HAL Id: jpa-00221826https://hal.archives-ouvertes.fr/jpa-00221826

Submitted on 1 Jan 1982

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

INTERACTIONS OF INTENSE MICROWAVESWITH RYDBERG ATOMS

P. Koch

To cite this version:P. Koch. INTERACTIONS OF INTENSE MICROWAVES WITH RYDBERG ATOMS. Journalde Physique Colloques, 1982, 43 (C2), pp.C2-187-C2-210. <10.1051/jphyscol:1982215>. <jpa-00221826>

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-187

INTERACTIONS OF INTENSE MICROWAVES WITH RYDBERG ATOMS

P.M. Koch*

Physias Department, Yale University, New Haven, CT 06511, U.S.A.

Résumé.- Nous présentons une revue d'expériences qui s'intéressent à l'ionisa­tion, l'excitation, et la structure quantique des atomes d'hydrogène et d'hélium très excités plongés dans un champ de micro-onde intense, ainsi qu'une comparaison avec les théories actuelles. Celles-ci sont capables d'expliquer qualitativement certaines structures étudiées, mais elles ne permettent pas encore d'expliquer quantitativement les mesures d'ionisation effectuées à l'aide d'un très grand nombre (plusieurs centaines) de photons micro-onde. Il existe des calculs par la méthode de Monte-Carlo basés sur une théorie classi­que des interactions qui donnent un accord raisonnable avec les expériences d'ionisation publiées. Ces calculs négligent l'effet tunnel (quantique) et des expériences sont actuellement en cours pour montrer si cet effet, négligé dans l'approche classique, joue un rôle important. D'autres expériences récentes montrent que la dépendance de l'ionisation en fonction de la puissance micro­onde d'Un atome de Rydberg d'hélium dans l'état triplet m=0 est très différente de celle d'un atome hydrogène. On est capable actuellement d'expliquer le seuil d'ionisation de l'hélium en termes d'interactions dynamiques aux croise­ments évités successifs du diagramme de potentiel Stark de l'hélium.

Abstract. - Experiments investigating ionization, excitation, and quantal structure of highly-excited hydrogen and helium atoms in an intense microwave field are reviewed in terms of existing theory. Quantal theory is able to explain some of the results of the structure studies but is not yet at the level of producing quantitative agreement with ionization experiments, which involve very large numbers (hundreds) of microwave photons. Published Monte-Carlo calculations based on a classical theory of the interactions are in reasonable agreement with the results of published ionization experiments, but additional experiments are being carried out to see if quantal tunneling, which is ignored by the classical theories, plays an important role. New experiments show that the power dependence of microwave ionization of m=0 triplet helium Rydberg atoms is very different from that of hydrogen atoms. We are able to explain the threshold region of ionization in helium in terms of dynamic interactions at the first (and subsequent) avoided crossings(.s) of adiabatic helium Stark potential curves.

1. Introduction. - This paper reviews briefly our present experimental and theoret­ical understanding about what happens when an atom with one of its electrons in a highly-excited level with principal quantum number n»l (to be called a Rydberg atom) interacts with an intense microwave electromagnetic field. We shall see that much remains to be understood about this problem, both qualitatively and quantitatively. We shall confine the discussion primarily to the one-electron hydrogen atom, the subject of nearly all the still rather meager amount of theory that has been done, and to a lesser extent, to the two-electron helium atom, the simplest multi-electron atom. We shall see that hydrogen and helium atoms prepared in Rydberg states with very nearly the same energy are ionized rather differently by the microwave field and shall explain, at least partially, why this is so.

Other review articles by the present author1'2 and by Bayfield3-5 have covered some of the material presented here. The latter articles1*"'5, especially, reviewed

*After 1 sept. 1982 : Physics Department, State University of New York, Stony Brook NY 11794 U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982215

C2-188 JOURNAL DE PHYSIQUE

a large body of the quantal and semi-quantal theories treating the interactions of atoms and molecules with intense electromagnetic fields. This material will not be reviewed in detail here; the interested reader should consult these earlier reviews.

A few, loose definitions are in order at the outset. We shall take "intense" to mean that the Rydberg atom undergoes multiphoton absorption and emission in the microwave field that can be of rather high order, as much as several hundred. This usually means that induced transitions to continuum states, multiphoton ionization, can complete successfully with those to lower- or higher-lying bound states, multi- photon (de-)excitation; spontaneous transitions generally play a negligible role. Viewed semi-classically, the oscillatory microwave electric field Focoswt has a peak strength Fo comparable to the mean Coulomb field binding the Rydberg electron to its ionic core.

The full problem may be sub-divided into the questions of (1) atomic "structure',' or what are the quantized energy levels of the atom in the microwave field and (2) atomic "transitions',' or what are the detailed electronic motions (in a classical sense) among the various atomic states, particularly to the continuum? We shall motivate the conclusion that, to data, quantal or semi-quantal theories have dealt better with various aspects of the structure question but less well quantitatively with the question of ionization (bound-free transitions) than have classical theories, which obviously have nothing to say about (quantal) structure but seem capable o f dea l ing w e l l with some aspects of the non-linear dynamics. Here there are exciting unanswered questions concerning possible "stochastic" ionization mechanisms6-ll. Of course, any purely classical theory ignores quantal tunneling processes, which play an important role in ionization of atoms by a static electric field. The importance of tunneling in ionization of atoms by oscillatory fields, however, is not yet well understood. Neither is the onset of stochasticity in quantal non-linear dynamical systems.

2. A "Reasonable Approximate Hamiltonian for Hydrogen.-Consider a hydrogen atom with an infinitely massive nucleus. For hi~hlv-excited levels H(hi~h n) with n>>l. - - . - . the binding energy of the Bohr levels scales as n-2, whereas the spacing between neighboring n-manifolds and the fine-structure and hyperfine-structure within a given n-manifold each scale as n-3, albeit with diminishing constants of proportion- ality. Quantum-electrodynamic splittings (Lamb shifts, etc.) decrease with n even more rapidly. In the rest of this paper, we ignore fine-, hyperfine-, and QED- contributions to the energies of hydrogen levels. They are much smaller than the Stark interaction energy associated with the oscillatory electric field in an intense microwave field. (In helium, however, we shall include, via a quantum defect formulation, the electrostatic fine-structure because it greatly modifies the atomic Stark structure, particularly for levels with low values of the magnetic quantum number, [ml=0,1. The (ml=0 Stark structure is especially non-hydrogenic. For the most part, the magnetic fine-structure can still be ignored.)

At a given point in free sp5ce an applied microwave field consists of+an oscillatory electric field ~(t)=~~coswt and an oscillatory magnetic field B(t)= -+ Bocoswt. For H(high n) atoms, it is not hard to show that the Zeeman interaction energzes with Zf(t) are generally much smaller than the Stark interaction energies with F(t). For a plane-wave microwave field, one can show that the ratio of the maximum Zeeman energy Wm associated with the electronic orbital magnetic moment and the maximum Stark energy We is of order

where a=(137)-' is the fine structure constant. For 1-1-27, this ratio is The Zeeman energy associated with the electronic spin-angular magnetic moment is even smaller.

If the microwave field is contained in a waveguide or cavity, its electric and magnetic components will have spatial dependences which depend on the particular mode being excited. There will exist regions where one field is much larger than the other. The experiments discussed in this paper have been perfzrmed only in waveguides or cavities in which the interaction of the atoms with F(t) was totally dominant over that with 3(t).

Thus it is reasonable to adopt as a Hamiltonian (in au)

which includes only the non-relativistic kinetic energy of the electron and its electric interactions with the point nucleus and with ?(t). This has been the starting point for all the non-relativistic theories treating hydrogen.

3. Static Electric Field Ionization.- As a prelude to describing the intense microwave experiments, it is useful to review briefly the results of our experiments at Yale on static electric field ionization of H(high n) atoms in resolved substates described by parabolic quantum numbers (n, nl, n2, Iml). Both kinds of experiments used similar fast-beam methods; in fact, micorwave ionization in a TM020-mode cavity was used as a "black-box," 100% efficient detector of H(high-n) atoms in the static ionization experiments.

SCHEMATIC OF APPARATUS (TOP VIEW)

GAS F R G E T MAGNET'C M,CROWA"E

BEAM I L--L---L---L-- f

CHARGE

MULTlPLlER SIGNAL

MULTICHANNEL LOCK-IN SCALER

REFERENCE (

Fig. la- A schematic diagram (top view) of the apparatus used for static and micro- wave ionization experiments with fast beams of hydrogen or helium atoms produced by ion-atom electron-transfer collisions. The various pairs of electrodes labelled F1-F5 produce static electric fields through which the atoms fly. Two continuous C02 laser beams may be directed either collinearly with the atomic beam or to cross it in F1 and Fj, respectively, where they drive transitions between Stark substates of different n-manifolds. A microwave cavity (or waveguide, not shown) is used to study microwave multiphoton ionization and excitation processes. Experimental analog signals are digitized and signal-averaged in a multichannel scaler.

Fig. la,b shows a schematic diagram of our apparatus. An H+ or ~ e + ion beam with laboratory kinetic energy Te3-20 keV undergoes electron-transfer collisions with target gas atoms, producing Rydberg atoms with approximately an n-3 distribu- tion and an unknown distribution of substates. With use of various combinations of static electric fields F1-F5, continuous laser beams (argon ion or C02), and techniques reviewed in Ref. 2, a beam of atoms could be prepared in an n-band (a narrowed distribution of n-manifolds between na and nb),in an individual n-manifold with unresolved substates, or in an individual (Stark) substate of a given n-mani- fold. Fig. lb shows the most recent, inclined laser beam-particle beam geometry we have been using for double-resonance C02 laser production of H(high n) and He(highn) atoms. Unlike our previous collinear-beam l2 the inclined geometry allows us to have each laser beam interact with the atoms in only one section of the beam apparatus. Fig. la shows a microwave cavity being used to ionize the Rydberg atoms for subsequent detection of the resultant fast ions. We also use ionization in a static axial electric field for detection (not shown in Fig. 1).

