PHYSICAL REVIEW A 68, 062110 ~2003!

Experimentally attainable example of chaotic tunneling: The hydrogen atom in parallel staticelectric and magnetic fields

Dominique Delande1 and Jakub Zakrzewski2

1Laboratoire Kastler-Brossel, Tour 12, Etage 1, 4 Place Jussieu, F-75005 Paris, France2Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagiellon´ski, ulica Reymonta 4, PL-30-059 Krako´w, Poland

~Received 11 July 2003; published 24 December 2003!

Statistics of tunneling rates in the presence of chaotic classical dynamics is discussed on a realistic example:a hydrogen atom placed in parallel, uniform, static electric, and magnetic fields, where tunneling is followed byionization along the fields direction. Depending on the magnetic quantum number, one may observe either astandard Porter-Thomas distribution of tunneling rates or, for strong scarring by a periodic orbit parallel to theexternal fields, strong deviations from it. For the latter case, a simple model based on random matrix theorygives the correct distribution.

DOI: 10.1103/PhysRevA.68.062110 PACS number~s!: 03.65.Xp, 05.45.Mt, 32.60.1i, 32.80.Dz

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I. INTRODUCTION

After many years of intensive research in the ‘‘quantuchaos area’’ it is now commonly accepted that the quanbehavior of complex systems may be strongly correlawith the character of their classical motion@1–5#. Even sucha purely quantum phenomenon as tunneling may befoundly affected by chaotic classical dynamics. For an~inte-grable! one-dimensional system, the tunneling rate depesmoothly on parameters. For an integrable multidimensiosystem, the eigenstates can be labeled with a set of gquantum numbers and tunneling can be studied with a simextension of the standard one-dimensional analysis, leato tunneling rates which are simple functions of the quantnumbers. In contrast, in the presence of chaotic motionmultidimensional system, the tunneling rates typicastrongly fluctuate when parameters are varied, the gamthen to identify both the average behavior and the statistproperties of the fluctuations.

Imagine the situation when the wave function is predomnantly localized in a region of regular motion. The tunnelito the chaotic sea surrounding the regular island, ca‘‘chaos assisted tunneling,’’ has been quite thoroughly stied @6–15#. It may be characterized by the statistics of tuneling rates, or directly measurable quantities such as tuning splittings between doublets of different symmetries@10#or tunneling widths@11,12# where the tunneling to the chaotic sea leads eventually to decay~e.g., to ionization ofatomic species!. Models based on random matrix theo~RMT! @16,17# show that distributions of both quantities aclosely correlated with both the splittings@10# and thesquareroots of the widths @11# having a common Cauch~Lorentzian-like! distribution with an exponential cutoff foextremely large events. Such a situation occurs for suciently small\ ~in the semiclassical regime! when the tun-neling coupling is much smaller than the mean level spacin a given system.

Another possibility occurs when virtually all accessibphase space~at a given energy! is chaotic: the tunneling occurs through a potential~rather than dynamical as in the prvious case! barrier. Then a standard RMT based answer leto the Porter-Thomas distribution of widths~or its appropri-

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ate many channel extension!, as applied in various areafrom nuclear physics@16#, mesoscopics@18#, or chemicalreactions@19# to name a few. Creagh and Whelan@20–23#developed a semiclassical approach to tunneling~for a semi-classical treatment concentrating on other aspects of tuning see, e.g., Refs.@24,25#! which enabled them to give aimproved statistical distribution of tunneling rates@26#. Thedistribution has been tested on a model system and showfaithfully represent the tunneling splitting distribution, provided the classical dynamics is sufficiently chaotic. Howevthis distribution fails for systems when scarred@27–31# wavefunctions dominate the process. In order to take into accoscarring, the same authors@23# developed a more complicated semiclassical theory which, in a model system, acrately describes the numerically observed tunneling rates

The aim of this paper is threefold. First, we proposesimpler approach to the effect of scarring than that in R@23#. Our approach is less general, as it is limited to the cwhen only one channel contributes to tunneling. Thishowever, a very frequent situation: because tunneling tycally decays exponentially with some parameter, most ctributions are often hidden by a single dominant one. Tformulas that we obtain are also much simpler. Second,consider the tunneling rate distribution in a challenging,alistic system—a hydrogen atom in parallel electric amagnetic fields. As mentioned by Creagh and Whelan,expects there the above-mentioned problems due to sdominated tunneling. Here again we test the proposed dibution on a vast set of numerical data. Third, in contrast wmost of the previous studies, we do not consider here a sation where tunneling manifests itself as a quasidegenebetween a pair of even-odd states, but rather the case wtunneling is followed by a subsequent ionization of the stem and manifests itself in the widths~ionization rates! ofresonances. The analysis for both cases is similar, butidentical.

II. THE DISTRIBUTION FOR TUNNELING RATES FORSCAR-DOMINATED CHAOTIC TUNNELING

Let us recall first shortly the analysis of chaotic tunneliused in Ref.@26#, which makes it possible to predict thdistribution of tunneling rates in terms of classical quantitiThis approach is based on the standard semiclassical ex

©2003 The American Physical Society10-1

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D. DELANDE AND J. ZAKRZEWSKI PHYSICAL REVIEW A68, 062110 ~2003!

sion of the Green function as a sum over classical or~which is used, e.g., in periodic orbit theory in the mannerGutzwiller!, but incorporates in addition some complex obits, that is, orbits where time, position, and momentumbe made complex. Such orbits may tunnel through the potial well and eventually lead to escape at infinity; they aessential for the calculation of tunneling rates. In the odimensional case, it is well understood that tunneling canquantitatively described using a single complex orbit knoas the instanton: the orbit propagates under the potentialwith a purely real position, and purely imaginary time amomentum, until it emerges in the real phase space whenpotential barrier is crossed~it can be visualized as a standareal orbit in the inverted potential!. The actionSI of theinstanton is then purely imaginarySI5 iK and the tunnelingrate is, not surprisingly, essentially described byexp(iSI /\)5exp(2K/\) contribution.

For a multidimensional system, the situation is somehcomparable, except that there are now several instantonbits. It also turns out that the structure of the tunneling coplex orbits can be extremely complicated@25,32#. However,because of the exponential decrease of the tunneling ratthe semiclassical limit\→0, there are cases when the istanton orbit with the smallest imaginary action will give tdominant contribution. Creagh and Whelan succeeded inpressing the tunneling rate in terms of the action and stabexponent of the instanton orbit@21#. They were able to describe the situation of a symmetric double well, where tuneling manifests itself through the existence of pairsquasidegenerate states, i.e., to calculate the splitting ofdoublets. Comparison with ‘‘exact’’ numerical results formodel system showed a very good agreement@20,21#. Theywere also able to describe the situation of tunneling outsidsingle potential well~with chaotic dynamics inside the wel!followed by ‘‘ionization,’’ that is, particle directly escapintoward infinity. The quantity of interest is the ‘‘weighteddensity of states, where the weight is given by the widthsGnof the resonances with energiesEn :

f ~E!5(n

Gnd~E2En!. ~1!

In the semiclassical approximation, it can be written—in tspirit of periodic orbit theory—as the sum of smooth aoscillatory terms:

f ~E!' f 0~E!1 f osc~E!, ~2!

