Ionization of excited hydrogen atoms by elliptically polarized fields: experiment vs. theories

Physica A 288 (2000) 98–118

www.elsevier.com/locate/physa

Ionization of excited hydrogen atoms by elliptically polarized elds: experiment vs. theories P.M. Koch ∗ , M.R.W. Bellermann Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800, USA

Abstract We present experimental results for the ionization of hydrogen atoms with an initial principal quantum number n0 in the range 31– 45, by an elliptically polarized (EP), !=2 = 9:904 GHz microwave electric eld with peak amplitude F. For this range, the scaled frequency n30 ! is relatively low, 0.04487– 0.1372. Some data show local maxima and minima in the F-dependence of the ionization probability Pion . Classical 3d Monte Carlo calculations carried out by Richards quantitatively reproduce the striking sensitivity of this behavior to EP, and classical theory shows it to be a resonant dynamical process. A dierent theoretical approach by Oks and Uzer uses quantal Floquet theory to show how this driven system is related to a hydrogen atom in crossed electric and magnetic elds. We show that their quantal theory does reproduce the position of some local experimental minima in Pion , but it also predicts some minima that do not occur and fails to predict others that do occur experimentally. Finally, we present EP experimental data for Pion vs. F that exhibit signi cant oscillations not anticipated by any theoretical approach. We c 2000 Elsevier Science B.V. All rights reserved. speculate on their origin. PACS: 32.80.Rm; 42.50.Hz; 05.45.Mt Keywords: Quantum chaos; Multiphoton ionization of atoms; Multidimensional nonlinear dynamics

1. Introduction Experiments on the ionization of hydrogen atoms with principal quantum number n0 by a linearly polarized (LP) microwave electric eld F sin(!t) have been a cornucopia of data. For recent reviews and commentaries that emphasize experimental data ∗

Corresponding author. E-mail address: [email protected] (P.M. Koch).

c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 4 1 7 - 9

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obtained at Stony Brook and compare them to other experimental data and classical, semiclassical, and quantal calculations, see [1–3]. One reason the experiments attract attention is because for a wide range of parameters the onset of ionization coincides classically with the onset of irregular (chaotic) trajectories. This system is a paradigm for the experimental=theoretical=numerical study of non-perturbative quantal dynamics in a time-dependent, low-dimensional, classically chaotic system. All previous LP experiments, as well as those in this paper for varying polarization, required the (net) absorption of many photons to reach the ionization continuum. (For the data in this paper, the number exceeds 100.) This, and many other quantal states being coupled by the strong driving eld, renders quantal perturbation theory questionable, if not useless, as a calculational method for simulating the experiments. Brute-force numerical integrations of the Schrodinger equation on large, fast computers are also used, but often the crucial insights into the complexities of the ionization dynamics have come from semiclassical or even fully classical theory and calculations. However, the microworld is not classical, so it is not surprising that some observed ionization behavior cannot be reproduced by classical calculations. This naturally spurs the development of non-perturbative, quantal theoretical methods. Dynamics is about time scales, which means comparing frequencies. Because of its special dynamical symmetries, the frequency of classical bound orbits of the unperturbed, non-relativistic 3d hydrogen atom at total energy E ¡ 0 depends only upon the principal action I0 . In atomic units (a.u.), 1 classically E = −(2I02 )−1 ; quantally, replace I0 with n0 . At E ¡ 0 there is only the Kepler frequency !K = @E=@I0 = I0−3 . Exemplifying Bohr’s correspondence principle, in the limit of n0 → ∞; !K is the frequency splitting between adjacent n-states, |
n| = 1. For large but nite n0 , !K is close to the average of the
n = ±1 splittings. When the excited hydrogen atom is driven at frequency !, the important frequency ratio is the scaled frequency !=!K = n30 ! ≡ 0 . Similarly, the scaled amplitude n40 F ≡ F0 is the ratio of the peak amplitude of the driving eld to the Coulomb eld at the Bohr (circular) orbit rn = n2 a.u. As was rst shown in Ref. [5], the resultant classical dynamics does not depend independently on !, F, and I0 ; it depends on the ratios 0 and F0 . However, a non-zero Planck’s constant ˝ spoils the classical scaling invariance in the quantal dynamics. Varying n0 ; !, and F in such a way as to keep 0 and F0 constant changes the size of the eective ˝; ˝˜ = ˝=n0 . For details see Ref. [1], which also describes the six dierent regimes of dynamic behavior so far discovered for LP microwave ionization of excited hydrogen atoms as n30 ! is varied. It is well known from pulsed laser experiments with tightly bound atoms that the polarization of an intense electromagnetic eld can strongly in uence atomic ionization when many photons must be absorbed. In a perturbation expansion 1

Atomic units (a.u.) are de ned by setting (2)−1 · Planck’s constant, ˝, the electron mass, me , the antielectron charge, e, and the electrostatic constant, 40 , all equal to one. De nitions and numerical values for a.u. are listed in Ref. [4]. When confronting experimental data obtained with real hydrogen atoms or its isotopes, one should use physical a.u. that result from substituting for me the reduced electron mass e .

