PHYSICAL REVIEW A NOVEMBER 1998VOLUME 58, NUMBER 5

H atom in elliptically polarized microwaves: Semiclassical versus quantum resonant dynamics

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

~Received 10 July 1998!

The dynamics of Rydberg states of atomic hydrogen illuminated by resonant elliptically polarized micro-waves is investigated both semiclassically and quantum mechanically in a simplified two-dimensional model ofan atom. Semiclassical predictions for quasienergies of the system are found to be in very good agreement withexact quantum data enabling a classification of possible types of motion and their dynamics with the change ofthe ellipticity of the microwaves. Particular attention is paid to the dynamics of the nonspreading wave packetstates, which are found to exist for an arbitrary microwave polarization.@S1050-2947~98!07711-7#

PACS number~s!: 05.45.1b, 32.80.Rm, 42.50.Hz

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I. INTRODUCTION

Consider a hydrogen atom, initially in a high Rydbestate with the principal quantum numbern0 illuminated bythe microwave field of a frequencyv close to the frequencyof the unperturbed Kepler motionvK51/n0

3. Quantum me-chanically speaking, a resonant periodic field coupstrongly severaln states due to almost resonant transition→n85n61 for n close ton0 . Since the driving field isperiodic, applying the Floquet theorem@1# one may findeigenstates of the atom-field system~the so-called Floquet odressed@2# states!. The eigenenergies are then referred toquasienergies of the system and are defined modulo\v. TheFloquet states, periodic in time, may be viewed as lincombinations of unperturbed system eigenstates. Whileconstruction of Floquet states is possible for both nonrenant and resonant driving, in the latter case they may hquite unusual properties, especially in a semiclassical lim

Classically a resonance between the driving frequeand the frequency of the unperturbed motion leads tstrong perturbation of the system and a creation of a stisland in the phase space centered on a periodic orbit offrequencyv. The motion in the island is locked to the mcrowave frequency due to the nonlinear resonance@3#. Semi-classically one then expects that the corresponding Flotime-periodic state will follow a classical trajectory~in thevicinity of the periodic orbit!, i.e., form a wave packet thawill not disperse in time.

States localized in the resonance island for such a periperturbation have been first considered more than 20 yago @4# and details of their semiclassical constructionsome one-dimensional~1D! model systems have been anlyzed @5,6#. The wave-packet character of the time evolutiof individual Floquet states has been realized only quitecently @7# for a hydrogen atom driven by linearly polarizemicrowaves. Independently, it has been shown that Gauswave packets may propagate almost without dispersion acircular periodic orbits in hydrogen atoms driven by circlarly polarized fields@8#. The fact that the harmonic approxmation implied in@8# and resulting the Gaussian wave packform is not a necessary condition for nonspreading properhave been discussed in@9,10# where it was shown that thharmonic expansion provides a good approximation for ex

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Floquet states of the system~which by definition, being timeperiodic, do not spread on a long time scale!.

The fact that one may construct in a nonlinear systwave packets that do not spread induced a flurry of activin the field. Some@11–16# concentrated on modifying thepotential so as to make the harmonic approximation as gas possible aiming at the construction of Gaussian nspreading wave packets. Claiming that the anharmonicrections are big for a hydrogen atom driven by circulapolarized fields those authors added a magnetic field perdicular to the microwave polarization plane@11–16#. Thishelps to minimize the unharmonicitiesin this plane, leaving,however, unaffected the terms along the magnetic field aThus the Gaussian wave packets still remain an approxition to the real dynamics and must disperse~although veryslowly in time!. Another approach aimed at optimizing thcoordinate system to the symmetry of the problem is given@17#.

As discussed by us elsewhere@9,10,18–21# much morefruitful is another approach, already outlined above. Namewe define the nonspreading wave packet as a single Flostate~for which the Gaussian packet may be merely anproximation!. Then a localization of the wave packet in thvicinity of a stable fixed point is assured by the correspodence principle provided the size of the surrounding islandthe phase space is comparable to\. One may construct anapproximate resonance Hamiltonian in the vicinity of tisland whose eigenstates will approximate well those Floqstates that are localized in the vicinity of the island~seebelow!. Simultaneously, time periodicity of Floquet eigestates assures that the exact Floquet states will not disp

The existence of such wave-packet Floquet eigensthas been proven by an exact numerical diagonalization ofproblem for both linear polarization~LP! @7# and circularpolarization~CP! @9,10# of the microwaves. To allow theirdetection one should consider the ways of populating efftively such states. For a CP, where the direct optical exction from a weakly perturbed low-lying state is impossib~since the wave packet is built from predominantly circuatomic states unaccessible from low-lying states due topole selection rules! one should first prepare the atomic cicular state@22,23# and then switch on the microwaves sufciently fast@9#. It has been shown that the wave-packet stamay be populated in this way with about 90% efficien

3974 ©1998 The American Physical Society

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PRA 58 3975H ATOM IN ELLIPTICALLY POLARIZE D . . .

