !511a

2 (n52`

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Vn~J,L !cos~nu1f2vt !

112a

2 (n52`

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Vn~J,L !cos~nu1f1vt !,

~2.7!

with

V0~J,L !523e

2J2, ~2.8!

Vn~J,L !51

nFJn8~ne!1L

JeJn~ne!GJ2 for nÞ0

~2.9!

and

e5A12L2

J2. ~2.10!

In Eqs. ~2.5!–~2.10! J is the principal action~a classicalanalog of the principal quantum numbern), u the corre-sponding angle,L the angular momentum~which is equiva-lent to the angular momentum projection on thez axis for the2D case studied! with the conjugate anglef ~which is theangle between thex axis and the Runge-Lenz vectorAW ).Jn(x) andJn8(x) are the Bessel function and its derivativrespectively, whilee stands for the eccentricity of the initiaorbit.

The pure Coulombic motion is characterized by the Kpler frequency

u5vK5]H0

]J5

1

J3, ~2.11!

while

f5]H0

]L50, ~2.12!

reflects the conservation of the Runge-Lenz vector inCoulomb problem. The primary resonances for the perturmotion satisfy

mu1f2vt5const ~2.13!

for the first part of the perturbation@Eq. ~2.7!# and

ku1f1vt5const ~2.14!

for the second part@proportional to (12a)/2#. Equations~2.11! and ~2.12! indicate that the above resonance contions are satisfied simultaneously, i.e., forJ5Jm5(m/v)1/3

them:1 resonance occurs due to the first part of the perbation while the second resonance condition is metk52m, Jk5Jm . Note that for positivem, the first reso-nance corresponds to the electronic motion rotating aro

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56 721H-ATOM IONIZATION BY ELLIPTICALLY POLARIZED . . .

the nucleus in the same sense as the first CP wave in~2.4!. Then the conditionk52m,0 signifies that the electron rotates in the opposite sense to the second CP wavEq. ~2.4!. We have called the former situation a ‘‘corotaing’’ resonance, while the latter was a ‘‘counterrotatingresonance in the previous analysis of CP microwave fiionization @13#. Let us immediately recall here that for loweccentricity orbits the corotating resonances strongly afthe electronic motion, while when the field rotates in topposite direction from the electron, the atom is more retant to the perturbation@13#. This will have important con-sequences below.

Let us apply now the secular perturbation theory aroungivenm:1 resonance. A canonical transformation to slow

varying variablesJ5J/m and u5mu2vt allows us to av-erage the perturbation over the remaining fast time variaSuch an approach is valid up to first order inF. ExpandingsimultaneouslyH0 around the resonance value of the prin

pal action J5 J2 Jm and leaving the terms quadratic inJyields the approximate resonance Hamiltonian

Hm523

2m2Jm4J21F

11a

2Vm~ Jm ,L !cos~ u1f!

1F12a

2V2m~ Jm ,L !cos~ u2f!. ~2.15!

It is crucial to realize that while the resonance island mot

is described byJ and the conjugate variableu, the remainingvariablesL,f are conserved~approximately, up to the firsorder inF) and characterize the initial orbit~compare alsothe discussion in@8#!. In particular,L, via the eccentricitye, describes the orbit’s shape, whilef ~being an angle between thex axis and the Runge-Lenz vector! describes itsorientation. Of course, on a long time scalef changes lead-ing to the precession of the electronic trajectory. Close to

center of the resonance island, where changes ofu are small,one can perform a further average overf to get an integrablependulum approximation for the effective secular motioHowever, our aim here is quite an opposite one: we areterested in the motion close to the borders of the resonaisland, where the overlap with the nearby resonances occ

Thusf andL may be considered as approximately coserved quantities, i.e., the parameters describing the intrajectory. The effective Hamiltonian,~2.15! may be ex-pressed as

Hm523

2m2Jm4J21FGm~ Jm ,L,f;a!cos~ u2b!,

~2.16!

whereb is a constant~up to the first order inF) and may beincorporated intou by the shift of the origin while

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Gm~ Jm ,L,f;a!5F S 11a

2VmD 2

12 cos2f11a

2Vm

12a

2V2m

1S 12a

2V2mD 2G1/2. ~2.17!