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polarization-dependent electric dipole selection rules determine pathways through unperturbed states, so small changes in polarization may dramatically vary the ionization rate for xed peak eld amplitude. References for a few examples from the laser multiphoton literature are in Ref. [6]. Summarizing all the results obtained so far, from both laser multiphoton- and abovethreshold-ionization experiments on strongly bound atoms and microwave ionization experiments on weakly bound (Rydberg) atoms, there is no universally applicable rule about how variation of the polarization aects the ionization. There are only “rules of thumb” applicable to particular cases. As a recent review [7] stated, “the matter of polarization : : : is more complicated than previously realized”. Stimulated by previous laser experiments and by microwave ionization experiments that began to explore the polarization dependence [8–11], we began at Stony Brook a series of experiments on the polarization dependence (linear: LP; circular: CP; elliptical: EP) of microwave ionization of excited hydrogen atoms. Unlike the LP and CP cases, the EP case has no integrals of the motion: It has three degrees of freedom and is not conservative. Moreover, ionization requires strong elds that couple many unperturbed states, so it is unlikely to be amenable to any simple selection rule analysis for LP vs. CP vs. EP driving. Our rst results were published in Refs. [6,12] and reviewed in Ref. [3], which contains numerous references to theoretical papers of a number of groups who have treated the polarization dependence of the microwave ionization of hydrogen atoms. Ref. [6] examined the polarization dependence for hydrogen atoms with n0 =70; : : : ; 98 at 9.904 GHz, which covers the range of scaled frequency 0 = 0:517–1:417. This includes the main, pendulum-like resonance zone centered at 0 = 1. The experimental and theoretical works showed that this resonance zone exerts a controlling in uence that makes the ionization dynamics near the onset of ionization independent of polarization when a certain classically derived, amplitude scaling of the eld is used. This was an important result for the following reason: Using lower case d (upper case D) to represent spatial (phase-space) dimension, one can use phase portraits (Poincare surface-of-section plots) to visualize in a 2D plot the stroboscopically sampled (1d + time) classical dynamics. Colloquially, such dynamics is said to have 1 12 degrees of freedom. To make contact with quantal dynamics, one can project onto phase portraits the Husimi distributions of computed (1d + time) quasienergy wave functions [13,14]. For a 3d atom in the time-dependent driving eld, the dimensionality depends on the polarization. For LP, the conservation of the projection of the orbital angular momentum on the eld polarization axis reduces the dimensionality to 2d + time, or 5D, or 2 12 degrees of freedom. Because of a separation of time scales [15], the LP dynamics is well approximated near the onset of ionization by the (1d + time) model. Qualitatively, the driven atomic orbits stretch out along the LP axis and become increasingly 1d. This explains why the surface-state-electron model, reviewed in Ref. [16], has been able to provide good estimates for onsets of ionization measured with 3d hydrogen atoms in an LP eld, at least for 0 up to 2.8 or so [17]. (At higher 0 , it has been predicted [18,19] that this will break down; see also Ref. [20].) For EP driving elds, however,

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the dimensionality is too high for projections of the full dynamics onto a 2D plane, so one loses the phase portrait as a powerful tool for visualization. This paper continues the presentation begun in Ref. [12] of EP results at low-scaled frequencies. It is organized as follows. Section 2 describes the experimental method. Section 3 presents experimental data and compares them with the results of classical theoretical calculations carried out by Richards [12,21]. His 3d classical Monte Carlo calculations quantitatively reproduce the EP behavior observed experimentally, and his approximate classical analytical theory focuses on the role played by a resonant process that aects the EP ionization dynamics. Section 4 compares experimental data with the results of a quantal theory presented by Oks and Uzer [22]. They used quantal Floquet theory to nd the quasienergy states of the hydrogen atom in the EP eld and showed how this problem is related to that for a hydrogen atom in crossed electric F and magnetic B elds. Their quantal theory does predict the position of some of the local experimental minima in Pion , but we show that it predicts some minima that do not occur and fails to predict others that do occur. In Section 5 we present one case of an experimental maximum in Pion that seems to be predicted by their theory. But then we show EP experimental data that exhibit striking oscillations in Pion vs. F that have not been anticipated by any theoretical approach, classical or quantal. In Section 6 we speculate on what causes the oscillations and nish with some conclusions.

2. Experimental method 2.1. Apparatus Because the details of the experimental method are reviewed elsewhere [1–3,6,12, 23,24], this discussion is brief. Ions extracted from a hydrogen ion source were accelerated, focused, and de ected in a mass-analyzer magnet tuned to transmit protons. 14:6 keV H+ –Xe electron-transfer collisions produced fast hydrogen atoms with an approximately n−3 -weighted distribution of states. A static electric eld & 100 kV=cm ionized all H(n ¿ 9) atoms, while those with n . 6 radiatively decayed in ight. With parabolic quantum numbers (n; n1 ; |m|) labeling Stark substates, in a 29:2 kV=cm static eld half of the (7; 0; 0) population was driven into (10; 0; 0) by a CO2 laser. Another CO2 laser drove the transition (10; 0; 0) → (n0 ; 0; 0) in a static eld whose strength ranged from a few to a few hundred V=cm. This created a beam of H(n0 ) atoms that was collimated by a 0.21 cm aperture before it traversed the 9.904 GHz cavity described in Section 2.2. This paper presents data for certain n0 -values between 31 and 45. A surviving atom signal Ssurv consisting of energy-labeled protons was produced by ionization of excited atoms in a voltage-labeled, rectilinear, 9.8 GHz cavity, followed by electrostatic de ection, transmission through an electrostatic lter lens, and detection in a particle multiplier (see Ref. [23, Section 13.3c]). A uniform, longitudinal 3:8 V=cm eld before the 9.8 GHz cavity caused only atoms with n-values below the n cuto nqc ' 110 to contribute to Ssurv (see Ref. [1, Section 2.3.3]).

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Reduction of the signal Ssurv was caused by H(n0 ) atoms being either ionized or excited to nal n-values n ¿ nqc in the 9.904 GHz cavity. Calling these two contributions to the reduction in Ssurv an “ionization” signal [1], the “ionization” and survival probablities are related as P“ion” = 1 − Psurv . Experimentally, we measured Psurv , but we interpret this in terms of P“ion” ; see Ref. [1, Section 2.3.4]. For clarity in the remainder of this paper, we no longer write “ionization” and “ion”, but the presence of quotation marks should be understood. 2.2. Microwave cavity The cylindrical brass cavity had inner dimensions length L = 2:57 cm and diameter D = 6:350 cm. The beam traversed 0.26 cm diameter holes in the 0.159 cm thick entrance and 0.476 cm thick exit endcaps, respectively. With the cavity excited via two spatially orthogonal coupling slots in the entrance endcap to resonate degenerate TE121 modes, one could vary at will the polarization of the 9.904 GHz cavity eld. The on-axis eld was ˆ sin(!t) + Yˆ sin(!t + )) ; F(t) = (t)F(X