Explicitly, the smooth term reads

f 0~E!51

2p

exp~2K/\!

A~21!d21det~W2I !, ~3!

whereK is the ~imaginary! action of the periodic instanton~that is the full orbit back and forth across the potential we!,d the number of freedoms of the system, andW the2(d21)32(d21) stability matrix of the instanton. The oscillatory term is not explicitly written by Creagh anWhelan, but it is rather simple to calculate it from their p

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pers. In the simple case where the classical orbit which isreal continuation of the instanton inside the potential wela periodic orbit~this is, for example, the case where thinstanton is along a symmetry axis of the potential!, it is notsurprising that the oscillatory terms will be governed by tproperties of this real periodic orbit. Indeed, it is known ththe eigenstates inside the well will be scarred by suchorbit, thus showing either an increased or decreased pability density at the point where the instanton emerges. Ithus reasonable to expect that scarred~antiscarred! states willshow an increased~decreased! tunneling probability. Themodulations of the weighted density of states are thus relato the action of the real periodic orbit. More specifically, ogets

f osc~E!51

pRe(

j 51

`exp@~2K1 i jS!/\#

A~21!d21det~WMj2I !, ~4!

whereS is the ~real! action of the periodic orbit in the weland M its stability matrix. The sum overj just takes intoaccount the repetitions of this orbit. This approach isstricted to a low tunneling rate when repetitions of thestanton give negligible contributions. The fact that, in tcontribution of the instanton, there is a 1/2p prefactor, halfthe prefactor for the oscillatory term, is not trivial, but eplained in Ref.@20#.

III. THE HYDROGEN ATOM IN PARALLEL FIELDS

We consider a hydrogen atom placed in static paramagnetic and electric fields. The Hamiltonian of the systis ~for infinite mass of the nucleus, neglecting relativistic aQED corrections, in atomic units!

H5p2

22

1

r2Fz1

g2

8~x21y2!1

g

2Lz, ~5!

whereg stands traditionally for the magnetic field in atomunits ('2.353105 T) while F is the static electric field~inatomic units of'5.131011 V/m! assumed to be orientedtogether with the magnetic field, along theOz axis. The sys-tem obeys cylindrical symmetry andLz is a constant of mo-tion. The last Zeeman term in the Hamiltonian gives thuconstant shift~for a givenLz) and will be omitted for sim-plicity.

Classically, the atom may ionize for energiesEcl.22AF. Note that ionization occurs in thez direction—thediamagnetic term provides a two-dimensional harmonoscillator binding potential in the perpendicular directions

The character of the classical motion depends on theergy as well as on the relative magnitude of electric amagnetic fields, as discussed long time ago in Refs.@33,34#.In fact, the system obeys the standard classical scaling l@35,36#. Explicitly, scaling with respect to the magnetic fieas r5rg2/3, p5pg21/3, e5Eg22/3, f 5Fg24/3, Lz5Lzg

1/3

leads to a new Hamiltonian dependent on two parameonly, the scaled energye, and scaled electric fieldf,

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EXAMPLE OF CHAOTIC TUNNELING: THE HYDROGEN . . . PHYSICAL REVIEW A68, 062110 ~2003!

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Later on we shall drop the; sign using classically scalevariables only.

With this scaling and forf 50 ~pure magnetic field!, themotion is predominantly regular for smalle,20.5; forlargere a gradual transition to chaos takes place so that,e.20.12, practically all the available phase space becochaotic@35,36#. This character of the motion is basically prserved forf .0, provided

e,e ion522Af , ~7!

i.e., below the classical ionization threshold.Quantum mechanics does not preserve the scaling. Ins

of finding, however, eigenenergies at given values of mnetic (g) and electric~F! fields, it is a celebrated traditionnow to consider scaled spectra@35,36#, i.e., choose values oexternal fields as to obtain eigenenergies at fixede. Thisprocedure is straightforward. Rewriting the Schro¨dingerequation, one may obtain a generalized eigenvalue probfor fixed e ~and f in our case! from which quantized fieldvaluesgn

21/3 are obtained. If one were to get back to toriginal problem, then a givengn value together with thedefinition of e yields the energyEn , which is an eigenvalueof the original Schro¨dinger equation for thatgn field value.The set ofgn obtained for fixed values ofe and f corre-sponds to the very same classical dynamics while differgn

21/3 play the role of different values of the effective Planconstant\eff .

One may then expect that to study quantum tunnelingthe semiclassical regime with a well-defined classical mchanics, it is sufficient to diagonalize a standard scaled plem. This is, however, not completely true: indeed, becatunneling implies that the electron ionizes, the energy sptrum is not a discrete spectrum of bound states, but racomposed of resonances. Far below the classical ionizathreshold, the widths of the resonances are extremely sand can be neglected. Then diagonalization of the Hamtonian in a convenient basis set may produce a discreteergy spectrum which very well approximates the true renances. On the other hand, far above the ionizathreshold, the spectrum is continuous and basically unsttured. We are interested in the intermediate situation, invicinity of the classical ionization threshold, where the resnances have a small but significant width due to tunnefollowed by ionization. The treatment of tunneling resnances necessitates a further standard extension, knownthe pure magnetic-field case above the ionization thresh@37,38#: a complex rotation approach@39–41#. The idea is toapply the following complex scaling~or rotation if viewed inthe complex plane!: r→r exp(iQ), p→p exp(2iQ) to theHamiltonian of the system, whereQ is a real positive parameter representing the complex rotation angle~typically of theorder of 0.1 rad!. The transformed Hamiltonian is no longea Hermitian operator, and its diagonalization yields compeigenvaluesEn2 iGn/2, whereEn is the energy of the reso

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nance andGn its width, i.e., the inverse of its lifetime. Welbelow the classical ionization threshold, Eq.~7!, the widthsshould be vanishingly small; with increasingf, some statesnotably those extended inz direction, i.e., along the ‘‘ioniza-tion direction,’’ should have increased imaginary parts incating tunneling ionization. Above the threshold, Eq.~7!,ionization becomes classically allowed and the widthsexpected to be large. Moreover, in the tunneling regime,widths should on an average exponentially decrease in mnitude with decreasing\ according to an exp(2K/\) law,with K being a characteristic tunneling action.

IV. NUMERICAL RESULTS—SHIFT OF THE EFFECTIVEIONIZATION THRESHOLD

The expectations described in the preceding sectionbased on a rough classical analysis of the ionization procIn order to test these ideas and the semiclassical predicfor the widths of the resonances, we have performed exsive numerical studies of the energy spectrum of the systThe matrix representing the complex rotated Hamiltoniana Sturmian basis set@42# is easily obtained, as matrix elements have strong selection rules and are all knownsimple analytic expressions. The matrix in then diagonalizusing the Lanczos algorithm@43#, producing several hundreds or thousands fully converged eigenvalues. We hcarefully checked that all eigenvalues presented in this paare fully converged. The only limitation is that the calcultion is performed in double precision, yielding about 12 snificant digits. This also implies that widths~tunneling rates!smaller than 10212 cannot be accurately computed.