The width of the resonance island is givenDm54m2Jm

2AFGm/3 @11#. Apart from the typical squareroot dependence on the perturbation strength representethe amplitude of the microwave fieldF, all the informationabout the interaction is contained inGm . A comparison ofEqs.~2.15!–~2.17! indicates that the electron is in resonansimultaneously with two waves, with strengths of the inteaction being characterized by (11a)Vm/2 and(12a)V2m/2, respectively. Both waves are coupled to eaother, the coupling strength being determined by the oritation of the electronic motion ellipse as given byf, i.e., theangle between thex axis and the Runge-Lenz vector. Threlevant range of changes inf is the@0,p/2# interval, whichmanifests itself in the form ofGm .

Following Eq. ~2.17!, Gm may be represented geometrcally as the length of the vectorGm5uaW 1bW u, whereaW de-scribes the ‘‘first’’ waveuaW u5(11a)Vm/2 andbW representsthe ‘‘second’’ waveubW u5(12a)V2m/2, the angle betweenboth vectors being 2f. In particular, forf50 ~correspond-ing to the situation when the Runge-Lenz vector is parallethe x axis, i.e., the direction of the large axis of the micrwave polarization ellipse!, the contributions of both resonances add constructively

Gm511a

2Vm1

12a

2V2m . ~2.18!

For f5p/4,

Gm5AF11a

2VmG21F12a

2V2mG2, ~2.19!

while for f5p/2, resonances actdestructively

Gm5U11a

2Vm2

12a

2V2mU. ~2.20!

Therefore, the effective strength of the perturbationstrongly dependent on the orientation of the electroniclipse. For the CP microwave field (a51) the second wavevanishes and the former results@13# are reproduced.

Having prepared the stage, we are now ready to applyChirikov overlap criterion to estimate the ionization thresolds in our system. Before doing so, one should realizethe original system studied, even in two dimensions,five-dimensional phase space due to the explicit time depdence of the Hamiltonian,~2.1!. Thus the Kolmogorov-Arnold-Moser~KAM ! tori do not divide the phase space anthe unbounded Arnold diffusion@15# is present in the systemeven if the resonances do not overlap. The Chirikov overcriterion yields the threshold for the diffusion ‘‘across th

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resonances,’’ whereas the Arnold diffusion ‘‘along the resnances’’ appears for arbitrarily small microwave amplitudHowever, the Arnold diffusion is quite slow@15#; moreover,it slows down considerably in the vicinity of KAM tori@16#.Thus one should not expect it to lead, by itself, to high ecitation ~ionization! in realistic time intervals. Classical ionization~occurring for several hundreds of microwave perioin experiments@8#! may thus be due only to diffusion acrosthe resonances, the threshold for which may be estimatethe Chirikov overlap criterion.

Applying the criterion to two neighboringm:1 andm11:1 resonances, together with the heuristic ‘‘2/3’’ ru@10,15,11,13#, we obtain for the threshold microwave amptude the analytic estimate

Fm,m11~L,f;a!5@~m11!1/32m1/3#2

3@~m11!2/3AGm111m2/3AGm#2v2/3.

~2.21!

The above prediction is the lower bound for the unbounddiffusion starting inJ somewhere on the border between tmth and the (m11)th resonance. The bound dependsboth the properties of the initial electronic orbit~its eccen-tricity e, or ratherL and its orientation in the real spacegiven byf) and the ellipticity of the incoming microwavradiation~as determined bya). Typically in experiments theionization thresholds are higher than the bound determiby the onset of the chaotic motion, even in the frequenrange where quantum localization@6# plays no role and classical and quantum theoretical simulations agree. This isto a finite interaction time in simulations and in experime@17#. On the other hand, close to the onset of the chamotion, the diffusion is slow due to bottlenecks to transposuch as cantori@18#, and very long times are required for thionization to occur.