JOURNAL DE PHYSIQUE

SCHEMATIC OF APPARATUS (SIDE VIEW)

C02 LASER BEAMS

Fig. lb- A side view of a part of the apparatus shown in Fig. la. It shows the geometry of the two continuous C02laser beams which cross the fast atomic beam in electric field regions Fl and F3, respectively.

Fig. 2 - Upper trace: A "single-resonance" Stark-tuned spectrum displaying C02 laser-driven transitions in F1 originating on a large number of H(n=lO) substates. Lower trace: A "double-resonance" spectrum displaying only those transitions out of the H(n=lO) Stark substate (10,0,9,0).

Fig. 2 shows two Stark-tuned spectra2 obtained with one collinear CO2 laser beam driving transitions between substates of two different pairs of n-manifolds: n=7 and n=10 in F1 and n=l0 and n=33,34 in F3. Each n-manifold has n(n+1)/2 Stark substates, The "single-resonance" spectrum (detected with one lockin ampli- fier) registered the many n=10 -+ n=33,34 transitions in F3 that were connected by appreciable matrix elements, The "double-resonancet' spectrum (detected simultane- ously with a second lockin amplifier) registered only those driven in F3 out of the single H(n=lO) substate (10,0,9,0). It was previously "tagged" by the laser-

driven transition (7,0,6,0) + (10,0,9,0) in F1. Obviously, the dcuble-resonance method allows isolation of individual substates.

Fig. 3- Dotted lines: Experimental TI(F) curves for the hydrogen Stark substates 1: {40,39,0,03; 2: {40,38,0,1); 3: 140,38,1,0); 4: {40,37,1,11; 5: I40,37, 2,O); 6: (40,36,2,1). The tic marks are discussed in Ref. 13. Solid lines: Theoretical curves calculated with use of a semiempirical, asymptotic formula (Eq. (6) of Ref. 14) with allowance for reduced-mass corrections. Squares: Numerical theoretical results calculated by Damburg and Kolosov for an infinitely massive nucleus.

These methods permitted the first precise measurement of the static electric fleld ionization rate rI(F) for resolved hydrogen Stark substates.13 Fig. 3 shows some experimental results (dotted lines) compared with numerical theoretical cal- culations (squares) and with calculations based on a semi-empirical analytic formula (solid lines] 1 4 . Since the six n=40 substates have large positive values of the "electric quantum number" nl-n2=34+39, they ionize at relatively large values of the scaled field, n4~=0. 2 - 0.3 au. Because of the quantal tunneling, ionization at rates ~10~s-l occurs at values of n4F about 20% below the classical ionization thresholds calculated and displayed graphically by Banks and Leopold.15 This behavior was also observed13 for a state (30,0,29,0) having a large negative value of(nl-n2)=-29. A n important conclusion of this work is that quantal tunneling lowers the observable ionization threshold in static fields (w=O) as much as 20% below the classical threshold.

4. Quanta1 vs. Classical Theories of Multiphoton Ionization.- It is an interesting and important question to determine whether tunneling would significantly lower the ionization threshold in an oscillatory field, wfO. We are explicitly investigating this question in current experiments. Theoretically, the answer to this question is unclear. ~ e l d ~ s h l ~ introduced a semiclassical model that attempts to treat non- perturbatively tunneling and multiphoton ionization as different limiting cases of the same physical process, namely ionization of an atomic electron by an intense electromagnetic field. He introduced the use of a parameter y that is the ratio of

C2-192 JOURNAL DE PHYSIQUE

the time required for the electron to tunnel through the instantaneous barrier to the period of the field; y much smaller and much larger than one correspond, respecitvely, to the adiabatic tunneling and multiphoton ionization limits. In atomic units, y=w/nFo for hydrogen. Various assumptions made in his early work with this model were improved by Perelomov et a1. ,17 but it is not obvious that the model has any validity for the case of very high-order microwave ionization of H(high n) atoms. The large density of Rydberg states (proportional to n3) means that a large number of multiphoton resonances between intermediate states are possible on the way to the continuum; the model does not handle these. N ~ ~ ~ ~ - theless, a prediction of this model in the adiabatic limit (y<<l) is that ionization in the oscillatory field has a lower threshold than in a static field. As will be seen below, this is also a prediction of calculations based on classical theory, which ignores tunneling.

Other papers of this Colloquium by Brandi and by Mainfray deal with theoretical and experimental investigations, respectively, of the validity of the Keldysh model of ionization. ~ e i s s l ~ has also reinvestigated it theoretically. It has not been specifically applied to the problem of microwave ionization of atoms.

Leopold, Percival, and Jones (Refs. 6-8, see also a review of their work in another paper of this Colloquium by Richards) developed an entirely classical theory of microwave ionization and excitation. The theory depends on two para- meters: (1) w/wat, the ratio of the frequency of the field to the atomic orbital frequency, and (2) n4~o, the scaled peak electric field strength. In addition, they introduced the use of a "compensated energy" for the orbiting electron in the oscillatory field to decide whether ionization did or did not take place. From considerations of the magnitudes of classical actions compared to R, they concluded that tunneling played a negligible role in the microwave experiments on n=66 atoms. l9 9 Since static ionization experiments show us that tunneling is very important in the w=O limit, however, it is unclear at what values of the scaled frequency w/wat and the scaled field strength n4FO tunneling ceases to become important.

5. Typcial Parameters for Microwaves Used in the Experiments.-Microwave sources and plumbing span the range from 1 GHz to more than 500 GHz. Generally, as the frequency goes up, the available output power of sources goes down, all components get smaller, and cost generally increases. There are a large number of microwave frequency bands; operation in a new band usually means that new cpmDonents must be acquired. For this reason, the microwave experiments reviewed in this paper have not yet been performed at a large number of widely separated frequencies. Most have been in the range 7-12 GHz, where sources and components were available.

Table 1

Fo for n4~,=0.1

Table 1 shows some useful numbers for 10 GHz microwave ionization experiments of H(high n) atoms. This frequency is near those used in most reported experiments. The columns show, respectively: (1) the principal quantum number, (2) the electric field strength Fo needed to produce a scaled field strength n4F0=0.1 au, approxi- mately the value observed empirically to produce appreciable ionization (notice that n4FO is the classical field strength parameter mentioned above); (3) the ratio of the binding energy of the level to the 10 GHz photon energy, or the number of photons energetically needed to get to the zero-field ionization limit; (4) the

ratio of the energy separation between the levels n and (n+l), or the number of 10 GHz photons needed to drive a An=l transition (this ratio is also the inverse of the classical scaled frequency w/wat mentioned above) ; (5) the approximate power that must be coupled into (and dissipated in) a typical TM020-mode cylindrical cavity [unloaded Q-lo4 and coupling voltage-standing-wave-ratio (VSWR) - 11 in order that the value of Fo on the axis of the cavity equal that shown in column 2; (6) the approximate power that must be transmitted through a TE1 0-mode X-band (8-12 GHz), rectangular waveguide in order that the value of Fo In the waveguide equal that shown in column 2.

If we adopt -10 W as an affordable upper limit on the power available from a tunable, continuous microwave source (such as the combination of a low-power oscillator and a traveling-wave-tube amplifier), waveguide ionization experiments are restricted to 11250 and cavity experiments to 11225. This means that inter-n transitions require on the order to ten photons, whereas ionization requires on the order of one hundred photons. Higher power, fixed frequency sources such as magnetrons are available in this and higher frequency ranges, but they would be best suited to experiments using resonant cavities. With them, one would be unable to study the frequency dependence of atom-field interactions. As we shall see, such studies are extremely important and have uncovered resonant behavior that is only partially understood.

6, Previous Ionization Experiments and Some New Results. - Bayfield and ~ochl' first reported eight years ago the observation of microwave ionization of H(high n) atoms in the n-band 63511269. Fig. 4 shows their data for the ionization probability at 9.91 GHz (solid circles), whirh corresponds to a scaled frequency w/wat=0.43. The atoms adiabatically entered and exited the microwave field inside a TM020-mode cavity and were exposed to about lo3 oscillations. The open circles show the classical, Monte-Carlo theoretical results of Leopold and ~ercival .6 9 Since, to date, these are the only calculations in quantitative agreement with the ionization experiments, the classical theory must be examined further as a viable model for the interaction of Rydberg atoms with intense microwaves. The horizontal axis is in units of the parameter y, which in the classical theory is the ratio (w/wat)./ (F,/F,~) of the classical frequency and field strength parameters. It is numerically equal to the Keldysh parameter, but its interpretation is very different because the classical theory obviously does not account for tunneling ionization.

It was found6-* that the classical electron orbits in the intense field could be grouped into four different classes C1-C4. Each could be recognized by its distinctive spatial trajectory or by a plot of its "compensated energy" Ec vs. time. The latter are shown in Fig. 5. Motions C1 lie on an invariant torus in phase space and may never ionize; C2 ionize rapidly and directly into the continuum; C3 ionize more slowly and pass through at least one intermediate, extra-highly-excited (EHE) state; C4 pass through one or more EHE but do not ionize during the time scale of the calculation. Notice that C3 and C4 exhibit de-excitation (downward) transitions as well as excitation (upward) transitions.

