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Kramers map approach for stabilization of

arXiv:chao-dyn/9303014v1 23 Mar 1993Kramers map approach for stabilization ofhydrogen atomin a monochromatic fieldD.L.SHEPELYANSKY(a)Laboratoire de Physique Quantique, Universit´e Paul Sabatier,118, route de Narbonne, 31062 Toulouse Cedex, FranceAbstract:The phenomenon of stabilization of highly excited states of hydrogen atomin a strong monochromatic field is discussed. Approximate description of dynamics by theintroduced Kramers map allows to understand the main properties of this phenomenonon the basis of analogy with the Kepler map.

Analogy between the stabilization and thechanneling of particles in a crystal is also discussed.Submitted to Physica D, February/March 19931

1IntroductionDuring the last years the phenomenon of stabilization of atom in a strong laser fieldattracted a great deal of attention [1]. While the existence of the stabilization of atom hasbeen clearly demonstrated in the numerical experiments the clear analytical criterion ofstabilization is still absent.

Usually it is assumed that stabilization condition is satisfiedif the energy of the laser photon is larger than the electron coupling energy and theamplitude of electron oscillations in the field is large in comparison with Bohr radius [2].However, the recent investigations of the corresponding classical problem demonstratedthat stabilization remains also in the classical atom [3, 4], where the above conditions areviolated. The physical explanation of this phenomenon and the condition of stabilizationwere given in [3, 4] but the detailed explanation of the effect still remains an open problem.For a better understanding of this stabilization I introduce here a one-dimensional atommodel which I will call Kramers model (having in mind that it arose from the Kramers- Henneberger transformation).

Numerical analysis of this model allowed to constructan approximate Kramers map which describes the process of energy excitation and givesconditions of classical ionization. In some sense the obtained Kramers map is quite closeto the Kepler map [5] which describes the motion in the limit of relatively small field.Indeed, even in the strong field the change of the electron energy happens only when theelectron passes near the nucleus while far from it the electron follows the Kepler orbit.The paper is constructed as follows.

In the section 2 a brief description of the Keplermap is given since analogy with this map can be useful in the stabilization regime. Inthe section 3 a qualitative explanation of the stabilization is presented.

The numerical2

analysis of the introduced one-dimensional Kramers model and the derivation of theKramers map are carried out in the section 4. In the section 5 I discuss analogy betweenthe stabilization and the channeling of electrons in the crystal.

In the conclusion thepossibilities of experimental observation of stabilization of Rydberg atoms are discussed.2Kepler MapAfter the pioneer experiments of Bayfield and Koch in 1974 [6] the problem of mi-crowave ionization of highly excited states of the hydrogen atom has been investigated bymany groups (see [7] and Refs. therein).

The fast ionization observed in the experimentswas really surprising since about 100 photons were required to ionize the atom. The typ-ical experimental conditions were the following: n0 ≈70, ǫ0 = ǫn40 ≈0.05, ω0 = ωn30 ≈1where n0 is the principal quantum number of initially excited state, ǫ and ω are thestrength and the frequency of microwave field (here and below we use atomic units).

Theclassical dynamics depends only on the rescaled values ǫ0 and ω0.For the understanding of the process of ionization in linearly polarized field it isconvenient to use the one-dimensional atom model [7, 8, 5]. The investigations of one-dimensional model showed that for high microwave frequency (ωn3 > 1) the dynamicsof the system, which originally is ruled by the continuous Hamiltonian equations, can bedescribed by the Kepler map [5]:¯N = N + k sin φ,¯φ = φ + 2πω(−2ω ¯N)−3/2(1)Here k = 2.58ǫ/ω5/3, N = E/ω has the meaning of the number of absorbed or emitted3

photons (E is the energy of the electron), φ is the phase of microwave field at the momentwhen the electron passes near the nucleus. The bar denotes the new values of the variablesafter one orbital period.The physical reason due to which the motion can be quite accurately [5] described bythe simple area-preserving map is the following: when the electron is far from the nucleusmicrowave field leads only to a small fast oscillations which doesn’t modify the averageenergy and the Coulomb trajectory of the electron.

The change of energy happens only atperihelion where the Coulomb singularity leads to a sharp increase of the electron velocity.Ionization takes place when the energy of the electron becomes positive after a pass nearthe nucleus N > 0. Then the electron goes to infinity and never returns back.