In the following sections we shall compare the predictiocoming from the analytical estimate for the threshold@Eq.~2.21!# with the results of numerical simulations for differeinitial atomic states and different microwave polarizationThis will allow us to gain some insight into the recent eperimental results@8#. We shall limit the discussion to thexperimental frequency range@8# around the primary resonance. The onset of the ionization will be determined byfield amplitude at whichm51, i.e., the 1:1 resonance, anm52, i.e., the 2:1 resonance, begin to overlap.

III. LINEAR POLARIZATION

We begin our discussion with the frequently studied cof linear polarization. Figure 1 presents the onset of chprediction based on the estimate,~2.21! for different initialorientations of the Kepler ellipse~as indicated by the valueof f in the figure! for different values of the initial angulamomentumL0. The results are rescaled toJ51, so L0P@0,1#. For the initial state with principal actionJ5n0 thehorizontal axis should be multiplied byn0. Similarly, themicrowave amplitude is expressed as a scaled fiF05Fn0

4. The corresponding scaled Chirikov overlap predtion value@Eq. ~2.21!# is denoted asFc in the following.

As expected, the initial orbits elongated parallel to tmicrowave polarization (f50) have the lowest diffusion

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threshold for all initial eccentricities (e5A12L02). For such

oriented orbits, however, the threshold is not very sensito L0 at least for L0,0.8. Surprisingly, the highesteccentricity orbits (L0'0) do not have the lowest diffusionthreshold; the shallow minimum appears aroundL050.7,corresponding to the eccentricitye'L0. For low eccentric-ity, in almost circular states, the threshold rises rapidly,dicating that these states do not contribute to the ionizayield a close to experimentally measured 10% thresholds

The difference between the minimum threshold andvalue for lowerL0 is quite small. This explains why onedimensional models@1,6# may yield a reasonable predictiofor the ionization onset. However, it is apparently not corrto say that the high eccentricity states elongated parallethe polarization axis are most susceptible to the perturbatIt is appropriate to mention here that the recently obtainexact quantum results for the 3D atom in the same frequerange@19# point in the same direction. Clearly, the validitof the one-dimensional model to describe the ionization pcess in linear polarization should be reexamined in a broafrequency range@20#.

For initial orbits inclined with respect to the polarizatioaxis, the minimum of the threshold becomes more pnounced and shifts to lower eccentricity states. For suchbits the Chirikov overlap threshold rises rapidly with eccetricity for e sufficiently large. This is due to both thdestructive effect of two waves rotating in the oppositerections @compare Eq.~2.20!# and the fact that asL0→0,Vm andV2m become equal. In effect, the Chirikov criteriobased on the effective Hamiltonian correct up to first ordeF predicts thatL050 orbits perpendicular to the polarizatioaxis (f5p/2) are stable against the perturbation.

To see the typical average behavior, for orbits of differeorientationsf one may use the resonance pendulum Hamtonian obtained by averaging Eq.~2.16! overf. The thresh-old values obtained from such an approach show also a

FIG. 1. Critical scaled field valueFc @Eq. ~ 2.21!# as a functionof the scaled angular momentumL0 for LP microwave excitation.The thick line gives the result obtained from the Hamiltonian avaged overf; other curves correspond to chosenf values as indi-cated in the figure. The initial configurations elongated parallethe polarization axis are most susceptible to perturbation. All curshow a minimum aroundL0'0.8, thus states of eccentricite'0.6 are the first to ionize.

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nounced minimum forL0'0.8 ~thick line in Fig. 1!. Thisminimum practically coincides with the diffusion threshofor f50, L050 orbits, i.e., for the one-dimensional modof a H atom.

The predictions presented in Fig. 1 are compared withresults of classical simulations as shown in Fig. 2. Forlatter we assumed the flat-top microwave pulse of the fo

f ~ t !5H sin2~pt/2t! for t,t

1 for t,t,T2t

cos2@p~ t2T1t!/2t# for T2t,t,T.~3.1!

While in the EP microwave experiment@8# a half-sine pulseis used, the flat-top pulse better corresponds to LP expments@1#. Typically we assumet525 microwave cycles.