The classical, non-linear dynamics of the ionization process is complicated. It is not clear how one would construct a quantum mechanical theory of microwave ionization that would reproduce this behavior and give physical insight into the different types of electronic motions. Stable and unstable (escaping) classical trajectories may co-exist for different initial conditions at the same values of the classical parameters, and both excitation and de-excitation transitions seem to be important. It appears from the plots in Fig. 5 that the electron may not end up with much kinetic energy after ionization, but this was not explicitly evaluated in the classical calculations.6-8 With use of a strong-field, quantal, perturbative theory, Gersten and Mittleman21 attempted to estimate how far into the continuum the electron would be ionized, but the validity of their theory for the microwave experiments was very uncertain. They tentatively concluded that, on average, twice the minimum number of photons would be absorbed. Since the Rydberg atoms in the experiments were bound by only tens of meV, their result, if true, means that the ionized electrons would have only this amount of kinetic energy. Recent laser multiphoton ionization experiments (reviewed in another talk of this Colloquium by Mainfray) have determined the relative probability for ionization of atoms by more than the minimum number of photons, but the theoretical interpretation of these

JOURNAL DE PHYSIQUE

Fig. 4 - Closed c i r c l e s : The measured p r o b a b i l i t y (Ref. 19) f o r i o n i z a t i o n of H(high n ) atoms i n t h e n-band 63211569 by a 9 .91 GHz microwave f i e l d a s a func t ion of t h e parameter y, which i s numerically equa l t o both t h e "Keldysh" parameter (Ref. 16) used i n a quanta1 theory and t o t h e r a t i o (u/wat)/(Fo/Fat) of t h e two parameters used i n a c l a s s i c a l theory (Refs. 6-8) of t h e i n t e r a c t i o n of hydrogen atoms with an o s c i l l a t o r y e l e c t r i c f i e l d . Open c i r c l e s : Resul ts of Monte-Carlo c a l c u l a t i o n s (Refs. 6,7) t h a t employed t h e c l a s s i c a l theory t o model t h e experiment of Ref. 19. ( taken from Ref. 6)

experiments i s s t i l l i n progress .

It may be p o s s i b l e t o use e l e c t r o n spectroscopy t o ga in informat ion about t h e d i s t r i b u t i o n of k i n e t i c energy of e l e c t r o n s ion ized by microwaves. Showing t h e feasi- b i l i t y of t h i s type of measurement was a by-product of a very r ecen t c o l l a b o r a t i v e experiment a t Argonne Na t iona l Laboratory undertaken by our Yale group and a group a t t h e Argonne Dynamitron Laboratory- More w i l l be s a i d a t t h e end of t h i s a r t i c l e about t h e experiment, whose d a t a a r e s t i l l being analyzed.

7. Fre uenc -de endent Ef fec t s . - The resonant response of H(high n ) atoms t o certainqrang:s o: microwave frequency was observed by Bayfie ld , Gardner, and ~och" . I n o rde r t o be a b l e t o sweep t h e frequency over t h e range -9.6-11.4 GHz, t h e atom beam was d i r e c t e d through an X-band, TE1,O mode waveguide. A s was explained e a r l i e r v i a Table I , a v a i l a b l e microwave powers < 20 W meant t h a t i o n i z a t i o n of atoms l a s e r exc i t ed i n t o i n d i v i d u a l n - l eve l s could be s t u d i e d f o r 11545. Fig. 6 shows i o n i z a t i o n curves f o r each l e v e l between n=45 and n=57. Fig. 7 shows, j u s t f o r n=48, an ioni- - z a t i o n curve (lower frame) and an e x c i t a t i o n curve (upper frame). The l a t t e r was taken i n a "two microwave f i e l d " exper inent t h a t was s e n s i t i v e only t o atoms t h a t were e x c i t e d t o n>48 bound l e v e l s , but not t o t h e continuum. Both curves i n Fig. 7 were taken a t a microwave power of 8 11, which corresponds t o F =80 V/cm. Each curve

t T " H ! 7 0 w

?

a ? Y 0 w

0

! Ion8zatron rco EHE I C 3 1 Fgml EHE slal t ICI)

Fig. 5 - P l o t s of t h e "compensated energy" Ec vs . time f o r four d i f f e r e n t types of o r b i t s C1-C4 obtained with Monte-Carlo c a l c u l a t i o n s (Refs. 6-8) t h a t were based on a c l a s s i c a l theory of t h e i n t e r a c t i o n of hydrogen atoms with an o s c i l l a t o r y e l e c t r i c f i e l d . The c a l c u l a t i o n s followed t h e e l e c t r o n during s e v e r a l hundred f i e l d o s c i l l a - t i o n s . When Ec < 0, t h e e l e c t r o n is s t i l l bound t o t h e proton. When Ec > 0, t h e e l e c t r o n i s ionized. EHE s tands f o r an extra-highly-excited Rydberg s t a t e with p r i n c i p a l quantum number about a f a c t o r of f i v e l a r g e r than t h e i n i t i a l one. (taken from Ref. 6 )

i n Fig. 6 was recorded a t a d i f f e r e n t microwave power empi r i ca l ly ad jus ted t o l i m i t t he f r a c t i o n of atoms ion ized t o 1-30%. The d a t a i n both f i g u r e s l ie i n t h e follow- ing ranges of t h e c l a s s i c a l parameters: (1) 0.13<w/w <0.26 ( o r i n the near- a d i a b a t i c frequency regime), and (2) Fo/Fa = n4FO = 8F59 au , o r s i g n i f i c a n t l y be-. low the c l a s s i c a l i o n i z a t i o n thresholds ~ ~ 5 % ~ = 0.13-0.38 au8'15.

The 2. 8mm diameter beam-entrance and -ex i t ho les per turbed t h e waveguide e l e c t r i c f i e l d d i s t r i b u t i o n s o t h a t Fo was wi th in %5% of i ts peak value only over a f r a c t i o n , perhaps h a l f o r less, of t h e 1 cm separa t ion between t h e opposing faces . This meant t h a t t h e a10 keV atoms experienced a transit-t ime-induced frequency spread of 0.05 GHz o r more. The 0.5-1.0 GHz widths of the resonances i n Figs. 6 and 7 a r e s i g n i f i c a n t l y l a r g e r than t h i s , s o they may be phys ica l ly important. The absolute r a t e s c a l e s i n Fig. 7 show t h a t multiphoton i o n i z a t i o n and e x c i t a t i o n may proceed a t comparable r a t e s ; which i s dominant depends s e n s i t i v e l y on t h e frequency.

These experimental r e s u l t s s t imula ted t h e o r e t i c a l a c t i v i t y . Delone et a l . l l advanced a d i f f u s i o n model t o c h a r a c t e r i z e t h e nticrowave i o n i z a t i o n , but t h e i r

JOURNAL DE PHYSIQUE

Fig. 6- The measured ion iza t ion of H(high n) atoms with each p r i n c i p a l quantum number n between 45 and 57 a s a funct ion of the microwave frequency between 9.6 and 11.4 GHz. The v e r t i c a l sca les a r e r e l a t i v e u n i t s d i f f e r e n t f o r each n. The v e r t i - c a l bars a r e worst-case est imates of t h e f luc tua t ions i n the s igna ls t h a t were caused by frequency-dependent var ia t ions i n t h e peak microwave e l e c t r i c f i e l d s t reng th experienced by the atoms i n s i d e t h e TEi O..mode waveguide. The microwave power incident on the waveguide was kept approxl&ately constant f o r each n-value but was readjusted with each change i n n t o keep the peak f r a c t i o n a l ion iza t ion i n the range 1-30%.

I -

10 5 Fig. 7- A comparison of t h e f r e - quency dependence and absolute r a t e s f o r microwave multiphoton /\- e x c i t a t i o n (a) and ion iza t ion (b) of H(n=48) atoms. The open

- (a ) c i r c l e s i n (a) a r e background s igna ls . Single- and double- headed arrows show t h e pos i t ion of multiphoton t r a n s i t i o n fre-

/ quencies between a f i e ld - f ree n=48 s t a t e and f ie ld - f ree n=49

C - and n=50 s t a t e s , respect ively, 19.5 10.5 11.5 1 f o r the number of photons shown I ! above t h e arrows.

FREQUENCY (GH2)

est imated d i f f u s i o n time was c r i t i c i z e d by Meerson et a l . l0 ,who proposed an i o n i z a t i o n model based on t h e s t o c h a s t i c i n s t a b i l i t y of a c l a s s i c a l , non- l inear o s c i l l a t o r . I n po in t ing out t h a t t h e i r own c l a s s i c a l , Monte-Carlo t r a j e c t o r y re- s u l t s were d i r e c t l y r e l a t e d t o t h e genera l problem of t h e onset of s t o c h a s t i c motions i n non-linear dynamical systems, P e r c i v a l e t a1.b-8 concluded t h a t "much more work needs t o be done.. . to o b t a i n a c l e a r p i c t u r e of t h e phys ica l process r e spons ib le f o r ionizat ion. ' : I n a r ecen t review a r t i c l e Zaslavsky ampl i f i e s and extends t h e s e p o i n t s of view and ". . .show(s) t h e p o s s i b i l i t y of t h e ex i s t ence of a s t o c h a s t i c mechanism f o r t h e i o n i z a t i o n of atoms." (Ref. 9 , p.236).

A p o s s i b l e , more fundamental problem concerns t h e n a t u r e of s t o c h a s t i c i t y i n quantum systems. Again quot ing Zaslavsky (Ref. 9, p.160):

A t p r e s e n t , t h e number of papers cons ide r ing s t o c h a s t i c i t y i n quantum systems is r a t h e r l a r g e . Not only i s t h e r e no unders tanding of t h e genera l s i t u a t i o n but t h e r e i s no more o r l e s s c l e a r answer t o even a s i n g l e , t h e "simplest", problem, e i t h e r . Among them a r e t h e f a c t t h a t no t even a s i n g l e model has been accura te ly solved completely, t h a t t h e r e a r e no exac t r e s u l t s , t h a t t h e r e i s t h e n e c e s s i t y t o ob ta in re- s u l t s t r u e f o r long t imes, and t h a t t h e r e a r e d i f f i c u l t i e s i n us ing computers.

Zewail concludes, s i m i l a r l y , t h a t t h e under lying t h e o r e t i c a l is " . . . t he quest ion of what p r e c i s e r e l a t i o n s h i p , i f any, e x i s t s between c l a s s i c a l chaos and quantum mechanical motion. C lea r ly we s t i l l do not know t h e p r e c i s e meaning of quantum e r g o d i c i t y . " (Ref. 22, p. 33) .

Since t h e c l a s s i c a l and quantum mechanical motions of t h e f i e l d - f r e e hydrogen atom a r e known e x a c t l y , t h e hydrogen atom i n an e lect romagnet ic f i e l d may be a use- f u l t e s t i n g ground f o r i n t e n s e - f i e l d s t u d i e s of t h e s e i s s u e s . One of them i s obviously what in f luence quanta1 tunne l ing may have on t h e onset of c h a o t i c and escaping motions i n non- l inear systems.