Thereforefor the map (1) ionization is equivalent to absorption of trajectories with N > 0.To find the chaos border in the Kepler map we can linearize the second equation in(1) near the resonant (integer) values of ωn3 obtaining the Chirikov standard map [9]:¯N = N + k sin φ,¯φ = φ + T ¯N(2)with T = 6πω2n5. The global chaos appears for K = kT > 1 that determines the criticalfield strength above which the classical atom is ionized.

In this regime excitation goesin a diffusive way with the diffusion rate D = (∆N)2/∆τ = k2/2 where τ measures thenumber of orbital periods of the electron.The first numerical and analytical investigations of the quantum one-dimensional atommodel [8] showed that quantum effects leads to the suppression of classical diffusion. In-deed, in the quantum case the variables (N, φ) becomes the operators with the commuta-tion rule [N, φ] = −i and the system is locally described by the quantum kicked rotator4

[10]. The photon number is analogous to the level number in the kicked rotator and theexcitation probability decreases exponentially with the number of absorbed photons sothat the ionization rate is proportional to WI ∼exp(−2NI/lφ).

Here NI = n0/2ω0 isthe number of photons required for ionization, lφ = D = 3.33ǫ2/ω10/3 is the localizationlength.For lφ << NI quantum ionization is exponentially small in comparison withthe classical value. However, for lφ > NI the diffusion is delocalized and the process ofionization is close to the classical one.In the 3-dimensional atom the Coulomb degeneracy leads to a slow motion alongenergy surface that allows to describe the excitation in energy also by the Kepler map witha small change of constant k. The motion along the energy surface has some additionalintegral of motion that explains the existence of localization in 3-dimensional atom [5, 11].Recently the existence of localization in the 3d case was reconfirmed in [12].Quantum localization of classical chaotic ionization has been observed in the mi-crowave experiments with hydrogen [13, 14] and rubidium [15] atoms.

Numerical simula-tions with the quantum Kepler map [16] reproduce the 10% ionization threshold obtainedin the laboratory [13]. The theoretical prediction for the quantum delocalization borderwas also observed in the skilful numerical simulations [12].Being very successful in the description of energy excitation the Kepler map, however,cannot be applied for the case of very strong field.Indeed, in its derivation it wasassumed that the change of energy after one kick kω is much larger than the energyof free oscillations ǫ2/2ω2.

This gives the condition of applicability of the Kepler map5

picture [5]:ǫ << ǫATI ≈5ω4/3(3)Let us note that this condition is independent on the initial state since n0 doesn’t enterdirectly in the expression for ǫATI.In the one-dimensional case for ǫ >> ǫATI a collision with the nucleus, being unavoid-able, goes in a fast way like with an elastic wall leading to a prompt ionization [5]. Inthe two-dimensional case for zero magnetic quantum number m such collision also alwaystakes place if the amplitude of free oscillations ǫ/2ω2 is larger than the unperturbed dis-tance between the electron and the nucleus in perihelion l2/2 (l is the orbital momentum).This gives the condition of prompt ionization for l > (3/ω)1/3 [17]:ǫ > ω2l2/4(4)where it was assumed that l is few times less than n. For l < (3/ω)1/3 ionization is ruledby the Kepler map and for ǫ0 > ω02/3/2.6 prompt ionization takes place after one orbitalperiod (see (1)).

Therefore, there is no stabilization of classical atom in the strong fieldfor m = 0. However, for high orbital momentum the atom remains stable up to very highfield values.3Stabilization BorderWhile for the magnetic number m = 0 ionization always takes place in a sufficientlystrong field the case of nonzero m is much more interesting.

Indeed, for the linear polar-ization of the field the projection m is an exact integral of motion and the created by it6

centrifugal potential gives a possibility to avoid a collision with the nucleus. To analyzethe motion in the strong field it is convenient to use the oscillating Kramers - Hennebergerframe [1] and cylindrical coordinates in which the Hamiltonian has the form:H = pz22 + pρ22 + m22ρ2 −1(ρ2 + (z −ǫω2 sin(ωt))2)1/2(5)If the frequency of the nuclear oscillations is large enough (the condition will be givenlater) then in first approximation the nucleus can be considered as a charged thread witha linear charge density σ slowly dependent on z: σ(z) = ω2/(πǫ(1 −(zω2/ǫ)2)1/2).

