Excited atomic and molecular states in strong electromagnetic fields

EXCITED ATOMIC AND MOLECULAR STATES IN STRONG ELECTROMAGNETIC FIELDS

J.E. BAYFIELD Dept. of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pa. 15260, USA.

~1~C

NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 51, No. 6 (1979) 317—391. NORTH-HOLLAND PUBLISHING COMPANY

EXCITED ATOMIC AND MOLECULAR STATES IN STRONG ELECTROMAGNETIC FIELDS

J.E. BAYFIELD Dept. of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pa. 15260, U.S.A. Received June 1978

Abstract: Large changes in atomic and molecular structure can occur when strong static or oscillating electromagnetic fields are present. Such fields also can ionize atoms. Strong oscillating fields induce multiphoton transitions between bound states as well as to the continuum. What is known about these phenomena is reviewed, with emphasis on theories and experiments concerned with the fieldsensitive excited and highly-excited states.

Contents: 1. Introduction 2. Quantum systems in static electric fields 2.1. The Stark effect 2.2. Static electric field ionization 3. Excited atoms in strong static magnetic fields 4. The alteration of static electric field phenomena by an additional static magnetic field 5. The structure of atoms in nonresonant oscillating electric fields 5.1. Energy absorption of an atom in an “isolated” state 5.2. Quasienergy states of dressed atoms and molecules 6. Atoms in strong nearly resonant oscifiating electric fields 6.1. The two-level problem 6.2. Resonance fluorescence and “two-photon” experiments 6.3. The low frequency limit for the two-level problem 6.4. Fine structure and Zeeman effects on atomic quasienergy levels

319 320 321 324 327 331 334 334 336 342 342 344 345

7. Dressed hydrogen and helium atoms 7.1. Hydrogen atoms in weak linearly polarized fields 7.2. Hydrogen atoms in circularly polarized fields 7.3. Hydrogen atoms in strong low-frequency linearly-polarized fields 7.4. Helium atoms in oscillating tields 8. Field-induced changes in atomic ionization potentials 9. The calculation of multiphoton transition probabilities between bound atomic states 10. Multiphoton electron detachment in negative ions 11. Multiphoton ionization of excited atoms 11.1. Semiclassical calculations 11.2. Perturbation theory 12. Multiphoton ionization of excited hydrogen and helium atoms 12.1. Excited hydrogen in ionizing oscillating fields 12.2. Ionization of excited helium atoms 13. Excited states of molecules in strong fields References

345

Single ordersfor this issue PHYSICS REPORTS (Review Section~ofPhysics Letters) 51, No. 6 (1979) 317—391. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 30.00, postage included.

347 348 351 353 355 355 357 362 371 371 373 378 378 382 384 386

J.E. Bayfield, Excitedatomic and molecular states in strong electromagnetic fields

319

1. Introduction The physics of atoms and molecules in intense externally applied electromagnetic fields is a research field just beginning a rapid period of development. Technological advances are partially the reason, for now focused pulsed laser beams with peak power densities of gigawatts per cm2 area are laboratory items. Such devices are strong enough to significantly distort excited atom electron distributions. In addition, new laser techniques have been developed for se1~ctivelyproducing highly excited atoms. Because of their weak binding, these atoms are very sensitive to external fields, and provide an excellent testing ground for our understanding of strong field atomic physics. One might wonder what the definition of an intense external field situation is. Certainly a field approaching the mean Coulomb field eZV2> within the atom should be called intense. However, it is customary to also define strong fields as those that mix field-free atomic stationary states to produce large mixing coefficients. We shall use this definition. The strong mixing of levels by time varying fields leads to nonlinear radiative phenomena; we would certainly want to include highly nonlinear situations as part of strong field atomic physics. The nonlinearity arises when multiple photon absorption becomes significant in comparison to single photon processes. Excited states involve a weaker Coulomb field and have a multiplicity of sublevels that are often almost degenerate in energy; both these factors make them more susceptible to strong mixing by a given applied field than are ground states. An increased realization of the importance of excited state processes has occurred because of present interest in gas discharges as high power laser media. It appears that the excited-state atomic and molecular collision processes so critical for such applications can be altered or actually controlled by externally applied fields. This promises to open up new dimensions in laser-induced chemistry and in field-induced properties of solids. The development of laser-produced plasmas is another important present day endeavor; strong fields are present during the production of such plasmas. Perturbations of excited states due to fields and collisions within partially ionized media are phenomena that can occur on time scales faster than the plasma buildup time. Such effects are also important to recombination in both terrestial and astrophysical plasmas in the steady state [1]. Intense field atomic physics is a somewhat fragmented subdiscipline at the present time, with many basic questions poorly developed or not really understood at all. The reason for this lies in the theoretical difficulty of treating the electronic motion under circumstances where neither the Coulomb attraction of the atomic nucleus nor the interaction with the external field can be treated using first order perturbation theory. Yet multiphoton processes are becoming everyday phenomena in the laboratory, especially those that involve strong coupling to continuum states. So it seems timely to write now a paper that attempts to present an overall picture of what is known at present. Also to be discussed are a number of stimulating but still somewhat speculative topics that represent recent probes into the many unknown areas within the subject. This paper is not comprehensive in the sense that all past significant works are referenced. Neither is this review balanced in proportion to the level of development of the various areas within strong field atomic physics. Instead, new advances are emphasized along with the interconnections between subfields. The most developed subfields have been recently reviewed; their details are therefore deemphasized somewhat here. This includes the two discrete level quantum system in an strong oscillating field, reviewed by Dion and Hirschfelder [2], as well as optical multiphoton ionization of atoms, reviewed by Bakos [3], Delone [4], Morgan [5] and Lambropoulos [6].

320

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

Atomic units with me = e 1 [71 are used throughout most of this paper. In addition, often an excited hydrogen notation is used to characterize the mean properties of an S-state atomic electron, i.e. ~fEn is the energy, J~the ionization potential, v,.~the mean electron velocity and F0 the 2) and F,, = n~3.The symbol F is mean nuclear force on thestrengths, electron, ethen = =energies = l/(2n reserved for electric field for 1, free-atom and ~ for perturbed energies and quasienergies of atomic electrons in external fields.

2. Quantum systems in static electric fields The electron in a hydrogenic atom exposed to an external uniform static electric field of strength F finds itself in the total potential V(r)

=

—Z/r + Fz.

The first term is the Coulomb interaction with the atomic nucleus, the second the field interaction with F = F±.As schematically indicated in fig. 2.1, the field couples the free atom states to produce changes in the atom’s structure, namely shifts in the stationary state energy levels as well as splittings of degeneracies where they exist [7]. In this paper such structure changes will be called “Stark effects”. Notice, however, that the field also must couple the bound states to the continuum states, for the electron can quantum mechanically tunnel through the barrier in the potential to produce field ionization. This leads to an ionization contribution to the width of energy levels and to the decay of states, an effect we normally include under the topic-heading “field ionization” rather than “the Stark effect”. This separation of effects becomes more and more artificial as the strength of the field is increased. One can view the “discrete” energy levels as being narrow bands of “dense continuum” when in the field, the bands becoming wider and wider as the field is increased. The nature of their blending at strengths where many bands strongly overlap is a subject not yet well understood. The potential V(r) has a saddle point at [Xe, Ye’ Z~]= [0, 0, —(Z/F)’/2] with the value V~ = —2(ZF)”2. Thus the atom can be ionized even in classical physics for field strengths larger than the classical critical field F~= E2/(4Z), Potential Energy Z~

Distance

~

Fig. 2.1. The interaction of an atomic electron with a constant external electric field produces a barrier in the effective potential along with a shift in the electron’s energy.

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

321

where E is the total energy of the electron in both Coulomb and external fields. If the field-induced structure changes are ignored, then E ~ E(n) —Z2ji2/(2n2), where ~.zis the reduced mass of the atom. This approximation has proven useful for predicting (within a factor of two) critical field strengths for ionization at rates 1 0~to 10b0 sec’ corresponding to level widths of 1MHz to 10 GHz. The approximate value of F,~is F~~ Z3~z2/(l6n4). The details of the Stark effect must be considered in a more accurate discussion of ionization. 2.1. The Stark effect The treatment of the low-field Stark effect using stationary state perturbation theory was one of the first applications of quantum mechanics [8, 9]. A few of the many existing treatments include the effects of relativistic atomic fine structure [10, 11]. In this case a secular equation is solved to obtain Stark energy levels for field interactions comparable in magnitude to fine structure splittings. Fig. 2.2 shows a typical result for the case of the n = 4 levels of hydrogen. By now such perturbation theory calculations have been made up through fourth order [12]. The simultaneous treatment of the Stark effect and field ionization has centered upon applications of the semiclassical WKB approximation [12—17] as well as upon the use of a completely quantummechanical power-series boundary condition method introduced for the ground state of hydrogen by Alexander [18]. The latter technique involves asymptotic expansions of the coupled equations for electron motion in parabolic coordinates, in powers of P,/F or its inverse, F, being a characteristic field strength for the free atom. Often 1~,is taken to be the expectation value ~Z(r2) = (Z/n)3 for S states, although other choices are possible. The coupled equations are solved in different spatial regions where different approximations are valid, after which the wave functions are matched at the ~4o

__________

I

I

2 Field

I

_______

I

Internuclear Separation

I

4

6

Strength (102 Volt/cm)

Fig. 2.3. Some pseudointersecting potential energy curves in the ZeZ’ molecular ion, after Ponomarev [27). The term eZ[n, n

Fig. 2.2. The static field Stark effect in then hydrogen atom, after Luders [11).

4 levels of the

1, n2, mJ does not cross the term eZ’[n’ (nZ’/Z) —r, n~,m’ = mJ ; the index r takes on values r = 1, 2, r~where r0 depends upon Z’/Z.

fl’1 = nj,

322

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

boundaries of these regions. This approach has now been applied to the excited states of hydrogen and is believed to be an accurate numerical approach to the problem [19—231. The development of accurate analytical theories is still an active area [24]. The theoretical work discussed so far considers only the coupling between the continuum and bound states having different / and m, but the same principal quantum n. However, at sufficiently high field strengths the Stark energies for neighboring values of n become comparable to the spacing En E,, 1 between them; then we expect additional strong mixing of these levels by the field. Such mixing does occur at field strengths lower than that needed for large ionization rates. The field coupling of states of different n can be viewed as a perturbation of the Stark effect levels for each n. For example, the n 4 energy levels in fig. 2.2 cross or intersect those for n = 5 at high fields, and we expect field coupling near such level crossings to be particularly important. Transition at crossings of molecular potential energy curves is a well developed concept, particularly in the theory of slow atom-atom collisions [25]. Indeed, the existence of such crossings in the hydrogenic Stark effect was predicted from the results of Born-Oppenheimer calculations of the excited state potential energy curves of the ZeZ’ molecular ion [26—281. At large internuclear separations the molecular energies become those of the Stark effect of an excited atom in the long range Coulomb field of an ion, the field being almost uniform over the electron distribution of the atom. Ponomarev [27] actually found that pseudocrossings of adiabatic levels can occur for some levels (see fig. 2.3), whereas actual crossings are other-wise predicted because of the hidden symmetry in hydrogen atom problems that is connected with the Runge—Lenz vector. More recent Stark effect calculations have concentrated on the consequences of this symmetry, including in particular a sorting out of which levels cross and which pseudocross [28, 29]. This sorting can be accomplished in terms of the three nodal quantum numbers n~for the wave function as a function of each parabolic coordinate. The Hamiltonian plus two of the corresponding nodal operators N~form a complete set of operators, leading to the result that states having the same eigenvalues for any two of the three JV~cannot be energy-degenerate for any applied field strength F. If one considers in addition the low field ordering of Stark energy levels, then the conclusion is that any two states a and b cannot cross if —

n’

—

~‘

n’ for all i

or

n7
These results for the strong field Stark effect in hydrogen have not yet been experimentally tested, although the development of laser-excited fast atomic hydrogen beams now makes the experiments possible [30, 31]. At still higher field strengths F we expect all the upper excited states and the continuum states to be strongly mixed. An interesting but still speculative question is whether a region of F’s exists where the levels in the field bear no resemblance to the low field levels while still their ionization widths are not completely dominant factors. Semi-classical considerations based upon a onedimensional model of the hydrogen atom [32] suggest the possibility of this, with the ultrastrong field energy levels becoming those of the triangle potential spaced by an n’~3rule [33, 34]. Experimental results for the static Stark effect in the highly excited states of Na also suggest such behavior, as we shall now see. Early optical absorption studies established that the field-free Rydberg spectrum of the alkalis is different from that of the hydrogen atom [35, 221]. Although possessing only one valence electron outside a core of closed atomic shells, the outer-electron excitation spectrum is considerably perturbed by the core because of nonzero overlap of the electron’s wave function with that of the core.

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

I 7s~

_

(n4-l)s

nd

‘E

~ l5d-~

~

500

1.00 4~4

np

(n—I)

d

6s

(n-I)p (n-2)d Fig. 2.4. An energy-level diagram for the field-free Rydberg levels (r ~. 1) of the sodium atom. Energy splittings are in units of the hydrogenic splitting IE,, — E~— ~I.Levels with orbital angular momentum I > 2 lie close to the d levels.

323

Electric Field

(

ky/cm)

Fig. 2.5. The static field Stark effect in the ml = 0 sub-states of sodium, in the vicinity of then = 15 high I states, after Littman et al. [371.