Figure 2~a! shows the ionization probability for orbitelongated parallel to the field (f50) as a function ofL0 fortwo different pulse durationsT. In both cases the behaviopredicted by the overlap formula, Eq.~2.21! and depicted inFig. 1 is reproduced qualitatively. Namely, the ionizatiyield obtained is not sensitive toL0 for L0 small and thensharply drops to zero for almost circular orbits. However,a finite time simulation, the transition to ‘‘excitation resitant’’ states occurs aroundL050.5 and is significantlyshifted with respect to theL050.7 prediction coming fromthe Chirikov criterion. This may be due to a finite interactitime as well as to the fact that in simulations we studyionization yield for a fixed microwave amplitude and not tabsolute threshold. Note that a higher ionization yield corsponds to a lower ionization threshold since, in most cathe ionization yield is classically an increasing functionF. However, the dependence of the yieldP on F may bedependent onL0. Therefore, the 10% threshold, i.e., the aplitude F value when the yield,P is 10%, may show aslightly different dependence onL0 from the absolute threshold. The determination of the real ionization threshold fogiven finite time would require determination of very loionization probabilities. To obtain then a statistically signicant result, a prohibitively large number of trajectories woube necessary. Thus a comparison with the overlap predicmay be only qualitative~also, a finiteT in simulations makesa quantitative comparison virtually impossible, as mentionabove!. For that reason we restrict ourselves mostly to coparisons of ionization yields.

Figure 2~b! shows the numerically obtained yield fostates elongated perpendicular to the polarization axis. Nthat no ionization yield is obtained for high-eccentricity~lowL0) states even for the quite highF0 value assumed in thesimulation. The yield drops for almost circular states,overall behavior being in qualitative agreement with the cresponding curve from Fig. 1.

Figure 2~c! shows the ionization yields for initial samplenot preselected with respect to the ellipse orientation. Tthey correspond to the thick line prediction in Fig. 1 obtainby averaging overf. Again observe the qualitative agrement of the numerical results with the theoretical predictia maximum yield occurs for states of medium eccentricfor all assumed microwave amplitude values. As previou

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FIG. 2. Ionization yieldP as a function of scaled angular momentum of the initial orbitsL0 for different orientations of orbitswith respect to the LP axis.~a! f50, v051.2. Circles correspondto a pulse of durationT5100 microwave cycles andF050.06,while squares correspond toT5300 andF050.036. In both casesthe initial sample consisted of 104 trajectories. Note that all orbitswith L0,0.4 lead to a comparable ionization yield.~b! f5p/2,F050.08, andT5300. Even when a large yield is obtained foorbits of largeL0, orbits elongated perpendicular to the polarizatiaxis do not ionize.~c! Data for initial microcanonical sample~allvalues of f are allowed!: v051.2, T5300, andF050.045, 0.038, and 0.033 for filled circles, open circles, atriangles, respectively. Note that orbits of medium eccentricity gthe largest ionization yield.

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one may observe a slight shift of the optimal yield wirespect to the position of the minimum threshold in the toretical predictions.

IV. ELLIPTICALLY POLARIZED MICROWAVES

When the polarization of microwaves is not linear, tstrength of the interaction between the field and the elecdepends on the mutual orientations of the direction of rotion of the electron on its orbit and the direction of rotatiof the microwave amplitude. The simplest situation occfor CP microwaves when only one (a51) of the two wavesin Eq. ~2.4! is present. For completeness let us recall h@12,13# that if the electron rotates in the same direction asfield it is much more vulnerable to perturbation and tthreshold for the ionization is then much lower than for topposite case. A transformation from one such situationthe other may be realized by changing the sign ofv or,equivalently, that ofL. As discussed in detail in@13#, thisasymmetry is directly related to an asymmetry of valuesVn(J,L) @Eq. ~2.9!# with respect to then→2n ~or, equiva-lently, L→2L) transformation. We referred to the resnances occurring for the field and the electron rotating insame direction as corotating~those are strong resonanceoccurring for vL.0) and the resonances occurring fvL,0 as counterrotating@13#.