8. New Hydropen and Helium Experimental Resul ts . - At Yale, our group has r e c e n t l y completed running some new experiments on microwave i o n i z a t i o n of hydrogen and helium Rydberg atoms. The goa l s were r a t h e r modest: We were i n t e r e s t e d i n com- pa r ing t h e Fo-dependence of t h e i o n i z a t i o n of a simple non-hydrogenic atom wi th t h a t of hydrogen. The apparatus was s i m i l a r t o t h a t shown i n Figs . l a , b , and a 9.91 GHz TM020-mode c a v i t y was used t o produce t h e i o n i z i n g microwave f i e l d . Analysis of t h e s e new experiments i s s t i l l i n p rogress , and t h e author apprec ia t e s t h e wi l l ingness of h i s col leagues a t Yale, D. Mariani and W. van de Water t o have some of our pre l iminary r e s u l t s and conclusions presented he re .

For a v a r i e t y of r easons , we chose t o work with n3s1 Rydberg s t a t e s of helium. We s h a l l compare r e s u l t s f o r atoms i n d i f f e r e n t n-manifolds by us ing t h e s c a l e d e l e c t r i c f i e l d n k 4 ~ , where n*=[n--6 (L,s) 1, t h e e f f e c t i v e quantum number, equa l s t h e d i f f e r e n c e of t h e p r i n c i p a l quantum number n and t h e quantum d e f e c t 6(L,S), which depends mainly on t h e o r b i t a l L and s p i n S angular momenta of t h e p a r t i c u l a r l e v e l . Helium n3s1 Rydberg l e v e l s have t h e l a r g e s t quantum d e f e c t , 6(0,1)=0.297. Since it w i l l be u s e f u l t o compare t h e i r microwave i o n i z a t i o n behavior t o t h a t i n a s t a t i c f i e l d , we show i n Fig. 8 a s t a t i c f i e l d i o n i z a t i o n curve f o r He(41 3~1) atoms pre- pared by double-resonance , C02 l a s e r techniques 2 y l 2 7 i n f i e l d s F1-F! i n Figs . l a , b Fig . 8 shows t h e f r a c t i o n of an %I2 keV beam of these atoms t h a t survlved passage through t h e 8 .3 cm-long i o n i z i n g f i e l d F5 i n Fig. l a . It was recorded using t h e same methods p rev ious ly used f o r p r e c i s e measurement of f i e l d i o n i z a t i o n r a t e s of hydrogen S t a r k subs ta t e s13 . W e n o t i c e a sha rp i o n i z a t i o n th resho ld (drop i n t h e curve) a t n * 4 ~ 2: 1/16 au , which i s t h e "saddle-point" i o n i z a t i o n theshold t h a t c h a r a c t e r i z e s i o n i z a t i o n of non-hvdrogenic, low-lml Rydberg s t a t e s t h a t fol low a d i a b a t i c pa ths t o t h e continuum23.

Fig. 9 shows microwave i o n i z a t i o n curves f o r H(n=41) atoms (wi th an unmeasured d i s t r i b u t i o n of s u b s t a t e s ) and He(41 3 ~ 1 ) atoms. Each curve shows a s i g n a l pro- p o r t i o n a l t o t h e f r a c t i o n of atoms ion ized dur ing t h e i r passage through t h e c a v i t y ;

JOURNAL DE PHYSIQUE

A (nf4 F)(A.U.)

.026 .053 .080 .I07 .I34 .I61 I I I I I I

(SADDLE POINT

I THRESHOLD)

- ln k z 3

m a Q

- J a z (3 ln

0 - I I 1 I I I

5 0 100 I50 200 250 3 0 0 FIELD (V/cm)

Fig. 8.- The measured f r a c t i o n of He(41 3~1) atoms i n a 12 keV beam t h a t su rv ive passage through t h e 8 . 3 cm-long i o n i z i n g f i e l d F shown i n

5 Frg. l a .

Fig. 9 - Microwave i o n i z a t i o n curves f o r H(n=41) atoms (dashed curve) and He(41 3~1) atoms ( s o l i d curve) . The h o r i z o n t a l s c a l e i s p ropor t iona l t o t h e microwave power i n c i d e n t on t h e 9 .91 GHz, TM020-mode c a v i t y ; t h e r e f o r e , t h e s c a l e of t h e peak amplitude of t h e microwave e l e c t r l c f i e l d Fo i s non-l inear . The abso lu te Fo-scale i s u n c e r t a i n by about 10%. The H and He atoms were exposed t o about 6 . 5 ~ 1 0 ~ and 1 . 3 ~ 1 0 ~ cyc les of t h e microwave f i e l d , r e s p e c t i v e l y . The s i g n a l s s a t u r a t e f o r nx4Fo 2 0.12-0.14 au.

t h e two curves have been approximately normalized. For n=41, t h e 9 .91 GHz f r e - quency corresponds t o a frequency r a t i o w/wat = 0.1, o r i n t h e "near-adiabatic" regime. Even though t h e d i f f e r e n c e i n ze ro - f i e ld binding ene rg ies of t h e atoms is very smal l , AE=6(0,l) /n3 auz0.95 cm-I = 23 GHz, o r f r a c t i o n a l l y only 1.4%, t h e two i o n i z a t i o n curves are ve ry d i f f e r e n t . The hydrogen i o n i z a t i o n th resho ld is approximately n4FO= 0.08 au, and t h e r e appears t o be some s t r u c t u r e i n t h e curve n a r th resho ld . The i o n i z a t i o n s i g n a l i s s a t u r a t e d ( r a t e > 10' s - l ) a t z n Fo 1 0.13 au. For helium, however, t h e i o n i z a t i o n s i g n a l rises a t much smal l e r va lues of nk4FO and is no t s a t u r a t e d u n t i l a somewhat h igher va lue than t h a t f o r hydrogen. Fig. 10 shows another hydrogen curve, t h i s time f o r n=27. For t h i s n- l e v e l , t h e frequency r a t i o i s w / w 0.03, o r even more i n t o t h e a d i a b a t i c regime. I n terms of t h e s c a l e d f f h d n 4 ~ o , t h e H(n=27) curve is very s i m i l a r t o t h e H(n=41) curve, except f o r t h e l a c k of p o s s i b l e s t r u c t u r e n e a r threshold.

n*4 ~ ~ ( a . u . 1 Fig. l o - A

0 0.05 0.10 I I I

microwave ion- i z a t i o n curve f o r H(n=27) atoms t h a t was taken under t h e cond i t ions descr ibed i n t h e cap t ion t o Fig. 9. The s i g n a l s a t u r a t e s f o r n4FoZ0. 12 au.

Let u s analyze t h e hydrogen curves f i r s t i n terms of t h e c l a s s i c a l , Monte Carlo t h e o r e t i c a l r e s u l t s 8 . Fig. 11 shows, f o r a number of va lues of t h e s c a l e d f i e l d Fo/F = n4FO, a composite of t h e i r c a l c u l a t i o n s of t h e percentage of ioniza- t i o n vs . t g e frequency r a t i o w/wat . (We po in t out t h a t more ex tens ive c a l c u l a t i o n s a t f ixed Fo/Fat = 0.08 and cn a f i n e r mesh of frequency r a t i o s w / w between 0 .3 and 1 .4 shows s e v e r a l resonances i n t h e i o n i z a t i o n p r o b a b i l i t y ( ~ e ? . 8 , Fig. 3) not un l ike those observed experimentally--see Figs. 6 ,7 above. Unfor tunate ly t h i s range of w/wat i s s l i g h t l y above t h e range 0.13-0.26 of t h e experiments 'O.) We n o t i c e t h a t f o r w/wat < 0 .1 ( o r t o t h e l e f t of lg(w/wat)=-1 on each h o r i z o n t a l s c a l e ) , they p r e d i c t t h a t n e g l i g i b l e i o n i z a t i o n t akes p lace u n t i l Fo/F > 0.1. Recal l t h a t t h e

a t measured hydrogen curves i n Figs . 9 , 10 show t h e onset of ionization s i g n a l s a t Fo/Fat = 0.08. A prel iminary a n a l y s i s l eads t o about a 10% uncer t a in ty i n our determinat ion of t h e abso lu te f i e l d s t r e n g t h Fo i n t h e c a v i t y i n terms of t h e measured power i n c i d e n t on it. Therefore , we have some p re l imina ry evidence t h a t t h e observed onset of i o n i z a t i o n occurs a t somewhat lower va lues of F,/F,~ than those p red ic ted by t h e c l a s s i c a l theory 6-8. It is h igh ly tempting t o view t h i s a s evidence f o r t h e importance of tunne l ing , but we s h a l l avoid t h i s c a t e g o r i c a l a s s e r t i o n u n t i l we have made a complete a n a l y s i s of our d a t a and u n t i l a l a r g e r

JOURNAL DE PHYSIQUE

Fig. 11 - A composite of the r e s u l t s of c l a s s i c a l Monte-Carlo ca lcu la t ions (Ref. 8) a t a number of values of the scaled f i e l d Fo/F of the percentage ion iza t ion of hydrogen atoms i n an o s c i l l a t o r y e l e c t r i c f i e 1 3 whose frequency w i s r e l a t e d t o the atomic o r b i t a l frequency through t h e r a t i o w / w t . Notice t h a t t h e ion iza t ion threshold f o r a near ly s t a t i c f i e l d i s near F,?F = 0.13 au and t h a t t h e ioniza- t i o n threshold f o r o s c i l l a t o r y f i e l d s with w/wa ag 1 is about a f a c t o r of two lower The c l a s s i c a l ca lcu la t ions ignore the e f f e c t s o! quanta1 tunnel ing, which lowers the observed s t a t i c ion iza t ion threshold (Ref. 13) by 10-20% depending on the p a r t i c u l a r Stark subs ta te s tudied. (taken from Ref. 8)

II*~F~(O.U.) Fig. 12- A

0 0.04 0.08 0.12 0.16 microwave I I I I I I I I I I ion iza t ion

curve f o r He(28 3 ~ 1 ) atoms t h a t was taken under the conditions described i n t h e caption t o Fig. 9 . The s i g n a l s a t u r a t e s f o r n * 4 ~ o > ~ . 16 au.