For states with low orbital angular momentum, the energy shifts due to core polarization and penetration effects are comparable to the hydrogenic excited state energy level separations, see fig. 2.4. As might be expected, recent studies show that core effects also alter the nature of the static field Stark effect. The first published experiments on the Rydberg states of sodium to employ pulsed dye-laser excitation techniques were those of Fabre and Haroche [36]. They employed quantum beat spectroscopy to study the Stark effect in the nD states, n = 10, 11, 12, at very low fields (F~25 volt/cm) where the Stark effect is quadratic because of the zero-field level splittings. The observations were consistent (to within 10% to 25%) with a hydrogenic model for the field-coupled levels corrected for their quantum defects j.t. The correction to the hydrogenic quadratic Stark energy —~cx0F2was estimated from perturbation theory, with the nF level being the main source of field mixing with the nD level. The result is 2 ~ct 2[3m2_ ~ 1)] 0F 2F where /, m are the electronic orbital angular momentum quantum numbers. The second polarizability factor a 2 is 5(n2 9)/35(IID 1~F) a2 = —9n when estimated using hydrogenic matrix elements and sodium energy eigenvalues. The quantities liD and 1-~~ are the quantum defects of the D and F levels respectively. The differences in the Stark effects in H and in Na can be summarized as follows. In H, all levels exhibit a linear Stark splitting, with the submanifolds with even and odd ml staggered. The level crossings at high fields are sharp. In Na, we have a quadratic Stark effect in the core-split fine structure states at low 1 and low F At these fields the lml submanifolds may coincide for even and odd ImI, while changing to hydrogenic behavior at high fields. The high field level crossings for low ImI and / become pseudocrossings because of coupling via the core, see fig. 2.5. These pseudocrossings break the low field degeneracy of the ImI = 0 and ImI = 1 levels. The level separations at the pseudocrossing field values are relatively large for coupling between a quadratic Stark s or p state and a linear Stark state, since one state is essentially spherical. On the other hand, the pseudocrossing separation for two oppositely sloped linear Stark states is small because the electron densities are ‘~~tark =

~cx

—

—

—

324

fE. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

pulled along the field direction but are on opposite sides of the atom, leading to little overlap and a small core-penetration Stark matrix element. This detailed picture of the Stark effect in Na is the outcome of the recent pulsed dye laser studies of Littman, Kleppner et al. and others [37—40]. It is important to note the behavior of the Stark effect structure at the highest fields in fig. 2.5. Although large numbers of field-free states are strongly coupled at these intense fields, still the structure in a way appears simple, possibly because of the reasons already mentioned. It should be pointed out that static field Stark shifts can be induced not only in bound atomic systems, but also can occur for discrete states embedded in a continuum, i.e. scattering resonances and autoionizing states. Recent examples of this are experimental studies in autoionizing states of H [41] and Sr [42]. 2.2. Static electric field ionization Let us now turn to the problem of the static field ionization of atoms, beginning with hydrogen. As was the case for the Stark effect upon atomic structure, a number of numerical quantum calculations [18, 19, 22, 23, 43, 44] as well as semiclassical calculations [14—16,24,45] have been performed for the field ionization of H. In those cases where direct comparisons between different calculations can be made, there is often a factor of two or more disagreement because of differing approximations; in some work there evidently are errors. As field ionization is a tunneling phenomenon, theoretical predictions are very sensitive to the nature of the wave function, especially under the barrier. The early WKB approach of Lanczos [14] was based upon a decoupling of the electron motion along different parabolic coordinates, a procedure now believed to be inaccurate for strong fields. Qualitative aspects of the theoretical predictions for hydrogen have been tested in fast atomic beam experiments using mixed state beams produced directly by charge exchange collisions [46— 50]. However, since the ionization probability is predicted to depend strongly on all the atom’s quantum numbers, definitive experiments in hydrogen await the further development of laser excitation techniques. The numerical calculations possible for hydrogen are generally difficult to carry through for multielectron systems with any hope of predictive accuracy. This explains why alternative, more analytic approaches continue to be pursued. The path integral theory of quantum mechanics originally worked out by Feynman [51] in a Lagrangian formulation has also been treated in Hamiltonian form [52]. Complex Newtonian electron trajectories arise in such theories as stationary values of semiclassical path integrands. As many-body classical systems can be accurately handled by a computer, it has been proposed that the path integral approach be used to develop many body quantum theory beginning with the exact classical solutions [53, 268, 269]. So far, however, this field has mainly developed in the direction of obtaining analytic approximations for tunneling based upon asymptotic (large) forms for the field-free atom wave function. Such work has been helpful in clarifying some of the qualitative features of static field ionization [54—58], primarily for the case of short range effective asymptotic atomic potentials. Those attempting an application of this approach to the long range Coulomb potential case have found the problem difficult [58, 59]. Some qualitative high field features of these attempts are interesting, especially for the case of time varying external fields that will be discussed later. The basic nature of field ionization is revealed by the following approximate one-dimensional path integral calculation for short range atomic potentials [56]. The Lippman—Schwinger equation for the wave function ‘I’(r, t) can be written

J.E. Bayfleld, Excited atomic and molecular states in strong electromagnetic fields

‘I’(r, t)

=

f

—i

325

dt’ fdr’ G(r, t; r’, t’) V(r’) ‘I’(r’, t’),

where the total Hamiltonian is decomposed so that V(r) is the free-atom electronic potential energy of an active electron and the Green’s function G is the propagator for a free electron in the external field G(r, t; r’, t’)

=

e(t_ t’)

f dp exp {i[11(t)

r

—

fl(t’) r’]

!.

f

~2

(T)

dT},

where 11(t) ~p ~p + f~Fdt’. Taking the lower limit of the time integration to be assumes an adiabatic switching-on of the external field. Assuming the ionization time is long compared to atomic electron orbit times and ignoring the natural radiative decay time of the initial state, we approximate the wave function inside the integral by an initial field-free form cb,,(r’) exp (—iE~t’). Inclusion of the Stark effect would be an improvement. If in addition the potential V(r) is short range, we will take it to be a delta function. Introducing the semiclassical approximation for G: G(r, t; r’, t’) = exp [iSi(r, t; r’, t’) t’)] —

—°°

A(t)

—

where S is the contracted action S mation ~i(r, t) ~ exp (—iE,,t)

E,,(t

—

f~[L(’r) + E,,] dr and L is the Lagrangian, we have as

f dt’fdr’ exp [i~(r,

t;

r’, t’)] ~(r’) ~~(r’)~ exp i[~(r, t; 0, t0)

—

an approxi-

E,,t].

This last step assumes that the integrand eiS is a rapidly oscillating function of time except near a saddle point t0 that corresponds to that moment when the atomic electron strikes the barrier to be penetrated. The point t0 is determined by aS(r, t; 0,

t’)/at’l~~ = 0. =

As a simple example, let us now consider only the motion along the direction of the applied field. In atomic units, Newton’s equation 5~= F for free electron motion in the field 2 t~)whenwe choose as initial conditions ~(0)has first 0 andand x(tsecond integrals k = Ft and x = ~F(t 0) 0. We evaluate 2 — , , 1 fx2(r)dr~~~~._[t3_(t)3] ~ F S(t, t)~En(t~ t)=-~—

and thus to = iv~/Fwhere v,, = (2E~)’/2= 1/n is the free atom classical electron velocity. The complex classical path in time for this barrier emergence problem is Im(t) A to

>Re(t)

and the complex saddle point t 0 makes an imaginary contribution to S. As a result, the transition probability is nonzero W(F)

llin l’I’(r,

2

t)l

~

exp

—

{r1LF%

2 Im ~(r, t; 0, t 0)}

=

exp {—2F,/3F},

326

fE. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

= l/~z~. A three dimensional zero-range S-state potential ~(r) (not the same where Fn F31t013 = as the 3-D delta function ~(r), [56]) leads to the same result for the so-called exponential factor in the transition probability. A more accurate function for V(r) leads to a field-dependent prefactor, often hard to calculate. Note that F 0 is a characteristic field strength different from F~ 4)1 for S states .t~,is the expectation value of the atomic Coulomb field (r2) within a constant factor. (1 6n An important aspect of the path integral approach is that the complex time formulation just illustrated is equally useful for the more difficult problem of time-dependent fields. Again qualitative results can be analytically obtained as will be discussed in sections 10 and 11. The path integral theory for short range potentials is relevant to the static field electron detachment of negative ions [49]. Fast ion beam experiments have investigated detachment in a number of weakly found negative ion states such as He(4P), C(2D), Si(2P) and Ar(’D), see fig. 2.6 and refs. [60—63, 49]. The comparison of such data with theory is of considerable interest. In addition to the path integral semiclassical efforts, several other model quantum calculations of field detachment of negative ions have been made either for S states [64] or more generally [16]. Fig. 2.7 is a useful plot of the dependence of the tunneling probability on electron affinity and applied field strength. Special attempts to understand the particular case of field detachment in the two electron W ion have also been made [60, 65]. Recent studies of field ionization have concentrated heavily on alkalai atoms, where pulsed laser excitation techniques make beams of single-state highly excited atoms available for experiments. Theoretically, quantum defect theory can be applied to explain deviations from hydrogenic models. An extensive investigation of Na (12 ‘~n~ 38,10, 1,2) has occurred over the past two years [37, 39, 66—71], but Ca (n > 30, 1 = 1) [72] and Ba [73] are also experimentally accessible. The principal conclusions of this work are the following. For S states at fields below the level crossing region, (see fig. 2.5), the ionization probability varies very strongly with F, essentially exhibiting threshold behavior at a critical field F,~that varies as C(n*)~,where n* is the quantum-defect corrected principal quantum number. The proportionality constant C is that determined by classical escape to within 30%; two thirds of this difference can be accounted for by the Stark level shift. For levels corresponding to high values of 1 where core effects are small, the WKB hydrogenic calculations [45]

.0

~

0

200

F

400

(ky/cm)

Fig. 2.6. Experimental static electric field ionization curves for C, He, Si~,A1, 0, liT and P ions. A fast negative ion beam of intensity i~is passed through a region with a strong axial electric field F, and a fraction j/jo survives. The fractk~ns for the various ions are plotted versus F, after Il’in et al. [49, 62, 63].

0.1 0.2

0.4 0.8 I

2

4

8

Electric Field (MV/cm) Fig. 2.7. The logarithm of the field ionization lifetime r (in seconds) plotted versus the field strength F for various negative ion binding energies el in eV. The results are obtained from a short-range potential model and assume a single bound level with I = 0, after Demkov and Drukarev [64].

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

327

l0~

T

108

J

-~

.~

\~..

I0~

I

5.4

I

5.6

I

15,8

I

6.0

E (ky/cm) Fig. 2.8. The ionization rate for a highly-excited sodium atom in the Stark state (n, n ~,n

2, m) = (12, 6, 3, 2) in the field region where a level-crossing occurs with the rapidly ionizing (14,0, 11, 2) level. The solid line is a best multistate close-coupling theoretical result, after Littman et al. [681. A number of field-free states with their important coupling matrix elements were simultaneously taken into account in the computer calculation of the theoretical curve.

are consistant with the data, often within experimental field strength inaccuracies of 3%. At higher fields where level crossings occur, mixing of the core-corrected hydrogenic Stark levels near crossing fields leads to strong deviations from the diabatic hydrogenic results, with the deviations sensitive to the nature of the core. Cases of a mixing-in of more easily ionized states to enhance ionization of other states are common, see fig. 2~8.Lastly, the effective ionization field thresholds are so sharp that they are resolvable for the n = 17, 1 2 sublevels with different m11 in Na, [69]. Field ionization has also been experimentally studied for highly excited He atoms [61] and in H2 molecules [74, 75] produced by charge exchange, as well as for highly excited laser pumped uranium atoms [76]. Ionization thresholds are always found to have classical values within a factor two. 3. Excited atoms in strong magnetic fields Much of the early interest in highly excited (n ~ 100) atoms arose because of the observation of n-changing radiative transitions in certain astrophysical regions of low particle density, intermediate temperature and fairly high ionization level. The Doppler-shifted emission of the resultant microwave line radiation from such H II regions has contributed to determinations of interstellar structure and the motion of astrophysical entities [77, 78]. The sensitivity of such highly excited atoms to external fields makes them in principle a useful probe of interstellar electric and magnetic fields. So far, a radiotelescope search for Zeeman splittings of the emission lines from five H II regions including Orion A has led to negative results; a splitting sensitivity of 3 Hz per microgauss and the use of the NRAO 43 meter telescope placed upper bounds near 2 milli-gauss for the magnetic fields in the regions studied so far [79]. Nevertheless, interest in excited atoms in external magnetic fields remains high, as regions near black holes, pulsars and neutron stars may have abnormally high magnetic fields [80]. Within such astrophysical objects themselves, magnetic fields are expected to be as high as 10” gauss. At such field strengths the nature of atoms and molecules is highly distorted

328

J.E. Bayfield, Excited atomic and molecular states in strongelectromagnetic fields

from that of field-free systems. The observation in emission of line radiation from the excited states of such distorted atoms would be interesting indeed. The problem of an one-electron atom in a constant external magnetic field has cylindrical symmetry. Hence parity and the component m of angular momentum along the field direction are quantum mechanical operators having good quantum numbers. The wave function has the form ~i(p, 0, z) = p~12f(p, z) eim1~~ in cylindrical coordinates. The time-independent Schroedinger equation reduces to a2f/ap2 + a2f/az2 +

(2,~/.?~2) [E

—

~wL12m

—

V(p,

z)]

f 0,

WL

eB/jic,

where the effective potential V(p, z) is a nonseparable sum of the centrifugal, Coulomb and quadratic Zeeman interactions V(p, z) =

~2

(m2

—

~)/p2

—

e2/(p2 + z2)’12 + ~jiw~p2.

This potential has the following features [81,82]: 1. For all energies, the motion along p is bounded by the Zeeman term at large p, and the centrifugal term at small p if m * 0. 2. For E m~~lwL ~‘ 0 and values of E such that E m?IWL> z) the electron can escape by traveling parallel to the z-axis. 3. For E mllwL ~‘ 0 and m * 0, the equipotential curves V(p, z) = E mi’~wLare pipes oriented along z with a bulge at z = 0 due to the Coulomb potential. 4. The ionization limit for m * 0 and B * 0 differs from the field-free value .4,. Thus, autoionizing states at E >.4, can exist, with widths affected by the Coulomb coupling between p and z motions and by the ionization limit shift. These effects have not yet been theoretically investigated in any detail, although they are important. Deviations from the usual Paschen—Back Zeeman effect were first predicted by Van Vleck [83] who recognized that there was an additional upwards diamagnetic shift which arises because the precessing atomic electron is not in an inertial frame. For p states, the additional shift was expected from perturbation theory to be proportional to (n*)4B2(l + m2), where n* is the quantum-defect corrected principal quantum number and B the magnetic field strength. A full quantum mechanical treatment of the field-mixing of states of different 1 and n was first done by Schiff and Snyder [84]. An adiabatic approximation was introduced, with a fast electron motion perpendicular to B assumed separable from a slow motion along B. The Coulomb interaction is ignored for the perpendicular motion. The use of such adiabatic basis states or variants of them is now the starting point for more detailed calculations that extend theory to field strengths above perturbation theory values [85, 86]. An intense field region is reached when the cyclotron radius aL = (cfl/eB)’/2 is smaller than the field-free atomic electron orbit radius a,,, or equivalently B > 2.35 X l09Z2n2 gauss. At very high fields the electron is confined to a tight pencil volume oriented along the field direction z, and the theoretical problem reduces to that of obtaining the energies for one-dimensional atom sublevels of each Landau level. Thus for each value of rn the electron binding energies are comparable to usual Rydberg energies, except for one deep even parity level located at Edeep = —4Z2 ln2 [(ZPm)~1], Pm (21m1 + l)”2Z”2aL —

—

—

V(p,

—

in Landau units. For very large m, Edeep depends little on ml. The quantum calculations have concentrated upon these deep levels for which variational lower bounds can be easily obtained for different trial wave functions, see fig. 3.1 and refs. [87—9II.