@24#. For a LP case, an addition of a static electric fieallows one to control the trajectory of the wave packet@18#,in particular wave packets moving along elongated, lowgular momentum trajectories may be created. Such wpackets may be accessible to a direct excitation from lolying states.

The next important question is the possible mechanismthe detection of these states. At least three possible wsuggest themselves. Two of them utilize the residual deof wave packets, either via the spontaneous emission~treatedboth for LP @25# and CP @26,21#! or the ionization. Theformer may not be efficient, since, at least for wave packmoving on circular trajectories, the corresponding spontaous emission rates are quite small@21#. In the ionizationexperiment, the population of the wave-packet state maydetected by a strong decrease in the ionization yield~sinceionization rates of wave-packet states are typically vsmall @9,10#!. On the other hand, these rates fluctustrongly ~the mechanism of their ionization, via ‘‘chaos asisted tunneling’’ is discussed in detail elsewhere@19,20#!—this may make their detection in the ionization yield quambiguous.

By far the most promising method is the Floquet spectrcopy @27#, i.e., probing, by a second weak microwave fiethe structure of Floquet~dressed@2# by microwaves! states.To this end a precise estimation of the quasienergieswave-packet states is necessary. An exact diagonalizatiothe problem gives all the Floquet states and a time consing inspection of individual eigenvectors is necessaryidentify the wave-packet states. This process may be omized by calculating properties of matrix elements of apppriately chosen operators but certainly it is desirable to hgood semiclassical predictions for the quasienergies. ForCP case those are given, to a very good accuracy fromharmonic approximation Hamiltonian@8–10#, this approachbeing, however, restricted to this particular system.

For a general case of periodically driven systems therno simple unitary transformation that removes the timependence~as it is in the CP case! and the correct approachto use approximate resonant Hamiltonians. The semiclasquantization of such a Hamiltonian gives not only the goestimate for wave-packet states but allows for the classifition of resonant states for systems of more than one dimsion. Recently, using such an approach we could discussresonant dynamics in a realistic three-dimensional~3D!model of a hydrogen atom in the LP case@28#. Similarly, wehave discussed the control of wave-packet trajectories uan additional static electric field@18#.

Until now the discussion of nonspreading wave-pacstates has been restricted to linear and circular polarizacases only. The aim of this paper is to treat a resonantnamics of a hydrogen atom, both semiclassically and qutum mechanically in a general case of an elliptical polarition ~EP!. Apart from generalizing the notion ononspreading wave packets to an arbitrary EP, we discthe full dynamics of quasienergies as a function of the elticity of the microwaves for a resonant case, being stimulaby recent ionization experiments@29#. Unfortunately, the ex-periments do not allow for a full selection of the initial staof an atom~states with all possible angular momenta a

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simultaneously excited! but this situation may improve in thenear future.

The EP case is highly nontrivial. For LP microwaves tconservation of the angular momentum projection ontopolarization axisLz makes the dynamics effectively two dmensional. For the CP case whileLz is not conserved, thetransformation to the frame rotating with the microwave frquency removes the explicit oscillatory time dependen~see, e.g.,@30–33#!. Both these simplifications are no longepossible in the general EP microwave field. In effect, texact quantum diagonalization approach for the part ofspectrum corresponding to the strongly perturbed atomicdberg spectrum would require very big computer memoLet us mention also that in the effective 2D LP case, the sof the art computations@7# consider initial atomic states withthe principal quantum number of the order of 20.