It turns out that, below the classical ionization threshoEq. ~7!, the widths of the resonances are usually very smMoreover, as we are interested in the situation whereclassical dynamics inside the potential well is chaotic,have to use a rather large value of the scaled enee—typically e520.1—which in turn corresponds to a rathsmall value of the scaled electric field at the ionizatithreshold, that is, typicallyf 50.0025 from Eq.~7!. For thesevalues, we observed that the numerically computed widare all vanishingly small, smaller than the 10212 accuracy ofthe numerics.

This can be understood from Eq.~3!. Indeed, the stabilityexponent of the instanton orbit is, in our specific case, enmous. The reason is that the potential in the vicinity of tsaddle point is very anisotropic. It is strongly bound by tdiamagnetic term in the transverse (x,y) plane but has a onlya very smooth potential maximum in the field~z! direction.The instanton can be seen as a real orbit propagating ininverted potential. This inverted potential has a shallminimum in thez direction but falls down very rapidly in thetransverse directions: the instanton moves along a sharptential ridge and is thus extremely unstable. We show inAppendix how to calculate the actionK and stability matrixW of the instanton. For smallf ~the regime we are interestein!, the following approximate expressions are sufficient:

K~e!522p~e12 f 1/2!

21/2f 3/4~8!

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D. DELANDE AND J. ZAKRZEWSKI PHYSICAL REVIEW A68, 062110 ~2003!

and

~21!d21det~W2I !5expS p

21/2f 3/4D . ~9!

For f 50.0025, the denominator in Eq.~3! is thus of theorder of 1043, which explains that the widths cannot be mesured in a numerical experiment.1

An alternative, equivalent, ‘‘quantum’’ explanation cangiven. The magnetic-field term in the Hamiltonian is resposible for a harmonic potential in the direction perpendicuto the fields. Thus the quantum-mechanical energy ofmotion in thex-y plane cannot be smaller than the energythe lowest state of the corresponding oscillator. ForLz5m50 the energy in question is the ground-state energyE05g/2 while for other ~conserved! m values it is Em5(umu11)g/2. Reaching the energy of the classical ioniztion threshold is thus not sufficient for the quantum systemionize. It requires the additional zero-point energyEm to beable to overcome the potential barrier. As the harmonictential is quite strong, this excess energy is rather highhas the effect of tremendously reducing the ionization prability. For the scaled problem, the energy shiftEm translateinto a shift of the scaled energy:

em5Emg22/35umu11

2g1/35

umu11

2\eff . ~10!

The equivalence of the two points of view can be estlished by noting that expression~9! has itself an exponentiadependence, which can be combined with the numeratoEq. ~3!. We obtain

f 0~E!51

2pexpF2p~e12 f 1/22\eff/2!

\eff21/2f 3/4 G , ~11!

which can also be written as

f 0~E!51

2pexpF2K~ e !

\effG , ~12!

where~here form50)

e5e2em . ~13!

The physical meaning of these equations is rather cleaeffect, tunneling can be described with a standard expontial ~action/\eff), provided the amounte05\eff/2 of energy issubtracted from the total available energye. The global ef-fect of the degrees of freedom transverse to the instanto

1This effect was not observed in the various numerical expments of Creagh and Whelan. This is because, in their casepotential varies quite rapidly along the instanton trajectory, havinshallow minimum in the transverse direction. This results indenominator being of the order of unity, several tens of ordersmagnitude smaller than in our case.

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nothing but a shift of the energy available for tunnelingthe zero-point energy in the transverse direction.

Such an analysis does not take into account the azimusymmetry around the field’s axis and the fact that the conbutions of the variousm values can be separated in the nmerical calculation. Not surprisingly, tunneling is much moeffective for m50 states which are not repelled from thezaxis by a centrifugal potential and thus feel more efficienthe instanton. In the quantum point of view, such states hthe lowest transverse zero-point energye0 . For otherm val-ues, the treatment of Creagh and Whelan has to be adainstead of considering the full semiclassical Green functof the system, one must expand it on the variousm subspacesand use only the relevant component in each subspacsimilar problem occurs in periodic orbit theory when oneinterested in the contribution to the density of states oforbit along thez axis. How to deal with such a problem habeen described in a general manner by Bogomolny@44,45#and in a specific case by Shaw and co-workers@46#. The ruleis that higher powers of the stability matrix come into plafor example,M umu11 instead ofM for the real orbit. Thesituation is similar for the instanton, resulting in the denomnator being raised to powerumu11. The net effect is againtaken into account by shifting the scaled energy byem5(umu11)e0 , i.e., the transverse zero-point energy in themsubspace. Thus results in an effective quantum ionizathreshold

e ionq 522Af 1

umu11

2\eff . ~14!

Because of the transverse zero-point energy, in the pence of magnetic field,larger electric-field strengths are necessary to observe the same ionization yield, or, converslarger scaled energy is required for a fixed electric fiestrength. We have thus performed numerical diagonalizaof the scaled Hamiltonianabove the classical ionizationthreshold, Eq.~7!. The results are shown in Fig. 1, where thwidths of the resonances are plotted versus the quantvalue ofg21/351/\eff , the inverse of the effective Planck’constant. At lowg21/3, i.e., large\eff , the transverse zeropoint energy is so large that the quantum ionization threold, Eq. ~14!, is far above the scaled energy of the stawhich consequently has a vanishingly small ionization raThis corresponds to the regiong21/3,28 in Fig. 1~a!, wherethe widths are smaller than the numerical accuracy. Asg21/3

is increased, the quantum ionization threshold decreasessignificant tunneling takes place, as observed in the ra28,g21/3,35. Finally, at sufficiently highg21/3 value, thescaled energy is higher than the quantum ionization threshand direct ionization takes place. There, the ionization raare large, comparable to the spacing between consecuresonances, and the tunneling regime is left. In the figure,g21/3 value where the quantum ionization thresholdreached is marked by the dotted line, and agrees withnumerical results. The two values ofe for m50 andm53have to be chosen quite different in order to observetransition within the range ofg21/3 available from numericaldiagonalization. Let us note that also in the pure magne

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EXAMPLE OF CHAOTIC TUNNELING: THE HYDROGEN . . . PHYSICAL REVIEW A68, 062110 ~2003!

field case, the statistics of level spacings in the vicinity ofionization threshold is sensitive to the very same quanthreshold law@38,47#.

V. NUMERICAL RESULTS AT CONSTANT MODIFIEDSCALED ENERGY

The behavior observed in Fig. 1 has important conquences. To study tunneling, we should consider onlyregion just below the threshold; this region is very small athe tunneling rate changes very rapidly withg. Thus, scaledspectroscopy is not appropriate for the analysis of statistproperties of tunneling. As is clear from the discussion aband the examples depicted in the figures, the proper pareter characterizing the spectrum is note but rather e5e2em . In order to overcome the difficulty described in thpreceding section, a simple solution is to scale the probfollowing the effective quantum ionization threshold insteof the classical one. One then gets rid of the huge denonator due to the transverse motion and may more easilycentrate on the interesting dynamics, namely, the interpbetween the instanton and the chaotic dynamics insidepotential well. We will thus solve the Schro¨dinger equation,not at constant scaled energye, but at constant modifiedscaled energye, Eq. ~13!. This results in the following generalized ~nonlinear! eigenvalue problem for the effectiv

FIG. 1. The ionization rates~widths! of resonances of the hydrogen atom in parallel electric and magnetic fields, as a functiothe effective principal quantum numberg21/351/\eff , which playsthe role of the inverse of the effective Planck’s constant. The dianalization is performed at a fixed value of the scaled energye andyields complex eigenvalues, their real part corresponds to quanvalues of magnetic fieldg for fixed values ofe andf. ~a! shows dataobtained for angular momentumm50, scaled electric fieldf50.0025, ande520.085, i.e., above the classical ionizatiothreshold as given by condition~7!. Observe very small ionizationwidths for g21/3,28, then a rapid, on an average increase asaturation forg21/3.33.3. The region of rapid increase corrsponds to tunneling, as explained in the text. The dashed verline gives the quantum ionization threshold, Eq.~14!. ~b! shows thedata form53, f 50.0025, ande50.005 showing a similar behavior. Now the threshold is atg21/3'19.