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

—

~::

Edeepi _____________________

329

________________________

3.6 _~“,,4~7

o’---—-—i~ 0

I0~

I 010

,

0’

‘a 0

1.10

~

0

Magnetic Field (G)

I

I

I

10

20

30

\40 47 I

40

E’ (t~WL)

1~ig.3.1. A comparison of binding energies (in eV) of the ground state of hydrogen from various variational calculations after Rau and Spruch [85]. —.—.—, Landau function x exponential in z; ——, Landau function x Gaussian in z; x x x x x x x, Gaussian in x Gaussian in z; Landau function x hydrogenic function in r; Slater orbitals of various I and powers of r.

Fig. 3.2. Plot of the energy level separation 6E/6n~versus the energy above threshold E’ E — in units of the cyclotron frequency ~ for a number of magnetic field strengths, after Starace [821.

p

The nature of excited state atomic energy levels in strong magnetic fields can be qualitatively ascertained by semiclassical arguments connecting limiting cases [87—891. The system has axial symmetry and the electronic motion in the z = 0 plane perpendicular to the field is that primarily affected. One then expects a smooth connection with varying B between the principal quantum number n good at low fields and the Landau quantum number n~,good at very high fields. This expectation is based upon the fact that there is a well-defined number of nodes of the radial wave function for an energy eigenstate, no matter what the field value. The corresponding quantum number n~,should exist and take on values consistent with n and nL in the two limits respectively. Semiclassically n~will be determined by a WKB modified Bohr-Sommerfield quantization condition [82,90] of the form 2 1/2 P2 rn2

f [2E_m11wL_-~-

+_-_~W~P2]

dpm(n~+~)ir,

where Pi and P2 are classical turning points. One can differentiate this expression to obtain a formula for the level spacing 5E/cSn~: 5E

(s;;;)

1

1 P2

_f

m2 2 [2E_rn~wL_~r+~-_*w~p2]

—1/2

dp,

which has been numerically integrated for energies E> m?k~~~ above the field-free atom ionization limit, see fig. 3.2. As expected, i5E/t5n~~~llWL as E ml~wL A less rigorous but generally useful procedure involves a brute force identification of n and nL as the same at intermediate field values and largen, where both free-atom and free-electron-in-field basis states are strongly mixed [89]. In field regime number 1 where the magnetic field interaction of the electron is a perturbation, the Coulomb and magnetic energies for a hydrogenic atom are —~ °°.

—

E~-,

E~,~*,.Lw~(x2 +y)

~2n4B2

4z2 ,

e2 ~

2

~.zelectronmass.

330

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

In atomic units e = m = 1 and the fine structure constant a e2/~cbecomes 1/c, which we here call ~ while using atomic units. In the strong field regime, number 2, the Coulomb field is a perturbation and ~

~

=

~

(—Ze2/r) = —ZV’~B/2nL= —4/B/2nLBO,

where B 0 ~ is a characteristic field strength equal to 2.35 X l0~gauss. The total electron energies for each case are 2/2n2 + ~2n4B2/4Z2, E~2~—Z\/~B/2nL+ E~’~ = —Z from which level splittings are computed as ~n3B2

~

~n

—2E~’~ + 4E~~ = —2E~1~ + 6E~~ n n n n’

=

n3

6E~2~Z 1~B1’1~1 ——l—I

bnL

~B = ~

—+

2 1 2J

n~2

E~2~ E~2~ E~2~3E~2~ = __+

2~L

nL

2nL

J~L

2nL

and fractional energies become =

[

~

6E(~

n — —

~

E~ ~ n

—

~ 1 + ~2B2n6Z~ —2 + ~2B2n6Z~

—

L’— + 3E~1~ t2nL 2E(2)J nL

— —

/~\ ~n /

1 + 2312n~’2~1/2B1/2Z1/2 (&flL

—2 + 23/2n~2~l/2B1/2Z~/2 ‘~nL

To obtain energy level matching at intermediate fields we can force either ~ ~ ôE(~)/6nor .SE(t)/E(i) to be identical, thereby obtaining results differing by numerical factors of order unity. Perfect matching can be achieved by matching the 6E(1)/E(’), which requires that nL ~~(~B/Z2)n4.

But, in addition, in the strong mixing region nL and n are the same, both really being n 1,. Thus

2/~B)’/3

nP strong nL mixing n (2Zregion. This same expression can be obtained within a numerical factor by in the setting either total electronic energy E(’~ E~’)+ Eg) equal to zero. For Z = 1 and B = 25 kilogauss, strong mixing occurs for n ~ 70. Further consequences of the above strong mixing condition are that, independent of basis,

~5E/E—l0~ne/ne and in the region E ~ 3Em (essentially equivalent to E ~ 0) cSE/5n~=4~B~

Thus, there should be equally spaced levels near the field-free atom ionization limit, with the spacing 3/2 the Landau level spacing. This last prediction is particularly accessible to experimental verification. Experimental study of strong magnetic field effects on atomic structure began in 1939 with the observation of the diamagnetic shift in the 32 S—n2 P characteristic lines of sodium atoms placed in a

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

Limit

_________

n

60 —_________

n

50

=

_________

__________

_________

n

331

_________

fl

40

40 (a)

(b)

Fig. 3.3. The strong field Zeeman effect in the highly-excited state absorption spectrum of the barium atom, (a) in zero field, and (b) in a field of 26 kilogauss. Note the equally-spaced field-induced energy levels above the field-free atom ionization limit. After Garton and Tomkins [93].

magnetic field, [91]. Traditional absorption spectroscopy was used. The shift was observed for n = 12 to 20. Deviations from Van Vieck’s predictions due to i-mixing were seen in the spectrum at n = 20 to 35, and evidence for some n-mixing at still higher n was found. Studies of these effects in the alkalis continues today using laser spectroscopy techniques [92]. Similar magnetic field experiments on the Rydberg absorption line series in Bal have been carried out under high resolution [93]. At 24 kilogauss, the a lines that correspond to quantized electron motion in the plane perpendicular to B exhibited a variety of effects as n varied from 26 on up to above the field-free ionization limit, see fig. 3.3. Diamagnetic shifts were easily observed for n = 26 to 38. Above n = 31, satelite lines due to 1-mixing were seen. In the region n = 40 to 45 the Zeeman splittings were comparable to field-free level separations of adjacent n, and the spectrum is quite complex. Let us lastly consider wavelengths corresponding to transitions to n 80 in the free atom. In this region the planar motion of the electron becomes dominated by the magnetic field, and the quasi-Landau spectrum with the spacing 41k.)L predicted above was observed. At least 14 energy levels above the field-free limit were resolved. This pioneering work on barium by Garton and Tomkins clearly demonstrated many important aspects of the strong field alteration of atomic structure. Experimental efforts to further pursue this subject are presently concentrating on the dye-laser spectroscopy of atomic beams passing through superconducting magnets [94]. Very recent experiments on the quasi-Landau spectra of Bal and SrI used such techniques to verify the coarse features predicted by semiclassical theory, including the B’13 scaling of n near the ionization threshold [283]. 4. The alteration of static electric field phenomena by an additional static magnetic field We discuss in this section two strong-field problems that have received some recent theoretical attention. First is the question of the stationary state energies of a hydrogen atom in combined static electric and magnetic fields [95, 96]. Second is the magnetic field alteration of static electric field ionization, modeling the atomic potential by a delta function [55, 97]. We recall that such modeling limits the possible applicability of results to the case of negative ion electron detachment.

332

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

In passing we note that double perturbation theory can be used to treat the combined Stark and Zeeman effects at weak field strengths [98, 99]. If a hydrogen atom is in uniform fields F and H with F>> l05n~V/cm, H>> l05n3 gauss, then the fine structure of the atom can be ignored and all the levels of given n considered degenerate. In 1923, Born showed that the electron orbit undergoes slow field-controlled motion, involving the independent uniform precession of the vectors 4nL c”) with angular velocities o.~2 = ~H T ~nF respectively [100]. Here (r> is the radius vector perpendicular to L averaged over the orbit. The recent work utilizes the 0(4) symmetry group for the free atom to solve the two-field problem, ignoring interactions between states of different n. We define the Runge—Lenz vector A

(—2H

12{4[pL]

—

[Lp]

—

r/r},

0)’

where H

0 is the field-free Hamiltonian. Then in parabolic coordinates one can simultaneously diagonalize H0, L5 and A~with (n,n2mJA5ln,n2m) = n2

—

n1.

For a given n, the operator z is also diagonal (n1n2mjzln,n2m>

=

4n(n1

and so in this subspace r H’Fr+

=

—

n2)

—~nA.The perturbation operator

HL/2c

can therefore in this subspace be written H’—~nFA + HL/2c. The quantities ‘1,2

~(L ±A) obey usual angular momentum commutation relations, and have

J~=I~j=j(j+l),

j~(n—l).

We can therefore write H’

=

w, I~+

0.12

If one denotes the projections of o.,, on.4 as n’, n”, then the first order energy shifts are ~En’w1

+n’w2.

Monozon and co-workers [95] carried this problem further termssituations accurate were to third order in 2/8c2. Somethrough degenerate found, but the fields, including the diamagnetic term (H X r) not treated using a secular equation. First, w~= w 2 if F• H = 0, leading to energy degeneracy; second, one w, is zero when P = Rand F/H = (3nc)’, another degenerate case. The 1970 perturbation calculations [95] have more recently been extended using a semiclassical calculation [96]. However, only the case of perpendicular fields F = Fj’ and H = H~was considered in the later work. After several transformations of the wave function an effective potential is found that exhibits a double well when 2e3/.I13. (mM/,.L)(hc/e2)i(F/H)(H/HL)h/3 >3, HL mc,~ This situation results from the competition between electric field formation of the usual barrier at smaller y and magnetic confinement of the electron at larger y to values near the cyclotron radius

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

aL

=

333

(h’c/eH)”2. The usual Coulomb well is shifted from the other (new) well by a distance Yo = —(rn +M)c2F/eH2.

The new well has a finite depth. The energy levels in the new well were investigated and a wave function sought assuming the usual separation of motion in cylindrical coordinates and taking the lowest Landau state for the radial wave functions, see section 3. The model calculations of magnetic field alteration of electric field ionization are particularly interesting, for they predict that it is possible not only to magnetically reduce field ionization rates, but even completely stabilize the system [97]. This possibility is also expected for normally autoionizing resonance states of field-free negative ions formed in a static magnetic field [101]. Let us consider a s-function atom in crossed uniform electric and magnetic fields. With H applied, a reduction of ionization is expected within a tunneling picture because a curving of the particle trajectory increases its length under the barrier, similar in effect to widening the barrier instead. Different regimes of semiclassical electron motion are delineated by the parameter cxtI/nF = v~H/mcF, a 1 / 137, where v,, is the electron orbit velocity in a.u., WL is the Landau cyclotron frequency and cot 2.4,(F/ F~),the classical escape time of field ionization mentioned in section 2. When ‘IL < 1 ionization is possible and increases with decreasing ‘IL’ When ‘IL 1, ionization does not occur for the model potential case. In the limit ‘IL ‘~ 1, semiclassical theory predicts an altered field ionization probability [55, 971 determined as the imaginary part of a complex energy eigenvalue ‘IL

E=

WL/Wt

—

F2/8~+ H2a2/24 t~ + irn,

[‘~(F, H) —1,,(F/2F,,)(l

—

~~/6)exp {~—~(F~/F)(1 + ‘y~/30)}.

Note the similarity with the low frequency ionization rate for an applied oscillating electric field (see section 10), where ‘y is instead the Keldysh parameter co/(nF). This is not too surprising as both a perpendicular H and a linearly oscillating F tend to spatially confine the electron, thereby reducing the ionization. In the stationary level limit ‘IL ~ 1 there are two cases, depending upon the relative size of the Larmor and atomic radii. If aL ~‘ a~,the energy is the same~asthe real part of the above expression for ‘IL =~ 1. Otherwise E ~ (a2H2/v,~)[1 + 1 .46IvnaL + f~ y2oa~/v,,’]. The end point of the continuous spectrum is found to lie near E=~mc2(F2/H2). In the transition region ‘IL ~ 1 where the end point is a little below the initial level in the well, the model semiclassical approach gives

r=i~(F/F,,)(l

—

~~)I/2

exp{—~(F~/F)G(’IL)},

where one result for G(’IL) is [97] G(7L)n~-~~-I [1 +(1 2’IL

—‘If) ~ 27L

1

‘IL] l+7L

A more extensive expression for G(’I) is available [55] for arbitrary field directions.