(2.1)

with 0661 and 066=2; 06(t)61 is the 152 cycle, half-sine pulse envelope seen by the atoms in their rest frame, and (X; Y; Z) are spatial coordinates in the laboratory frame. The ratio F=F of the peak amplitude F to the amplitude F of the Yˆ -component depends on the polarization. With an attenuator in one arm, to control , and a phase shifter in the other, to control , we used the atoms to ne-tune the polarization. Extinguishing the power in one arm (60:0002) created LP. To create EP one could either (i) keep  = 1 and vary  or (ii) make  ¡ 1 and keep  = =2 or (iii) vary both  and . Experimentally, it was more precise to use scheme (ii), but for comparisons with theory to be√made below, it will be easier to characterize the EP √ via scheme (i), for which F=F = 2 cos(=2) varies from 1, at CP, to 2, at LP. For EP elds we could set  to ±0:01. We could determine the absolute amplitude of F (or F) to 5% accuracy [3,6,12]; see also Refs. [25,26]. 2.3. Ionization curves Taken for LP, CP, and two values of EP, Fig. 1 shows representative ionization curves, Pion vs. F0 = n40 F, for two nearby n0 values. For both cases, the scaled frequency 0 is low, below 0.14 (or in the Regime-II described in Ref. [1]). Little polarization-dependence occurs near the onset of ionization. This is no longer the case well past the onset of ionization, where the CP curves “stretch out” and reach Pion = 1 only at a signi cantly larger value of F0 than do the other curves. Why this happens will be explained in Section 3. What produces the structure evident in the =0:45 EP curve for n0 = 42 will be explained classically in Section 3 and quantally in Section 4.

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Fig. 1. A survey of the polarization dependence for experimental ionization curves for n0 =42 and 45 (inset).

3. Polarization dependence at low 0 To explain why the CP ionization curves in Fig. 1 stretch out compared to those for other polarizations when 0 is low [12,21], we recall well-known features of ionization of excited hydrogen atoms by a static, an LP, and a CP eld. Hereafter using a.u. (see footnote 1) unless explicitly noted, the Hamiltonian in the laboratory coordinate frame (X; Y; Z) is H(t) = p2 =2 − 1=R + R · F(t). For a static eld, R · F(t) = ZF, the system is separable in parabolic coordinates (; ; ) [27]. Below the onset of ionization, all three classical actions I ≡ (I ; I ; Im ) [quantum numbers (n1 ; n2 ; m)] are conserved. Classically, there is a sharp threshold eld Fcrit (I) below which the motion with these actions remains bound; from the least robust orbit [quantally, m = n1 = 0; n2 = (n − 1)] to the most robust orbit [n1 = (n − 1); n2 = m = 0]; n40 Fcrit varies between 0.13 and 0.38. Tunneling through the -barrier allows ionization for F ¡ Fcrit ; for interaction times near 10−8 s and n0 ' 40, it lowers thresholds by 10 –15% [1,28]. For an LP eld, R · F(t) = (t)ZF sin(!t + ), with an initial phase; again, separability leads to conservation of m. The dynamics is quasistatic [1,29] if 0 is suciently small and away from exponentially sharp resonances discussed theoretically for 1d in Ref. [30]. The spatial reversal of F on each half-cycle interchanges n1 and n2 ; therefore, as Fig. 1 of [29] shows, for a uniform mixture of substates of a given n0 , classically a microcanonical ensemble, the ionization probability Pion rises from 0 to 1 as the scaled amplitude F0 varies between 0.115 and 0.17. This is just the beginning

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of the wider classical range, F0 = 0:13–0:38, lowered 10 –15% by tunneling, discussed above for the static eld case. In a frame (x; y; z) rotating at frequency !, the Hamiltonian is given by K(t) = p2 =2 − 1=r + (t)Fy + !Lz + 0:5(t)F[(y sin 2!t − x(1 − cos 2!t)] sin 1 + 0:5(t)F[x sin 2!t − y(1 + cos 2!t)] sin2 (1 =2) ;

(3.1)

where 1 = =2 −  measures the deviation from CP. For CP the Hamiltonian in the rotating frame consists of the rst four terms in Eq. (3.1), which describe the free hydrogen atom perturbed by a static eld term and the Coriolis term !Lz . It has the same form as that for the hydrogen atom in static crossed electric F and magnetic B elds if B is weak enough for the diamagnetic term (˙ B2 ) to be neglected; see, e.g., Ref. [31] and references therein. (It is also close to one known to give integrable motion [32–34].) For ! low enough the Coriolis term should have little eect on the ionization dynamics. Indeed, Ref. [21] showed that for

0 . 0:1, classical ionization curves calculated with (the CP case) and without (the static eld case) the !Lz term were very similar. Therefore, at low 0 the lack of interchange of n1 and n2 (see above) causes quantal CP ionization curves to stretch out and approach Pion = 1 at a higher value of F0 than for the LP case, as Fig. 1 shows. Now note in Fig. 1 the dierent shape of the n0 = 42 EP curve for  = 0:45. Fig. 2 shows a striking EP dependence for n0 = 42 over a ne range, from  = 0:42 to 0:5, with an experimental uncertainty
=0:01. Here there is a remarkable local variation of Pion with  at xed F0 . Over about the same range of , similar EP dependence was observed for n0 = 43 and 41, as will be shown later in this article in Figs. 6 and 7, respectively. For values of n0 just above and below this range, i.e., for n0 = 44 and 40, respectively, this hyper-sensitivity of ionization to EP went away. In the remainder of this section we brie y review the discussion presented in Ref. [12] to gain classical insight into the physical origin for the sensitivity of n0 = 41– 43 at 9.904 GHz to EP for values of  close to CP. Mathematical details of the classical theory are given in Ref. [21]. Section 4, will give a quantal explanation for the EP sensitivity. We begin with 3d classical Monte Carlo calculations (3dCL), the results of which [12,21] are shown in the insets to Figs. 2 and 3 and in Fig. 4. The 3dCL results were obtained by numerical integrations of Hamilton’s equations of motion. Except for being entirely classical, these calculations simulated all the important parameters of the experiment, such as the uniform substate distribution for atoms prepared in the initial n0 level, the driving eld parameters and pulse shape, and the n-cuto nqc . Allowing for the ±0:01 experimental uncertainty in , there is impressive agreement in Fig. 2 between the experimental data and 3dCL; similar agreement (not shown here) was obtained for the neighboring n0 values, 41 and 43. Also see below for n0 = 31. This agreement indicates that classical theory will be useful for understanding local hyper-sensitivity of the ionization dynamics to EP. But we need to understand what