For that reason we shall consider not the realistic fullymodel of an atom but rather the restricted 2D model in whthe electronic motion is restricted to the polarization plaStudy of such simplified models has been most successfuthe past both for LP~where one-dimensional model has bea main source of quantum results for a long time@34,35,6#!and in CP@30–33# where also the 2D, polarization planrestricted model has been utilized. An additional argumfavoring the 2D model comes from our classical studydynamics in EP microwaves@36,37# where comparison of2D @36# and 3D@37# analysis shows the similarity of physcal phenomena in both cases. Simply put the perturbatiomost effective if the polarization plane coincides with tplane of Kepler motion.

II. SEMICLASSICAL VERSUS QUANTUM APPROACHES

The Hamiltonian of the hydrogen 2D model atom drivby an elliptically polarized electromagnetic field reads in tdipole approximation and in the length gauge~in atomicunits!

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wherer 5Ax21y2 while F andv denote the amplitude anthe frequency of the microwave field, respectively.a definesthe ellipticity of the microwaves witha50 (a51) corre-sponding to a LP~CP! limiting case.

Using the Floquet theorem the solution of the quantproblem is equivalent to diagonalizing the Floquet Hamtonian

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]t Dcn5HFcn5Encn ~2.2!

with En being the quasienergies whilecn time-periodic Flo-quet eigenstates.

The details of the numerical method are described inAppendix. In short, the calculations proceed by expressthe Floquet eigenvalue equation in the scaled semiparabvariables

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3976 PRA 58KRZYSZTOF SACHA AND JAKUB ZAKRZEWSKI

where L is an arbitrary scaling factor. This allows oneremove the Coulomb singularity and cast the Schro¨dingerequation into the generalized eigenvalue problempolynomial-like operators. Standard harmonic oscillator cation and annihilation operators allow for simple evaluatof matrix elements. The approach closely resembles thathe 2D atom in the CP field@33# except that the explicit timedependence is treated by the Fourier expansion. It is wstressing that using the complex scaling parameterL oneeffectively realizes the complex rotation in the system tenables the exact treatment of the coupling to the continu~ionization!; for details see@7,33# for LP and CP cases, respectively. In this paper, however, we shall concentratethe Floquet level dynamics only for the sake of the compson with the semiclassics. The analysis of the ionization pnomenon is left for a future publication.

The semiclassical quantization of resonant dynamclosely resembles the similar procedure applied by uscently for the LP case@18,28#. That in turn originates from ageneral prescription for EBK quantization of Floquet spec@38#.

Starting with the Hamiltonian~2.1! we remove the ex-plicit time dependence by going to the extended phase s@3#. Defining the momentumpt conjugate tot ~time! variablewe get the new Hamiltonian

H5H1pt , ~2.4!

which is conserved during the motion. The quasienergiethe system will be then the quantized values ofH.

As the next step we express the Hamiltonian in actiangle variables of the unperturbed Coulomb problem@36#.For the 2D model atom those are, e.g., the canonically cjugate pairs (J,u) and (L,f). J is the principal action~cor-responding to the principal quantum number,n0). The cor-responding angleu determines the position of the electron oits elliptic trajectory.L is the angular momentum~equal toLz for the 2D motion in thex-y plane! while f is the con-jugate angle~the angle between the Runge-Lenz vector athe x axis, i.e., the main axis of the polarization ellipse!.

We shall consider below the case of the resonant drivi.e., when the frequency of the Kepler motionvK51/J3 isclose to the microwave driving frequencyv. Applying thesecular perturbation theory@3# to average over the nonresonant terms one obtains an approximate resonant Hamilto~in the frame rotating together with the electron! of the form

Hr521

2J22vJ1FG~L,f;a!cos~ u2d!1 pt , ~2.5!

where u5u2vt while pt5pt1vJ. Hr yields the pendu-lumlike principal action motion with the strength and thequilibrium position determined byG(L,f;a) and d5d(L,f;a), respectively. The pendulum Hamiltonian is otained by additionally expandingHr around the center of theresonance island given byJ5v21/3 up to second order inDJ5J2v21/3 but we do not apply this expansion. BothGandd depend on the initial shape and orientation of the eltronic ellipse~via L,f) as well as on the ellipticity of mi-crowaves,a, and are given by@36,39#

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~12a!V211~11a!V1tan f, ~2.7!

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V61~J,L !5v22/3FJ 18~e!6A12e2

eJ1~e!G . ~2.8!