334

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

Numerical calculations [97] not employing a semiclassical approximation give the following expectations for a negative ion with 0.030 eV binding energy: at F = 180 V/cm, a magnetic field of 360 gauss should reduce the ionization rate to zero. At a higher value ofF = l0~V/cm, 106 gauss is needed. 5. The structure of atoms in nonresonant oscillating electric fields We saw in section 2 that a static electric field simultaneously shifted and mixed atomic energy levels and in addition caused ionization, i.e. transitions to the continuum. This situation holds true when the field oscillates periodically in time. The two field effects, atomic structure distortion and induced transitions, are intimately related, with one having a strong influence on the other especially when the field strength is high. Nevertheless, a separation of the problem into atomic structure and transition rate parts is generally useful, for each part corresponds to a different kind of experiment in the laboratory. Measurements on transition rates involve the preparation of a known initial fieldfree state, the subsequent application of the field, and the final analysis of the amplitudes for all possible final field-free states. On the other hand, experiments on atomic structure involve spectroscopy (i.e. characteristic photon emission or absorption) of the atom while in the applied field. Such atoms have loosely been called “dressed atoms”. From a dynamic point of view, such structure experiments involve two-or-more-photon “Raman like” processes, with one photon being at a frequency used for the spectroscopy, and at least one photon being from the dressing field. The field strength at the spectroscopy frequency is kept low, so that there are no strong field (nonlinear) effects except those due to the dressing field. In a typical atomic structure experiment the frequency of the spectroscopy field is swept; in a typical transition rate experiment some property of the strong dressing field is swept. 5.]. Energy absorption of an atom in an “isolated” state As an introduction to the subject of the distortion of atomic structure by strong fields, let us qualitatively consider the absorption of energy from the field for a simple nonresonant case [102]. Consider an essentially isolated nondegenerate atomic energy eigenstate ~ with energy E0. Let us place such a simple quantum system in an external low frequency oscillating electric field F(t) = F cos cot. The interaction between atom and field will be taken as V = d F(t), where d er is the dipole moment operator. If the field were static, we would have a perturbed statefrom with 2 canstationary be obtained energy E0 + ~12) where the second order Stark energy shift ~ ~aF second order time-independent perturbation theory. If instead the field oscillates very slowly with time, in general we expect the energy to follow with an oscillating Stark shift ~E(2)(t)= ~cxP2cos2 cot. The state is no longer stationary, but rather an adiabatic oscillating atomic orbital ~i(r,

t)

=

0

2 cot’dt’}

0(r) exp (iEot

+~_F2

f cos

=

2))) exp ~{~(2) sin 2wt}

~i 0(r)exp {i(Eo + ~.E(

If we perform spectroscopy on such an oscillating orbital, then we will observe a spectrum given by a Fourier decomposition of ~i: +°~

0(r,

t)

=

0o(r) exp{i[E

2)]t} 0 + ~E(

~

( ) 4w

e~1~t.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

335

2nw with relative internsities J,~[AE~2~/4w]. The spectrum contains components at [E0 + ~~E~2)]± The photon replica or satelite levels (n *0) become important when co ~ ~E’~2~ = ~aF2. Thus at sufficiently low frequencies or sufficiently high field strengths, nonresonant photon absorption becomes important. Two changes in the spectrum result, namely 1. AC Stark shifts and splittings, 2. Photon replica state populations. Let us now investigate these phenomena more carefully, first using the usual time dependent perturbation theory, and then by introducing quasienergy states. In the rest of this work we shall be primarily concerned with the effects induced in atomic and molecular systems by time-varying strong external electromagnetic fields. Our attention will focus upon the particular “harmonic” case of a periodic time variation with a single oscillation frequency co; sometimes a relatively slow amplitudemodulation of the field will be included. Although important, the general treatment of more complicated multifrequency problems is beyond the scope of this paper. We are now concerned with solutions of the time-dependent Schroedinger equation + H~1)(r,t) — i~/~ .~-j]~Iffr, t)= 0, where the field-free atom Hamiltonian 11(0) and the field-atom interaction H(’~(r,t) are Hamiltonian operators in the vector space of ~ and we restrict our attention for now to a field perturbation of a free-atom bound state. We introduce a switching functionf(t) by H(’~(r,t)

=

f(t) V(’)(r,

t)

and impose the boundary condition t

-+

oo)

—~

0

and correspondingly ‘I’(r, t —°°) ‘P~°)(r, t) 0~°kr) exp {—(i/1~)E~°~t}. The range of choices of f(t) includes the sudden perturbation [103] and adiabatic switching [104] cases. Time dependent perturbation theory is useful at low strengths of the interaction H(~)(T,t). It can be formulated in terms of an expansion using the free-atom (nondegenerate) stationary states as orthonormal basis states 0~°kr) [105] —~

‘I’(r,

W~(r,t),

t)

-+

‘I’(”)(r,

t)

~

a,~”)(t) 0~°kr) exp

[~E~0)t}

The corrections ~j,(n), n > 0, are presumed to make smaller and smaller contributions to ‘I’ as n is increased. Inserting this ansatz into the Schroedinger equation gives [u0(r)

— ill

[.110(r)

— ill

‘I’~°)(r, t) = 0,

~~-J ‘I’(’~(r,t) + H~)(r,t)

~

— 1)(r,

t)

=

0,

n > 0.

Multigrating these equations gives an infinite set of coupled differential equations

aarkr

= ~

~

exp

[— ~

1E~0)t]

lH~°(r, t)l Ø,(°)exp

[—

~E~0)t})ar1

—

336

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

with the boundary conditions a~°~(t) = aj~’)(t-+---oa)0,

n*0.

If the interaction is harmonic H~’~(r, t) = 2 V 0(r) [e~~~t+ e_t(~3t]f(t), and ideal adiabatic switching assumed forf(t), [106], then after the switching is over the fmal results of perturbation theory for a single discrete nondegenerate zero-field level in a nonresonant field through second order is [105] a,~’kt)

11

)

(O(o)(r)lV(r)I0,~o)(r)){e~ [i(w + w,~)t]+ exp [—i(w—co~)t] for all k, w,~+w

2)(t) = (1l)_2

~

c4

+ a~)(t)=

exp (icOimt) co,,0

(~)_2

~ex~ [i(2w + w,~)t] + exp [—i(2w w~)t] (2w+w,~ 0)(w+w~)(2w—co,~~)(w—w,0) —

[(w+co10)

—1

—(w—w,0)

exp 2 (i2wt) ~ l(O°lV0I0~°~)l( 2w(w +w

—1

]j~

k*0

+ exp (—i2cot) +

it

2w(w — co10)

(co + w,0)

it

—

1=0

10)

(co

—

w10)

2— ~(co — w —

~(w + w,0~

10)_2).

Here the field-free atoms energy level splittings are related to the w10 by 11w,0

E~°) —~

5.2. Quasienergy states of dressed atoms and molecules The above perturbation expressions for the ‘I’,,(r, t) are not readily interpreted in a physical way as they stand. The spectral content of the perturbation results can be revealed by a judicious rearrangement and recombination of terms, a procedure equivalent to the introduction of a new set of basis states called quasienergy states (QES) [107—109]. The full Schroedinger operatorH(t, A) is broken up in a new way H(t, Aj’I’

[~io+ AV(t)

—

iJ~ä~-]\I’(r,t, A) = 0,

~e(O) ~jj(O)

— ih

~.,

H(t, A)~.t’(°) +AV(t),

where A is a small real expansion parameter. Since the Hamiltonian jj(o) + AV(t) for our problem is periodic in time, it is reasonable to search for periodic solutions 1J.I(r, t, A) for the full Schroedinger equation: ‘I’(r, t, A)

=

~t,(r,t, A)e_is(~)tm,

1P(r, t + ~!-,

x)

=

~‘(r,t, A),

where ~(A) is the quasienergy for the QES ~ti(r,t, A). To develop a QES perturbation theory, “unperturbed” QES are then ~fr(r,t,A0)0(r,t,A0)exp

[—i/llg(A0)t],

H(t, A)0

=

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

337

where .*‘~°)Ø(r, t, 0)

~(0) Ø(r,

t,

0).

This equation has solutions of the form t,

0)

Ø~°)(r) ~iq(~~t

=

~

=

E~°~ + qilco.

The total unperturbed QES ~i(r, t, 0) is some linear combination of~,q(r, t, 0) that depends upon the switching function f(t). Ideal adiabatic switching in the case of nondegenerate free-atom states correlates a specific value q,, of q with each state 0~(r)and JI(r, t, 0)

=

[~~C,~0,~°kr)

exp

(—~[E~°~+ q,1~!lw]t}]exp [—i/-fl6(0)t],

where the complex coefficients C,, are the probability amplitudes for the stationary states ~ with energies ~o)• Considering again a single discrete field-free level k = 0 in a harmonic field, and ideal adiabatic switching such that ,j4°)(r,t) we seek a perturbed QES ~i(r, t, A) that is periodic in co. The periodicity requirement necessitates [105, 107] (P(A)llI1(A))

1,

(i,1i(0)I0(A))

(~‘(A), ~‘(0)).

These requirements must be combined with a perturbed QES Schroedinger equation

[~r(°)+ AV(t)

— g(t, A)] ,(i(r, t, A)

0

and the quasistationary state requirement [(~i(0)~i,(i(A)>/(~(A)Iip(0)>] ~0. The resulting QES are obtained by a transformation and have an additional phase factor exp {(i/1l)O(t, A)} which is determined by the equations (1) ~

+ AV(t) —

ao(t, A)

—

~(t, A)] {i,(i(r, t, A) exp [(i/1I)O(t, A)]}

=

0,

(2) O(t, A) periodic in T = 2ir/w and real, ao 1 +T/2 (3)-~-=~(t, A)—7 g(t, A)dt,

f

-T/2

+T/2

(4)

f

O(t,A)dt=0,

— T/2 (5) d(t, A)

=

1(0) + ARe [(~‘(A)lV(t)l~tI(0)>/(%t’(A)hIJ(0)>].

We now make a QES perturbation theory expansion for the QES wavefunction ~i(r, t, A)

[~A”Ii~(r.

t)] exp{—(i/li)[ g(t, A) + O(t, A)]t}

338

I.E. Bayfleld, Excited atomic and-molecular states in strong electromagnetic fields

and likewise expand the quasienergy ~(A) and the functions ~ (t, A), O(t, A): 1(A) _=~An1(n),

g(t, A) ~

O(t, A) ~~AnO(n)(t).

1(n)(t),

The result through second order is i,1,(r,

t,

A)

=

where i~(t, A) E1

{Oo + A[0~ ~

+ 0~e~~] + A2[0~ ei2~t+ ~

e~”t+ 20~~] exp [—(i/i1)~(t, A)]},

E

2E 0t + 2AE1 sm cot + 2A 2 (t + sin 2wt)

(ø~°~i V0I~°~), E2

V~k/.~ + 0~?),

~

V(t)

2V~cos cot.

oW remain to be found by solving the time-independent ~equations

The component QES [1J(0) — E0 ±11w] ~

+ [V0 — E1] ~°)

=

0;

similarly — E0 [H(0) —

±211w]Ø.~)+

2~ + ~[ V

E0]0~

[V0 — E1]oW — E20~~ = 0,

0 — E1](Ø~’~) + 0~) E20~°~ = 0. —

We are now in a position to rewrite the results of usual time-dependent perturbation theory in terms of the component QES 0~,OW, O~ Note from the equations above that 1’~, ~i(2)(r,t) = O?~ei2wt + ~ ~ + 20~2), 0~°)(r, t) = O,~0), i,1i~’~(r, t) = O~?eic~)t+ 0~e thus the desired QES decomposition is [105]; ‘I’(°)(r,t)

=

i4s(°)(r)exp

[— ~.5

‘I’(’)(r,

=

[~1J(1)(r,t)

!_ç~’)

t)

—

(2 sin cot)

~i~0)fr)]

~‘.Jr(2)(r t) = (~i(2)(r,t) — ~

._~__

exp

g(1)

[—i- s~0)tJ

(2 sin wt)~W( 1.t) 2 sin cot)}2~,(o)fr)) exp

[t + (sin 2wt)}O(o)(r)

~

[_~ g(0)t]

[~~i~(

1~2~=2E 2,

1~=O

These results for the QES formulation of perturbation theory deserve some inspection. Focusing first on the new phases linear in time, we see that in the “one level” problem a field-induced energy shift occurs in second order; it is equal to 1(2) = 2E2. Photon replica contributions to the full wave function occur in all the orders, through a number of terms. In first order the absorption and emission of one photon from the field is responsible for e±t(~)t terms, while in second order two2)/cocharacterize photon processes occur in addition. The field strength parameters ~~‘~/coand ~~ the size of many of the multiphoton absorption terms. In the static limit co 0, ~ = 0~ ~ and ~ = = 0~ Ø~.Thus the QES results smoothly go over to the static field results -~

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

0) 0) -÷E

—~

—~

—~

339

+ (2A)2O~2~ + . . .] exp (—ifl(t, A)/h), [~(°~+ 2AO~1~ 2 (o~°~l v

0 + 2A(O~°~l V0IO~°~) + (2A)

0l~~+

The question arises as to whether QES are better basis states for practical calculations than are the field-free states. Considerable work has established a reasonable level of mathematical rigor for theoretical work based upon the introduction of QES at the outset. Developed are QES variational principles, Hellman—Feynman theorem, and hypervirial theorem [107]. Existence, orthonomality and completeness theorems exist for some problems [110]. The asymptotic nature of QES perturbation theory has been investigated [107, ill]. The question of errors introduced in truncating basis sets in practical problems remains unsettled in general. As the applications discussed below suggest, such errors may be smaller at intermediate field strengths when QES rather than field-free basis states are used. The induced electronic electric dipole moment d(t)

=

e(’P(r, t)Ir~’I’(r,t))

has been obtained to third order using the QES formulation [105] and produces all the terms one expects: static zeroth and second order moments, induced first and second order moments, fielddependent corrections to the polarizabilities, and second and third order harmonic-generation terms. The usual probability interpretation of the amplitudes obtained in perturbation theory assumes that the eigenvalues of the basis-defining Hamiltonian constitute an observable energy spectrum. As we shall see, spectroscopy on atoms in strong fields can be performed and photon-replica energy levels directly observed. An old topic has been the question of choice between the distance (er F) and velocity (eA p)/c forms for the interaction Hamiltonian [112, 113, 270—272]. In the long wavelength, uniform field (electric dipole) approximation the two forms give equivalent results if no truncation of basis sets is made [112]. In the case of bound state interactions, truncation with convergence seems to be easier with the er F form. Work involving continuum states usually is based on eA p. At strong field intensities nonresonant phenomena require calculations better than that obtained with the rotating wave approximation, where 2 cos cot is replaced by exp (icot) [273]. Also, quadrupole and higher order terms in the multipole expansion can be important [114]. The question of er F versus eA p at strong fields is unresolved, although the new development of a perturbation theory that is gauge invariant in every order is directed towards this problem [116]. The above perturbation theory of nonresonant field distortion of atomic structure has been applied to several problems. An example is a molecule with a permanent electric dipole moment d0 in a strong linearly-polarized low-frequency field [116]. If the interaction energy is small compared to the molecule’s rotational splittings wj,, then the total angular momentum is conserved as well as its projection m1 along the field direction P. The wave function for the molecule in state lJm1) becomes field-modified by the factor 2 (t + sin 2cot/2co)], a~(t)= exp [(id0F/co) sin cot + ~ia0F where d 0 (Jm1IdlJm~>is the static dipole moment and 2/co~, is the polarizability. a0 ~ RJm1ld[im1)l

340

I.E. Bayfleld, Excited atomic and molecular states in strong electromagnetic fields

The photon replica spectrum is obtained by Fourier analyzing a1(t): a1(t) = exp (iaoF2/41t ~

~$n) =

~

exp ~

a~f)e_i~~~)t,

sint’ +

~

sin 2t’ + ~t’)

dt’