For a general EP case, as discussed in Sec. III, the etron is in resonance simultaneously with two CP wavesopposite helicity, i.e., one of these resonances is of the ctating type while the other is of the counterrotating type. TcorrespondingVn(J,L) coupling terms in the Fourier expansion~2.7! are multiplied by (16a)/2 coefficients. For the LPcase, considered in Sec. III,a50 and both terms come withequal weights. Then, as expected, the sign ofv ~or L) playsno role.

In the following we shall assume thataP@0,1# andv.0. Then the electronic orbits withvK'v andL.0 willbe in a strong, corotating resonance with the field, the renant Fourier expansion coefficient being (11a)V1(J,L)/2and the effect of the counter-rotating resonance being sdue to both the (12a)/2 factor and the fact thatV21,V1for L.0 @13#. On the other hand, forL,0, the difference instrength of theV61 (V21.V1) terms may be partially canceled by the (16a)/2 terms. This may lead to a great varieof possible behaviors of the diffusion threshold field valuea function ofL, the orientation of the electronic ellipse witrespect to the main axis of the microwave polarization ellip~as given byf; compare Sec. II!.

Figure 3 presents the thresholds forL.0 in the L0-fplane ~numerical data are presented in scaled variablesbefore!. The threshold surface is symmetric aroundf5p/2and the range up top is plotted for better visualization of thsurface. For all values ofa ~as indicated in the figure!, thelowest threshold value, for a givenL0, is obtained forf50. The sensitivity of the threshold to changes off is thebiggest for high-eccentricity~low L0), strongly elongated or-bits. This is easy to understand: the electron moving overy elongated ellipse~degenerating into a line forL050)‘‘feels’’ mostly the microwave field component polarizedthe direction of its major axis~the Runge-Lenz vector!. Onthe other hand, the circular orbits are insensitive tof. Note

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a

that for all a values~the degree of ellipticity of the micro-wave polarization! orbits of medium eccentricity have thlowest threshold for unbounded diffusion. Following the linf50 for differenta values in Fig. 3 it is easy to notice thathe minimalFc value occurs forL0'0.8 for arbitrarya.

Typically, for L0,0 the thresholds, as predicted by E~2.21!, lay significantly higher than forL0.0, as shown fora few smallf values in Fig. 4. This is expected from thdiscussion presented above; forL0,0, the leading termsV21 andV22 and are multiplied by (12a)/2 thus the widthsof the resonance islands grow slower than forL0.0. Sur-prisingly, however, when one allowsf to take all possiblevalues ~Fig. 5!, a spectacular, double-peak stability regicentered aroundf5p/2 is revealed. The eccentricity of thorbits corresponding to the stability region depends stron

FIG. 3. Chirikov overlap prediction for the onset of unbounddiffusion in the plane spanned byf and the scaled angular momentum L0 ~for L0.0). The values of parametera determining theellipticity of the microwaves are indicated in each panel.

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in-di-


on the degree of the microwave field ellipticitya; comparethe plots in Fig. 5.

The stability has a quite simple origin, although its apearance seems to be, at first glance, quite unexpected.sider again Eq.~2.17!, which gives the width of a givenresonance island, and in particular Eq.~2.20!, correspondingto f5p/2. The contribution of two waves to the resonanisland act destructively there. Since, forL0,0, V2m.Vmfor m.0, there exists ana value ~dependent on the resonance numberm and the angular momentumL0) where thetwo terms cancel, leading to the vanishing width of theland. At the cancellation point, the threshold value obtainfrom the Chirikov criterion is maximal. For the overlap btween 1:1 and 2:1 resonance zones, for a givena, the widthof the 1:1 resonance shrinks to zero at a different valueL0 from that of the 2:1 resonance island~compare Fig. 6!,resulting in the double-peak structure in Fig. 5.