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Planck’s constant\n and the eigenstatesfn(r ):

S 2\n

2

2D1\n

umu11

22 e2

1

r2 f z1

x21y2

8 Dfn~r !50,

~15!

with D being the Laplace operator.This equation is solved by expansion over a Sturm

basis and a modified version of the Lanczos algoritadapted to such a generalized eigenvalue problem@48#. Wehave been able to obtain a few thousands of resonawidths for a givenm value, all lying in the tunneling regimeAn example is presented in Fig. 2.

As expected from Eq.~12!, the ionization rate shows aoverall exponential decrease with 1/\eff . The rate of thisdecrease is directly related to the tunneling action ofinstanton: the prediction of Eq.~12! is shown as a dasheline in the figure. Obviously, the agreement is excellentshould be noted that the semiclassical prediction is entiobtained from classical ingredients and free of any paraeter. Note also the existence of very large fluctuationsexpected for chaotic tunneling—around the mean value.

In order to make a more quantitative test, we removeglobal exponential decrease and define, following Ref.@21#,a rescaled width

yn5r0~En!

f 0~En!Gn , ~16!

wherer0(E) is the density of states.

of

-

ed

d

al

FIG. 2. Widths~ionization rates! of the resonances of the hydrogen atom in parallel electric and magnetic fields, computed at c

stant modified scaled energy, Eq.~13!, e520.1005, i.e., 0.0005below the quantum ionization threshold, Eq.~14!. Parameters arem50, f 50.0025. The widths~in logarithmic scale! are plotted vsg21/351/\eff . The data show the exponential decrease of thefor \eff→0 characteristic for tunneling process. The dashed linethe average behavior predicted from the semiclassical analysis~12!. One may also notice periodic short-range modulations ofionization rates~with a period close to 0.4!; this is a manifestationof scarring by the periodic orbit along the fields axis and is dcussed in Sec. VI.

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D. DELANDE AND J. ZAKRZEWSKI PHYSICAL REVIEW A68, 062110 ~2003!

From its definition,yn should have average value unitythe semiclassical limit. The distribution ofyn ~same data asin Fig. 2! is shown in Fig. 3, on a linear scale. It has velarge fluctuations—several orders of magnitude with a laproportion of very small ionization rates—but we checkthat the average value ofyn is constant across the spectruwithin a few percent~althoughGn themselves vary over fiveorders of magnitude! and equal to 0.9560.03. This is onlyslightly smaller than unity. The difference might be duedeviations from harmonicity of the potential in the vicinity othe saddle point~an assumption made in our calculation!.Another plausible source of deviation is the assumptmade in the calculation of Creagh and Whelan that evelectron which tunnels through the barrier will eventuaionize; although this is very likely to happen, the channalong thez axis may also reflect a small part of the electronwave function, even after tunneling took place. This woumanifest itself by theyn being smaller than unity.

The main point remains that semiclassics are able todict quantitatively the average behavior of the ionizatirates in the tunneling regime.

VI. FLUCTUATIONS OF THE IONIZATIONRATES—EFFECT OF SCARRING

Beyond the average behavior discussed in the precesection, we are also interested in the fluctuations of theization rates. The most probable origin of these fluctuatiis the fact that the classical dynamics inside the potenwell is chaotic. This implies that the resonance wave futions in the well display apparently erratic fluctuations frostate to state. States with a large probability density nearclassical saddle point are more likely to tunnel and ionizeones with small probability density. As a simple approximtion, the tunneling probability and thus the ionization rateproportional to the squared overlap between the eigens

FIG. 3. Rescaled widths~ionization rates! yn , Eq. ~16!, of theresonances of the hydrogen atom in parallel electric and magnfields ~same data as in Fig. 2!. As expected from the semiclassicanalysis, theyn values have an average value close to unity, wlarge fluctuations. Note especially the large proportions of vsmall widths, characteristic of chaos assisted tunneling with a siopen channel.

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al-

hee-ste

and a wave packet optimally tuned for tunneling, i.e., builtfollow the instanton trajectory. Creagh and Whelan hashown how to explicitly build such a wave packet@23#. For aquantum chaotic system, the simplest model for describthe statistical properties of the energy spectrum and eigstates is to use RMT@16,17#. There, it is assumed that anunknown matrix element will be statistically described byGaussian distribution. In our case, although the system istime-reversal invariant~because of the magnetic field!, it hasa generalized time-reversal symmetry and the Gaussianthogonal ensemble of random matrices must be used. Tthe matrix element is purely real and its square, and conquently the ionization rate, will be described by a PortThomas distribution@16,49#:

P~y!51

A2pyyexp~2y/2y! , ~17!

wherey is the mean value ofy, unity in our case. In Fig. 4,we show the numerically obtained distribution for them

tic

yle

0 1 2 3 4

y1/2

0.0

0.5

1.0

1.5

P(y

1/2 )

10-6

10-4

10-2

100

102

y

10-4

10-2

100

102

104

P(y

)

FIG. 4. Statistical distribution of the rescaled ionization ra~widths! y of the resonances of the hydrogen atom in parallel eltric and magnetic fields. The data are taken at fixed modified sc

energye, Eq. ~13!, and rescaled according to Eq.~16! in order totake into account the exponential decrease due to the tunnethrough the potential barrier. These data are obtained for them53 series at scaled electric fieldf 50.0025. In order to improve the

statistics, several distributions obtained for various values oeslightly below20.1 are used. The inset shows the distributionP(y)on a double logarithmic scale, with ay21/2 behavior at smally,characteristic of ionization with only one open channel, andexponential tail at largey. The numerical results are shown by thhistogram, the dashed line is theuniversalPorter-Thomas distribu-tion, Eq. ~17!, predicted by random matrix theory, while the solline is the prediction of Eq.~22! obtained by taking into account thscarring of the eigenstates by the orbit along the field axis. Notethere is no adjustable parameter. The main figure shows the dbution of Ay, which is not singular asy→0, on a double linearscale. The prediction of random matrix theory is a pure GaussFor them53 series, the effect of scarring is small and both theretical distributions agree well with the numerical results.

0-6

nisictictio

onyuy

-

ivf ta-uboarbuf

e

tediinq2

areon

er

ssi-dso

bitsry

th

in

r—ha

ehav-the

ell.

ly

-are

e

ichherep-er

icdese

EXAMPLE OF CHAOTIC TUNNELING: THE HYDROGEN . . . PHYSICAL REVIEW A68, 062110 ~2003!