=

~(—l)~

~+

25

(~)~(g~)~

The result contains as special cases Townes andWe Schawlow in the the interesting limits d0 = 0 2/8cothe~ results I (s = 0ofdominates). consider [117] some of (n = —2s only survives) and a0F special cases now. (1) At very low field strengths, d 2/4 -~co. Then only a5°)is important, and the only observable effect is the energy 0F~ shift co E and a0F 2/4. = a0F (2) Intermediate case A assumes a large 2molecular polarizability, i.e. d 2/4 ~ co. 0F -~ co and a0F The level shift remains E 2/8co) all important for n 2/4co. 2, but there are photon replicas weighted by J_2,, (a0F ~ n2 a0F case B assumes a large dipole moment, i.e. d (3)1 Intermediate 2/4 ~ co. The shift is 0F ~ co and a0F still E 2, but there are replicas now weighted by J,,(d0F/w) all important for n ~ n3 2/4. d0F/co. A number of (4) “High” field case A again assumes a large d0, i.e. d0F>> co and also d0F>> a0F important replicas lie in the region up to n 2/4co — cl 4 a0F 0F/w. Using the saddle point method to evaluate the integral for a1 leads to the result that the width ~n of the region of important replicas is fractionally small, z.~n~ n~and is not peaked near n = 0. The energy shifts here are predominantly linear [116]. (5) “High” field case B is for a0F~/4 d~F~‘ co. All replicas up to n4 are important, and large linear and quadratic energy shifts occur. Combining cases (3) and (4) with what we know about the static-field linear Stark effect, we can construct a qualitative picture of the frequency dependence of the perturbed molecular spectrum; this is shown in fig. 5.1.

high F wro

,~iiiIi~,

______

I

Ill

Iii

I

Ii

Ii I

high F

h:gherw low F high w

Energy Fig. 5.1. Crude qualitative features for an energy level ofa I = 1/2 polar molecule in an electric field of strength F and frequency w. The photon-replica spectrum ranges from the static-field limit at the top to the weak field situation at the bottom. AC Stark shifts are not shown, but are superimposed upon the replica spectrum. As discussed in section 7, the n = 2 excited states of the hydrogen atom should exhibit similar behavior at an intermediate range of values of F and w.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

341

Table 5.1

Polarizabilities and components Xj~x

11

of the nonlinear susceptibility tensor of rank four for the ground states of a number of atoms. Values are for the static field (w = 0), Nd-glass laser and ruby laser frequencies. Xo x1(c~,= 0). Numbers in parenthesis show powers of ten: (n) = 10(n)• Values of and x11 for Cs at w = 0.0656 are not given because of the presence of the two-photon resonance 6S9D. After ManakovetaL [118] Atom

H He Ne Ar Kr Xe Li Na K Rb Cs

c~= 0

“~d= 0.043

0.0656

a

X0

a

x1

x11

4.5 1.49 5.2 13.1 18.2 27.1 166 162 290 318 396

222.1875 9.5 51.0 390 800 2020 2.7(5) 1.04(6) 3.48(6) 5.42(6) 1.18(7)

4.35 1.49 5.25 13.2 18.3 27.4 277 234 607 692 1117

82.967 3.18 16.9 130 292 705 —8.8(4) 2.5(5) 1.3(7) 1.1(8) —6.1(7)

251.577 9.55 51.1 408 898 2150 2.4(5) 1.0(6) 2.6(7) 1.7(8) —4.4(8)

x1 4.618 1.49 5.27 13.3 18.5 27.9 2.46(3) 575 —1320 —1206 —775

85.78 3.24 17.1 140 338 800 —5.5(8) 9.7(7) 7.8(7) 6.2(7)

264.62 9.76 51.9 442 1065 2460 —1.7(9) 1.1(8) 1.5(8) 1.3(8)

—

—

A second application of non-resonant QES perturbation has been computation of at 4 term theory coefficients) for the ground state atoms dynamic polarizabilities and hyperpolarizabilities (F the ruby laser (w~= 0.0656 a.u.), Nd-glass laser (wNd = 0.043) and static (co = 0) frequencies [118]. The dependences on co are found to be small (see table 5.1) except for alkalai atoms, where the laser photon frequencies are not so much smaller than the energy gap between ground and first excited states. The effect of field elliptical polarization has also been investigated for nonresonant processes at low frequencies. A major result is that the ellipticity e enters into the argument of the Bessel function photon-replica factors as J~[(1

—e2)E 2/2co1.

For circular polarization 1 — e2 = 0 and no replicas occur. Experimental verification of the strong nonresonant electric field distortion of atomic structure exists, although detailed studies have not yet been made on very many specific systems. None of the experiments has directly addressed the question of non-adiabatic effects. An early observation of photon replicas was made in the studies by Townes and Merritt [119] of molecular spectra perturbed by strong microwave fields. More recently replicas have been seen in the emission spectrum of He in a CO2-laser field [129, 121] and in the absorption spectrum of highly excited H in microwave fields [122—124]. A most interesting case is CO2-laser induced replicas about the elastic scattering peak in 2, free-free the energy loss spectrum for e—Ar scattering At a laser flux watt/cmreplicas induced multiphoton transitions of order up to[125, three126]. were responsible forof thel0~ observed shown in fig. 5.2. Semiclassical theories nonperturbative in the field strength [127, 128, 188, 189] are able to explain the observations. Related nonresonant strong field phenomena involving spin states in strong oscillating magnetic fields have of course been well studied in microwave spectroscopy using both molecular beam techniques [129] and the optical pumping of vapours, see the references in [130—132]. It would be well to summarize some aspects of strong field studies at this point. In general, quasi-

342

I.E. Bayfield, Excitedatomic and molecular states in strong electromagnetic fields

liii” —2

0

2

—2

0

2

Energy (Units of Loser Photons)

(a)

(b)

Fig. 5.2. The energy loss spectrum for 11 eV e—Ar scattering at 153 degrees, (a) without and (b) with a ~ 10.6 microns, after Weingartshofer et al. [126].

2 laser field at

watt/cm

energies (QE) are not those observed in a spectroscopy experiment. There must be a Fourier analysis of the complete wave function in order to compute the spectrum for the system of the atom interacting with a strong field. At weak fields the QES each have only one important Fourier component that adiabatically correlates with a field-free energy eigenstate; in this case the QE do directly determine the strong lines in the atom’s spectrum.

6. Atoms in strong nearly resonant oscillating electric fields 6.1. The two-level problem Let us now consider two nondegenerate stationary states Ot, 02 of an atom that have corresponding eigenergies E 1 and E2. Consider a nearly resonant applied oscillating field of frequency co, for now assuming one-photon resonance: E2 — E1 ~ 11w. The component QES having quasienergies (E1 + 11co, E2) and (E1, E2 — llco) are pairwise nearly degenerate. Let us assume a field-switching that populates the two unperturbed QES u~°kr, t) = 01(r),

g(0)

E1,

=

u~°kr, t)

=

O2(r)e_~.~)t,g~O)= E2 —11w.

To first order we must now solve a secular equation for the degenerate QES [107, 118] to obtain ~‘+fr, t,

A)

=

[C1~u1(°~ + C2+u?~]exp (_~.[~(E1+ E2 —11co) +

~~i)J)

where

2] 1~’2, ±[& + I(011V0102>1 2+1C 2=1. 1C1±1 2±1

~

(E 1

—

E2 + llco)/2X,

If the remaining arbitrary phase is set by assuming lated stationary state is [133, 107]

i~1i(t=

0)

R~

=

(OIVIO)

C =

E~’~ —

~

0~,evolution of the initially popu-

I.E.

~i(r, t, A)

=

Bayfield, Excited atomic and molecular states in strong electromagnetic fields

exp [—(i/211)(E1+ E2 R~ R

11co)]

—

______________________

+ {[R÷exp [—i(A/1c)E~)t]

—

343

R exp [—i(A/11)E~’~t]]O1

+ e~t[exp [—i(A/11)E~)t]— exp [—i(A/fl)E(’)t]]~J. This wave function contains spectral information obtainable by analysis. It also can be used to compute a transition probability, a topic discussed primarily in the latter parts of this paper. If one computes the probability for the system to be in the state 02(r) at the time t, one obtains the wellknown Rabi formula: 21<0 21r2 [sin co~’~t/co~’~]2, coW As/fl.

P2(t) = A

11V0102)1

These results and generalizations of them that include the possibility of spontaneous decay of one or both field-free states are well established using theories that do not necessarily introduce QES explicitly [134, 2, 108, 109]. Other names for periodic QES are Floquet modes and dressed atom states. Calculational difficulties arise within the two level problem when field strengths are reached where multiphoton transitions are rapid, thereby populating many photon replica levels. Various numerical procedures based upon Fourier expansions of the QES have been developed and adequately reviewed [2]. The amplitudes for the (Fourier) component QES satisfy an infinite set of algebratic equations which have continued fractions as solutions [2, 107, 135—139]. The numerical approaches seem capable of “solving” to any accuracy the problem extended to any reasonable finite number of bound states [138, 140] and are being extended to include continuum QES [141]. Semiclassical [142] and full quantum [137, 143, 138] treatments of the external field give identical results for various phenomena. The results of two-level calculations are exemplified in the pioneering work of Autler and Townes [135]. Figure 6.1 shows the calculated energy level spectrum (along the vertical scale) computed for the fixed ratio of coupling strength to field-free energy level separation E 2 — E1 57r(011d Fj~)and plotted as a function of photon energy 11w in units of E2 — E1. In the resonance region E2 — E1 ~flco,

2T~~~•

I

,,

0

0.2 0.4

~ 0.8

ic~/( E2— E1

1.2

)

—~

Fig. 6.1. The frequency dependence of the Autler—Townes effect, i.e. of the spectrum ofa two level atom in a strong oscillating electric field. For a discussion, see the text and the original reference [135]. Levels associated with the upper state symmetry are shown dashed, while the others are solid lines. The width of each curve is a measure of the relative amplitude for that level.

344

I.E. Bayfield, Excited atomic and molecular states in strongelectromagnetic fields

a vertical cut in the figure exhibits two peaks displaced to each side of each of the field-free energy values. The symmetry character of the states associated with each pair of peaks is predominately that df the field-free state corresponding to the nearby energy level. Above the resonance region, broad lines with energies near the field-free values exist but with a separation less than E2 — E1. The figure also shows the movement of the spectral lines as the photon energy is reduced to the value 0.2 (E2 — E1), well into the photon-replica region discussed in the last section. The replicas having appreciable intensity lie in a band of total width approximately 27r(011d centered near each field-free level. For the chosen coupling strength the replicas halfway between E2 and E1 are of low intensity as are those well outside E1 and E2. The motion of the levels is in accord with the Von Neumann non-crossing rule which states that energy levels having the same symmetry properties can not cross, i.e. be energy degenerate [144]. The “splitting” of levels in the resonance region has in recent years often been called “the” Autler—Townes effect, alias the optical or dynamic Stark shift. It would seem appropriate to call photon-replica formation at lower frequencies the Autler—Townes effect also, [124]. The splitting of levels near resonance [275, 276] is also termed the production of Jaynes—Cummings doublets [274]. 6.2. Resonance fluorescence and “two-photon” experiments A large number of optical experiments on the dynamic Stark effect in the one-photon resonance region have been carried out recently. A principal difficulty in such work has been the experimental limiting of the system to the interaction of only two levels; otherwise one must include three or more levels in the theory. Once one or the other of these two steps has been taken, theory is in good agreement with experiment for the field strengths investigated so far, i.e. for those corresponding to strengths less than that assumed in2.fig. The Stark utilize effect is clearly observable for visible andbe IR The6.1. experiments a weak probe transition which can laser fluxes as low as one watt/cm either spontaneous decay to a third level or an induced transition to such a level. The case of spontaneous decay is called “resonance fluorescence” [145, 146, 140], see fig.- 6.2. Induced probe

E:3}~.~ 2}IIT2MH 3 Level 2 Level l

(a)

—200

0

200

Scattered Spectrum (MHZ) (b)

Fig. 6.2. Observation of the near-resonance dynamic Stark effect in a resonance fluorescence spectrum (b) produced by dye laser excitation of the D 2, accordingspectrum to Schuda et al. [145]. 2 transitions in Na shown in (a). The fluorescence is plotted for different amounts oflaser detunmg from resonance, at a fixed laser flux of 406 ±100 mW/cm

I.E.

Bayfield, Excited atomic and molecular states in strongelectromagnetic

fields

345

transition experiments are usually called “two-photon”, a term that includes the possibility of having the probe field strong. Recent two-photon experiments utilize two lasers. Examples are the dye-laser transitions 32S~2— 2P 2P 2S~ in Ne [148], and 3 3p, 3 312—5 2in Na [147], the and He—Ne transitions the IR xenon laser transitions at 3.51 4.54laser microns between 5p55d—5p56p sublevels in Xe [149]. The theory [150] for two-photon experiments in Na has been modified to include hyperfine effects [15 1] in attempts to explain the data. A similar multilevel story has unfolded for resonance fluorescence [140, 152], although a symmetry in some experimental Stark triplets remains unexplained [277, 288]. 2P4—3~2,

2~2—2P4

6.3. The low frequency limit for the two-level problem In the low frequency limit co ~ co 12 ~1F’(E2 — E1), many photons must be absorbed for a transition to be induced between the two field-free energy levels E1 and E2. As we have seen in fig. 6.1, the spectrum for the two-level problem tends to that for two of the isolated or one-level problems of section 5, assuming the field strength is not too large, i.e. V0 (011d FI~)-t11w. One can develop a simple theory even for V0 ~ co12 by using two adiabatic oscillating orbitals as basis states, [135, 153]. Formulas using such an adiabatic approximation have been developed by Perel’man [154]. Calculated is the transition rate W12(v) for a nonperturbing probe laser transition at a high frequency v connecting a third far-away level to the photon-replica levels lying in the energy region encompassing E1 and E2. The quantity W12(p) has a number of terms corresponding to different field-shifted multiphoton Raman transitions: W12(v)

=

(V~/2w)jUI2 ~

6(S

—

v/co —

where the quasienergies E~ i~coS enter into the energy conserving functions, and the factor U(v/w, co12/w, V0/co) is a contour integral evaluated for various special cases using the method of steepest decents. Two of the basic strong field effects are predicted, namely (1) photon replica structure and (2) a decreasingly strong dependence on the increasing strength F of the field at frequency co. This latter effect is related to the behavior of the transition rate for the multiphoton transition rate from the level E1 to level E2 due to the field at co alone. Such topics are discussed primarily in sections 9 through 12 below. 6.4. Fine structure and Zeeman effects on atomic quasienergy levels The theory of strong resonant field distortion of structure can be extended to include the effects of atomic fine structure [155—158]. As is the case for static fields, the zero-field atomic coupling scheme is destroyed at large field strengths, changing the nature of the Stark shifts and splittings as well the work considers the effects on multiplets 2P as the selection rules [157], see table 2S~6.1. Most of 2D 2S~ n’ 112,312 one-photon coupled to an n 2state, or n’ 312512 two-photon coupled to n 2.For LS coupling, P and J5 have good quantum numbers at low fields, while only L5 and S~have at high fields, with 2 along P. The strong field 2S~ decouples L and S, splitting levels with different 1m11. Multiphoton transitions involving the n 2state 2S~ 2P~, then couple only m1 = 0 states of the various n’ 21 levels. For the case of one-photon n 2— n’ 2,312coupling, an eight order secular determinant is solved

346

I.E. Bayfleld, Excited atomic and molecular states in strong electromagnetic fields Table 6.1 Electric dipole selection rules for one-photon effects upon atomic states with fine structure, [157J. LS, if and /1 coupling cases are considered in various low and high field limits

LS-coupling

V’~VLs

V~’VLs

~J=0,±l ~L=0,±l

~S=O

=

=

0

&W0,±1

f/-coupling

/1-coupling

v,~v/j

vff.~v’~YjS

~J0,±1 410,±1 ~.M=0,±1

4f~0,±1 ~m~m 1”0,±1 ~m,=0

V.~VKS

VKS.~V.~I~I,

~J0,±1

~K0,±1

~M

~S=0 ~.M ~mE

0 ~S0,±1

v~,.vls

~S0 ~M=~in1=0,±1 = 0

V~’V, ~S=0

0, ±1

~M ~in5 = 0,

±1

~m50

~in1 = 0

to obtain the quasienergies [156]. In the strong field limit these QE become, for m5 2)], = E~3/2~ E2) = E~1/2+ ~[E~,3,~2 — E~112][1 + 4m5Re(F~~F0~/IFoI =

=

+~,—~:

±~&2IFoI[fs/(2m~.Esp)]1/2

+~{ES/ +E~, +11w —~[E~

2)]}.