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Fig. 2. The detailed elliptical-polarization dependence of the ionization probability for n0 = 42 at 9.904 GHz in steps of = = 0:01 [= = 0:02] from experimental data [from 3dCL calculations (inset)]. The eight lled circles and one cross (hard to see in the upper right-hand corner of the gure) are symbols marking the values of F0 at which a quantal theory described in the text predicts minima in the ionization curves for the relevant values of . As explained in the text, the vertical location of each symbol has no meaning other than to guide the eye to the relevant curve.

produces the structures that include local maxima and minima in some ionization curves. With the details of the complexities of the multidimensional dynamics being given in Ref. [21], the key point is that all perturbations vary little during an unperturbed Kepler period, so one can average over this fast motion, reducing the number of degrees of freedom, and then write the resulting equations for the mean motion in terms of the vectors X =√L − A and Y = L + A where L is the orbital angular momentum and both X and A = (p × L − r=r)= −2E is the Runge–Lenz vector. For a CP eld (1 = 0), q

Y rotate uniformly about the eld direction with the scaled frequency ! = !S2 + 02 , where !S = 3F0 =2 is a classical Stark frequency associated with the scaled amplitude F0 . For the ranges of F0 and 0 considered here, ! ¡ 13 . Perturbations of frequency 2 0 and strength proportional to 1 , which appear for an EP eld, √ can resonate with the mean motion when ! = 2 0 , i.e., when F0 ' F0r ≡ 2 0 = 3. For 0 = 0:1116 this gives F0r = 0:13; equivalently, this is F0r dropping from about 0.145 to about 0.13 as = increases from 0.42 to 0.5. This qualitatively agrees with the variation of the local maxima in Fig. 2. We use “2 0 resonance” to label this dynamics. The averaged equations of motion give a model for understanding how the 2 0 resonance aects the bound-state dynamics. With the full richness of the details being

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Fig. 3. The detailed elliptical-polarization dependence of the ionization probability for n0 = 31 at 9.904 GHz from experimental data [from 3dCL calculations (inset)]. The six double- lled circles and four double crosses (hard to see in the upper right-hand corner of the gure) are symbols marking the values of F0 at which a quantal theory described in the text predicts minima in the ionization curves for the relevant values of . As is explained in the text, the vertical location of each symbol has no meaning other than to guide the eye to the relevant curve.

given in Ref. [21], ionization is added post hoc to the model via the time dependence of Fcrit (t) = Fcrit (I(t)), which can be used to mimic the classical escape over the barrier when Fcrit (t) ¡ (t)F. (Recall that 06(t) is the pulse envelope.) This model, compared with exact calculations in [21], shifts one’s attention to understanding the temporal evolution of the classical actions I(t) and critical elds Fcrit (I(t)). Though the details are complicated, the analysis shows that if Im ¿ 0 [ ¡ 0] then Fcrit (t) decreases [increases] as (t)F increases through the resonance by an amount which increases as d=dt decreases. Numerical calculations show that the mean over Im ¿ 0 and Im ¡ 0 is dominated by the behavior of the former, thereby producing the observed local maximum in the ionization probability [12,21]. Data for n0 = 31–33 also exhibited extreme sensitivity to EP, but for lower values of  than for the 2 0 resonance. Fig. 3 shows results for n0 = 31 for eight values of  between 0:28 and 0:45 (F=F between 1.28 and 1.08). Observe the clear similarity between Figs. 3 and 2. The inset shows that the 3dCL results, obtained for the same eight values of , again reproduce the experimental polarization dependence. Though K(t), Eq. (3.1), has harmonic terms only at 2 0 , if we q extrapolate the frequency-matching condition to a “4 0 resonance”, viz., 4 0 = ! =

(3F0 =2)2 + 02 ,

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Fig. 4. The detailed elliptical-polarization dependence for n0 = 31 at 9.904 GHz for experimental data (solid curves) and for 3dCL calculations (inverted lled triangle: 422 orbits; upright lled triangle: 864 orbits; lled square: 1728 orbits).

this gives 0 = 0:39F0 . The classical static eld ionization threshold, F0 = 0:13, gives

0 = 0:05; at 9.904 GHz this gives n0 = 32, the middle of the three n0 -values where extreme sensitivity to EP was observed [12,24]. This result predicts that successive application in the rotating frame of classical perturbation theory will lead to resonances at even powers of 0 . The classical derivation has been accomplished, so far, only for the 2 0 resonance [21], but see also [22]. The top three curves in Fig. 3 show another rise in Pion near F0 = 0:17. For ve values of = in steps of 0.01, Fig. 4 compares experimental data in this region with the results of 3dCL. The rise, subsequent plateau, and even the mild drop in the data for = = 0:30–0:32, as well as its disappearance for higher values of =, is quantitatively reproduced by 3dCL. Ref. [12] speculated that this behavior is caused by a “6 0 resonance”. These low-frequency experiments showed that certain ranges of parameters gave hyper-sensitivity of the ionization to EP. The ionization data were reproduced quantitatively by 3d classical Monte Carlo calculations, and classical theory was able to provide an explanation for some of this behavior via a 2 0 resonance. The quantal Floquet theory [22] described in the next section shows that there are series of “even-photon” resonances in a rotating frame (also “odd-photon” resonances for some conditions), though, as far as we are aware, how the actual quantal dynamics evolves