Then,for small z and ρ the Hamiltonian of averaged motion takes the form [3, 4]:Have = pz22 + pρ22 + m22ρ2 + 2σ(z) ln(ρω2ǫ )(6)The constant under the logarithm takes into account that for ρ >> ǫ/ω2 the couplingenergy becomes much less than ω2/ǫ. From (6) one easily finds the position of the potentialminimum ¯ρ =pπǫ/2m/ω and the frequency of small oscillations Ω= 2√2ω2/(πǫm) (forz << ǫ/ω2).

The depth of the potential or the energy required for ionization of atom isapproximately I ≈2ω2L/πǫ with L = ln(2ǫ/(eπω2m2))/2. The minimal distance betweenthe nucleus and electron is determined by the condition I = m2/2(ρmin)2 giving:ρmin = m2ωrπǫL(7)The physical reason for the growth of the minimal distance with the field strength isthe following: with the increase of the field the amplitude of the field oscillations growsleading to the decrease of attractive Coulomb force while the centrifugal potential remainsthe same.7

The averaged description of the motion (6) is correct if the frequency of field oscilla-tions ω is much larger than the frequency Ωof oscillations in ρ. In that case the averagedHamiltonian (6) is the constant of the motion with adiabatic accuracy and ionization ofatom doesn’t take place.

This gives the stabilization border [3, 4]:ǫ > ǫstab = β ωm(8)where β is some numerical constant. The same estimate can be obtained from the con-dition that the change of energy ∆E during the collision between the electron and thenucleus is smaller than I.

Indeed, the change of the momentum is ∆p ≈∆t/ρmin2 ≈ω/(ǫρmin) and the change of the energy ∆E ≈ω2/(ǫ2ρmin2) is less than I if (8) issatisfied.It is interesting to note that the stabilization border (8) can be written asvn = ǫ/ω > vmax where vn is the typical velocity of the nucleus and vmax = 2/m is themaximal velocity of the electron in the atom without the external field.Another condition intrinsically used in the derivation of (6) and (8) is ρmin < ǫ/ω2which gives ǫ > m2ω2. Also, there are two qualitatively different situations depending onthe ratio between m and (3/ω)1/3.

In the case m << (3/ω)1/3 (stabilized atom regime)we have m2ω2 << 5ω4/3 << βω/m = ǫstab. For small field amplitude (3) the excitationis described by the Kepler map and the complete ionization after one orbital period ofthe electron takes place for ǫ0 > ω02/3/2.6 [5].

Between this border and above chaosborder ǫc0 = 1/49ω01/3 ionization goes in the diffusive way which is also relatively fast.However, for the more strong field (8), when the Kepler map picture is not valid (see(3)), atom becomes stable. The case of opposite inequality is less impressive.

Indeed,for m >> (3/ω)1/3 (stable atom regime) we have βω/m << 5ω4/3 << m2ω2 and atom8

remains stable (nonionized) up to ǫ ∼m2ω2 as it was in (4) (l ∼m). Above this valuea significant portion (order of half) of atoms will remain stable since condition (8) issatisfied.

Finally, ionization takes place only when the value of ρmin (7) becomes largerthan the size of the atom 2n02 and the electron cannot be captured in the stable regionduring the switching of the field. This gives the destabilization borderǫdestab ≈16Lω2n02πm2(9)This border is also valid for the case m << (3/ω)1/3.

Of course, in that case the stabiliza-tion can be observed only for the time of field switching Tsw less or order of one orbitalperiod of the electron. Otherwise a collision with nucleus will take place at field strengthǫ < ǫstab and atom will be ionized.The results of numerical simulation of ionization process of system (5) are presentedon the Fig.1.

The stabilization probability Wstab = 1. −Wion is given for different fieldstrengths ǫ0 and frequencies ω0.The ionization probability Wion was defined as therelative part of the trajectories with positive energies after field pulse.

The initial distri-bution of 100 trajectories (25 for ω0 = 1000.) corresponded to a quantum state with fixedspherical quantum numbers (fixed actions and equipartition in conjugated phases).