J1(x) andJ 18(x) denote the ordinary Bessel function and

derivative, respectively, whilee5A12L2/J25A12L2v2/3

stands for an eccentricity of an electronic ellipse. For copleteness let us mention that a similar approximation onpurely quantum level, leading to Mathieu equation has bperformed for the CP case only in@40#.

The semiclassical quantization of Eq.~2.5! is straightfor-ward and follows closely the procedure described in deelsewhere@28# for arbitrary m:1 resonance. For 1:1 resonance considered here, the trivial quantization ofpt , explor-ing the time periodicity of the system, yields additive termkv to quasienergies~different values ofk correspond to dif-ferent Floquet zones!. As discussed first in@41#, see also@28#, the orbital motion in (J,u) variables~along the per-turbed Kepler ellipse! is much faster than the modification othe ellipse shape and its movement~precession! as describedby the motion in (L,f). Thus the slow and fast motions mabe adiabatically decoupled. Making Born-Oppenheimerproximation and using standard WKB rules, the fast (J,u)motion is quantized taking the Maslov indexn52 ~corre-sponding to librations, i.e., we quantize states inside the renance island!. Being interested in resonantly localized statwe shall consider later the ground state of the radial motionly. For the slow angular (L,f) motion we take the Maslovindex m50 or 2 for a rotational or librational motion, respectively.

Similarly, as in the LP microwaves@28#, it is easier toquantize first the slow motion generated by constant valof G(L,f;a) and later treat the fast motion. Such a procdure is justified since quantizing the fast motion one taG(L,f;a) as a constant quantity and thus the order ofquantization does not matter.

The existence of the resonance island in (J,u) space en-sures that the radial motion is localized. So the remainanalysis should concern the angular (L,f) motion, whichreflects the slow evolution of the Kepler ellipse. The struture of the (L,f) space influences values of quasienergieswell as the structure of corresponding semiclassical eigstates. Trajectories in the (L,f) space are determined bconstant values ofG also responsible for the size of the resnance island in (J,u) space~recall that the island’s size isdetermined approximately byAFG).

Before presenting the results let us define scaled vables, typically used as a convenient parametrization ofmicrowave ionization problem~since the dynamics scale

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PRA 58 3977H ATOM IN ELLIPTICALLY POLARIZE D . . .

FIG. 1. Two-dimensional hydrogen atom illuminated by resonant,v051, elliptically polarized microwaves. The effective scaled pturbationG05G/(n011/2)2 is plotted as a function of the scaled angular momentum,L0 , and the angle,f, between the Runge-Lenz vectoand the main axis of the polarization ellipse—upper row. Bottom row shows equipotential curves of the angular partG of the HamiltonianHr , Eq. ~2.5!, representing the slow evolution of the Kepler ellipse. The curves correspond to semiclassical states originating fn0521 hydrogenic manifold. Columns correspond to the different ellipticity of the microwavesa50.1 ~left! anda50.6 ~right!.

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classically@42#!. The common choice is to link the scalingthe initial energy of the electron@35#. For an unperturbed 2Dhydrogen atom the eigenenergies are given byEn521/2(n11/2)2 with n being non-negative integer. This allows usdefine scaled variables via the principal quantum numbethe initial staten0 via

v05v~n011/2!3, ~2.9!

F05F~n011/2!4, ~2.10!

L05L/~n011/2!. ~2.11!

In particular note that the scaled angular momentumL0 maytake values in the@21,1# interval with extremal values corresponding to circular orbits on which the electron movestwo opposite directions.

III. RESULTS AND DISCUSSION

To compare the semiclassical predictions to the exquantum calculations, we consider then0521 manifold ofour 2D model atom. This value is a compromise betweenrequirement to be in the semiclassical, largen0 regime andcomputer memory limitations~size of the Floquet matrix tobe diagonalized!. For resonant driving we take the micro

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wave frequency to bev5vK51/(n011/2)3, (v051).In Fig. 1 we present values ofG as a function of the scaled

angular momentumL0 and thef angle for two differentvalues of the degree of the field ellipticity given bya. Equi-potential curves ofG corresponding to semiclassical statoriginating fromn0521 hydrogenic manifold are also showin the figure. Due to the specific form of the resonanHamiltonian~2.5! those lines areindependentof the micro-wave amplitudeF.