312_Ep1/2][l +4m5Re(F~F0~/IFoI

The terms proportional to m 5 exhibit a polarization complex field 4~ exhibit the expected linear dependence optical Starkthrough shift. Athe generalization to amplitude two F0. The four levels E,~’ coupled doublets has also been discussed [155]. For the case of two states with total angular momenta j, J, (V — /1 < 1) and one-photon resonance, quasienergies have also been found for an arbitrary elliptic field polarization and for an external static magnetic field B 0 [1581. Fig. 6.3 shows these for the case of linear polarization with F along B0 and =2 A i(F~F~~, - F0~F~)/IF0I as functions of V/~ dF/p.~g 2B0 and of the quantum number m1 for the component J~off along B0.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

? ~

347

M~M~M

3[

,/

,/

3

~

8=0

8=~/2 V/L~

4_,,, Bi~

3[_

8=3~/2

~

Fig. 6.3. Quasienergies form

1 = 0, ±1sublevels of two l = 0,1 atomic states one-photon coupled by an electric dipole matrix element V. The free-atom level splitting is called ~, and the frequency deviation from resonance is called 6. After Zon and Urazbaev [158].

7. Dressed hydrogen and helium atoms In a sense the excited states of one-electron atomic systems are special because of their small fine structure due only to relativistic and quantum electrodynamic effects rather than multielectron Coulomb effects. Except for quite low field strengths, the energies of the stationary states of H with given n (but different 1 and m) are essentially degenerate. As in the static field case of section 2, oscillating electric fields produce large perturbations in hydrogen since even for small field strengths the interaction is essentially first order. For off-resonant fields the degeneracy in the field-free atom is handled by solving a QES-state secular equation. Near resonance there are then two types of degeneracy within QES theory, since the energy degeneracy of the component QES also occurs. The latter of course is just a way of describing atomic resonance with the external field frequency. Despite the 1, m1 degeneracy complication, H is an interesting system to study using QES theory. Accurate theoretical calculations are possible in principle because both the free-atom and the lowintensity static field Stark wave functions are known to high accuracy [7]. As might be expected, the problem of hydrogen atom structure in an oscillating electric field is most easily addressed in certain limiting cases. At low fields where V/co 1, both low (co -~E1, — En+ ~) and high (co >>En) frequency limits can be treated [134, 159—161, 118, 162]. In addition, one can discuss a low frequency (co <
348

I.E.

Bayfield, Excited atomic and molecular states in strong electromagnetic fields

7.1. Hydrogen atoms in weak linearly polarized fields

Let us first consider the linear polarization case. One interesting prediction of perturbation theory in the high frequency limit is that the H-atom field-free energy level ~ splits into ~n(n + 1) sublevels all having a quadratic shift [161, 118]. The QES are superpositions of q = 0 harmonics (i.e. the fundamentals) in this limit. The frequency-dependent complex polarizabihity 13k approaches a limiting value —4/co2 a.u. and the energy shift 6k accordingly approaches —j3~F2/4= F2/co2, just the kinetic energy of a free electron in the field. Perturbation calculations of $3k at Nd-glass and ruby laser frequencies [161] for n ~ 6 show an approach to the high frequency limit, either as co is increased or as n is increased. This occurs for all m and both parities, except in those cases where a resonantly coupled state of low 1 exists, see table 7.1. An example of the latter situation is the NdTable 7.1 Polaiizabilities of the levels of the hydrogen atom for n ~ 6 and linearly polarized radiation at the Nd-glass laser (w = 9440 cm1) and ruby laser (w = 14 400 cm~)frequencies, after

Zon et al. [161]

N

L

m

~=9440cm~ Rep

1 2

r .~ 1

e e o o

0 0 0 1

e o o e e

0 0 1 1 2

I

e

0

o

0

4

5

Imp

Rep

0

1

e e o o

1 2 2 3

e

0

o

0

o e e

1 1 1

e

2

o o

2 3

—442.6

0 0 0 0 0 0 0 0 0 0 141.7 201.8 222.6 71.56 139.3 66.52 148.8 89.21 50.04 28.81 7.382 100.2 73.41 107.2 44.82 68.24 37.06 7.067 71.07 42.06 6.250 27.35

—573.6

15.65

e

3

—545.8

4.744

—233.4

e

4

—560.9

2.646

—236.0

C

4.550 181.6 336.5 242.1 1—2036 1 —727.0 —1404 —833.7 —1734 —937.0 —599.0 —525.4 f —568.6 —374.4 1 —580.3 1 —399.8 —601.5 —616.9 —480.0 —612.1 ~ —526.4 —395.7 —551.5 —457.2 —333.2 —565.5 I —360.7 —528.6 —462.7

‘14 400 cm~

f —578.1 1 —535.1

4.618 887.9 1751 1253 —2227 —232.6 —1239 —264.4 —1678 —287.9 —309.9 —238.6 —293.6 —216.4 —248.0 —219.8 —295.2 —252.4 —229.7 —246.3 —236.8 —230.0 —2491 —250.6 —220.9 —240.8 —222.9 —248.3 —230.4 —242.1 —231.5 —228.9 —239.1

Imp 0 0 0 0 86.01 98.60 132.7 87.19 69.29 42.19 32.84 44.03 53.11 7.949 33.44 7.508 26.37 15.79 5.817 3.343 25.00 0.632 14.32 26.67 4.953 16.42 4.679 13.01 0.607 7.722 0.530 3.551 2.032 0.404 0.225

I.E.

N

6

Bayfield, Excited atomic and molecular states in strong electromagnetic fields

L

Table 7.1 (cont.) 1 ca., 9440 cm Rep Imp

m

~ —548.6 { —530.1 60.01 g —536.8 ( —169.7 64.47 1 —556.6 —537.2 —30.09 1 —531.5

e

0

o

0

o

1

e

1

1 —153.6

e

2

~

o

2

o e e o o

I —235.0 1 —558.8 1 —540.5

3 4 4

—542.5 —552.2 —534.4 —547.2

—561.9 —538.5

5

44.55 5.867 54.89 0.684 53.88 33.77 38.52 0.665 21.91 5.630 39.71 23.37 4.952 60.74 16.03 9.143 0.511 3.743 2.084 0.379 0.207

=

349

14 400 cm~

Rep —233.7 —230.9 —237.7 —231.6 —224.2 —239.0 —231.7 —225.5 —237.3 —237.4 —231.1 —237.9 —231.8 —231.9 —229.5 —236.1 —232.3 —232.9 —234.4 —232.8 —233.5

Imp 15.30 0.546 7.301 0.042 3.103 15.13 0.041 2.913 9.257 7.381 0.524 4.364 0.459 0.037 2.180 1.244 0.031 0.349 0.194 0.023 0.013

glass laser coupling of n = 6 to n = 3. In dipole approximation the n = 3 states do not couple to states of higher n that have high 1. Thus the n = 6 high-l containing levels have asymptotic values of ~k while the n = 6 states with low 1 do not. An interesting close connection may exist between the F2/co2 shift here due to photon processes in the applied radiation field and certain quantum electrodynamic shifts that involve virtual photon processes in the vacuum field instead [134, 279—282]. The external field shift can be written e2n 7/2mw, where n7 is the photon number density. It appears like a mass renormalization (in a quantum field theoretic development) that is finite rather than infinite; relativisticahly 2 + e2n 2. m* = (m 7/co)’/ The existence of an external field mass shift is experimentally unverified, and the concept of m* should therefore be used with caution. Perturbation calculations for linear polarization have also been carried out at CO 2 laser frequencies [162]. The level shift is obtained from 2, z~E —2.072 X l0’~ 13(a.u.)[F(V/cm)] where the values of j3 forn ~ 6 are given in table 7.2. The calculated splitting of some H 6 lines 2 linearly polarized CO 1) is shown in fig.(n7.1. = 6 ni = 2s) in a 1 MW/cm 2 laser field (Mw = 1000 cm At this flux level the optical Stark shift is as large as 0.1% at these relatively low frequencies. At 10 MW/cm2 there would be a 1% effect in n = 6, which is beginning to be large enough to have applicability in experiments requiring field-dependent frequency tuning. The QES polarizability seems to increase about as n5, so at n = 10 one expects a 12% effect. —~

I.E. Bayfield, Excited atomic and molecular states in strong electromagneticfields

350

Table 7.2 The polarizabilities p for hydrogen atoms with principal quantum numbers n ~ 6 in a CO

2 laser field. The

field polarization is here assumed linear. After Zon [162]. n

I

m

p

n

1

m

p

n

I

m

1 2

e e o o e

0 0 0 1 0

5

e

0

3.024 (4)

6

e

0

o e o e e

0 1 1 2 0

0

0

e 0

1 1

e o o

2 2 3

4.500 (0) 1.204 (2) 2.168 (2) 1.565 (2) 1.019 (3) 2.449 (3) 1.669 (3) 2.173 (3) 1.167 (3) 1.419 (3) 5.434 (3) 1.158 (4) 8.590 (3) 1.582 (4) 1.010 (4) 5.835 (3) 1.490 (4) 6.657 (3) 1.221 (4) 7.809 (3)

3

4

p —2.943 (5)

6.436 (4) o

0

e

1

o

1

e

2

0

2 3 3 4

e o e

1.105 4.908 8.304 5.410 1.063 3.149 7.738 3.455 9.397 6.248 7.358 3.916 4.530

—9.775 (4)

(5) (4) (4) (4) (5) (4) (4) (4) (4) (4) (4) (4) (4)

o

0

e

1

o

1

e

2

o

2

e o

3 3

e o o

4 4 5

—4.932 —3.757 —2.084 —6.746 —2.201 —4.686 —3.345 —1.004 —6.541 —1.081 —3.997 —2.474 —5.926 —2.896 —1.212 —4.907 —1.405 —3.488 —1.674

(5) (5) (5) (5) (5) (5) (5) (5) (5) (6) (5) (5) (5) (5) (5) (5) (5) (5) (5)

An early experimental search for CO2 laser induced shifts in H was conducted by Dubreuil and Chapelle [168]. A hydrogen gas discharge was used; the effect upon the visible spectral lines was observed in emission with a spectrometer. Laser-induced shifts and partially resolved splittings of the H8 lines were observed, but the resolution was insufficient to resolve the effects in fig. 7.1, see 2 fig. 7.2. There was evidence for significant but incomplete laser field ionization even at 0.5 MW/cm

~, 0

17~

I

0.60.40.2

0

~o~j 0

0.4

0.8 cm

—

0

(

Wavelength Shift A Fig. 7.2. Partially-resolved experimental splitting of the H 6

Fig. 7.1. Calculated 2 linearly-polarized splittings of some CO H6 lines of hydrogen by [162). a 1 MW/cm 2 laser field, after Zon

lines of hydrogen atoms in a gas discharge and in a CO2 laser field, after Dubreuil and Chapelle [168].2 —~—-—~— The laser4fluxes MW/cm2 for the three curves and—5 MW/cm2. are — — — 2.5 MW/cm

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

351

for n = 6, 7, 8 and 9. However, atomic collisions of the photo-excited higher levels may have contributed to this. Also, these experiments were complicated by laser induced excitation and dissociation processes in H2 occurring along with those in H [169]. The quantitative comparison of theory with these experiments is further difficult because the laser was operated multimode, which leads to mode-beating time-dependent intensity variations. 7.2. Hydrogen atoms in circularly polarized fields Let us turn now to a perturbative treatment of the circular polarization case, considering H (n first. We seek the quasienergy states W~(r,t)

=

2)

e_iStu~(r,t),

where g is the quasienergy and the u, are periodic u,,(t + 2ir/co) = u~(t). For this special case of circular polarization, the time dependence of u~can be removed by transforming to a rotating coordinate system u,(r, t) = exp (—iecoLt)~~(r), where the problem of finding qS, is time-independent: [H°(r)+Fx+ecoL5

—g]~~(r)=0.

Here F is the amplitude of the oscillating field now along x in the rotating frame, and the axis 2 of quantization is along the propagation direction of the field rather than along x as in the linear polarization case. The quantity e takes on values ±1corresponding to left and right handed polarization. The field in this case couples states with different eigenvalues m, m1~of2Po L5 state as well as requiring undergoes ito change. Thus the 2s and electric dipole in coupled while the only nonresonant effects. The2p~states 2Po shift ~are can be written terms of hypergeometric functions [118]; ~ mE~°) —

4

=

~a

2, 2~0F~,

±~

[IE~°~I~ co]V2,

P(p)

a2~0(co) [P(v~) + P(vj — l]/co (2 +

v)2(3 — v) 2i’1[l~ —2—v; 4—v;

(~~

The coupling between the ~, in this problem enters in second order, with frequency-dependent composite QES matrix elements being involved: W

2 (ilxIk)(kIx[f)/(~J(°)— 11mF IC’ ~ if ~

g(0))

where the subscripts i, / take on values 0, 1, —l for the 2s, 2p~ 1,2p_1 states respectively and the sum is over all possible intermediate states. The W11 can be written in terms of 2F1’s. The are the shifted zeroth-order energies of unperturbed QES component states g(0)

g’(o)

gJO) + emICco.