53 series compared with the Porter-Thomas prediction odouble-logarithmic scale which is more convenient to dplay the large fluctuations. The agreement is excellent, whproves that the distribution of ionization rates in our realisproblem can be quantitatively predicted, using a combinaof semiclassics~for the mean value! and random matrixtheory ~for the fluctuations!.

It should be emphasized that large fluctuations of the iization rate should also be observed in multidimensional stem where the classical dynamics is regular. Indeed, in sa case, the eigenstates are described by series of energels labeled by several quantum numbers~one good quantumnumber per dimension!. The tunneling rate is a smooth function of these quantum numbers, implying that the spectrumcomposed of several interleaved series of states. In a gseries, the ionization rates vary smoothly, but, because ointerleaving of the various series, wildly fluctuating ioniztion rates are observed for consecutive energy levels. In sa case, the statistical distribution of the ionization rates isno meansuniversal, but depends on the specific propertiesthe tunneling actions in the various series. On the contrrandom matrix theory predicts a universal statistical distrition of the ionization rates, which is thus characteristic ochaotic classical dynamics.

The results for them50 series are shown in Fig. 5. Thoverall agreement is rather good, with a cleary21/2 behaviorat smally and an exponentially small tail at largey. How-ever, a significant deviation is clearly visible at intermediavalues. What is the origin of this deviation? We have beable to show that it is directly related to the unstable perioorbits inside the chaotic potential well. The quantitativeterpretation is based on the semiclassical prediction, E~1!–~4!. A clue is provided by a careful inspection of Fig.

0 1 2 3 4

y1/2

0.0

0.5

1.0

1.5P

(y1/

2 )

10-6

10-4

10-2

100

102

y

10-4

10-2

100

102

104

P(y

)

FIG. 5. Same as Fig. 4, but for them50 series. The effect ofscarring is much more important and large deviations fromPorter-Thomas~random matrix theory! distribution ~dashed line!are observed. In contrast, the agreement with the model takingaccount the scarring by thez orbit, Eq.~22! and solid line, is muchbetter. This proves that our model—with no adjustable parametedescribes properly the physics of tunneling and ionization of cotic states in the presence of scarring.

06211

a-h

n

-s-chlev-

isenhe

chyfy,-

a

enc-s.

which shows that fluctuations around the average trendnot random but clearly display a short-range oscillati~about 100 oscillations in the covered range!. The simplestway of measuring this oscillation is to perform a Fouritransform with respect to 1/\eff , a standard tool in periodicorbit theory. We define

g~s!52p

D (n

Gn exp~K/\n!exp~2 is/\n!, ~18!

where the sum is taken over some range of 1/\ of lengthD.The functiong(s) is shown in Fig. 6 both form50 andm53. As expected,g(0) is very close to unity@this proves thatthe actual average width is well predicted by the semiclacal formula, Eq.~12!#. g(s) has a very large peak arouns/2p52.655, with harmonics at integer multiples, but alsmaller peaks at other values. From Eq.~4!, the peaks areexpected to take place at the actions of the periodic orinside the well, which are real continuations of the imagina

e

to

-

0 2 4 6 8 100

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

s/2π

Fou

rier

sign

al |g

(s)|

(a)

(b)

FIG. 6. Fourier transforms of the distribution of widths~ioniza-tion rates! with respect to 1/\eff , Eq. ~18!, obtained forf 50.0025

and e520.1005 in them50 ~a! andm53 ~b! series. It displays alarge peak~hardly visible! at s50 with amplitude being unity, asexpected from semiclassics; this peak describes the average bior of the widths. The other peaks correspond to modulations ofwidth associated with periodic orbits inside the inner potential wThe peaks are much more pronounced for them50 series~a! thanfor the m53 series~b!, because the former are more strongscarred by the orbit along the field axis, with actionSclas/2p52.655. The repetitions of this orbit are clearly visible, with amplitudes forming a geometric series. The heights of these peaksdirectly related to the instability of the periodic orbit along thfields axis. In~a!, the amplitude of the first peak is 0.58960.005 inexcellent agreement with the semiclassical prediction 0.592, whinvolves exclusively classical properties of the periodic orbits. Tother peaks, appearing at actions clustered slightly below theetitions of the main orbit, correspond to other orbits in the innpotential well, which are slightly off thez axis but approach thesaddle point. In~b!, the centrifugal potential prevents the periodorbits from strongly scarring the quantum states, and the amplituof the peaks~especially the ones associated with repetitions of thzorbit! are much smaller.

0-7

cual

-o

eorh

os

u

n,lor

cam

hem-iesledex-

,

ap-

lten-the

9%ck

e

re

g

aren

isd

fi-ift

i

they a

ef.w-

rics ineti-

not

m

e

D. DELANDE AND J. ZAKRZEWSKI PHYSICAL REVIEW A68, 062110 ~2003!

instanton. In our case, this orbit is entirely along thez axisand its classical actionSclas is easily computed. We findSclas/2p52.655 for the parameters of the figure, in perfeagreement with the numerical quantum calculation. It shobe noticed that we use for the classical calculation the scenergy20.1005, i.e., the value of themodifiedscaled energyof the quantum calculation.2 The fact that both agree validates our correction and fully confirms the important rolethe zero-point transverse energy.

The semiclassical formula~4! also predicts the amplitudof the peak that should be observed in the Fourier transfg(s). There is however an important subtlety here. Tmonodromy matrix of the real orbit along thez axis entersthe formula. It turns out that, because the orbit is very clto the saddle point~reached ate522Af ), it undergoes aninfinite series of bifurcation ase→22Af . At closely spacede values, the orbit loses and regains stability. At each bifcation, a new periodic orbit is born, which is off thez axis,but close to it. Such a phenomenon is well known wheparticle either approaches a saddle point~see, for examplethe Henon-Heiles model in Ref.@50#! or explores a channewith a long-range potential, as, for example, is the case fRydberg series converging to an ionization threshold@51#.We show in Fig. 7 the trace of the stability matrix of thezorbit as a function of the scaled energye for f 50.0025,which clearly shows this series of stable-unstable bifurtions. However, we also plot in the same figure the deno

2For the\eff values used in our calculation, the scaled energyfar abovethe classical threshold.

0.11 0.108 0.106 0.104 0.102 0.1Scaled energy ε

4

2

0

2

4T

race

FIG. 7. Trace of the stability matrixM of the periodic orbitalong the fields axis, as a function of the scaled energy~dotted line!.The orbit is stable when the trace is in the@-2,2# range, unstableotherwise. There is a series of stable-unstable bifurcations acculating at the saddle-point energye520.1 (f 50.0025 in this plot!.In contrast, the contributionuA2det(WM21)/A2det(W21)u,plotted as a solid line, is a smooth function ofe. It is precisely thiscontribution which enters the denominator of the semiclassicalpansion, Eq.~4!.