The question remains, however, whether the stabilitygion discussed above is an artifact of the Chirikov overcriterion analysis valid in first order inF. The results of thenumerical simulations, performed for severala values, indi-cate that indeed the stability region exists and its positiowell correlated with the predictions of Eq.~2.21!; compareFig. 7. As could be expected, the Chirikov criterion overetimates the stability of the motion in this case. After all testimation of the resonance island is based on the expanin F in first order. The disappearance of the island in fiorder does not preclude existence of the real island up toorders inF. The primary resonance becomes the secondresonance with an understandably smaller width. It wouldmost interesting to extend the analysis to higher-order renances to get an analytic prediction for the threshold inregion, but this is beyond the scope of the present pape

To complete the presentation of the EP case, we shoFig. 8 the thresholds obtained by averaging the Hamiltonoverf together with the results of the appropriate numerisimulations. Keeping in mind that the high numerical ioniztion yield corresponds to a low threshold value~much lowerthan the value ofF used in the simulation!, i.e., that theoret-

FIG. 4. Chirikov overlap prediction for both positive and negtive L0 in EP microwaves ofa50.7 for orbits whose main axis isclose to the main axis of the polarization ellipse. The values ofare indicated in the figure. Notice that orbits withL0,0 requirehigher microwave amplitudes for the overlap to occur.

-on-

-d

of

-p

is

-

iontallryeo-is

innl-

ical and numerical simulation curves may be roughly copared if the numerical result is reflected with respect tohorizontal axis, the corresponding curves show nice quative agreement. This indicates again that the Chirikov crrion captures the essential features of the ionization threold.

V. IONIZATION THRESHOLDSFOR THE MICROCANONICAL SAMPLE

After studying in detail the onset of diffusive excitatiofor different, well-defined initial states, it is now possiblerelate the predictions based on Eq.~2.21! to the results ofrecent experiment of Bellermannet al. @8#. As mentioned inthe Introduction, in the experiment, atoms entering the cahave a well-defined principal quantum numbern0. However,

FIG. 5. Same as Fig. 3, but forL0,0. Values ofa are indicatedin each panel. Notice double-peak structures, indicating ancreased stability of the orbits for the corresponding initial contions. For a discussion see the text.

ma

foirg

et,ab

llro

alere-hir-

entin-leovn-

-on-ro-re

eld

th

to

Fth

ien-a

ld

mi-


the sample is a mixture of different angular momentustates. The atoms interact with a half-sine-shaped microwpulse~while passing through the cavity! of a duration of 153microwave cycles. Clearly, this is not a situation suitablethe quantitative comparison with the predictions of the Chikov overlap criterion. For the latter, a flat-top, very lonpulse would be more appropriate.

Still, it may be worthwhile to compare qualitatively thexperimental results~as well as 3D classical simulations thaaccording to@8#, reproduce quite well the experimental dat!with the Chirikov overlap estimates. The latter may be otained based on Eq.~2.21! which yields the onset of diffusiveexcitation for a trajectory with a givenL andf. Since in a2D atom allL values are equally probable, similarly to af values, it is a straightforward procedure to find the mic

FIG. 6. Widths of the resonances~denoted in the figure! as afunction of L0 for a50.7 ~full lines! and a50.3 ~dashed lines!.Note that the minima of resonances widths weakly depend onresonance order.

FIG. 7. Ionization yield as a function ofL0 for orbits orientedperpendicular (w5p/2) to the main axis of EP field witha50.7.The flat top pulse of durationT5300 cycles andv051.2 was ap-plied to 5000 initial conditions. Filled circles correspondF050.09, open circles toF050.19. Observe the narrow minimumcorresponding to the stable double-peaked structure in Fig. 5.finite times and due to the statistical errors for a finite samplesplitting into the doublet cannot be observed.

ve

r-

-

-

wave amplitude for which the threshold is reached forgiven s% of the microcanonically distributed sample, whithe remaining part of the sample is still below the corsponding threshold. This will then be interpreted as the Cikov overlap prediction for thes% ionization yield~assum-ing a very long interaction time!.

Let us note that the approach outlined above is differfrom the application of the Chirikov overlap criterion used@8#. The authors of@8# found the threshold field by determining the minimum of the overlap with respect to all possibtrajectories in the microcanonical sample. Since the Chirikcriterion may yield the qualitative estimate of the experimetal threshold only~for the reasons stated above!, the simplerapproach of@8# is fully justified for the threshold determination. However, this approach cannot give predictions ccerning the dependence of the ionization yield on the micwave field value and its polarization, while our proceduyields also this information.