2 + 9F2)’12 the energies of the mixed 2s, 2p~ In terms of the WI, and j3

(w

1levels in the field are

352

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

E~°~ — ~

2

(W11 — W_1 ~) — ~(W11+ W1_1) + (3F/213)2 [W11+ W_1_1 + W11

=

+ W_~1— 2W00], ~~3~Er)—

2[W

~3=—W00—(3F/213)

11+W_1_1+W1_1+W_11—2W~].

In the how frequency limit co ‘~ IE~~ — ~ n * 2, the frequency co can be ignored in calculating the W,,,. Then one obtains as co 0 the static electric field Stark energies for the parabolic states (n = 2, n1, n2, mj) quantized along P. —~

2[l0+27F2/j32]-+ ~36F

0~[n1n20,mp±l),

~478F~~[n1n20,mp

=±l] ~=o~3 2

4 levels, when co -~4nF they

Thus while generalstatic the oscillating field completely n coalesce intoin nearly levels numbering only ~n(n splits + 1) the = 3. In the high frequency limit co ~ F, ~i2

-+

±co ±~F2 —

W~ 1±1(co),

~

-~

—W~(co).

Figure 7.3 shows the quasienergy shifts ~, 2(F) for n = 2, first for a nearly static case (co = 20 cm’) and then for the case of CO2 laser radiation (co = 1000 cm’). As expected, at higher frequencies the energy levels shift less. For the circular polarization case, we have seen then that a change in character of the system atom-plus-field occurs with increasing field, with the QES changing from states quantized along F to ones quantized along k. The nature of this transition in field mixing can be clarified by considering low frequencies (co ~ E,~°)—E,~°)) and completely ignoring levels with n’ * n. The quasienergies become [118] 2P’2, = ~(°) + (n÷ + n_)co [1 + (3nF/2w) where the good QES quantum numbers n+, n_ in this approximation are for the operators J~ ~(L ±A) along the directions of~2~ wz ±~nF~ they take on valuesn÷,n_= —4(n 1),.. .~(n 1). A is the Runge—Lenz vector [170]. It is clear from the expressions for ~ and En,n+,n that as 3nF/2w increases, S~rotates from 2 to x and as a result the behavior of the quasienergy changes. It —

—

~I~7

~

io~

106

F [V/cm] Fig. 7.3. The quasienergy shifts ~ 1,2 of then = 2 states of the hydrogen atom as a function of the strength F of an applied oscillating electric field, on a double log scale, as determined by Manakov et al. [118). The solid line is for an applied frequency of 20 cmt, and the dotted line for 1000 cm1.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

353

is interesting that the critical parameter (y’) = (3nF/2co)2 = is essentially the ratio of energies for two limiting physical situations [122]: E~cis the (peak) energy of a free electron in the oscillating field, i.e. with the atomic Coulomb force turned off. On the other hand, J, is the ionization potential for the atom with the oscillating field turned off instead. When ‘y’ ~‘ 1, a given QES becomes a single parabolic function with mp = n~.— n_ and there is a linear, static-like Stark effect. In the other limit ~y’-~ 1, the spatial part of the QES becomes a single parabolic function with m~= —(n÷+ n_). 7.3. Hydrogen atoms in strong low-frequency linearly-polarized fields Let us now consider what is perhaps the most interesting of the simple cases we are discussing, namely a low frequency (co <
and the field does not induce a static-like “permanent” dipole moment. At high fields, numerical calculations find that 1~and ~ become the same. A better representation at high fields (y’> 1) is to introduce new amplitudes B 1, 2 that are Stark mixtures B12 =~(A~ ±A21,).

If ~

~(a~+ a2~0),in the Stark representation the coupled 2 equations 2cos2 cot ~(a cos2 cot become 1IB ~ (B1\ = fd2 cos wt—~&F 2~—a~)F 1 2 cot —d 2cos2 cotJ~B at ~B2’ L~(o~21,0_a~)F2cos 2Fcoswt—~&F 2

The new off-diagonal elements are smaller in magnitude than the diagonal ones by a factor of about (a2~— a~)/(a2~0 + a~)~ Thus to a reasonable approximation (but certainly not exactly), the Stark representation decouples the problem, with both Stark states having the same dipole moment Id21 and same static polarizability & Ignoring the off-diagonal terms means that the full wave function is a linear combination of two oscillating orbitals of exactly the form of the molecular dipole problem of section 5, differing only in the signs ford2: 2 .oF2 = [Ci exp ~ F I .d2F sin Wt) ±C2exp I.d2F sin cot,j exp IiâF ~—~---+i-~-~-sm2cot .

y—

.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

354

If at

t

=

0 the atom is in the 2s state, then c1

c2 = ~. The 2, analysis of section interest. 5 now completely is of particular The resultant applies; its limiting case number 4, d2F>> co and d2F>> should ~F linear level shifts and narrow bands of photon replicas occur at CO laser within frequencies at 2, a2value reachand of the field strengths of 5 X l0~ V/cm, i.e. power fluxes of 5 X 1012 watt/cm largest focused pulsed laser systems. However, the basic parameter y’ is proportional to nF/co. Thus intense field microwave or far-infrared experiments in this linear shift regime may be possible at large n. The interesting graphical results of Zon and Shokalov [116] for the photon replicas of the 2s state in weaker CO 2 laser fields are shown in fig. 7.4. Some microwave highly excited hydrogen atom (n = 48) spectroscopy experiments that show strong photon replica formation have been proposed [122] and recently completed [123, 124]. A fast beam of atoms was formed by charge exchange of a keV energy proton beam to form an H(n = 10) component. This was subsequently excited ~o n = 48 photon replica states using Doppler-tuned CW CO2 laser excitation [31]. The n = 10 atoms were excited up to the n = 48 replica levels while passing through a linearly-polarized field within a 9.911 GHz microwave TM010 cavity with its axis along the beam direction. Doppler-tuning over q = 0, 1, 2, 3 and 4 replica energy levels was achieved by sweeping the atomic beam energy through the corresponding energies 9.60, 8.23, 7.15, 6.04 and 5.10 keV respectively. The widths of all five resonances were the same, 900 MHz, possibly arising from unresolved Stark splittings of the QES. Although many Stark sublevels exist for n = 48 and their populations were undetermined in the experiments, it is interesting to note the similarity of the measured field dependences of the n = 48 photon replicas (see fig. 7.5) with the predicted Bessel function behavior for n = 2 at higher frequencies (see fig. 7.6). The microwave experiments at n = 48 show strong replica formation at only 3% to 10% of the field strength required for appreciable ionization. This is just what is expected. The field ionization at nJ = 48 and co = 1.2 X 106 is about the 4)’. Thefor Bessel function classical ionization field F~= (1 6n 1(p) has its first maximum near p = 1, or at a field defined by =

Pi —9n(n1 — n2)F1/co= 1.

q

0.3

+ I Photon Replica

0.2 ~ 0.6

0.4 0.2 —

o.~

\

q +2

+~

,.

+ 2 Photon

0.1

\.-‘~—T~,-—~~----.

~

)c__’~.~_ ~

Fig. 7.4. The relative amplitudes of the photon replicas with q = 0, ±2,±4for the 2s state of hydrogen in a CO2 laser field, plotted as functions of the field strength in i0~a.u., after Zon and Shokalov [116).Notice that the curves for replicas (—q, +q) do not coincide because ofsecond order terms. The basic Bessel function behavior seen here is discussed in section 5.2.

2

J ~ 1

0

.0

2.0

x)

3.0

Fig. 7.5. The q = +1 and q = +2 experimental photon replica signals for n 48 hydrogen atoms dressed by a 9.9 GHz microwave field, plotted as a function of a field strength parameter x. The variable x is an effective value of p determined by normalization of the q = +1 replica to the function

I~(x).

I.E. Bayfield, Excited atomic and

molecular states in strong electromagnetic fields

355

Experimentally F~/F~onize = 4 X 1 0~,which agrees with the theoretical estimate 3co/3(n 2 ~‘~/“~onize [32n 1 — n2)] if the induced permanent dipole strength factor n 1 — n2 is taken to be 22, a reasonable value that can be considered to be determined by experiment to within a factor of two. Thus it seems that it will be possible to use highly excited atom techniques to study photon replicas in the strong field regime where there is strong coupling of levels of different n. The effect of continuum states on strong-field bound-state structure alteration is not only an experimental concern, but also a theoretical one. Bound state processes can strongly affect ionization and vice versa at high field strengths. Ionization processes will be discussed in sections 10 through 12. We only mention here that there exist general methods for determining the ionization widths of field-induced energy levels. The Green’s function techniques of Kovarski and Perel’man [1641 and the more recently developed techniques of Chu and Reinhart [141] both address this problem for the specific case of the hydrogen atom. 7.4. Helium atoms in oscillating fields Turning now from hydrogen to helium, we find experiments easier but theory more difficult. An early pioneering work on strong microwave field effects in He dipole was that of Hicks,were Hess studied and Cooper 1F—2’P electric transitions in [172]. The allowed 4’D—2’P and forbidden 4 emission, with the excited atom exposed to microwaves at frequency co as well as a Zeeman-tuning static magnetic field. Photon replicas were observed at ±2qcoand ±(2q+ 1 )co for the allowed and forbidden transitions respectively, producing “comb-like” structures in the visible spectrum. As the amplitude of the microwaves was increased, more and more replicas were observed and the two combs were shifted in opposite directions by the quadratic AC Stark effect. All these observations are consistent with the theoretical ideas already discussed for hydrogen. More recently, experiments in helium have been done in the infrared using CO 2 lasers. Dubreuil and Chapelle have extend their earlier work2 with hydrogen discharges to those of helium [173]. A caused a laser-induced 0.0 19 A shift of the 43S—23P line CW unfocused beam with a flux 43 kW/cm of Hel at 4713 A. Quantum-defect based perturbation calculations give results near 0.018 A for the shift, in agreement with the observations. Similar experiments at higher laser fluxes (F ~ 6 X l0~V/ cm) have also been carried out for this same spectral line in He by Prosnitz and George [120, 121]. Stark shifts were observed, as well as certain photon replica lines that were enhanced by the nearly resonant 43P intermediate state. Again the results could be interpreted adequately using perturbation theory. Since CO 2 are now possible using pulsed high power 2 laser fluxesexperiments approachingin1014 laser amplifier chains [176], the watt/cm strong field regime in He would now appear possible. Atomic beam techniques could be employed in order to avoid gas breakdown effects. Experiments at 1.06 micron and l016 watt/cm2 using Nd-glass lasers [177] would also be interesting.

8. Field-induced changes in atomic ionization potentials The field-induced shifts of bound states can be combined with similar shifts of the continuum states to produce what one might call an overall shift in the “resonant” frequency of a bound-free transition. At this level of sophistication a number of authors have discussed the question of field-

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

356

induced changes in “ionization potentials” [178—181, 282]. It has not, however, been generally remembered that a series limit becomes ill-defined because of strong field mixing of continuum states with the higher excited states; an excellent example of this is the static magnetic field alteration of Rydberg states discussed in section 3. This section ignores such effects although they are clearly important for the oscillating electric field case also. It was mentioned in section 6 that QES perturbation theory predicts a high frequency limit for the polarizabihity of —4/co2, and thus an energy shift in this limit of ~ = F2/co2 I3~,the positive energy of a free electron in the field. This effect tends to increase ionization potentials. However, the shifts of the bound excited states (again obtained from perturbation theory) can be larger in magnitude and have either sign, depending upon details. Consider a linearly polarized electric field perturbing a hydrogenic atom. One can envisage a semiclassical picture where a given classical electron orbit is altered by the field, producing new excursions about the old orbit in the low field limit. An expansion of the instantaneous potential energy about the field-free orbit location behaves like V(r+i~r)[l +~rV+~(z~r’V)2+...]V(r). Welton [182] discussed the Lamb shift in terms of quantum mechanical fluctuations in position being caused by virtual photon interactions with the radiation field; the shift is obtained by averaging V(r + &) over the fluctuations to obtain the correction to the Hamiltonian. This approach can also be used for an excited hydrogen atom in an applied field [179, 181] , by taking V(r) = —Z/r. Then to lowest order the effective static potential becomes i~V(r)= *(l3~ V)2 V(r) = ~Z{-~7ri3~6(r)+ (~3~/r~)(l — 3 cos2 O)} where i3~= Fco2 has a magnitude equal to the classical excursion of a free electron in the oscillating field. Of course there remain time-dependent terms that can be important [1801. We have seen that the field mixing of the states of given n and different 1 is a treatable problem in hydrogen; the approximate approach of Schiff and Snyder [84] treats them as a bundle of states that undergo a mean energy shift [1791 (—Z4/l5n3)~. This result assumes no coupling between states of different n, i.e. ~ =

-~

E,~— En — or 1

—(Z4/15n3)(F/co2)2 -~Z2/n3, which implies that ~ -~ 4/Z. Thus certainly f~x-~a~is assumed, where a~= n2/Z is the mean radius of the free-atom electron orbit. So in this low-field, high-frequency approximation (F -~co2), since f~x =

~EC/I.~EhI~

=n3co2/15Z4 =

(i3x/cmn)2(1~n/l5)

is small i.e. the continuum energy level shift in the free electron approximation is relatively small. This result has been assumed in more recent work [180, 181], and thus little is known about ionization potentials for the region 13x ~ 1, much less f3~~ a~.For experiments on highly excited states (n ~ 30) in microwave fields insufficiently strong for ionization, one has F 1 0~a.u. and co ~ lO_6 a.u.; then ~ ~ 1 o~and an ~ 1 o~and we see that the unstudied regions are experimentally accessible. Several attempts have been made to transform strong oscillating field problems into new repre-

I.E.

Bayfield, Excited atomic and molecular states in strong electromagnetic fields

357

(a.u.) 0

0

1.0

2.0

—0.1

Fig. 8.1. The ionization potentials of then after Gersten and Mittleman [181].