For a50 the (L,f) space is symmetric with respect ttheL050 axis since, in the LP case, dynamics is not affecby the direction of the rotation of an electron. The left coumn in Fig. 1 presents the results fora50.1, i.e., the casevery close to the LP problem. Note the presence of fostable fixed points: two of them,L0'0 andf5p/2, 3p/2,correspond to almost straight line orbits oriented perpendilarly to the main axis of the polarization ellipse, the othtwo fixed points correspond to circular orbits,uL0u'1 ~forsuch orbitsf is a dummy variable!, with an electron rotatingin the same or opposite direction to the direction of thetation of the field vector. In the vicinity of these fixed pointhe electronic motion is ‘‘shape’’ localized@since the motionremains in the vicinity of a given (L0 ,f) point the eccen-

tricity of the orbit,e5A12L02, as well as its orientation with

respect to the polarization ellipse, given byf is approxi-

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3978 PRA 58KRZYSZTOF SACHA AND JAKUB ZAKRZEWSKI

mately preserved#. Whether the motion is well localized inits radial motion in (J,u) variables, i.e., along the ellipsedepends on the size of the resonance island, given byAFG asmentioned above. If the island in (J,u) space is sufficientlylarge ~comparable in size to\ or larger! the quantum statewill be localized in the island. On the contrary, a too smresonance island cannot lead to localization in (J,u) space,and possible quantum states will spread over the wholelipse ~then also the semiclassical approach used becomeviously inadequate!. Thus to observe interesting, nonspreaing wave packets the resonance island must be large eno

In this respect, two fixed points corresponding for smalato almost circular orbits,uL0u'1, lie at the maxima ofG~compare the left top panel in the figure! and for sufficientlylarge F may enable a strong radial localization. The othtwo stable fixed points, corresponding to elongated orbitsat local minima ofG. There exist also two unstable fixepoints aroundL0'0 andf50,p; they form the origin of theseparatrix dividing the space.

Whena increases the fixed points situated initially arouL0'0 and f5p/2, 3p/2 move in the direction of greatenegative values ofL0 ~see the right column in Fig. 1!, thus,the eccentricity of the corresponding orbits decreases.circular orbits do not change the shape but the resonaisland in (J,u) space associated withL0'1 becomes largewhile that associated withL0'21 becomes smaller as reflected by the greater and smaller values ofG, respectively.Note also that the islands in the (L,f) space containing li-brational states shrink with the increase ofa, hence, they cansupport fewer and fewer semiclassical states—whilea in-creases librational states vault over the separatrices andcome rotational. In the limit ofa51 the islands disappeaand there exist only rotational states.

Consider the level dynamics corresponding to the chaof a. Figure 2 shows the level dynamics for the groupstates originating from then0521 hydrogenic manifold. Wetake the scaled field amplitudeF050.03, which for the CPcase corresponds already to a significant ionization yield@33#and may be, therefore, considered a quite large value. Fo

FIG. 2. Two-dimensional hydrogen atom driven by resonav051, elliptically polarized microwaves. Level dynamics, vsa~i.e., the degree of the field ellipticity!, of the semiclassical quasienergies~full lines! of the states originating from then0521 hydro-genic manifold forF050.03 compared with the exact quantum rsults ~dotted lines!.

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LP case, i.e.,a50, all states are doubly degenerate. Thosethem that correspond to rotational states in the (L,f) spaceare degenerate because the change of the rotation of antron does not affect dynamics in the LP problem@the (L,f)space is symmetric with respect to theL050 axis#. The re-maining librational states are degenerate as the two islaaround L050,f5p/2 and L050,f53p/2, are identicaland support identical states. Of course the degeneraciesbe removed due to tunneling processes, for instance,wave packets localized on circular orbits~the highest levels!rotate in the opposite directions and belong to distinct seclassical states but quantum mechanical states are themetric or antisymmetric linear combination of them. Thtwo wave packets propagating on a circular orbit in the oposite directions correspond to a single eigenstate~thesewave packets were already discussed in@43# for the limitingLP case only!. Similarly tunneling effects affect the librational states, i.e., quantum states are the linear combinatof solutions in each island.