- I I 1 I I I I I I

0 400 800 1200 1600

Fo (Vlcrn)

number of c l a s s i c a l c a l c u l a t i o n s a r e a v a i l a b l e i n t h e th resho ld region.

Now we move t o a d i scuss ion of t h e helium microwave i o n i z a t i o n d a t a . Fig. 12 shows an i o n i z a t i o n curve f o r He(28 3 ~ 1 ) atoms. S imi la r t o t h e He(41 3s1) curve shown i n Fig. 9, t h e i o n i z a t i o n s i g n a l r i s e s a t very low va lues of n * 4 ~ , but t h e He(28 3s1) curve has a p l a t e a u (and even a s l i g h t d i p ) between about n*{~, = 0.05-0.1 au , rises again above t h i s va lue , and does n o t s a t u r a t e u n t i l about 0.16 au. Fig . 1 3 shows an i o n i z a t i o n curve f o r He(28 3 ~ 1 ) over a much smal l e r

Fig. 1 3 - A microwave i o n i z a t i o n curve f o r He(28 3~1) atoms t h a t emphasizes t h e th resho ld , low-Fo reg ion of Fig. 12. The o r i g i n of t h e "bump" is explained i n t h e t e x t . The s i g n a l f o r nn4FO > 0.02 au i s no t s a t u r a t e d ; it is a p l a t e a u un- observable on t h e s c a l e of Fig . 6.

range of n*4F0-values nea r t h e onset of i o n i z a t i o n . Notice t h a t Fo s c a l e s of t h e two f i g u r e s a r e s o d i f f e r e n t t h a t t h e ve ry i n t e r e s t i n g th resho ld s t r u c t u r e observ- a b l e i n Fig . 1 3 i s compressed and unobservable on t h e s c a l e of Fig. 12. We s e e t h a t t h e i o n i z a t i o n s i g n a l i n Fig. 1 3 begins ve ry sha rp ly a t nx4Fo = 0.011 au , o r Fo = 100 V/cm, r i s e s t o an l o c a l maximum, dec reases , r i s e s again more s lowly, and then reaches a l o c a l p l a t eau .

We a r e a b l e t o exp la in t h e "bump" n e a r 100 V/cm i n terms of known f e a t u r e s of t h e helium spectrum. Fig. 14 shows a "Stark map" of m=O t r i p l e t Rydberg l e v e l s . The c a l c u l a t i o n was performed on an (n,k,mk) b a s i s by D r . van de Water of our group using matr ix-diagonal izat ion programs adapted from those k ind ly supp l i ed by Dr. ~ i m m e r m a n ~ ~ . The magnetic f i n e - s t r u c t u r e of helium is n e g l i g i b l e on t h e s c a l e of t h e map and was i ored i n t h e c a l c u l a t i o n . Not ice t h a t t h e a d i a b a t i c cont in- ua t ion of t h e He(28 z l ) l e v e l undergoes i ts f i r s t avoided c r o s s i n g wi th t h e upper- most m=O t r i p l e t l e v e l (which is t h e a d i a b a t i c con t inua t ion of t h e (n=27, R=26, mR=O) l e v e l ) r i s i n g from t h e n=27 manifold nea r 110 V/cm, which is wi th in es t imated exper imental e r r o r of t h e p o s i t i o n of t h e th resho ld "bump" i n t h e i o n i z a t i o n curve. Since t h e frequency r a t i o u/uat 0.03, t h e o s c i l l a t i o n of t h e appl ied e l e c t r i c f i e l d is a d i a b a t i c wi th r e spec t t o t h e motion of t h e e l e c t r o n . The "Stark map" i n t h i s case i s analogous t o t h e molecular p o t e n t i a l curves t h a t one ob ta ins i n a Born-Oppenheimer sepa ra t ion of t h e slow i n t e r n u c l e a r motions from t h e f a s t e l e c t r o n i c motions i n a molecule. I n a molecule, o r i n a quasi-molecular t r e a t - ment of an atom-atom c o l l i s i o n , t h e i n t e r n u c l e a r motion produces dynamic terms which couple s t a t e s belonging t o d i f f e r e n t a d i a b a t i c molecular p o t e n r i a l curves. I n t h e p resen t case , t h e o s c i l l a t o r y f i e l d moves t h e system back and f o r t h a long t h e S ta rk p o t e n t i a l curves and may cause t h e system t o jump (make d i a b a t i c , o r Landau-Zener t r a n s i t i o n s z 5 from one p o t e n t i a l curve t o another .

JOURNAL DE PHYSIQUE

FIELD (V/cm)

Fig. 14- A pre l iminary "Stark map" f o r some m=O t r i p l e t Rydberg l e v e l s of helium t h a t was c a l c u l a t e d by D r . W. van de Water a t Yale wi th use of matrix-diagonaliza- t i o n programs adapted from those k ind ly supp l i ed t o us by D r . M.L. Zimmerman (Ref. 24) . It was computed on an (n,ll,mL) b a s i s of 106 s t a t e s i n t h e n=26-29 manifolds. Magnetic f i n e - s t r u c t u r e is unobservable on t h e s c a l e of t h e f i g u r e and was neglected.

It appears , t h e r e f o r e , t h a t p a r t i a l l y d i a b a t i c t r a n s i t i o n s a t t h i s f i r s t and a t o t h e r avoided c ross ings t r a n s f e r populat ion from t h e i n i t i a l l e v e l onto o t h e r S t a r k s u b s t a t e s . The remarkable t h i n g is t h a t t h e system i s then a b l e t o proceed a l l t h e way t o t h e continuum a t t h i s r a t h e r low f i e l d s t r e n g t h . Since t h e atom experienced approximately lo3 o s c i l l a t i o n s of t h e f i e l d , however, i t had many o p p o r t u n i t i e s t o make such t r a n s i t i o n s . We n o t e t h a t t h e f i r s t avoided c ross ing between a d i a b a t i c l e v e l s i n neighboring manifolds , n and (n-1), occur a t an F,- value t h a t s c a l e s a s n-5.

I n a d d i t i o n a l experiments wi th n=26,29 ~ e ( n 3~1) atoms, we observed threshold "bumps" and "plateaus" s i m i l a r t o those shown f o r n=28 i n Figs . 6,7. For t h e s e o t h e r n-values, which a l s o have a d i a b a t i c frequency r a t i o s w/wa t , t h e bumps show up a t F -values correspondin? t o t h e f i r s t avoided cross ing. We d i d not observe a threshofd "bump" wi th He(41 S1) atoms, however. It i s tempting t o exp la in t h i s behdvior by t h e less a d i a b a t i c frequency r a t i o w/wat " 0 . 1 f o r n = 41. I f t h i s is so, one would have an e f f e c t analagous t o t h e breakdown of t h e Born-O~penheimer a d i a b a t i c approximation i n moIecules.

Though our p re l imina ry a n a l y s i s exp la ins , a t l e a s t p a r t i a l l y , t h e o r i g i n of some th resho ld bumps, we cannot exp la in q u a n t i t a t i v e l y t h e l o c a t i o n of p l a t e a u reg ions over which some i o n i z a t i o n curves a r e l o c a l l y independent of t h e va lue of Fo. Our experiments on microwave i o n i z a t i o n of non-hydrogenic atoms have d i s - played f a s c i n a t i n g behavior t h a t i s very d i f f e r e n t from t h a t observed i n hydrogen atoms having n e a r l y t h e same i n i t i a l b inding energy. One may specu la te t h a t t h e hydrogen-like, high-lml s t a t e s of a Rydberg atom l i k e helium may d i sp lay microwave i o n i z a t i o n behavior more l i k e t h a t of hydrogen. Since t h e e l e c t r i c f i e l d i n t e r - a c t i o n conserves lmQl ( a t l e a s t i n t h e l i m i t of n e g l i g i b l e magnetic f i n e - s t r u c t u r e ) ,

t h e o s c i l l a t o r y e l e c t r i c f i e l d would not induce t r a n s i t i o n s between d i f f e r e n t ImQI- s t a t e s . S t ray f i e l d s o r c o l l i s i o n s could do t h i s , however.

We o f f e r an important conclusion: Nearly a l l t h e t h e o r i e s t r e a t i n g microwave i o n i z a t i o n have been based on a n o n - r e l a t i v i s t i c t reatment of hydrogen. Since we have observed dramatic d i f fe rences between hydrogen and non-hydrogenic S t a r k s t a t e s , i t i s inappropr ia te t o use non-hydrogenic Rydberg atoms i n experiments designed t o t e s t t h e o r i e s based on hydrogen. Nevertheless , t h e non-hydrogenic e f f e c t s w i l l be i n t e r e s t i n g t o s tudy i n d e t a i l i n t h e i r own r i g h t , but one w i l l need t o develop a d d i t i o n a l theory.

9. S t r u c t u r e of H(high n) Atoms i n an In tense Microwave Field . - Thus f a r , w e have emphasized quest ions of atomic t r a n s i t i o n s dr iven by i n t e n s e microwaves, from one bound s t a t e t o another o r from a bound s t a t e t o t h e continuum. I n t h i s s e c t i o n , we consider the problem of atomic s t r u c t u r e , o r what a r e t h e (quasi-)bound energy l e v e l s of t h e hydrogen atom i n t h e i n t e n s e microwave f i e l d . ~ l o c h i n z e w ~ ~ f i r s t inves t iga ted t h e quanta1 s t r u c t u r e of t h e Hamiltonian Eq. (2) . More recen t t r e a t - ments of t h e "AC-Stark e f f e c t " have included those of Aut ler and ~ o w n e s ~ ~ , of Bakshi e t a l V 2 $ , of and Hicks e t a ~ . ~ ' , and of a number of o the r t h e o r i s t s whose work has been reviewed recen t ly by Delone e t a1.3 and by ~ a y f i e l d ~ , ' . For the degenerate s t a t e s of t h e hydrogen atom, these t h e o r i e s p r e d i c t t h a t one e f f e c t of an i n t e n s e , l i n e a r l y po la r i zed o s c i l l a t o r y f i e l d is t o produce quasi-energy s a t e l l i t e s ( a l s o c a l l e d dressed s t a t e s , photon r e p l i c a s , e t c . ) d i sp laced by i n t e g r a l numbers of photons + p?~w above and below t h e ze ro f i e l d l e v e l . The ampli- tude of t h e s a t e l l i t e of order p is propor t iona l t o

t h e value of t h e Bessel funct ion of o rde r p whose argument i s t h e r a t i o of t h e peak d ipo le energy dFo t o t h e photon energy hw. A s i n t h e s t a t i c S t a r k e f f e c t , t h e per- manent d ipo le moment d is c rea ted from opposi te-par i ty atomic states t h a t a r e mixed by t h e f i e l d . The Bessel funct ion weighting of t h e s a t e l l i t e s i s a genera l f e a t u r e of in tense - f i e ld problems: they appear, f o r example, i n s p i n systems i n i n t e n s e , o s c i l l a t o r y magnetic f i e l d s 3 2 and i n electron-atom c o l l i s i o n s i n t h e presence of an i n t e n s e laser f i e l d 3 3 .