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as the eld amplitude changes during the pulse envelope has not yet been worked out. This is important because the Monte Carlo and other calculations presented in Ref. [21] showed that the classical dynamics at the 2 0 resonance is strongly aected by changing the pulse envelope; moreover, whether the ionization increases or decreases, respectively, as the system passes through the resonance depends on whether the atomic electron is counter-rotating or co-rotating with the eld. Because of the quantitative success of the 3d classical Monte Carlo calculations, the classical theory needs to be extended to cases beyond the 2 0 resonance. The observed resonances are a striking example of the kind of the richness in the higher-dimensional dynamics, here (3d + time), of a driven quantal system. As was emphasized in Ref. [12], this EP-induced resonant dynamics even provides a mechanism for controlling the ionization. We emphasize that because our experiments recorded an ionization signal, two conditions must be ful lled to observe the resonant behavior explained either classically (this section) or quantally (next section): (i) the resonant dynamics must occur for the range of experimental parameters being explored; (ii) the microwave amplitude around the resonance must be such that “some”, but not zero nor complete, ionization takes place. However, recent work [35] in our laboratory showed how a resonant dynamical process in the ionization of 3d hydrogen atoms by collinear static and LP microwave electric elds can give n-selective resolution. With further development this will lead to a practical detector that would allow for recording the eect of EP-induced resonant dynamics among the 3d hydrogen bound states at EP amplitudes below the onset ionization. This would expand the range of parameters over which the EP-induced resonances could be observed and studied and give much more detailed information, e.g., how the distribution of surviving n-states changes when experimental parameters are varied.

4. Quantal theory Exploiting and extending theoretical results published earlier (see, e.g., Refs. [36,37] and references therein), Oks and Uzer recently presented “a non-perturbative analytical solution to the problem of the Rydberg atom ionization in a strong elliptically polarized microwave eld with frequency much lower than the Kepler frequency”. 2 Their work is based on the use of an averaging method combined with Floquet theory for nding the quasienergy solutions of a time-periodic Hamiltonian. They work in a rotating frame, but not in the uniformly rotating frame described in the previous section. Instead, they work in a reference frame that rotates with the variable frequency of the EP electric eld, ’(t) ˙ = !(cos2 !t + 2 sin2 !t)−1 ; 2

Subsequently extended to high-frequency EP elds [38].

(4.1)

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where  = tan(=2) relates the ellipticity parameter  [LP:  = 0; CP:  = 1; EP: 0 ¡  ¡ 1] to the phase angle  [LP:  = 0; CP:  = =2; EP: 0 ¡  ¡ =2] introduced in Section 2.2 for parametrizing the EP eld. (For the CP case the frame rotation frequency ’(t) ˙ becomes the constant frequency !.) Ref. [22] presents expressions for the quasienergies in the frame rotating at ’(t); ˙ these are related to splittings of atomic energy levels in crossed electric and magnetic elds. The quasienergies depend on the EP parameter(s) and on the reduced eld strength v = 3n0 F=(2!) = 3F0 =(2 0 ) ;

(4.2)

where F is the peak eld amplitude, F0 = n40 F and 0 = n30 ! are the scaled peak eld amplitude and scaled frequency, respectively, for level n0 . Ref. [22] indicates that the results derived in the rotating frame are valid for 1¿¿v−1=2 ; these conditions were ful lled for all graphical comparisons presented in that paper and are ful lled for those presented later in this paper. The analysis in Ref. [22, Section 2] then proceeds back to the laboratory frame, making use of an averaging approximation for large phases and an expression valid for states within a xed n-shell that expresses matrix elements of the coordinate in terms of those for the angular momentum L. (It is unclear to us what limitation(s) their expression for the matrix elements places on the subsequent steps in the theoretical development.) This leads to expressions for an eective “electric interaction”, which is parametrized by an eective electric eld El; phys in the polarization plane, and an eective “magnetic interaction”, which is parametrized by an eective magnetic eld Bl; phys perpendicular to the polarization plane and whose dimensionless strength is within the range [ − 1; 1]. The expression for El; phys is so complicated that we do not write it here, but we do present their expression for the eective magnetic eld, Bl; phys = (2=) Arcsin[sin(vJ1 (v)=2)] ;

(4.3)

where J1 is the ordinary Bessel function. How these eective elds relate to ionization of the hydrogen atom by the EP eld is explained as follows. Increasing [decreasing] El; phys (the driving “force”) should increase [decrease] the ionization while increasing [decreasing] Bl; phys (a con ning “force”) should decrease [increase] the ionization. The decrease in ionization should be sharpest where El; phys passes through a minimum while |Bl; phys | passes through a maximum, and vice versa. In Ref. [22, Figs. 1 and 2] graphical comparisons were made for the EP experimental data for n0 = 42 and n0 = 31 that were published in Ref. [12]; see also Ref. [3]. We now discuss and extend those comparisons. According to Ref. [22], the maximal value |Bl; phys | = 1 occurs when |vJ1 (v)| = 1= = 1=tan(=2) :

(4.4)

Fig. 5 shows a graph of the function |vJ1 (v)| up to v = 12, which is high enough for the purposes of this article. The values of v that solve Eq. (4.4) occur whenever

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Fig. 5. A graph of the function |vJ1 (v)| from v = 0 to v = 12. The reduced eld strength v is de ned by Eq. (4.2). The values of v that solve Eq. (4.4) occur whenever the horizontal line at value 1= = 1=tan(=2) crosses the plotted function. (The parameters  and  characterize the ellipticity of the driving eld.) The dashed line corresponds to  = 0:48 ( = 0:9391), which is an EP eld close to CP. Notice that for this case two solutions occur for each vertical “lobe” of the plotted function. Each solution gives a dierent value of v that solves Eq. (4.4). According to the theory of Ref. [22], these solutions can correlate with minima in the EP ionization probability. Filled circles [crosses] are symbols marking solutions from the left-hand [right-hand] side of a lobe. The number of symbols stacked vertically show the number of the lobe producing the solution.

the horizontal line at value 1= = 1=tan(=2) crosses the plotted function. The dashed line in the gure corresponds to  = 0:48 ( = 0:9391), which is an EP eld close to CP. Notice that for this case two solutions occur for each vertical “lobe” of the plotted function. Each solution gives a dierent value of v = 3F0 =(2 0 ) and, therefore, a dierent value of the peak electric eld amplitude F at which |Bl; phys | takes on its maximal value 1. According to the theory of [22], minima in the EP ionization probability can correlate with these values of v. In the discussions below of the graphical comparisons between experiment and the results of the theory of [22], we will use the graphical symbols present in Fig. 5. A lled circle, or cross, will correspond, respectively, to a solution giving |Bl; phys | = 1 from the the left-hand or right-hand side of a lobe. The number of symbols give the number of the lobe. Note that with decreasing  (i.e., further excursion away from CP, for which |vJ1 (v)|= 1), there comes a point at which the dashed line rises above the rst (second, third, : : :) lobe of the function. Then, the values of F0 that produce |Bl; phys | = 1 will come only from lobes farther to the “right”.