An increase ofa removes the semiclassical degeneracyrotational states as one can see in Fig. 2—observe splittof upper levels in the manifold. It corresponds to the broksymmetry with respect to change of a direction of rotationan electron. Quasienergies of states corresponding to thetion in the opposite sense to the rotation of the field vectoi.e., corresponding to negative values ofL0 , move downwhile those with positiveL0 go up. It is a consequence of thbehavior ofG, i.e., the greaterG the greater the quasienergvalue, see Fig. 1. Observe also that the degeneracy olibrational states~lower levels in Fig. 2! is not immediatelyremoved after a slight change of the field ellipticity froma50 but is removed successively during an increase ofa.This reflects the shrinking of the corresponding islands~thesituation mentioned above!, which causes the librationastates to vault over the separatrix and become rotational.levels with the smallest energy difference correspond tolibrational and rotational orbits closest to the separatrix. Tnarrowing of the level spacing in their vicinity is just a cosequence of the slowing down of the classical motion@44#.In the limit of a51 there is no degeneracy in the manifol

Note that the energy splitting of the manifold is the largein CP case while the smallest for LP. It is associated withcorresponding strength of the perturbation and simplypresses the dependence ofG on a.

Exact quantum results coming from the numerical diagnalization of the Floquet Hamiltonian are presented alsoFig. 2. Among the multitude of Floquet states appearingthe same energy range only those with the largest overlathe initial manifold are plotted for clarity. One can see ththe agreement with the semiclassical predictions is very gexcept in the region of broad avoided crossings~with otherlevels—partners in the crossing eliminated by the overselection!, which appear in the upper part of the figure. Tsemiclassical method does not take into account the intetion of the consideredn0521 manifold with other manifolds~the method describes only a single resonant manifold!, thussuch avoided crossings have no chance to appear insemiclassical calculations. For a more accurate comparwe plotted, in Fig. 3, quasienergies for two different valuof a separately. The agreement between the semiclasand quantum results is very good for high-lying leve

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PRA 58 3979H ATOM IN ELLIPTICALLY POLARIZE D . . .

~greater values ofG! while for lower levels some differenceappear as expected due to small size of the resonance is

In the limiting casea51, for the CP case, another analsis of resonant dynamics is possible, namely, a harmoapproximation around a stable equilibrium point in the roting frame@8# ~corresponding to the expansion about a stafixed pointL051, f arbitrary in our picture!. It is interestingto compare both approaches. The present semiclassical qtization of the resonance Hamiltonian is certainly a mogeneral approach valid for a general EP case and forwhole manifold. The harmonic expansion is limited to tCP case and valid for highest-lying states in the manifonly. On the other hand this expansion is quadratic in detions from the equilibrium point but nonlinear in the micrwave fieldF strength while the resonance Hamiltonian aproach is a first order inF expansion. Clearly in the deesemiclassical regime~largen0) and for sufficiently largeF0the harmonic expansion approach yields a better approxtion for wave packet states in the CP than the present rnant Hamiltonian analysis~yielding, however, little informa-tion on the energies of other states in the resonant manifo!.On the other hand, quite surprisingly, we have found, bdirect comparison of numerical values, that forF0 around0.01 or 0.06 andn0 up to 30–40 the resonant Hamiltoniayields semiclassical values closer to exact quantum datathe harmonic expansion. Thus for an intermediate rangen0 and F0 values the resonant Hamiltonian quantizationsurprisingly good in accuracy, yielding, at the same time,predictions for the whole resonant manifold and for an artrary polarization.

To complete the picture we show in Fig. 4 the level dnamics of the samen0521 manifold versus scaled field amplitude, F0 , for a50.6. Quantum results are presentedgether with the semiclassical ones. Again, for the resonaisland in orbital (J,u) motion sufficiently large to capturequantum states, semiclassical results reproduce quanones quite well even beyond a classical chaos border~ofcourse except avoided crossings!.

Exact quantum Floquet matrix diagonalizations, p

FIG. 3. Comparison of the semiclassical quasienergies~circles!originating from the unperturbedn0521 manifold forF050.03 andv051 with the exact quantum values~crosses!, at different valuesof the degree of the field ellipticitya50.1 ~a!, 0.9~b!. Integer indexp counts consecutive states in the perturbed manifold.