Rather than giving the absolute microwave scaled fi

e

ore

FIG. 8. Comparison of~a! the overlap prediction with~b! resultsof numerical simulations for samples averaged over possible ortations of electronic orbits~i.e., overf); both are represented asfunction of the initialL0. The ellipticity parametera50.263 corre-sponds to experimental data of@8#. In numerical simulationsF050.045, the pulse durationT5300 andv051.2. Note that themaximal yield@in ~b!# nicely corresponds to the minimal threshovalue in ~a!. States of medium eccentricity withLv.0 are mostsusceptible to perturbation and determine the threshold for thecrocanonical initial sample.

ovby-

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n

obeemto-ivfot-Cfoarthio3ity

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ter-itasnsedschza-alfalloreld.

EPthe

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ed

el-veor-a-lyro-hea-ldeex-EP

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values corresponding to a givens% yield for different polar-izations of the microwaves as coming from our Chirikoverlap analysis, we plot their ratio in Fig. 9. DenoteRL(s%)5F0

LP(s%)/F0CP(s%) the ratio of the scaled LP mi

crowave amplitude leading tos% yield according to theChirikov prediction to the corresponding amplitude for tCP field. Similarly setRE(s%)5F0

EP(s%)/F0CP(s%), where

a50.263 as in@8# for the EP microwave field. It has beefound experimentally that RL(10%)51.41 andRE(10%)51.26. For largers%, RL(s%) decreases whileRE(s%) remains roughly constant@8#. Our 2D Chirikovoverlap analysis consistently overestimates bothRL andREwhile rather nicely reproducing the trend withs% ~compareFig. 9!.

For a comparison we show also in Fig. 9 the resultstained from a fully numerical 2D model simulation for thmicrowave pulse of the shape and duration closely resbling the experiment@8#. The numerical results are closerthe experimental data@8# than the Chirikov overlap estimates. That suggests the difference in the chaotic diffustransport above the onset of the chaotic motion occurringLP and CP cases. Specifically sinceRL is higher for the onseof diffusive motion than for a finite time simulation, we conclude that the diffusion just above the threshold for thecase is considerably slower than the diffusion occurringthe LP case. This suggests that the remnants of regulacting as bottlenecks for transport are more important inCP case. The remaining differences between our simulatin two dimensions and the experimental results and thesimulations of@8# may be due to the reduced dimensionalin our approach.

The results of the previous sections make it easy, onother hand, to interpret the behavior ofRL andRE as a func-tion of s%. Recall that the threshold field value strongdepends on the direction of rotation of the electron on

FIG. 9. Ratio of the LP microwave amplitude to the CP micrwave amplitude, both leading tos% ionization yield, denoted asRL , and a similar ratio for EP microwaves, witha50.263, to theCP amplitudeRE as a function ofs%. Full circles~triangles! cor-respond toRL (RE) obtained from the Chirikov overlap criterionwhile open symbols represent the results obtained numericallythe half-sine microwave pulse with a duration ofT5153 micro-wave cycles, frequencyv051.2, and the cutoffnc5110 @8#. Thedifference between the two reflect the dependence of the diffuspeed on the microwave polarization.

-

-

er

PrityensD

e

s

ellipse and the helicity of microwaves. The strongest diffence appears for the CP case@12,13# when the corotatingresonances overlap for much weaker fields than the counrotating ones. In the initial microcanonical sample, half ofcorresponds to initial orbits rotating in the same directionthe field, while the other half rotates in the opposite seand is weakly perturbed by the microwave field which leato ionization of the other half. Obviously, there is no sudifference in the LP case. In effect, to get, say, 30% ionition in a CP case, one must ionize 60% of the corotating hof the sample, while in the LP case the 30% comes fromthe possible trajectories. Stated differently, there are morbits resistant to perturbation in the CP than in the LP fieIn effect, the ionization yield increases withF more slowlyin the CP case than in the LP situation. For the generalfield the situation is somewhat in between two extrema,CP and LP cases.