06211

tlded

f

me

e

r-

a

a

-i-

nator of the semiclassical contribution, Eq.~4!, to the ioniza-tion rate, whose calculation is detailed in the Appendix. Tfact that what counts is not the real orbit itself but its cobination with the instanton completely eliminates the serof bifurcations and gives a smooth contribution as the scaenergy is varied, as observed in the numerical quantumperiment. Moreover, the semiclassical formula~4! predictsfor g(s) a peak atSclas/2p52.655 with amplitude 0.592while the numerical result is 0.58960.005, in excellentagreement. Similarly, the harmonics of the peak formproximately a geometric series with amplitude'0.59j forthe j th repetition of the primitive real orbit. The physicainterpretation is clear: because the motion in the inner potial well is chaotic, each time the quantum particle leavesvicinity of the saddle point~along thez axis!, it exploressome part of the chaotic phase space and roughly only 5of the electronic density is reflected by the nucleus baalong thez axis.

It is important to remark that the oscillations of thwidths, Eq.~4!, induced by the orbit along thez axis and allits repetitions addcoherently. Indeed, if we assume fosimplicity3 that thej th repetition contributes with amplitudRj ~with R'0.59 in our case! and phasej f, the series, in-cluding the smooth termf 0 can be summed exactly, leadinto the following contribution to the ionization width:

f ~\!51

2pexp~2K/\!

12R2

11R222Rcos~Sclas/\2f!.

~19!

This, in turn, predicts that the average normalized widthsnot uniformly distributed, but should follow the distributio

y~\!512R2

11R222Rcos~Sclas/\2f!. ~20!

The physical interpretation of this distribution is simple. Itnothing, but the function giving the intensity transmittethrough a Fabry-Perot optical cavity with reflection coefcientsR for the combination of the two mirrors, phase shf at the reflections, and optical lengthSclas/\ in units of thewavelength. This is of course aperiodic function of the vari-able 1/\ with period 2p/Sclas. It has maxima atSclas/\2f equal to an integer multiple of 2p—where the value ofthe function is (11R)/(12R)—and minima at half-integermultiples where the function is (12R)/(11R). If R is large,

s

3Strictly speaking, the amplitudes and phases associated withrepetitions do not form a geometric series, but should be given brather complicated formula, as explained in Appendix A of R@23#. Only for smallR does it reduce to a geometrical series. Hoever, it turns out that although ourR value is not very small~0.59!,the full result does not differ very much from a simple geometseries—as is also visible in the heights of the repetition peakFig. 6—which has the advantage that the summation over reptions can be explicitly carried out, see Eq.~20!. In any case, thedeviation from an exact geometric series is small and doesaffect our statistical analysis.

u-

x-

0-8

browth

daml

eu-e

e

.e

-aleee.

aela

onal

ove.otic

ew,

p-rm

msc-vis-si-dicre

eits

rgeby

ait.m-ed,nitensekeyr-s ofing

ic

linicicra-

-ionto-

nticonsumenotal-tes

rseri-ery—icalho-re

-eonA

dees

n

EXAMPLE OF CHAOTIC TUNNELING: THE HYDROGEN . . . PHYSICAL REVIEW A68, 062110 ~2003!

the maxima are sharp peaks. The analogy with the FaPerot cavity is more than formal: it actually describes hthe electronic density can be resonantly trapped insideinner potential well along thez axis, resulting in enhancetunneling amplitude and ionization rate. Because the dynics is chaotic, such a resonant enhancement is only partiaRmust be smaller than unity!, resulting in scarring of thewavefunction rather than complete localization.

In order to test whether this distribution adequately dscribes the numerical result, we have ‘‘folded’’ all the nmerical valuesyn inside a single ‘‘free spectral range’’ of thFabry-Perot cavity, by plotting them against

xn5Sclasgn

21/3

2p~mod 1!5

Sclas

2p\n~mod 1! , ~21!

wherea(mod 1) denotes the fractional part of the numba. The result is shown in Fig. 8. Clearly the largestyn valuesare grouped around a well-definedx value, as expectedLarge fluctuations still exist; in order to smooth them, wplot also the running average~over 100 values! which clearlypresents a resonant behavior aroundx50.54. The semiclassical prediction, using theR value deduced from the classicstability of the orbit, is shown as a dashed line and agrfairly well with the numerical result. This proves that thorbit along thez axis plays the dominant role in our problemTo be completely honest, we must mention that the phf50.5432p, which is directly related to the position of thmaximum in the plot, has not been extracted from the csical dynamics but fitted to the numerical data.

The last step is to characterize precisely the fluctuatiof the individual widths that appear on top of the glob

FIG. 8. Rescaled ionization ratesyn , Eq. ~16!, of the m50series forf 50.0025 ande520.1005, as a function ofx, Eq. ~21!.x represents~within a multiplicative 2p factor! the phase accumulated by the wave function along the periodic orbit in the innpotential well. There are large fluctuations, as expected in a chasystem, but the average behavior is obviously dependent ox,which proves that the periodic orbit strongly affects the width.pure random matrix approach predicts a uniform distribution inpendent ofx. The solid line is a running average which smooththe fluctuations and clearly shows the resonant behavior of theav-eragewidth with x. The dashed line is the semiclassical predictioEq. ~23!, which incorporates the effect of the periodic orbit.

06211

y-

e

-(

-

r

s

se

s-

s

exponential decrease and the modulations discussed abSuch fluctuations are thought to be unavoidable in a chasystem, and display universal properties~described by RMT!in the presence of chaos. In a semiclassical point of vithey can be seen as the effect of the whole set of~unstable!periodic orbits in the inner potential well. Each orbit aproaching the saddle point contributes an oscillatory teanalogous to Eq.~4! to the width, and the individual widthsare just the result of the superposition of plenty of such terwhich oscillate rapidly: it results in apparently random flutuations around the mean value. A number of peaks areible in the Fourier-transformed spectra in Fig. 6; semiclascal theory tells us that they appear at the actions of perioorbits. In our specific case, it is visible that most of them aclustered at actions slightly smaller than the action of thzorbit and its repetitions. Physically, they correspond to orbmainly located close to thez axis, born from thez orbit at thebifurcations discussed above. There is actually a very lanumber of such orbits with similar shapes, but differingsmall details. Thus, it is in general difficult to associatepeak in the Fourier transform with a single periodic orbExcept for the lowest members, we could not assign unabiguously such peaks. This is not a simple problem: indemany orbits are very close in phase space and, for a fivalue of \eff , cannot be considered as isolated in the sethat the saddle point approximation around each orbit—aingredient of periodic orbit theory—is not valid. In such cicumstances, it is not possible to separate the contributionthe various orbits which have to be grouped together usfor example, a uniform approximation@52#. The origin of thedifficulty is that the neighborhood of the simplest periodorbit along thez axis is rather complicated. In Ref.@22#, ithas been shown that it can be treated also using homocorbits. Note that this is a fundamental difficulty of periodorbit theory, not a practical problem related to the prolifetion of the number of orbits.