Twenty yea rs ago, Baranger and ~ o z e r ~ ~ proposed using s p e c t r a l - l i n e - s a t e l l i t e s produced by t h e AC-Stark e f f e c t a s a d iagnos t i c probe of high-frequency e l e c t r o - magnetic o s c i l l a t i o n s i n a plasma. This motivated a s i g n i f i c a n t amount of experimental and t h e o r e t i c a l work ( f o r u s e f u l lists of references s e e Refs. 35 and 36). The Ph.D. t h e s i s of Hicksz9 inc ludes t h e repor t of a n i c e o p t i c a l spectroscopy experiment on a hydrogen plasma, but t o t h i s au thor ' s knowledge, i t is a v a i l a b l e only a s a l abora to ry repor t . He measured t h e Balmer spectrum emit ted by t h e hydrogen plasma i n s i d e a Ge i ss le r tube t h a t was c o l l i n e a r wi th t h e microwave e l e c t r i c f i e l d i n s i d e a rectangular , TMOl0-mode cav i ty . He was a b l e p a r t i a l l y t o reso lve t h e microwave-induced s a t e l l i t e s on each s i d e of t h e H,,H H H l i n e s and t o ob ta in e x c e l l e n t r e s u l t s wi th a theory B Y . y ' 8 based on t h e work of Blochmzew.

The agreement between experiment and Blochinzew-type theory has genera l ly been much l e s s impressive f o r experiments which used i n t e r n a l f i e l d s i n s i d e a plasma, p a r t i c u l a r l y a tu rbu len t one, t o produce t h e s a t e l l i t e s . An e x t e r n a l l y app l i ed f i e l d was used i n t h e publ ished helium experiment of Hicks et a1.30 and i n t h e hydrogen experiment j u s t mentioned29. Nee and ~ r i e m ~ ~ observed internal-plasma-field- induced p = f l s a t e l l i t e s of n=13+12 and n=12+ll i n f r a r e d t r a n s i t i o n s i n hydrogen but found t h a t the absence of higher-order s a t e l l i t e s could no t be explained by a Blochinzew-type theory based on a single-frequency, constant amplitude e l e c t r i c f i e l d F ( t ) = Focoswt. They developed a theory t o model t h e multimode f i e l d s of a tu rbu len t plasma and obtained reasonable agreement wi th t h e i r experimental r e s u l t s . We no te t h e exper imental condi t ions descr ibed i n Eq. (4.7) and Table I of Ref. 35 corres- ponded f o r n=13 t o Fo/Fat = 0.03 au and w/wat = 0.05. These values a r e no t too f a r below those needed f o r i o n i z a t i o n processes t o become important. Their theory35

JOURNAL DE PHYSIQUE

ignored such processes. Sanchez and ~ e n ~ s t o n ~ ~ observed Stark-broadening and s a t e l l i t e s of l i n e s i n a turbulent helium plasma and found e x i s t i n g theory incapable of explaining t h e i r r e s u l t s quant i t a t ive ly .

10. East-beam Studies of Hydrogen Spectra i n Intense Microwaves.- ~ a ~ f i e l d ~ ~ f i r s t proposed a fast-beam, l a s e r experiment f o r production and detect ion of s a t e l l i t e l e v e l s of H(high n) atoms i n a s i n g l e microwave f i e l d . The present author39 demon- s t r a t e d t h e f e a s i b i l i t y of t h i s c l a s s of experiments with an important modification: The fast-beam was passed through two spat ial ly-separated regions of microwave f i e l d whose s t reng ths could be adjusted independently. C02 l a s e r photons were used t o d r i v e t r a n s i t i o n s from i n i t i a l n i = 1 0 l e v e l s with an unmeasured d i s t r i b u t i o n of sub- s t a t e s t o f i n a l high nf l e v e l s i n t h e presence of the f i r s t microwave f i e l d , which produced t h e s a t e l l i t e s t r u c t u r e being invest igated. The second f i e l d was used t o de tec t by microwave ion iza t ion those atoms t h a t were pumped by both t h e l a s e r and microwave photons i n t h e f i r s t f i e l d . Thus, the satel l i te-producing f i e l d could be weaker than t h a t needed t o ion ize t h e atoms. The frequency of the continuous C02 l a s e r was Doppler-tuned i n t h e atomic r e s t frame by varying t h e beam ve loc i ty . Since t r a n s i t i o n s t o nf 2 30 l e v e l s were s tud ied , we expect t h a t i f the e f f e c t i v e dipole moment d of the atoms i n the f i r s t f i e l d sca led a s n2, t h e moment d(n ) of t h e f i n a l l e v e l was a t l e a s t an order of magnitude l a r g e r than t h e moment $(ni=lO) of t h e i n i t i a l l eve l . Therefore, the r a t i o s of dFo/-liw t h a t e n t e r i n t o the Bessel funct ions i n Eq. (3) were much l a r g e r f o r t h e f i n a l l e v e l , and i ts s a t e l l i t e s played a more important r o l e than those of t h e i n i t i a l l eve l .

The r e s u l t s of t h a t f i r s t unpublished experiment39 were included i n a recent review2, from which Fig. 14 is taken. The hatched curves show experimental da ta f o r the production of t h e pf = +1, +2 s a t e l l i t e s of H(nf=48) i n a 9.91 GHz f i e l d with peak amplitude Fo(A). The abscissa of t h e f igure i s a quadrat ic Fo(A) s c a l e (pro- por t iona l t o t h e power d i ss ipa ted i n the microwave cav i ty ) normalized t o t h e f i e l d Fo(ion) " 90 V/cm t h a t was observed t o ion ize t h e H(nf=48) atoms. We see t h a t t h e pf = 1, +2 s i g n a l s peak a t values of Fo(A) l e s s than 10% of Fo(ion). The dashed curves show t h e o r e t i c a l s i g n a l curves obtained using t h e form of t h e amplitude given by Eq. (3) . +1 dashed curve was normalized by ad jus t ing t h e magnitude and arg2:;f magn~tude = of d f ) of t h e f i r s t maximum of ~~2 t o agree with the pos i t ion and height of the peak of t h e experimental curve. The same normaliza- t i o n parameters were used with t h e funct ion J~~ t o p l o t the dashed pf = +2 curve. The f i t t e d dipole moment is df = 2 . 6 ~ 1 0 ~ au, o r scaled down by n2, df2/nf2 1 1 f o r n~ = 48. This is a r a t h e r reasonable r e s u l t ; t h e permanent dipole moment i n the microwave f i e l d was about one scaled atomic u n i t (au/n2). According t o Delone e t a1.31, the l i n e a r i n t e r a c t i o n of t h e atoms with t h e o s c i l l a t o r y f i e l d dominates the quadrat ic i n t e r a c t i o n when n 4 ~ , << 0.06 au, o r Fo << 60 V/cm f o r nf = 48. Since i n Fig. 14 Fo(A) 5 10 V/cm, t h e experiment f u l f i l l e d t h i s condition.

Bayfield e t a1.40 recen t ly published the r e s u l t s of a more extensive study of the s p e c t r a of H(n =44-67) atoms i n a 5.9-8.0 GHz microwave f i e l d . They used a f "two-field" technique s imi la r t o t h e one mentioned above. Fig. 15 shows two of t h e i r s p e c t r a f o r e x c i t a t i o n of ni=10 atoms t o s a t e l l i t e s of nf=43-45. The higher- power spectrum i n the lower frame e x h i b i t s more complicated, overlapping s e r i e s of s a t e l l i t e s (up t o ]pf I=15) compared t o t h e lower-power spectrum i n t h e upper frame.

Since t h e zero-f ield hydrogen l e v e l s (pf=O s a t e l l i t e s ) a r e unequally spaced (n t o (n+An) spacing equal t o dn/n3 au) and the s a t e l l i t e s of a given n-level a r e equally spaced, t h e s a t e l l i t e pa t te rns of d i f f e r e n t n- levels a r e out of s tep. I n t h e l i m i t of a huge number of s a t e l l i t e s being appreciably populated i n a s t rong f i e l d , one would expect the empty spaces between the zero-f ield l e v e l s t o f i l l up with a kind of "lumpy continuum." The analogy20 between t h e problems of multiphoton e x c i t a t i o n and ion iza t ion of hydrogen atoms by a s t rong microwave f i e l d and of multiphoton d i ssoc ia t ion of polyatomic molecules by an in tense , i n f r a r e d l a s e r f i e l d should be mentioned here. The lumpy continuum i n t h e present hydrogen problem is s imi la r t o the quasi-continuum of upper l e v e l s i n t h e molecular problem (see Ref. 22 f o r a usefu l list of references) .

Fig. 15.- Cross-hatched a reas : Experimental s i g n a l s f o r CO laser-induced t rans i - 2 t i o n s from the pi=O s a t e l l i t e l e v e l of H(ni=lO) t o t h e pf=+l, +2 s a t e l l i t e l e v e l s , respect ively, of H(n -48) i n a 9.91 GHz e l e c t r i c f i e l d whose peak f i e l d s t reng th was Fo(A). Fo(ion)=PO Vfim was the f i e l d s t reng th observed t o Ion ize t h e H(nf=48) atoms. Dashed curves: Theoret ical curves based on Eq. (3) t h a t a r e proport ional t o the square of t h e Bessel funct ions of order p f . The normalization procedure is discussed i n t h e t e x t .