Theinitial value of orbital momentum was equal to l/n0=0.3 and its projection was equal tom/n0=0.25. The time of field switching (on/off) measured in the number of field periodswas chosen to be equal to Tsw = ω0 (one unperturbed orbital period of the electron).The pulse duration of the field was Tint = 500ω0 (500 orbital periods).

The data clearlydemonstrate the stabilization of atom for field strength larger than some critical value. Itis convenient to define the stabilization border as the field strength ǫstab0(20%) for which9

Wstab = 0.2. The dependence of ǫstab0(20%) on ω0, extracted from the data of Fig.1 canbe well fitted by the theoretical expression (8) with β = 12 in the wide frequency range(see Fig.2 of [3]).

This dependence continues up to ω0 = 1000 where we enter in the stableatom regime with m > (3/ω)1/3 and where stabilization disappears in agreement withabove theoretical arguments (see Fig. 1).

However, the stability of atom in that case isof the other nature than it was in [17] since the condition (4) is strongly violated. So, forsuch strong fields the stability of atom is based on the same physical grounds (8) as inthe stabilized atom regime for m << (3/ω)1/3.

The numerical check of the dependenceof stabilization border ǫstab0(20%) on m as well as the destabilization border (9) on ω0demonstrates good agreement with the theory (8)-(9) [3].4Kramers MapIt is important to stress that according to (8) the stabilization can take place evenwhen the size of electron oscillations α = ǫ/ω2 is much less than the unperturbed size ofthe atom n20. An example of the motion in this case is presented on the Fig.2.

In sucha case the electron follows the usual Kepler elliptic orbit and his energy (the size of theorbit) can be changed only during his fast passage near the nucleus. In this sense we canexpect that the motion can be effectively described by some map analogous to the Keplermap.To construct such kind of map let’s introduce a simplified one-dimensional Kramers10

model given by the HamiltonianH = pρ22 + m22ρ2 −1(ρ2 + ǫ2ω4 (sin γ + sin(ωt))2)1/2(10)This model is obtained from the Hamiltonian (5) by neglecting the changes of z andconsidering z = −ǫ/ω2 sin γ as a constant. In other words electron collides with the lineρ = 0 always at the same z value.

The physical reason for that is the following. Thecollision of the electron with the nucleus is analogous to a collision of a fast heavy particlewith the light electron.

In such collision the change of energy (velocity) takes place mainlyin the perpendicular ρ-direction, while the velocity in z-direction remains practically thesame. According to this physical picture the model (10) mainly presents the changes inρ-direction.

In that sense it is quite different from the well known one-dimensional atommodel of Eberly [1] which implicitly takes into account the change of energy (velocity)only in z-direction. Also in [18] the authors considered the velocity change only in z thathas led them to a higher stabilization border than (8), while the estimate for ρmin hasbeen found correctly (see (7)).According to the analogy with the Kepler map we can expect that the change of theelectron energy in the model (10) will take place only when the electron passes near thenuclear and that it will depend only on the phase of the field φ = ωt at that moment.If also the size of the orbit is much lager than the size of the nucleus oscillations (α =ǫ/ω2 << n02) then the change of the phase is given by the Kepler law and is the same asin (1).

Basing on these arguments we can assume that the dynamics of energy excitationis governed by the Kramers map of the following form:¯E = E + Jh(φ),¯φ = φ + 2πω(−2 ¯E)−3/2(11)11

with E = ωN, where as in (10) N is the photon number, the maximum change of theenergy is given by a constant J and the unknown function of the kick h(φ) varies in theinterval [-1,1].To check the validity of this map I integrated the continues equations of motion forthe model (10) and plotted the change of energy as a function of the field phase at themoment when the value of ρ took one of its minimal values (pρ = 0). Such approachallows to find the kick function h(φ) the examples of which are presented on Figs.3,4.The numerical results clearly demonstrate that the function h exists.

However, it hasa quite unusual property. Indeed, some values of φ never appear (even if the numberof periods was increased in 20 times).These values of φ are approximately equal toπ + γ, 2π −γ and correspond to that values of the field at which the nucleus passes viathe point of collision z = −ǫ/ω2 sin γ.