An interesting illustration of this problem may be obtained by launching a bunch of trajectories from the sectz5z0 close to the saddle point. Following the real trajecries ~all started with positive momentum inz) until they hitagain the same planez5z0 with positive momentum one caget a feeling of the relevant dynamics. For a fully chaosystem one could naively expect that a plot of, say, acticalculated along the trajectory versus the initial momentalong thez axis will not show any structure. This is not truin our system as visualized in Fig. 9. Observe a stronglyergodic behavior with allowed actions forming almost parlel strips. A clear accumulation of actions in strips correlanicely with peaks in the Fourier transform, Fig. 6~a!, in therange just below the second repetition at 2Sclas/2p55.31 ofthe straight line periodic orbit. Each strip contains of coumainly nonperiodic orbits, but also periodic orbits with vaous shapes~and actions!, and homoclinic orbits. As all thesorbits have very similar actions, they all contribute vesimilarly to the semiclassical formula. It is not clear to usand to some extent irrelevant for the present statistanalysis—which semiclassical approach, either usingmoclinic orbits or sets of periodic orbits, would be moadapted for dealing with our specific situation.

rtic

-

,

0-9

aoelioetu

e

u-

Itacarg

r

. I,

in

c

er-there-

ded,

--

thep-

ula-

ahe

itstruthn

the

into

he

D. DELANDE AND J. ZAKRZEWSKI PHYSICAL REVIEW A68, 062110 ~2003!

The basic assumption, usual in studies of quantum chis that the effect of long orbits is to create fluctuations wdescribed by random matrix theory. As—see above—theization rate appears as the square of some real matrixments, the simplest hypothesis is to assume that the fluctions are described by a Porter-Thomas distribution, Eq.~17!.However, the mean valuey is now taken as predicted by thsemiclassical theory, i.e., Eq.~20!. As thexn values are uni-formly distributed, this results in a global statistical distribtion:

P~y;R!5E0

1 1

A2pyy~x!expS 2

y

2y~x!D dx, ~22!

where

y~x!512R2

11R222R cos 2px. ~23!

This distribution is plotted in Fig. 5 as a solid line.clearly, very significantly, improves over the Porter-Thomdistribution and is in excellent agreement with our numeridata. Especially, it correctly describes the excess of laionization widths.

The same approach can be used for the data in othemseries. However, as is obvious in Fig. 6~b!, the contributionof the z orbit is much smaller in, e.g., them53 series. Asmentioned above, this is well understood semiclassicallysimple words, as the centrifugal term is more importantkeeps the electron away from thez axis and strongly dimin-ishes the contribution of this orbit. TheR parameter for them53 series can be extracted from the Fourier transformFig. 6~b! and is close to 0.1. The semiclassical predictiowhich can be calculated in the spirit of Refs.@45,46#, is@R(m50)#4'0.12 in reasonably good agreement. For su

FIG. 9. Distribution of actions of trajectories launched fromplane z5z058.82 towards the saddle point as a function of tmomentum along thez axis for m50, f 50.0025, ande520.1005. The action is calculated along a given trajectory till it hthe same plane also with positive momentum. Observe a clear sture of strips. Arrows indicate actions corresponding to peaks inFourier transform of the widths. The circle indicates the secorepetition of the orbit along thez axis with maximum action2Sclas/2p55.31 and momentumpz50.2643.

06211

s,ln-le-a-

sle

nit

n,

h

a small R value, the deviation of distribution~22! fromPorter-Thomas is very small. This explains why the PortThomas distribution correctly reproduces the results ofnumerical experiment, see Fig. 4. We have also obtainedsults for them51 andm52 series, shown in Figs. 10 an11. Significant deviations from Porter-Thomas are observalthough smaller than for them50 series. Again, the modified distribution, Eq.~22!, agrees very well with the numerical results.

An alternative approach to the statistical properties ofionization widths is possible. From the semiclassical aproach, we know both the average trend and the modtions; we can thus subtract~or rather divide! these factors in

c-ed

0 1 2 3 4

y1/2

0.0

0.5

1.0

1.5

P(y

1/2 )

10-6

10-4

10-2

100

102

y

10 -4

10 -2

100

102

104

P(y

)

FIG. 10. Same as Figs. 4 and 5, but for them51 series. Theeffect of scarring is intermediate and some deviations fromPorter-Thomas~random matrix theory! distribution ~dashed line!are observed. In contrast, the agreement with the model takingaccount the scarring by thez orbit, Eq.~22! and solid line, is muchbetter.

0 1 2 3 4y

1/2

0.0

0.5

1.0

1.5

P(y

1/2 )

10-6

10-4

10-2

100

102

y

10-4

10-2

100

102

104

P(y

)

FIG. 11. Same as Figs. 4 and 5, but for them52 series. Theeffect of scarring is small and only marginal deviations from tPorter-Thomas~random matrix theory! distribution ~dashed line!are observed.

0-10

t

ern-avtu

mnim

tioel

ny

in-

eere

e

theEq.

s-et i

gorr

nale of

thethe

l of

EXAMPLE OF CHAOTIC TUNNELING: THE HYDROGEN . . . PHYSICAL REVIEW A68, 062110 ~2003!

order to concentrate on the fluctuations. We thus rescalenumerical data to the expected average1oscillatory behavior,that is, define

zn5yn

y~\n!~24!

with y is defined in Eq.~20!.The statistical distribution of thez variable is shown in

Fig. 12 for them50 series. As can be seen, it agrees vwell with a pure Porter-Thomas distribution. This fully cofirms that, once the average and oscillatory behavior hbeen properly taken into account, only the standard fluctions described by random matrix theory persist.

Finally, we have studied a slightly less realistic systethe two-dimensional hydrogen atom in parallel electric amagnetic fields, obtained from the previous system byposing that the motion takes place in the (x,z) plane. Theclassical dynamics isexactly the same as form50 states~obviously the motion is planar for such cases!. One couldthus naively expect the same properties for the ionizawidths for the quantum system. This is however not entirtrue for the following two reasons.

~1! The zero-point transverse motion is now in one dimesion instead of two. Thus, the shift energy is reduced bfactor of 2 compared to Eq.~10!:

e2D5 14 g1/35 1

4 \eff . ~25!

The modified scaled energy, Eq.~13!, must be modified ac-cordingly.

0 1 2 3 4

z1/2

0.0

0.5

1.0

1.5P

(z1/

2 )

10-6 10-4 10-2 100

102

z

10-4

10-2

100

102

104

P(z

)

FIG. 12. Statistical distribution of the ionization widths, recaled according to Eq.~24!, to take into account tunneling and theffect of the periodic orbit along the field axis. A good agreemenobtained with the Porter-Thomas distribution, Eq.~17!, shown as adashed line. The data are the same as in Fig. 5. The fact that aagreement is obtained indicates that our model describes scaand tunneling in a satisfactory manner.

06211

he

y

ea-

:d-

ny

-a

~2! The stability matrix is a 232 matrix instead of a 434 matrix. As explained in the Appendix, this resultsdenominators in Eqs.~1!–~4! to be square roots of the threedimensional results form50.

The net effect is that the instability of the real orbit in thpotential well is significantly reduced, simply because this less space for the electron to escape far from thez axis.The analysis is similar to the three-dimensionalm50 case,with the parameterR being now taken at power 1/2, i.e.,R'0.77 instead ofR'0.59. Stronger deviations from thPorter-Thomas distribution are thus expected foryn . Figures13 and 14 show that it is indeed the case. Once more,agreement with the modified Porter-Thomas distribution,~22!, is very good.

s

oding

0 1 2 3 4

y1/2

0.0

0.5

1.0

1.5

2.0

P(y

1/2 )

10-6

10-4

10-2

100

102

y

10-4

10-2

100

102

104

P(y

)

FIG. 13. Same as Figs. 4 and 5, but for the two-dimensiohydrogen atom in parallel electric and magnetic fields. Becausthe reduced dimensionality, the motion transverse to thez periodicorbit is less unstable than in the three-dimensional atom, andeffect of scarring is enhanced. Very large deviations fromPorter-Thomas~random matrix theory! distribution ~dashed line!are observed, but the improved model, Eq.~22! ~solid line!, repro-duces well the numerical results.