Bayfield e t a1.40 measured e x c i t a t i o n curves s i m i l a r t o those shown i n Fig. 14, and saw no o s c i l l a t i o n s pas t the f i r s t maximum t h a t one would expect according t o the extrema and zeroes of Bessel funct ions, Eq. ( 3 ) . They ascr ibed t h e deviat ion between t h e " theore t ica l" and experimental curves a t higher values of Foas due t o the n f - s a t e l l i t e l e v e l s having a spread i n the value of the dipole moment d f . This would cause a spread i n the value of t h e argument of each Bessel funct ion amplitude, Eq. ( 3 ) , and f o r each d i f f e r e n t df-value, the squared Bessel funct ion curve would peak a t a d i f f e r e n t value of Fo. Some kind of average df-value can adequately explain t h e low-Fo p a r t up t o the f i r s t maximum of each curve, but the spread i n df-values changes the shape of each curve a t higher Fo-values. For nf- levels between 44 and 67, and f o r f requencies 5,9-8.0 GHz, they found t h a t t h e power a t

which each pf=-1 s a t e l l i t e peaked was proport ional t o w (3.0+0.S)n(-4.0C0.2) ; t h e corresponding value of d var ies a t n2u-k. f

S teh le used a quanta1 forward s c a t t e r i n g method t o model t h e o r e t i c a l l y the two- frequency ( l a s e r + microwaves) e x c i t a t i o n of H(ni) atoms t o n f - s ta tes . (The present author appreciates rece ip t of a pre-print on t h i s work from Professor Stehle before i t s publicat ion.) H i s theory makes a p red ic t ion t h a t could be t e s t e d i n fu ture experiments: The e x c i t a t i o n amplitude f o r each s a t e l l i t e i s s t i l l weighted by i t s pth-order Bessel funct ion, Eq. (3), but a l l sa te l l i t e -ampl i tudes share a common f a c t o r J o ( 2 d F o ~ w ) . ~ h i s new f a c t o r would make a l l s a t e l l i t e s disappear together a t values of Fo corresponding t o the zeroes of Jo. I f t h e experimental s i t u a t i o n includes a spread of df-values, these would have t o be taken i n t o account i n applying t h e theory. We mention t h a t modification and even vanishing of t h e Lande g-factor i n atomic magnetic resonance experiments i n a s t rong o s c i l l a t i n g magnetic f i e l d B1 was observed and explained t h e o r e t i c a l l y using the "dressed-atom" formalism

JOURNAL :DE PHYSIQUE

Atom Energy ( k e V ) plus 0 . 2 5 keV

Fig. 16.- Exci tat ion s p e c t r a (Ref. 40) i n hydrogen atoms f o r C02 laser-driven t r a n s i t i o n s from s a t e l l i t e l e v e l s of t h e ni=10 manifold t o those of the n =43, 44,

S or 45 manifolds. Assuming t h a t only t h e pi=O s a t e l l i t e s of t h e ni=10 manlfold contr ibute , t h e add i t iona l numbers near the peaks l a b e l t h e orders pf of t h e s a t e l l i t e s populated i n t h e n -manifolds. The fixed-frequency C02 l a s e r rad ia t ion f was Doppler-tuned i n the a t o m c r e s t frame with v a r i a t i o n of the k i n e t i c energy of the atomic beam. Since lower energy atoms spent a longer time i n t h e waveguide f i e l d region, t h e lower energy resonance peaks a r e more intense. The p -0 resonances con- t a i n l a r g e contr ibut ions from atoms pumped outside the waveguide f i e l d region. (taken from Ref. 40).

by Cohen-Tannoudji and Haroche many years ago (Ref. 32, Section 1I.B.) The e f f e c t i v e g-factor ( o r , equivalent ly, the e f f e c t i v e magnetic moment) i n the in tense f i e l d a l s o displayed a zeroth-order Bessel funct ion dependence on B1.

It is c l e a r t h a t fu ture experiments w i l l be much improved i f one were t o use H(ni=lO) atoms prepared with a s i n g l e value of di, t h a t i s , i n a s i n g l e S ta rk sub-

s t a t e . These methods have been developed f o r s t a t i c f i e l d s t ~ d i e s l r ~ ~ * ~ ~ and must now be applied t o t h e microwave s tud ies . It i s not yet known, however, what happens t o a hydrogen atom prepared i n a s i n g l e , s t a t i c S ta rk s t a t e when i t passes i n t o a microwave e l e c t r i c f i e l d . I f the microwave e l e c t r i c f i e l d is c o l l i n e a r with t h e s t a t i c f i e l d d i r e c t i o n , t h e Iml-quantum number w i l l c e r t a i n l y be conserved, but i t i s not c l e a r t h a t the value of the s t a t i c dipole moment w i l l be. Perhaps a cleaner s i t u a t i o n w i l l be t o superimpose c o l l i n e a r s t a t i c and microwave e l e c t r i c f i e l d s , but some of the same u n c e r t a i n t i e s about whether the value of the dipole moment remains constant w i l l s t i l l have t o be addressed.

11. Mixed-field Studies.- I f t h e hydrogen atom were exposed t o a microwave f i e l d plus a d i t i o n a l s t a t i c o r o s c i l l a t o r y , magnetic o r e l e c t r i c f i e l d s , t h e problem would obviously be made more complex. Such a s i t u a t i o n i s beyond t h e scope of t h e present paper, but a few experimental and t h e o r e t i c a l papers have been published which po in t t h e way t o f u t u r e s tud ies . We have already mentioned above t h e plasma physics experiments which inves t iga ted modifications of f i e l d - f r e e atomic spec t ra by i n t e r n a l , plasma f i e l d s . Some of these f i e l d s can be described a s quas i - s ta t i c and o thers a s dynamic. Gallagher and ~ e v i n e ~ l reported observation of s a t e l l i t e s of H - l ines i n B a turbulent hydrogen plasma t h a t were ascr ibed t o resonance e f f e c t s between ortho-

gonal components of dynamic and quas i - s ta t i c i n t e r n a l e l e c t r i c f i e l d s . These resonance e f f e c t s , which produce s p e c t r a dramatical ly d i f f e r e n t from t h e S ta rk spectrum of each f i e l d a c t i n g independently, were f i r s t inves t iga ted numerically by Cohn e t a l .42 More recen t ly , Gavrilenko and ~ k s ~ ~ have a n a l y t i c a l l y confirmed and extended the t h e o r e t i c a l work of Cohn e t a l . 4 2 They f i n d t h a t f i e l d s with f ixed and random phases, respec t ive ly , exer t d i f f e r e n t resonant ac t ion on t h e hydrogen s p e c t r a l l i n e s t h a t s t a r t from l e v e l s with 1123.

An important s e r i e s of fu ture experiments w i l l use co l l i s ion- f ree beams of hydrogen atoms laser-exci ted i n t o ind iv idua l ( resonant - ) sa te l l i t e s t a t e s produced by mixed, appl ied f i e l d s . This w i l l be a cleaner experimental s i t u a t i o n f o r studying t h e f i e l d - e f f e c t s themselves than is provided by the i n t e r n a l f i e l d s i n a turbulent plasma . 12. Ldc i t iona l Fast-beamExperiments.- Our group a t Yale (Mariani, van de Water, and Koch) recen t ly joined with a group (Gemel, Kanter, Schneider, e t a l . ) a t the Dynamitron Accelerator a t Argonne National Laboratory t o i n v e s t i g a t e f u r t h e r the influence of Rydberg atoms on measured energy s p e c t r a of "convoy electrons" produced by f a s t ions incident on a t h i n f o i l ( f o r an extensive review of convoy e lec t ron s t u d i e s , see Ref. 44). These e lec t rons (ca l led "cusp electrons" i n ion-atom c o l l i s i o n s i n a gas t a r g e t ) emerge from t h e c o l l i s i o n with near ly t h e same ve loc i ty and d i rec t ion a s t h e unscat tered ions o r forward-scattered ions t h a t have captured t a r g e t e lec t ron(s ) i n t o a bound s t a t e ( s ) . For example, convoy e lec t rons produced i n 0.5 MeV @-fo i l i n t e r a c t i o n s emerge with a k i n e t i c energy about a f a c t o r (m / ) e "'P smaller , o r near 272 eV. The Argonne group showed previously45 t h a t e lec t rons captured i n t o bound Rydberg s t a t e s could be ionized by e l e c t r i c f i e l d s i n t h e e lec t ron spectrometer and elsewhere i n t h e apparatus i n such a way a s t o con t r ibu te t o t h e apparent convoy-electron peak.

Since the ana lys i s of t h e experiments is s t i l l i n progress , we do not go i n t o any d e t a i l here. We used a 9.91 GHz microwave cav i ty t o ion ize Rydberg atoms a f t e r t h e i r point of production i n a t h i n (2-5 micrograms/cm2) carbon f o i l and before t h e i r entrance i n t o t h e e lec t ron spectrometer used t o observe t h e convoy-electron spectra . Because of i t s high frequency, t h e microwave e l e c t r i c f i e l d neg l ig ib ly a f fec ted the "true" convoy e lec t rons . The f o i l could a l s o be e l e c t r i c a l l y biased t o s h i f t the energy of the convoy e lec t rons r e l a t i v e t o those produced by microwave ion iza t ion of t h e Rydberg atoms. We were ab le t o resolve c l e a r l y these two sources of e lec t rons i n c o l l i s i o n s of 0.5-2 MeV H+ and ~ e + atomic ions and H + molecular ions with t h e f o i l . We a r e i n t e r e s t e d t o see i f an ana lys i s of t h e j a t a w i l l give us preliminary experimental information on the continuum e lec t ron energy d i s t r i b u t i o n produced i n microwave ion iza t ion of hydrogen and helium Rydberg atoms.