A more close consideration of motion near thesespecial φ-values shows that the electron remains during some small time interval (withincorresponding phase interval ∆φ) near the nucleus making one (Fig.3) or two (Fig.4)oscillations in ρ of very small amplitude so that the value of ρ remains practically (butnot exactly) the same.This gives correspondently two (or three) values of the phaseφ with the same change of ∆E since the value of E was determined in the aphelion.This of course puts the question about the derivation of the Kramers map in some othersynonymous form. However, the main properties of the motion can be derived alreadyfrom the approximate representation (11) where we will define the function h in the emptyintervals by connecting the last points at the ends of the interval by straight line.Defined in such a way the Kramers map has the properties quite similar to the Kepler12

map. Indeed, the function h is close to cos φ and the approximate chaos border in (11)can be defined by the linearization of the second equation giving:K = 6πωJn5 > 1(12)where we used substitution E = −1/2n2.

According to this criterion and in agreementwith the numerical data the motion is chaotic for the cases of Figs. 3,4.

If to introducek = J/ω, which will give the number of absorbed photons after an orbital period, we willget the same formulas for the diffusion rate D = k2/2, the localization length (l = D) andthe ionization time τion = N 2I /D as in the Kepler map. In this sense the most importantproblem is the definition of the dependence of J on the parameters of the system.According to the results of the previous section the amplitude of the kick J mustdecrease exponentially with the increase of the stabilization (adiabatic) parameter S =ǫm/ω ∼ω/Ω(see (7),(8)).

This expectation is in agreement with the results presentedon Fig.5. Indeed, the exponential decrease of J with the field strength, and thereforestabilization, are evident.

Let us at first discuss the properties of J for nonzero values ofγ. Even though the value of energy for the cases of Fig.5 was quite small neverthelessthere is some dependence of J on energy.

An example of such dependence is presentedon the Fig. 6.

We see that lnJ depends on energy E approximately in a linear way andgoes to a constant value for E = 0. This means that in the limit n2o >> ǫ/ω2 the valueof J is independent from n0.

This result is consistent with the above arguments that thechange of the energy takes place only in the small vicinity of the nucleus. However, in thedifference from the Kepler map it is necessary to have quite strong inequality −αE << 1to neglect the dependence of J on E. We will try to explain this fact later.

In this regime13

of small energies the main change of the phase of the field between collisions (the secondequation in (11)) is obviously given by the Kepler law.To determine the dependence of J(E = 0) on the parameters it is convenient to fixthe stability parameter S that allows to eliminate the strong exponential dependence andto find the factor before the exponent. The numerical results are presented on Fig.7.

Thevalues of J(E = 0) were obtained from nonzero energies by linear extrapolation to E = 0(see Fig.6). The numerical data clearly show that for the fixed S the value of J(E = 0)is independent on the frequency and is inversely proportional to m2.

In principle thefactor 1/m2 gives simply the correct dimensionality however the independence on theother dimensionless parameter ν = mω1/3 is not so obvious.Combining all the obtained numerical results we can present the dependence of thekick amplitude J on the parameters for | E | ǫ/ω2 << 1 in the following form:J = g1sinγm2exp(−(g2 −g3ǫE/ω2)ǫm/ω)(13)where g1,2,3 are some functions weakly dependent on γ. For γ = 0.6 we have from Figs.5-7 that g1 ≈0.13, g2 ≈0.19, g3 ≈0.08.

The numerical data for other values of γ showthat the fitting parameters vary not more than in 2 times for practically the whole intervalof γ. For example, g1 = 0.1 and 0.2, g2 = 0.13 and 0.21, g3 = 0.045 and 0.1 for γ = 1.2and 0.3 correspondingly.To understand the numerically obtained formula (13) for J it is possible to makethe following estimate.

Taking the partial time derivative from the Hamiltonian (10) weobtain the expression for the change of the energy after one orbital period:∆E = ǫ2ω4Z cos(η + φ)(sin γ + sin(η + φ))dηρ3/2(η)(14)14

where η = ωt and in the denominator we neglected the term with ǫ2/ω4 in comparisonwith ρ2. We can assume that near the nucleus the time dependence of ρ is the same asfor a free electron with momentum m that gives ρ2(t) = ρ20 +v2t2 where ρ0 is the minimaldistance from the center and v is the velocity of the electron far from the center.