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4.1. Discussion of graphical comparisons In Fig. 2, the single lled circles on eight of the curves show the values of F0 that give a solution of Eq. (4.4) from the left-hand side of the rst lobe of Fig. 5, for n0 = 42; ! = 2 × 9:904 GHz, and values of  decreasing from 0:5 (cp) to 0:43. No such solution exists for  = 0:42, the value for the top-most curve in Fig. 2. The right-hand side of the rst lobe leads to only one solution that will t into the gure, viz., F0 = 0:2248 for the cp case,  = 0:5. (The single cross for this case is hard to see in the gure.) No values of F0 coming from solutions from the second and higher lobes of Fig. 5 will t into Fig. 2. Important: Only the horizontal positions of the lled circle and cross symbols in Fig. 2 and other gures have meaning. They mark the values of F0 found by the solutions. The vertical positions have no meaning because the theory of Ref. [22] was not used to calculate ionization probabilities. Therefore, to help the reader, we placed each symbol vertically in the gures to be near the experimental curve with the relevant value of . As was commented in Ref. [22], where the rst such comparison was made for our n0 = 42 EP data from Ref. [12], a number of the lled circles showing where values of |Bl; phys | = 1 occur in the theory do lie near experimental minima in Pion . The agreement is impressive for four out of the eight cases and less so for the others. No experimental minimum appears near either symbol for the cp case. There is no obvious local minimum in the experimental curve for  = 0:42; the theory agrees, giving no solution that would lead to |Bl; phys | = 1 within the gure. Fig. 6 compares our experimental data for n0 = 43 (not previously published) with the eight theoretically predicted minima that fall within the gure. Five out of the six single- lled circles lie near local minima in the measured Pion . The predicted minimum for  = 0:49 fails to locate an experimental minimum. Nor were any minima observed near the two single crosses, though the experimental curves there are very near Pion = 1 and are close to the maximal value of F0 that was used. Fig. 7 compares our experimental data for n0 = 41 (also not previously published) with the eight theoretically predicted minima that fall within the gure. Here the agreement is poor, with no local experimental minima apparent. Further work will be needed to understand why there is fair or good agreement for minima for the data for two of the three n0 -values (viz., 42 and 43) near the 2 0 resonance discussed in Section 3, but poor agreement for the third, n0 = 41. We now make comparisons near the 4 0 resonance mentioned in Section 3. 3 For n0 = 31 Fig. 3 only double symbols appear within the range of the gure, six [ ve] from the left-hand [right-hand] side of the second lobe in Fig. 5. Because the double crosses are lost in the congestion of the upper right corner of the gure, we defer discussion of them to two paragraphs below. Three of the double- lled circles lie near local minima in the experimental Pion , two are signi cantly displaced, and one is near 3

See also Ref. [22, Fig. 2] and the discussion accompanying it.

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Fig. 6. The detailed experimental elliptical-polarization dependence of the ionization probability for n0 = 43 at 9.904 GHz in steps of = = 0:01. The six lled circles and two crosses are symbols marking the values of F0 at which a quantal theory described in the text predicts minima in the ionization curves for the relevant values of . As explained in the text, the vertical location of each symbol has no meaning other than to guide the eye to the relevant curve.

no local minimum. 4 The experimental curve for  = 0:30 has a local minimum near F0 =0:165, but Eq. (4.4) gives no solution that would predict a minimum there. (Similar remarks may also be possible for the  = 0:28 case.) Now we look back at Fig. 4 to see that this is just the region where the experimental rise [lack of rise] in Pion near 1 and F0 & 0:17 for  = 0:30; 0:31; 0:32 [0:33; 0:34] was quantitatively reproduced by 3d classical Monte Carlo calculations [12,21]. The  = 0:31 and 0:32 experimental curves (and perhaps also the  = 0:30 curve) have local minima. The “high-statistics” 3d classical Monte Carlo calculations for the  = 0:31 case likely reproduce the local minimum. In Fig. 8, which extends a bit beyond the range of  in Fig. 4, six double symbols appear within the range of the gure, two [four] from the left-hand [right-hand] side of the second lobe in Fig. 5. Both double- lled circles are near apparent experimental local minima in Pion , but the double crosses are not. For the experimental curve with the clearest “high-F0 ”-local minimum ( = 0:32), the double cross is closer to the local maximum. We may sum up the comparisons made above by remarking that the theoretical condition, Eq. (4.4), that maximizes |Bl; phys | has had some signi cant success in locating 4

These conclusions dier somewhat from those expressed in Ref. [22].

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Fig. 7. The detailed experimental elliptical-polarization dependence of the ionization probability for n0 = 41 at 9.904 GHz in steps of = = 0:01. The four lled circles and four crosses are symbols marking the values of F0 at which a quantal theory described in the text predicts minima in the ionization curves for the relevant values of . As explained in the text, the vertical location of each symbol has no meaning other than to guide the eye to the relevant curve.

the values of F0 at which Pion should pass through a local minimum. However, we need to understand why this success is not universal by doing more theoretical and experimental work on this fascinating and complex problem. 5. A local ionization maximum and some surprises In Section 4 we remarked that Ref. [22] predicted that experimental parameters taking El; phys through its maximum could produce a local maximum in Pion as F0 was varied. 5 The authors of that work remarked [one paragraph below their Eq. (31)] that at v = 3:83; El; phys reaches its absolute maximum at ellipticity  = 0:28. These parameters are close to those for an ionization curve we measured for n0 = 34 at  = 0:2, which gives  = 0:32. Fig. 9 shows this and three other ionization curves. Note the local maximum in the n0 = 34 curve near F = 540 V=cm, which corresponds to v = 3:56. This is near the value of v = 3:83 just mentioned. Moreover, the subsequent 5 Look at the many maxima displayed in, e.g., the experimental ionization curves in Figs. 2, 3 and 6. The authors of Ref. [22] consider these maxima to be of “geometric” origin, i.e., the result of a local maximum that must appear between two adjacent, physically caused minima. The author of Ref. [21], however, nds these same maxima to gure importantly in his theoretical treatment and discussion of the classical 2 0 resonance dynamics, which also produces the subsequent minima.