13. Conclusions.- It i s usefu l t o summarize here t h e conclusions presented throughout t h e paper. The i n t e r a c t i o n s of Rydberg atoms with an in tense microwave e l e c t r i c f i e l d a r e not yet wel l understood. At the same f i e l d s t reng th , multi- photon e x c i t a t i o n and ion iza t ion can proceed a t comparable r a t e s , and each displays a d i f f e r e n t pa t te rn of frequency-dependent resonances i n hydrogen. To da te , c l a s s i c a l theory has furnished t h e only l imi ted , quant i t a t ive agreement with hydrogen microwave ion iza t ion experiments. It a l s o gives i n s i g h t i n t o some of the complicated motions of t h e atomic e lec t ron in t h e o s c i l l a t o r y f i e l d , but it ignores quantal tunnel ing processes t h a t may be important i n some cases. Tunneling has been shown t o be important i n s t a t i c f i e l d ion iza t ion . There may be fundamental questions about a " s tochas t ic mechanism" f o r ion iza t ion , but t h i s has only been examined i n a c l a s s i c a l context which t r e a t s t h e hydrogen atom i n the in tense f i e l d a s a non- l i n e a r , c l a s s i c a l dynamical system. I f quantal e f f e c t s a r e important, new quest ions a r e raised because t h e na ture of t h e onset of s t o c h a s t i c i t y i n quantum systems i s poorly understood. Because of t h e i r very d i f f e r e n t behavior i n an in tense microwave f i e l d , non-hydrogenic Rydberg atoms should not be used t o t e s t hydrogen theory. Though we a r e able t o explain a t l e a s t some threshold fea tures of the microwave ion iza t ion of helium Rydberg atoms, we a r e not yet ab le t o explain most of t h e s t r u c t u r e i n the ion iza t ion curve.

C2-208 JOURNAL DE PHYSIQUE

Quantum mechanics d e a l s w e l l wi th some aspec t s of atomic s t r u c t u r e i n t h e o s c i l l a t o r y f i e l d , but i t has n o t y e t been inves t iga ted c a r e f u l l y f o r f i e l d s s t rong enough t o cause appreciable i o n i z a t i o n . Recent theory p r e d i c t s e f f e c t s such a s t h e disappearance of t h e s a t e l l i t e spectrum assoc ia ted with a given n-level a t c e r t a i n values of t h e f i e l d s t r e n g t h Fo. This could be i n t e r p r e t e d a s t h e field-induced modif icat ion and vanishing of t h e atomic e l e c t r i c d i p o l e moment a t c e r t a i n values of Fo, and, t h e r e f o r e , t h e e l e c t r i c resonance analog of t h e modif icat ion and vanishing of t h e Lande g-factor i n magnetic resonance of atoms i n a s t rong o s c i l l a t - ing magnetic f i e l d .

Future microwave experiments w i l l use a v a i l a b l e techniques f o r production of hydrogen atoms i n ind iv idua l Rydberg s u b s t a t e s . These w i l l allow more d e t a i l e d s t u d i e s of t h e atomic s t r u c t u r e i n t h e i n t e n s e f i e l d a s more re f ined e x c i t a t i o n and ion iza t ion experiments.

14. Acknowledgments.- The author apprec ia tes having shared many i n t e r e s t i n g d i s - cussions on a number of t h e s u b j e c t s t r e a t e d i n t h i s paper with h i s pas t and present Yale col leagues , p a r t i c u l a r l y J.E. Bayfie ld , D.R. Mariani, and W. van de Water. He is g r a t e f u l t o t h e U.S. Nat ional Science Foundation f o r support of much of t h i s work and t o t h e A.P. Sloan Foundation f o r award of a Fellowship. He apprec ia tes t h e work of t h e o rgan ize r s of t h i s Colloquium and t h e support given i t by t h e French C.N.R.S.

REFERENCES

KOCH P.M., i n ATOMIC PHYSICS, Vol. 7, D. Kleppner and F.M, Pipkin, eds. (Plenum, New York, 1981).

KOCH P.M., i n RYDBERG STATES OF ATOMS AND MOLECULES, R.F. Stebbings and F.B. Dunning, eds. (Cambridge Univers i ty Press , New York, t o be publ ished 1982).

BAYFIELD J.E., i n MJLTIPHOTON PROCESSES, J.H. Eberly and P. Lambropoulos, eds. (Wiley , New York, 1978).

BAYFIELD J.E., Phys, Rep. 51 (1979) 317.

BAYFIELD J.E., Prog. Ouant. Elec. 5 (1980) 219.

LEOPOLD J . G . and PERCIVAL I . C . , Phys, Rev, Le t t . 41 (1978) 944.

LEOPOLD J.G. and PERCIVAL I.C., J. Phys. ~12 (1979) 709.

JONES D.A., LEOPOLD J.G. , and PERCIVAL I .C. , J . Phys. B G (1980) 31.

ZASLAVSKY G.M., Phys. Rep. 80 (1981) 157,

EfEERSON B.I., OKS E.A., and SASOROV P.V., Pis'ma Zh. Eksp. Teor. Fiz . 2 (1979) 79 [Sov. Phys.-JETP Le t t . 9 (1979) 721.

DELONE N.B., ZON B.A., and KRAINOV V.P., Zh. Eksp. Teor. Fiz . 75 (1978) 445 [Sov. Phys.-JETP 48 (1978) 2231.

KOCH P.M., and MARIANI D.R., J. Phys. B g (1980) L645,

KOCH P.M., and MARIANI D.R., Phys. Rev. L e t t . 46 (1981) 1275.

DAMBURG, R.J. and KGLOSOV V.V., 3. Phys. B g (1979)2637.

BANKS D. and LEOPOLD J.G., J. Phys. B z (1978) L5, 37.

KELDYSH L.V., Zh. Eksp. Teor. F i z . 47 (1964) 1945 [Sov. Phys.-JETP 2 (1965) 1307 I .

PERELOMOV A.M., POPOV V. S. , and KUTNETSOV V.P., Zh. Eksp. Teor. F iz . 54 (1068) 841 ISov. Phys.-JETP 11 (1968) 4511.

REISS H.R., Phys. Rev. A 2 (1980) 1786.

BAYFIELD J.E. and KOCH P.M., Phys. Rev. L e t t . 33 (1974) 258.

BAYFIELD J.E., GARDNER L.D., and KOCH P.M., Phys. Rev. L e t t . 2 (1977) 76.

GERSTEN J. and MITTLEMAN M.H., Phys. Rev. A z (1975) 1103.

ZEWAIL A.H., Phys. Today 2 (1980) 27.

JEYS T.H., FOLTZ G.W., SMITH K.A., BEITING E.J., KELLERT F.G., DUNNING F.B., and STEBBINGS R.F., Phys. Rev. L e t t . (1980) 390.

ZIMMERMAN M.L., LITTMAN N.G., KASH M.M., and KLEPPNER D., Phys. Rev. A s (1979) 2251.

RUBBMARK J.R., KASH M.M., LITTMAN M.G., and KLEPPNER D., Phys. Rev. AZ (1981) 3107.

BLOCHINZEW D., Phys. Z. Sowjetunion 5 (1933) 501.

AUTLER S.H., and TOWNES C.H., Phys. Rev. @ (1955) 703.

BAKSHI P . , K . G., and COHN A., Phys. Rev. L e t t . 31 (1975) 1576.

HICKS W.W., Lawrence Berkeley Laboratory Report LBL-2470 (Dec. 1973) unplubl ished.

HICKS W.W. HESS R.A., . a d COOPER W.S., Phys. Rev. A2 (1972) 490.

DELCNE N.B., ZON B .A., KRAINOV V.P. and KHODOVOI V.A., Usp. F i z Nauk 120 (1976) 3 [Sov. Phys.-Usp. 2 (1976) 7111.

COHEN-TANNOUDJI C . , and HAROCHE S., J. Physique 30 (1969) 153.

WEINGARTSHOFER A . , HOLMES J . K . , CAUDLE G., CLARKE E.M., and KRUGER H., Phys. Rev. L e t t . 2 (1977) 269.

BARANGER M . , and MOZER B., Phys. Rev. 123 (1961) 25.

NEE T.-J.A., and GRIEM H.R., Phys. Rev. A s (1976) 1853.

PROSNITZ D., WILDITAN D.W., and GEORGE E.V., Phys. Rev. A 1 3 (1976) 891.

SANCHEZ A., and BENGSTON R.D., Phys. Rev. L e t t . 38 (1977) 1276.

BAYFIELD J.E., Abs t rac t s of I n v i t e d Papers , European Study Conference on Multiphoton Processes , S e i l l a c , France, A p r i l 1975 (unpublished).

KOCH P.M., Post-deadline paper presented a t t h e I n t ' l Conf. on Multiphoton Processes , Rochester, NY, June 1977 (unpublished).

BAYFIELD J.E., GARNDER L.D., GULKOK Y . Z . , and SHARMA S.D., Phys. Rev. A42 (1981) 138.

41. GALLAGHER C.C., and LEVINE M.A., Phys. Rev. L e t t . 30 (1975) 897.

C2-210 JOURNAL DE PHYSIQUE

42. COHN A., BAKSHI P . , a n d KALMAN G., Phys. Rev. L e t t . 9 (1972) 324 [Erratum: Phys. Rev. L e t t . 3 1 (1973) 6201.

43. GAVRILENKOV.P., a n d OKs E.A., Zh. Eksp. T e o r . Fiz . 80 (1981) 2150 [Sov. Phys.- JETP 53 (1981) 11221.

44. BREINIG M. ELSTON S.B., HULDT S . , LILJEBY L., VANE C.R., BERRY S.D., GLASS G.A.. SCHAUER M., S e l l i n I.A., ALTON G.D., DATZ S. , OVERBURY S. , LAUBERT R., a n d SUTER M., Phys . Rev. A s (1982) 3015.

45. VAGER Z . , ZABRANSKY B.J., SCHNEIDER D. , KANTER E.P., ZHANG G.Y., a n d GEMMEL D.S., Phys. Rev. L e t t . 5 (1982) 592.