Forthis free motion with the fixed momentum m we have the relation: ρ20 = m2/v2. Forthe velocity it is possible to use the following expression: v2 = ω2/Cǫ + 2E.

Where thefirst term takes into account the fact that the energy must be measured in respect to theminimum of the effective potential (see (6)) and C is some unknown constant. It is easyto see that C determines the minimal distance ρ20 = Cǫm2/ω2 for E = 0.

In principal thevalue of C depends on γ.After substitution of all these expressions in (14) we obtain the following estimateJ ∼S2m2(m3/2ω1/2) sin γexp(−CS/(1+2CǫE/ω2)), h(φ) = cos(φ), S = ǫm/ω(15)Of course, the presented derivation is not exact. However, it reproduce quite well theexponential dependence (13) (while the factor before the exponent is not in agreementwith the dependence obtained from the numerical simulation).

Indeed, numerically h(φ)has maxima near 0 and π. Comparison of (15) with (13) gives g2 = C and g3 = 2C2.

Thevalue of C can be defined directly from the numerical simulation of the one-dimensionalKramers model for different γ. The comparison of the g2 with C is presented on the Fig.8showing the good agreement with the prediction.

The ratios of the numerical value ofg3 (see above) to the theoretical value 2g22 are equal to 1.13, 1.1, 1.33 correspondinglyfor γ = 0.3, 0.6, 1.2 and are also in good agreement with the theoretical estimate. In thefuture estimates we will use the expression (13) with the theoretical substitution for g215

and g3. Let us also mention that for γ = 0 we get from (14) that g2 = 2C (from Fig.5the ratio to the theoretical value is approximately 1.1) and h(φ) = sin(2φ) that is quiteclose to the numerical data.

Further theoretical analysis is required to obtain the factorbefore the exponent in (13).While still there are some unclear questions with the construction of the Kramersmap the approximate consideration made above and the analogy with the Kepler mapallow to understand the main properties of motion. If the number of photons required forthe ionization is large then, as it was for the quantum Kepler map, it is possible to havediffusive excitation and quantum localization of chaos.

However, due to high values ofthe frequency it is also quite easy to have a situation when one photon can already leadto ionization. In this case for k = J/ω < 1 the one-photon ionization rate (per unit oftime) is given by the perturbation theory and as for the Kepler map (see [5]) it is equal:Γ ≈J28πn3ω2(16)For J > w approximately a half of probability is ionized after one orbital period ( in (11)as in the Kepler map the orbit is ionized if after a kick E > 0).

From (16) it is clearthat we may have long living states if the field is sufficiently strong. From the quantumview point one of the most interesting cases is the case of small m. In this case we needto make substitution m →m + 1 since as it is well known the correct quasiclassicalquantization leads to the appearance of the effective centrifugal potential even for zeroorbital momentum.

That gives the stabilization border ǫ > 10ω for m = 0.Finally let us mention that in (11) we assumed that J is independent on energy. Totake into account this dependence we need to put in the first equation J = J( ¯E) and in16

the second equation to add the phase shift ∆φ = dJ/d ¯Ef(φ) with h(φ) = −df(φ)/dφ. Inthat way the map will remain canonical.5Channeling AnalogyHere I would like to discuss the analogy between the phenomenon of stabilization ofatom in strong field and the channeling of particles in a crystal (see for example [19]and Refs.

there in). Let’s consider the electron moving in the crystal with the velocityv ≈c = 137 (we will consider nonrelativistic case).

Then in the frame of the movingelectron its interaction with the protons in the crystal lattice will have approximatelythe form (5) if to take into account the interaction only with a nearest proton. On thegrounds of that analogy we find that the effective distance between atoms in the crystala and the velocity of the electron are equal to:a = ǫω2 , v = ǫω(17)The frequency of perturbation is ω = v/a so that ǫ = v2/a.

Since in the crystal thedistance between the atoms is approximately the same in all directions the analogy is validfor ǫ/ω2 > n02. The necessary condition of channeling is that the critical injection angle θmust be much less than one that implies: θ ≈v⊥/v ≈1/(v√a) ≈(ω4/3/ǫ)3/2 << 1.