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Fig. 8. The detailed experimental elliptical-polarization dependence of the ionization probabilty Pion near 1 for n0 = 31 at 9.904 GHz. The two double- lled circles and four double crosses are symbols marking the values of F0 at which a quantal theory described in the text predicts minima in the ionization curves for the relevant values of . As explained in the text, the vertical location of each symbol has no meaning other than to guide the eye to the relevant curve.

minimum in the n0 = 34 curve occurs near F = 570 V=cm, or v = 3:76, which is far from any maximum in |Bl; phys | that would serve to suppress the ionization. (For the experimental parameters of interest here, the lowest value of F leading to a maximum in |Bl; phys | is F = 2:3 × 103 V/cm, which does not occur until the fth lobe of Fig. 5.) Therefore, we may have some evidence for a maximum in the experimental ionization yield being correlated with a maximum in the eective electric eld El; phys introduced in Ref. [22]. We need more data than just this one case. The other three experimental curves in Fig. 9 reveal surprises not anticipated by any of the theoretical calculations discussed above. Notice the distinctive oscillations that appear in the ionization curves we measured for n0 = 31–33 at  = 0:2. The many local maxima and minima in these curves are unlikely to be associated with any of the dynamics we have discussed earlier in this article. We also have indication of oscillations of some other ionization curves (not shown here), such as at  = 0:2 for n0 = 30 and 29 and at  = 0:3 for n0 = 32. 6. Conclusions Let us begin with the last observation we discussed above, namely the unexpected oscillations of curves in Fig. 9. What could be causing these oscillations? Earlier, He

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Fig. 9. Experimental 9.904 GHz ionization curves for four dierent n0 values, all taken for the same value of  = 0:2. The location of the maximum in the n0 = 34 curve is compared in the text to the predictions of a quantal theory [22]. The oscillations in the experimental curves for n0 = 31–33 have not been anticipated by any theory for microwave ionization of excited hydrogen atoms by EP elds.

Rydberg atom experiments [26,39] in our laboratory saw Stueckelberg oscillations in experimental signals recorded as a function of the peak microwave electric amplitude F of a half-sinewave-shaped pulse of microwaves lasting a few hundred eld oscillations. The oscillations constitue an interference eect that occurs when quasienergy states are coherently excited on the rising edge of the pulse-envelope and then recombine on the falling edge. These involve Landau–Zener transitions at an avoided crossing(s) of Floquet states and accrual of (dynamical) phase dierence(s) by each of the dierent quasienergy states. Obviously, the situation is simplest when only one avoided crossing dominates [26]. Theory could help in two ways to test our speculation that the oscillations in Fig. 9 are Stueckelberg oscillations. First, if they are the result of a quantal interference phenomenon, such oscillations cannot show up in 3d classical Monte Carlo simulations of these experimental curves. These calculations should be carried out for the cases in Fig. 9. Second, such oscillations should show up if suciently accurate quantal Floquet calculations could be done to supply the quasienergy potential curves needed for the EP-driven hydrogen atom. The Floquet methods of Ref. [22] could, perhaps, begin this process, but that work did not take into account any pulse-envelope-driven dynamics that would lead to Landau–Zener transitions and could produce Stueckelberg oscillations. Nor did it account for the continuum, though

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other 3d Floquet calculations with complex dilation have done this with LP elds [40] and CP elds [41]. The fully three-dimensional and time-dependent quantal dynamics at issue here will severely test any quantal calculations, whether they use Floquet methods or use a brute-force integration of the time-dependent Schrodinger equation. We hope such calculations will be carried out. Our observation of these oscillations and our earlier observation of the hyper-senstivity of the driven hydrogen atom to EP elds in a certain range of the parameters were unexpected. They are clear examples of the kinds of surprises in store when one studies the quantal and classical dynamics of a strongly driven, multidimensional Hamiltonian system. We are grateful that our work has stimulated the work of theoretical groups, and we echo the opinion that “it is increasingly recognized that multidimensional problems are the next challenge in the fascinating transition between classical and quantum mechanics” [38]. Finally, as be ts a paper appearing in a statistical physics journal, we conclude with some comments directly related to that eld. One focus of quantum chaos studies has been the exploration of the correspondence (or lack thereof) between the classical dynamics of Hamiltonian systems and the statistical properties of the energy spectra (wave functions, too) of their quantal counterparts. Random matrix theory has played a key role in this exploration. The Gaussian orthogonal ensemble (GOE), which applies to systems that are time-reversal invariant or have any other antiunitary invariance, is the ensemble of random matrices that has been of most relevance to studies involving atomic physics. Sacha et al. [42,43] however, presented what they believed to be the rst case of an experimentally realizable example of a quantal system that would not have such an invariance and, therefore, be applicable to the Gaussian unitary ensemble (GUE). The system is the hydrogen atom perturbed simultaneously by a static electric eld and a resonant, elliptically polarized microwave eld with arbitrary mutual orientation. Their initial studies have focused on microwave frequencies near the 1:1 (n30 ! = 1) and 2:1 (n30 ! = 2) pendulum-like resonances and small eld amplitudes that give a classical dynamics with mixed character, i.e., coexisting regular and irregular regions in the phase space. For eld orientations that break any antiunitary symmetry of the system, they see a clear eect on the statistical distribution of computed quasienergy levels. A subsequent paper [44] extends this work to the non-resonant case. Extending these calculations to treat the case of EP+ static elds strong enough to cause ionization of the 3d hydrogen atoms will de nitely be challenging. It will also be challenging to do such experiments. There is clear incentive to try them.