VI. CONCLUSION

Using the Chirikov overlap criterion we have estimatthe onset of unbounded chaotic diffusion in the phase spof the hydrogen atom irradiated by the microwave fieldelliptic polarization. The obtained analytic prediction@Eq.~2.21!# is valid for an arbitrary polarization of the microwaves and the overlap between anym:1 and (m1):1 reso-nances. On the other hand, the prediction is obtained bysuming the expansion in the first order inF. In the discussionthat follows we concentrate mainly on them51 case forwhich recent experimental results@8# are available.

We discussed first the LP case showing that the Chirikoverlap criterion predictions indicate that not only quasi-ondimensional orbits pointing in the direction of the polariztion axis are most susceptible to ionization. While 1D a2D threshold values may agree, the orbits that contribmostly to ionization may have quite small eccentricity. Thindicates that 1D models may oversimplify the physics ofLP microwave ionization of H atoms. Let us mention heagain that similar conclusions may be reached basedquantum-mechanical results in the same frequency reg@19#. Clearly, additional studies that will compare quantuand classical results in different frequency regionsneeded here especially since the optimal classical approwould utilize the second-order invariant@19# for proper clas-sification of initial orbits. Our results for the CP case@13#indicate, however, that this optimal choice would not affestrongly the conclusions reached here.

Equation~2.21! allows us to study the onset of unbounddiffusion for orbits of different shape~eccentricitye) ori-ented in different ways with respect to the polarizationlipse. The initial orbits most susceptible to perturbation habeen identified. At the same time we have found an imptant family of initial orbits that are resistive to the perturbtion. Numerical simulations indicate that the stability is onapproximate; still, these orbits ionize at much larger micwave amplitudes than the typical orbits. Interestingly, teccentricity of ‘‘stable’’ orbits is dependent on the polariztion parametera. Thus irradiating the microcanonicasample by EP microwaves of sufficiently high amplitumay lead to an almost complete ionization of all atomscept those of the eccentricity related to the polarization of

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aahe

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microwaves. This may serve as a crude way of producatoms in states of a given eccentricity while ‘‘destroyingthe rest of the microcanonical sample.

Finally, we compared the Chirikov overlap predictionsapplied to the microcanonical sample with experimental d@8# providing the explanation of the slower increase of tionization yield withF for elliptically and even slower forcircularly polarized microwaves. This slower increase is dto the existence of orbits less affected by the microwafield, orbits on which the electron rotates in the directiopposite the microwave field vector. We related the disagment between the overlap prediction and finite time simu

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scant.

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tions to the fact that the former is conclusive for very lotimes only. The discrepancy provides information aboutdifference in diffusion times in the systems compared.

ACKNOWLEDGMENTS

We thank Andreas Buchleitner and Dominique Delanfor discussions and Robert Ge¸barowski for supplying hisclassical simulation code. K.S. acknowledges the supporthe Polish Committee of Scientific Research under YouResearcher Project. J.Z. is supported under Project2P03B03810 by the same committee.

sen,

-

rds,za-

nWe

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either a genuine ionization or excitation above some critiprincipal quantum numbernc'110, which leads to subsequeionization by the electric field outside the microwave cavity

@10# B. V. Chirikov, Phys. Rep.52, 265 ~1979!.@11# J. E. Howard, Phys. Rev. A46, 364 ~1992!.@12# R. Gebarowski and J. Zakrzewski, Phys. Rev. A51, 1508

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K. S. Mogensen, and K. Taulbjerg, Phys. Rev. Lett.72, 1612~1994!, and references therein.

@15# A. J. Lichtenberg and M. A. Liberman,Regular and ChaoticDynamics, 2nd ed.~Springer, New York, 1992!.

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in Aspects of Electron-Molecule Scattering and Photoionition, edited by A. Herzenberg, AIP Conf. Proc. No. 204~AIP,New York, 1990!, p. 45, suggest that real 3D atomic motioshould be taken into account for large-scaled frequencies.stress that our results, are obtained nearv/vK'1; see also J.G. Leopold and D. Richards, J. Phys. B19, 1125~1986!.

H-atom ionization by elliptically polarized microwave fields: The overlap criterion

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