0 1 2 3 4

z1/2

0.0

0.5

1.0

1.5

P(z

1/2 )

10-6

10-4

10-2

100

102

z

10-4

10-2

100

102

104

P(z

)

FIG. 14. Same as Fig. 12, but for the two-dimensional modethe atom in parallel electric and magnetic fields~data as in Fig. 13!.Again, scarring is well accounted for by our model.

0-11

thg

r bnalsidtntsa

vetra-

n

temr-lanarncoit

onsic

foldieeiclap

ee

icacall

its

hee-thea-

ted

xis,n inge,

gi-

pe-

is

.

canheom-theec-

at-

D. DELANDE AND J. ZAKRZEWSKI PHYSICAL REVIEW A68, 062110 ~2003!

VII. CONCLUSION

In this paper, we have studied the widths~ionization rates!of resonances of a realistic system, the hydrogen atomparallel electric and magnetic fields, in conditions whereclassical dynamics is chaotic. We have shown that, usinsemiclassical approach without any adjustable parameteonly with classical ingredients, we are able to predict alytically the average behavior of the widths. We have ashown the existence of a modulation of the average wassociated with a periodic orbit and have calculated quatatively its properties, again using only classical ingredienFinally, the residual fluctuations have been shown to becurately described by a random matrix model. This prothat a proper combination of semiclassics and random matheory canpredict the behavior of the system versus ioniztion.

Our results are comparable to the ones obtained omodel system in Ref.@23#. For example, their Fig. 1 isclearly comparable to our Fig. 8. Note however that, duethe specificities of ourrealistic system, the expressions wobtain have a simpler form. On a different model systeBies et al. @31# observed also deviations from the PorteThomas distribution. Part of the deviation is due to the retively small value of the effective Planck constant, but aother part is certainly due to scarring. Their Figs. 4 and 5again very comparable to our Fig. 8. Because they doconsider a scaling system, the classical dynamics—andsequently the properties of the periodic orbits—change wenergy which makes a comparison with our distributirather difficult. We however have little doubt that the baprocess at work is similar to ours.

ACKNOWLEDGMENTS

We are grateful to Niall Whelan and Stephen Creaghan initial stimulation to look at tunneling in the parallel fieproblem, and anonymous discussions. We thank W.E. BL. Kaplan, and E.J. Heller for their permission to use thdata for our manipulations. Support of KBN under ProjeNo. 5P03B-08821~J.Z.! is acknowledged. The additionasupport of the bilateral Polonium and PICS programs ispreciated. Laboratoire Kastler Brossel de l’Universite´ Pierreet Marie Curie et de l’Ecole Normale Supe´rieure is UMR8552 du CNRS. CPU time on various computers has bprovided by IDRIS.

APPENDIX: CLASSICAL DYNAMICS NEARTHE SADDLE POINT

In this appendix, we discuss how the various classquantities which enter the semiclassical formula can beculated in our specific system, the hydrogen atom in paraelectric and magnetic fields.

The Hamiltonian of the system is given, in scaled unby Eq. ~6!. The saddle point is located along thez axis atposition

06211

ineaut-

ohti-.

c-six

a

o

,

--eotn-h

r

s,rt

-

n

ll-

el

,

zsaddle51

Af, ~A1!

with energye ion522Af .As we are interested in highly excited states lying in t

immediate vicinity of the saddle-point energy, it is convnient to expand the Hamiltonian at second order aroundsaddle point. The normal modes of this harmonic approximtion are along thez axis and in thex-y plane. In thex-yplane, the saddle point is a potential minimum associawith a vibration frequency

vr5A114 f 3/2

2. ~A2!

Because of the azimuthal symmetry around the field athis mode is degenerate. In order to have a chaotic motiothe inner potential well, the scaled energy must be lartypically of the order of20.1, which in turns implies thatf israther small. In most cases, one can thus forget thef depen-dence in Eq.~A2! and use the approximation

vr'1

2. ~A3!

Along thez axis, the saddle point is a potentialmaximum.It is thus associated with an eigenmode with purely imanary frequencyivz , where

vz5A2 f 3/4. ~A4!

The corresponding imaginary period is nothing but theriod of the instanton. Alternatively,vz can be viewed as thevibration frequency around the saddle point in theinvertedpotential. In a harmonic potential, the action of an orbitsimply ~within a 2p factor! the ratio of its excitation energy~with respect to the equilibrium point! to the frequency. Thisyields the~imaginary! action of the instanton given by Eq~8!.

The harmonic approximation around the saddle pointalso be used for the calculation of the stability matrix of tinstanton. Indeed, as the harmonic potential separates cpletely in a transverse and a longitudinal component,monodromy matrix of the instanton in each transverse dirtion, after propagation during timet, is simply of the form

S cosvrt 2sinvrt

sinvrt cosvrt D . ~A5!

The stability matrix of the instanton is obtained by evaluing the monodromy matrix at the period of the instantont52ip/vz :

W5S cosh2pvr

vz2 isinh

2pvr

vz

isinh2pvr

vzcosh

2pvr

vz

D . ~A6!

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EXAMPLE OF CHAOTIC TUNNELING: THE HYDROGEN . . . PHYSICAL REVIEW A68, 062110 ~2003!

In our case, the ratiovr /vz is very large, so that thehyperbolic trigonometric functions can be approximatedan exponential, yielding

A2det~W2I !'expS pvr

vzD . ~A7!

For the three-dimensional hydrogen atom, the stabimatrix is a 434 matrix which actually splits in two 232identical blocks~along thex andy directions! of type ~A6!.Thus, contribution~A7! must be squared to get the corresemiclassical contribution. In contrast, for the simplifitwo-dimensional model, there is only one such contributiIf one uses the approximate value~A3! in Eq. ~A7!, onefinally gets contribution~9!.

The last ingredient in the semiclassical approximationthe stability matrix of the real periodic orbit in the inne

A

Z.

. E

ys

,

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y

.

s

potential well. As explained in the main text, the seriesbifurcations taking place in the vicinity of the saddle-poienergy implies that this matrix changes rapidly withe. Onthe other hand, whene is varied, the dynamics inside thpotential well is only weakly affected: the main effect is ththe electron spends less or more time in the immediatecinity on the saddle point. As the transverse potentialrather steep there, the stability matrix varies a lot. Thmodifications are essentially described by a multiplicationa matrix similar to Eq.~A5!. A small variation of the periodof the orbit is enough to affect strongly the matrix. Howevit is the product of the stability matrices of the instanton athe real periodic orbit which describes the semiclassical ctribution, Eq. ~4!. As it is the very same matrix type~A5!which contributes to the two stability matrixes, it turns othat themodulusof det(WM2I ) actually depends weakly one as shown in Fig. 7.

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