Thisis the condition of unapplicability of the Kepler map (3). From the stabilization condition(8) it follows that channeling takes place for electrons with momentum m > 10/v.

Thisis always satisfied for fast electrons with v ≈137. The existence of channeling for veryenergetic electrons (that corresponds to strong field for stabilization problem) gives one17

more evidence for existence of stabilization of atom in strong field in the regime when onephoton frequency is larger then the ionization energy.6ConclusionBasing on the Kramers map (11) and using its analogy with the Kepler map weobtained the estimate for the one-photon ionization rate (16). This ionization rate sharplydecreases with the stabilization parameter S = ǫm/ω.

Such stabilization for excited stateshas some interesting advantages in comparison with the stabilization of atom in the groundstate. Indeed, in this case stabilization can take place with ǫ << 1 and ω << 1.

Thisleads to a large energy difference δE between the excited states and the ground state.So, the energy in an exited state is approximately (ǫ/ω)2/2 >> 1 while the energy ofthe ground state remains as in the unperturbed atom (it is not the case for ǫ, ω >> 1when the ground state is also stabilized since there the electron has the same energy offree oscillations). Due to that in the case of Rydberg stabilization it is possible to haveradiative transitions to the ground state with the radiation of X-ray photons.

For thefrequency of CO2 laser ω ≈1/300 (0.1 ev) and m=0 (or 1) the stabilization will takeplace for ǫ ≈1/30 (1.6 108 V/cm). The size of the atom will be larger than the sizeof the field oscillations α = ǫ/ω2 for n > 40.

According to (13) and (16) for the field ǫ= 5 108 V/cm and n=60 the life time of the atom will be about 5 105 orbital periodsor 10−6 seconds (we take for the estimate the case with γ = 0.6). Of course, to obtainsuch states the time of field switching must be less than the time of orbital period as wediscussed above.

Since recently it was predicted that the Rydberg atoms can form long18

living states (bands) in the solid state [20] (giving very high density of excited atoms) itwill be interesting to consider a possibility of stabilization not only for a separate atombut also for such Rydberg solid state.I had started to be interested in the problem of microwave ionization of hydrogenatom in far cold 1980 during the first visit of JeffTennyson to the group of Boris Chirikovat Novosibirsk. Now, after many years of researches by different groups, this problem stillcontinues to live by its own life as well as the memory about Jeffcontinues to live amongthe people who met him in Siberia.19

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Figure captionsFig. 1: Stabilization probability Wstab = 1.

−Wion is given for different field strengthsǫ0 and frequencies (ω0=0.3 (◦), 1. (∗), 3.

(+), 10. (✸), 30.

(△), 100. (✸), 300.

(△), 1000 (•)).Fig. 2: Example of trajectory for ω0 = 300, ǫ0 = 20000 with initial l/n0 = 0.3 andm/n0 = 0.25; 105 field periods are shown.Fig.

3: Example of numerically obtained kick function h(φ) in Kramers map (11) forǫ = 3 104, ω = 125, m=0.2, γ=0.6, E=-0.125 (so that effective n0=2), J=1.1 10−4.Near 200 orbital periods (points) are shown.Fig. 4: The same as Fig.3 with ǫ = 4104, γ=1.2 and J = 5.810−4.Fig.

5: Dependence of the kick amplitude J in (11) on stabilization parameter S = ǫm/ωfor ω = 125, m = 0.2, γ = 0 (◦); ω = 1000, m = 0.1, γ = 0 (+); ω = 125, m = 0.2,γ = 0.3 (open squares); ω = 125, m = 0.2, γ = 0.6 (points); ω = 125, m = 0.2,γ = 1.2 (full squares). For all cases E=-0.125.

Lines are drawn to adopt an eye.Fig. 6: Example of dependence of J on energy | E | for ǫ = 2.2 104, ω = 125, m = 0.2,γ = 0.6 (points).Fig.

7: Dependence of J on m for fixed stabilization parameter S = 35.2 and E=0; 9cases are shown for ω in the interval [10,1000] and m in the interval [0.05,0.6]. Thestraight line shows the dependence J ∼1/m2.22

Fig. 8: Dependence of C = ρ0ǫ/S2 on γ (full line).

Points give values of g2 to demon-strate theoretical relation C = g2.23

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