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-

formed using the Lanczos code yields not only the eigenvues but also the corresponding eigenvectors. Their time elution may be visualized@45# to confirm directly that indeedthe Floquet states localized both in (J,u) and in (L,f)spaces correspond to the nonspreading localized wave pets. Consider again the level dynamics presented in FigAs discussed above the higher-lying state in the manifcorresponds to the wave-packet motion of the electron oncircular orbit in the direction coinciding with the direction orotation of the electric field vector. And indeed such a mtion is revealed by the plots of the corresponding Floqstate~see Fig. 5, upper row!. This Floquet state changes littlwith change of the ellipticity of the microwaves,a, and inthe limiting case of CP becomes a well-known CP nospreading wave packet@8,9#. In the opposite limit ofa50i.e., the LP microwave case, this state becomes almostgenerate with another one corresponding to the differentrection of the electron rotation. The two, almost degenerexact Floquet states are linear combinations of the two wpackets~at least in 2D! as discussed in@43#.

It seems interesting to see what happens to the secmember of the pair as polarization is changed from lineaelliptical ~a increases!. The two states separate fast ina andeach of them represents a distinct motion. As mentionabove the state going down in energy corresponds to a wpacket moving on the circular orbit in the direction opposto the field. It undergoes a series of avoided crossings wother states of the manifold losing progressively its localizcharacter. Still, fora not too large, and far from avoidecrossings its wave-packet character is clearly visible~com-pare lower row in Fig. 5!. This state loses its wave-packcharacter when the librational islands~compare Fig. 1! movesufficiently far down so that rotational states with large netive L0 disappear since there is no ‘‘space left’’ for them dto the finite value of\. As may be seen by comparison oFigs. 1 and 2 this transition occurs arounda50.45 for n0521, manifesting itself in the quantum data~Fig. 2! by

FIG. 4. Level dynamics of the exact quantum quasienerg~dotted lines! in the vicinity of the resonant manifold emerginfrom n0521, compared with the semiclassical prediction~fulllines!, for F050, . . . ,0.06 anda50.6,v051. Note that the maxi-mum field amplitude clearly exceeds typical ionization threshomeasured in current experiments at the principal resonance.semiclassical prediction accurately tracks the exact solution acrolarge number of avoided crossings.

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3980 PRA 58KRZYSZTOF SACHA AND JAKUB ZAKRZEWSKI

relatively broad avoided crossings encountered by the wapacket state. The transition point is, of course,\ dependent,for smaller \ ~larger n0) the rotational, wave-packet-likestate with large negativeL0 exists for largera. It is worthstressing that fora below this critical,n0- ~and thus drivingfrequency! dependent value there are two distinct Floqustates corresponding to two wave packets moving in oppodirections.

IV. CONCLUSIONS

To conclude we have shown, by comparison with tquantum numerical results, that the proposed semiclasmethod, based on the first-order resonant Hamiltonian, ggood quantitative predictions. Numerical calculations pduce nothing but numbers and an advantage of a semiclcal analysis is an understanding of what physics is hidbehind them. Apparently the behavior of the H atom placin the EP microwave field, in the range of parameters uabove, is determined by the underlying classical dynamespecially, one may build wave packets that follow classtrajectories without spreading. We have shown that the wpacket discovered in CP microwaves@8,9# corresponding tothe motion of the electron on a circular orbit in the samdirection as the rotation of the microwave field exists alsoEP case. For a sufficiently small ellipticity of the microwapolarization,a, there exists also another wave packet sta

FIG. 5. Nonspreading wave packets for a 2D hydrogen ailluminated by the elliptically polarized microwaves with the amplitude F050.03, frequencyv051 for n0521 and ellipticity a50.4. Top row: the exact Floquet state corresponding to a nspreading wave packet moving on a circular orbit in the directionthe rotation of the microwave field at timesvt50,p/4,p/2 from leftto right. This wave packet corresponds to the one known forcircular polarization and may be obtained from the latter bychange of the microwave polarization. Bottom row: another exFloquet state corresponding to a nonspreading wave packet moin the opposite direction~shown at the same timesvt50,p/4,p/2from left to right!. Note that while the former almost preservesshape in temporal evolution, the latter becomes significantlytorted. Still being an exact Floquet time-periodic eigenstate itgains its shape~as depicted, e.g., in the bottom left corner of tfigure! every period of the microwave. The size of each box6800 Bohr radii in bothx andy directions.

e-

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,

corresponding to electron rotating in the opposite directionthe microwave field. In the limiting LP case both wave pacets coalesce, the exact Floquet states correspond semiccally to linear combinations of two wave packets propaging in the opposite direction on the circular orbit@43#.