Note added in proof In Section 4 of this paper we followed the lead of Oks and Uzer in [22] and focussed on the use of maxima in the eective magnetic interaction (Eq. (4.3) herein;

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Eq. (23) in [22]) for explaining or predicting local minima in experimental EP ionization curves. This ‘magnetic’ approach was successful in [22] for the set of experimental EP data available to the authors of that paper. But the comparisons made herein with an expanded set of experimental EP data show its shortcomings. Oks and Uzer extended their work after receiving a copy of our manuscript and present in E. Oks and T. Uzer, J. Phys. B 33 (2000) L533–L538 ‘guiding principles’ for using both the eective magnetic and electric interactions to explain and predict local extrema in experimental EP ionization curves. Acknowledgements We appreciate continuing nancial support from the US National Science Foundation and, early on, from Schlumberger–Doll Research. We appreciate helpful correspondence or conversations with E. Oks, D. Richards, and T. Uzer. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

P.M. Koch, K.A.H. van Leeuwen, Phys. Rep. 255 (1995) 289. P.M. Koch, Physica D 83 (1995) 178. P.M. Koch, Acta Phys. Pol. A 93 (1998) 105. W.E. Baylis, G.W.F. Drake, in: G.W.F. Drake (Ed.), Atomic, Molecular & Optical Physics Handbook, American Institute of Physics, Woodbury, 1996, pp. 1–5 (Chapter 1). J.G. Leopold, I.C. Percival, J. Phys. B 12 (1979) 709. M.R.W. Bellermann, P.M. Koch, D.R. Mariani, D. Richards, Phys. Rev. Lett. 76 (1996) 892. H.R. Reiss, Progr. Quantam. Electron. 16 (1992) 1. Panming Fu, T.J. Scholz, J.M. Hettema, T.F. Gallagher, Phys. Rev. Lett. 64 (1990) 511. T.F. Gallagher, Mod. Phys. Lett. B 5 (1991) 259. C.H. Cheng, C.Y. Lee, T.F. Gallagher, Phys. Rev. A 54 (1996) 3303. C.Y. Lee, J.M. Hettema, C.H. Cheng, C.W.S. Conover, T.F. Gallagher, J. Phys. B 29 (1996) 3401. M.R.W. Bellermann, P.M. Koch, D. Richards, Phys. Rev. Lett. 78 (1997) 3840. R.V. Jensen, M.M. Sanders, M. Saraceno, B. Sundaram, Phys. Rev. Lett. 63 (1989) 2771. L. Sirko, A. Hamans, M.R.W. Bellermann, P.M. Koch, Europhys. Lett. 33 (1996) 181. G. Casati, I. Guarneri, D.L. Shepelyansky, IEEE J. Quantum Electron. 24 (1988) 1420. R.V. Jensen, S.M. Susskind, M.M. Sanders, Phys. Rep. 201 (1991) 1. G.P. Brivio, G. Casati, L. Perotti, I. Guarneri, Physica D 33 (1988) 51. D. Richards, in: A. Herzenberg (Ed.), Aspects of Electron–Molecule Scattering and Photoionization, American Institute of Physics, New York, 1990, pp. 55–64. D. Richards, J. Phys. B 25 (1992) 1347. A. Buchleitner, D. Delande, Phys. Rev. A 55 (1997) R1585. D. Richards, J. Phys. B 30 (1997) 4019. E. Oks, T. Uzer, J. Phys. B 32 (1999) 3601. P.M. Koch, in: R.F. Stebbings, F.B. Dunning (Eds.), Rydberg States of Atoms and Molecules, Cambridge University Press, London, 1983, pp. 473–512. M.R.W. Bellermann, Ph.D. Thesis, Physics Department, State University of New York, Stony Brook, 1995. B. Sauer, K.A.H. van Leeuwen, A. Mortazawi-M., P.M. Koch, Rev. Sci. Instrum. 62 (1991) 189. S. Yoakum, L. Sirko, P.M. Koch, Phys. Rev. Lett. 69 (1992) 1919. D. Banks, J.G. Leopold, J. Phys. B 11 (1978) 2833. P.M. Koch, D.R. Mariani, Phys. Rev. Lett. 46 (1981) 1275.

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[29] B.E. Sauer, S. Yoakum, L. Moorman, P.M. Koch, D. Richards, P.A. Dando, Phys. Rev. Lett. 68 (1992) 468. [30] P.A. Dando, D. Richards, J. Phys. B 26 (1993) 3001. [31] J.C. Gay, in: G.W. Series (Ed.), The Spectrum of Atomic Hydrogen: Advances, World Scienti c, Singapore, 1988, pp. 367–446. [32] M.J. Rakovic, S.-I. Chu, Phys. Rev. A 50 (1994) 5077. [33] M.J. Rakovic, S.-I. Chu, Phys. Rev. A 75 (1995) 1358. [34] M.J. Rakovic, S.-I. Chu, Physica D 81 (1995) 271. [35] E.J. Galvez, P.M. Koch, S.A. Zelazny, D. Richards, Phys. Rev. A 61 (2000) 060101-1. [36] E.A. Oks, V.P. Gavrilenko, Opt. Commun. 46 (1983) 205. [37] E.A. Oks, Plasma Spectroscopy: The In uence of Microwave and Laser Fields, Springer, Berlin, 1995. [38] E. Oks, J.E. Davis, T. Uzer, J. Phys. B 33 (2000) 207. [39] L. Sirko, S. Yoakum, A. Hamans, P.M. Koch, Phys. Rev. A 47 (1993) R782. [40] A. Buchleitner, D. Delande, J.-C. Gay, J. Opt. Soc. Am. B 12 (1995) 505. [41] D. Delande, J. Zakrzewski, A. Buchleitner, Europhys. Lett. 32 (1995) 107. [42] K. Sacha, J. Zakrzewski, D. Delande, Phys. Rev. Lett. 83 (1999) 2922. [erratum: Phys. Rev. Lett. 84 (2000) 200]. [43] K. Sacha, J. Zakrzewski, D. Delande, Ann. Phys. (N.Y.) 283 (2000) 141. [44] K. Sacha, J. Phys. B 33 (2000) 2617.