The analysis presented is restricted to the twdimensional model atom~similar assumption is implicit in@43#!, its validity for the real three-dimensional atom is aopen question. Certainly, in the limiting LP case, due totational symmetry with respect to the field axis, the two wapackets moving in the opposite directions in 2D lead in 3Da doughnut shaped localized function oscillating betwenorth and south poles of the sphere~assuming a vertical polarization of LP microwaves! @28#. For CP, on the otherhand, the wave-packet motion remains essentially twomensional@10#. It will be most interesting to see how ththird dimension affects the dynamics also for a general cof EP microwave field since already classical studies@36,37#indicate that some qualitative differences may appear. Tsubject is left for future studies.

Still the present 2D analysis shows that the nonspreadwave-packet states localized on circular orbits are notstricted to circular polarization of microwaves only. Thus tperfect circular polarization is not essential for a possiexperimental observation of nonspreading wave packets

Interestingly, in the EP case, there appears a possibilitthe angular localization in minima of the effective potentG with the position of the minima beinga dependent~com-pare Fig. 1!. If the creation of localized wave packets in suminima were possible it would allow one to control thshape of the trajectories on which the wave packets progate by a change of microwave polarization, and not byadditional static field as proposed in@18#. Study of the cor-responding wave functions indicates, however, that duethe small resonance island width the localizationalong theellipse @i.e., in (J,u) space# is not very effective at least inthe range ofn0 andF0 studied by us.

ACKNOWLEDGMENTS

We are grateful to Dominique Delande for numerous dcussions and the permission to use his Lanczos diagonation routines. The support of the KBN under Project N2P302B-00915~K.S. and J.Z.! is acknowledged. Numericacalculations were performed at the Academic Computer Cter Cyfronet in Krako´w with the help of Grant Nos. KBN/S2000/UJ/067/1998 and KBN/S2000/UJ/068/1998. K. Saacknowledges financial support from the Foundation for Pish Science.

APPENDIX

Consider the Floquet HamiltonianHF as defined in Eq.~2.2! with the Hamiltonian given by~2.1!. For the sake ofefficiency of the numerical diagonalization it is convenientrewrite the Floquet Hamiltonian in the velocity gaugeapplying a standard unitary transformation@2,39#

HF52 i]

]t1

px21py

2

22

1

r1

F

v@pxsin vt2pya cosvt#

1F2

4v2~a211!. ~A1!

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--

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PRA 58 3981H ATOM IN ELLIPTICALLY POLARIZE D . . .

Next we turn to scaled semiparabolic coordinates~2.3!,which allows us to express the Floquet eigenvalue equa

HFuun~ t !&5Enuun~ t !& ~A2!

as a generalized eigenvalue problem

~A2EnB!uun~ t !&50, ~A3!

where

A5~u21v2!HF

and ~A4!

B5u21v2,

with

HF5pu

21pv2

2L2~u21v2!2

2

L~u21v2!2 i

]

]t1

F

Lv~u21v2!

3@~upu2vpv!sin vt2a~vpu1upv!cosvt# ~A5!

~with the additive pondoromotive term omitted!.Floquet states are periodic in time~with the periodT

52p/v), thus we may expanduun(t)& in the Fourier series

uun~ t !&5 (K52`

1`

e2 iKvtuunK&. ~A6!

This allows us to cast Eq.~A3! into the equivalent set ocoupled equations

.

ys

ev

.

n Fpu21pv

2

2L2 22

L2Kv~u21v2!G uun

K&1~Ws2Wc!uunK21&

2~Ws1Wc!uunK11&5En~u21v2!uun

K& ~A7!

where

Ws5 iF

2vL~upu2vpv!,

~A8!

Wc5aF

2vL~vpu1upv!.

All terms in the above equation have the polynomial formcoordinates and momenta. This suggests using the harmoscillator basis for an efficient evaluation of matrix eleme@39#. The method becomes then analogous to the treatmof the circular polarization case discussed in detail elsewh@33#. The resulting generalized eigenvalue equation is dianalized using the Lanczos code, which allows for an extrtion of eigenvalues in a selected energy range and the cosponding eigenvectors. For completeness let us mentionthat the Floquet Hamiltonian is invariant under the geneized parity transformation, i.e., the parity transformaticombined with the translation in time byp/v. ThusA andBmatrices may be split into uncoupled matrices that aretimes smaller. This makes the numerical calculations mefficient.

.

. E

ys.

es

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