=

1 and n

=

2 ~ states of hydrogen as a function of the classical excursion parameter ~

sentations that clarify the dressed-atom structure problem and/or simplify the picture for induced multiphoton transitions in atoms. An appropriate unitary transformation is presumed to be the key. However, the momentum translation method [183] appears to have little applicability [184, 185], and the Kramers—Henneberger transformation [186, 187] leads to an effective static potential that requires further time-dependent corrections of equal magnitude [180] except in the high frequency limit. Nevertheless the usefulness of quasienergy or dressed states (see section 5) and the simplicity of strong field structure (see section 3) both suggest that unitary transformations to strong field pictures may be of value. This has been the case for charged particle scattering in a strong oscillating field [188, 189]. Calculations have recently been carried out for the shifts in the ionization potentials of H (n = 1, 2) in the high frequency limit using the Kramers—Henneberger transformation and a variational approach [181]. Circular polarization was assumed. The transformation is to a frame where the electron would be at rest if free in the field. It then sees the atomic nucleus driven around it at the applied frequency co, assumed large compared to electron orbit frequencies. After time-averaging, the electron effectively sees the field of a ring of charge at radius ~ Some field-perturbed binding energies are shown in fig. 8.1 for 0 ~ ~ 2.5. 9. The calculation of multiphoton transition probabilities between bound atomic states The rest of this paper is concerned with the calculation and measurement of transition probabilities for bound—bound and bound—free atomic transitions induced by a strong externally-applied oscillating electric field. There is of course a close relationship between this subject and that of atomic structure in the field already discussed in earlier sections. The field alteration of the structure is due to induced multiphoton transitions; on the other hand, rates for tranSitions between initial and final field-free states are affected by structure. In this and the following sections we emphasize observations made in single field experiments that bring the atom out of the field before entering a detector for state population analysis. This constrasts with atomic structure e~xperimentswhere a second weak probe field [resonantly] initiates the detection process while the atom is still in the field. The analysis of such two-field experiments can of course be viewed as the calculation of two-field transition probabilities for the various order Raman processes that occur. The topic of transition probabilities is concerned with dynamical effects as well as spectroscopic factors. We are now interested in the detailed time evolution of the system as the atom responds to

358

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

the field. As in other dynamical problems such as atomic collisions, the S-matrix will play a central role. It is often true that at low field strengths field-switching details are relatively unimportant and transition probabilities are low enough for a transition rate W [independent of time] to exist: P~Wt,

Wt~l.

At long enough times of field exposure or at higher field strengths, there is saturation P(t)

=

1 —

ewt.

As we shall see, a number of strong field situations are expected where a rate W as defined above does not exist and one should talk rather in terms of P(t) only. Let us begin by ignoring spontaneous decay, induced energy-level shifts and widths, and explicit dependence on field polarization. Then usual time-dependent perturbation theory gives the following formula for an induced N-photon transition probability between an initial state i and a final state f[6] (fIr~I ) (a Ir~~~Ii) 2 W~= 2ir(271.a)NJN_ J(w)co~’[ ~ — — °~I — —

]

1

°N— 1

~

[coON_

1

co1

(N

l)w]

[co01 co1

co]

where the total photon flux is I, the photon flux per unit bandwidth I(co), the energy levels 11w0, the summation is over all intermediate states a1 (including the continuum), and energy conservation requires cof — co~= Nco. Sometimes the summation can be truncated while maintaining accuracy. For the hydrogen atom, Karule [190, 191] has used the Coulomb Green’s function to obtain analytic expressions for W~in terms of hypergeometric functions. The numerous optical experiments on two and three photon bound-bound transitions in atoms are referred to by Lambropoulos, [6].

If one of the energy denominators in the expression for W(-~above is zero, then the rate is enhanced by an energy resonant intermediate state. The energy denominator for the resonant terms must then be treated more carefully by including either spontaneous decay, or at larger fields the energy-shift function Rr = Sr + ~Fr,where S~is the field induced level shift and Fr the field induced decay rate out of resonant intermediate state r [192]. The total transition amplitude can then be separated into resonant and non resonant (background) parts that can quantum-mechanically interfere to produce a variety of resonant line shapes in the local frequency dependence of W~.The range of line shapes ranges from Gaussian-liketo dispersion-like as is well known from the early work of Fano on the interaction of a discrete state energetically embedded in a continuum [193, 194]. To first order in time-dependent perturbation theory, the introduction of phenomenological spontaneous decay rates ‘y~for excited states leads to the following amplitudes for the intermediate state in a resonant two-photon excitation process [195]; (1)

Air (t)

—1/

~1. +

I —Ir i~2.

exp

[i~2irt]

{exp (_Yrt)

—

exp [—(7k+

where ~Zjr co — cojr is the energy detuning for the absorption of the “first” photon. The first term in A~)(t)decreases with the decay rate Yr of the intermediate state and does not undergo oscillations; its effect on ~ leads to “step-wise” absorption via the “real” intermediate r. The second term is oscillatory at large ~jr and decays with ‘y~rather than ‘Yr; its effect heads to true two-photon absorp-

I.E. Bayfield, Excited atomic and molecular states in

strong electromagnetic fields

359

tion i.e. absorption via a virtual intermediate state. The two photon transition probability to the final state fis [195, 196]

2IVj~I2IJ’~fI2 ReI1—~—_ 1 w(2)= (Y~ Yr)~+ ‘~~r I LYr Yr + Yi ~

—

rl

1

+1——

t.2Y~ Yr + 7i

1

+

I 1~2irJ

1

—

1~2irj~)ff + 7~+

1 ‘Y~+ Yj ~(~~ir + ~rf)

The terms (Yr + ‘y 1 ±~~ir)’ inside each square brackets arise from the interference between the step-wise and two-photon amplitudes. On resonance the interference meaningful separation 2ir’~ Yi’ Yr’ ~ makes is thea sum of transition of ~ into two parts impossible; far off-resonance, ‘~ probabilities for the two processes: -~ ‘~r’’~f’ ~

&2~

(2Yr (‘yr

Yr + + Yf)2 +

‘Y~’

~rf

+

2’yi (Y~+

+ + (air +

Yr

-~-

7~)2

The dependence on the sum ~2ir + ~2rf = 2co — ~ is characteristic of the “true” two-photon process. When one proceeds beyond perturbation theory to an intense-field strong coupling description of the above three-level problem, the separation of terms in the amplitudes into step-wise and twophoton parts is always meaningless, for strong driven interference among the saturated component transitions heads to a lack of identification of the latter. This is consistent with the structure of the system atom-plus-field bearing no resemblance to that of the field-free atom, see further section 5. In section 5, we also mentioned that the two-state problem in the low frequency regime was a relative simple case where the strong field saturation of photon-replica structure could be observed unobscured by other strong field effects. Let us consider this problem again, calculating the transition rate using adiabatic oscillating orbitals and the time-dependent WKB approximation [197, 198]. As before, let the field free states have opposite parity, so that the coupling matrix element (0 11r F0 cos cot ~I~2) is nonzero. Let the field-free stationary state energies be ~e/2. The QES phases involve 2 cot] Y 2FoIz 2(t) = R[l + 72 sin 12I/e. 1/2

1,

The transition probability can be obtained using the complex trajectory technique introduced in section 2. The complex turning points tq in the upper half of the complex-time plane are the points of degeneracy of the (complex) quasienergy phases: 1 (‘y~), q = 0, ±1,±2 Si(tq) 22(tq), tq = qir/co + (i/co) sinh In the transition rate limit P -~1, the contribution to P from the zeroth order branch point t 0 is [54] p~)= exp [_2 Im f

[~1(t)

—

g2(t)] dtj

exp [—2(e/w) (1 +

~2)l/2

D( 1 +

72 )1/2]

where D is a complete elliptic integral of the third kind and we call i~ c/co an “adiabaticity parameter”. The quantities ,~and a parametrize the problem along with the order k0 of the multiphoton transition. The total amplitude A12 for a transition from state Ø~to state ~2 is a coherent sum over contributions aq arising not only from t0 but all the tq• Each contribution is proportional to ~

JR

360

BqJ’fle~f,£*cU~dv~oiMc0W rnolçcular ~t4tI~insfi~ngtle.ctibmagneric fields

and has a wave-like additzo~1lactnr i~adlngto ~ diffr~ctioi~-grating type formula when we include N turning points {~MS #)17slr~4G~—er), A~2I2= ~?2°~su~2~ where ~

-

is the adiabatic ~ctIos~a~Cijfhu1âted betw~tztii~ighbo~mg tutning pdints Since the latter are equally spaced in time ~ */~i

~ta1 ti~~bt1 tis

•‘

1.

T=irN/w. In the many oscillatioli limil, N- o~,sin2 Nx(N sin2 x)becothes ai~energy-conserving delta function For the linear re~io~. lit tJ~nethe transition rate be~on1e~

2IT—~

W

12 = IA1~I

~

k 0~...,l,,

where k0 = 2q + 1, q Q, ~, ~, 3, is tijé odd nuthbe~ot ab~otbedphotons whose summed energy must equal the adi~batlcalI~’ ~tark-~hifted tra itionetterg~Sw/it ~tresonance In the low field 2 ~ 1, the ~at~ ofl~teaonan~ 1~c~mes hmit k07 w? 2/7r)(e7/4)~, 2 = (2w which agrees with t~ !estiTl; of tithe-ttep~nden~ pertu~atf~eo~y

W~= 2j 2(h~yJ4)~f~/e,,~ I)!’1~, once Stirling’s formula (valid when k,) 1) is used

W

ecfe4~t1oi~to h’~to first order in k 2 gives 07

12~W~2exp[.-~-~k,3~1~

and predicts that lat~rfiekIstedace~t1ieIrangitioir rate T~That~edicted by perturbation theory, at least for high order i~iu1~t4~hotoh tr~n~taon~ wttlInt a twt~1eV~~tstemThe range of ~yfor which our assumption W12 ‘~ I £~‘yahddepe’n~on ~ ‘~ <~~ of k0 The above anaIysi~~y ~acøtski ~i1f1~kniiii~ is int~re~trA~ in thi~4tseems to account for photon replica brightening, ie~the~~t~iration Ot low Ordet ttâhsltlófis to adjacent virtual levels between the mitial and final state energ~t~ve1~ Such saturation is e~pecte~I tø Occur in the region ~y~ 1 and should reduce the field depèp4enc~ol the ti~m~ition rate i e the effective or apparent multiplicity k becomes less than k0 as s1u~WitIn fi~9 1 It should be tefle~bcf~tI, however, that the results shown in the figure are only qi~at4tht4~ve for app1t~atio’n~ in tti4 fegiqn ~ ~ 1, where in practice W12 is often not small Such a liftntatio~a~enisto~c~ut with n1~sttt~t~tka1attempts to treat the strong field region, as we shall s~e “ The above ~idiabMte tWo-kate cakuiations ha* been g~hei~aH#~td to the strong coupling case where repopulation of ti iiutW state i~important ~19~iT*~semiclassical theory is quite parallel to that for low en~t~y atom—atom level-trossing c~ofl~siffi~$-, ~ di~batic(i e zeroth order) states of the transiently fbrm.d thtcjl~uIebe~thnvenergy de~fl~t~ at certain points along the (complex) collision trajectOry, ai~d~tahsition~ bètWeei~4dtal~aJw itat~sO~t1rs’ia non-adiabatic couplmg in the regions near those ~oittts 1itt~iepresent o~il1atingfle1~p~obleinrth~ diabatic states can be taken to -

J

JE Bayfield Excztetjoiomk ~ndmolecular srireg ~nwwsg electronia$neThcf~k4

36i

0

-

1dFI~O769

I

~

0 25 I

-2 0

12

~

~g a...

I-’-.

.4 Fig 9 1 The dependence of the logarithm ~ X 2]of the transition rate W In [rrWj 2/2~, 12 fQr~niUJp~ho~ç,p nonresonant bound—bound transition plotted jspa t2t~ field strength parameter ~y= 2F0Iz I2~/~ aftec~*ret*dai~d Kramov [197]

Fig

,~

I

O.8~. 1.2 -.

O~4

I

1.6

-Frequency

~ 2 Strong-cciuplhig c cgla.tl9as of the time and phase

av~ra~ed t~nsitionproj t$If thr a bound—bound transition for two jx~1lngitr~~il~~ (4~~1 ?1ot~edversus the frequency 9f the appH4Jield aft~Meilpn~yaM Mçath [199] The idbl 0 ~ curve *ciW q~~, 3 ~ncL5multiphoton resonance pe~k~ tiss WF1 ~$ 769 ei*~vvp~akefor q = 1 3 5 and 7 Eaily cx~oz~~i 1~t’~ t281 4bs ~ofmse]u~urves as these is 4lscwped

be component QES that are tui~edtnto energy 4egen~rac~ ~t a point in tIxflo dti~ingeach oscillation by the adiabatically oscillatmg St*rlj ~ineracdionThe L~ttexalso..(1) pr’qvi9~sthe couplmg that splits the diabatic state degeneracy in fc&rzfl3ng the adiabatic st~teslu~d(2) pivd~~c~s ~he non-adiabatic coupling between adiabatic states tl)~t~a4s to a transitioii~nstews~huithe adiabatic representation The semiclassical two state S-ma$$,~~ Of f3~ form ~R~’e~1] De~2

—

S_[Re1s2

where Si

2

J’

—

g~ 2(t)

i e~p

dt

I t~(t)

To lowest order m nonadiabatiç Q~*1pbJ3g ri S~e~11

LR

-R 1+i&’

~

.

.,

.

,.~-

where 2F~oI4e the “static” is ~ ~(~/c~)L~. —)c~w+ StarkNshift is For adetuning largenumbexN~foS~il~*iqns, the~~J~id S t~ry~ 1$the ~ st~c” h,4e li~iz4 -400 a multiphoton = Iz12I Rabi equation is obtained for th~ti~zisiUo,1 probability [l~Ø},.~ee ~l~o ~cticu~5

R2 P(t) R2 + (~/2)2sin{fR+(~t2)~I~)(/T}, ~

which descnbes the oscillatmg trai~1tionpr~iabilityat lal’go t~imesin tlw weak coi~plinglimit Further work within the adiabatic represen~at1opiismg the fW&S.~iiia uc.a~1~1 ~ tinite number of oscillations should contribute ~o our unçIerst,~ndmgof the st~ong’fieldIow-frcquency region This is particularly true for the three-l~ve}probl~inwhere an intermediate I eLperpnts resonant multiphoton enhancement of the rate ~i9~ As already discussed in section 6, ~ rnln~e~’ Of QES techxuque~(a1ia~~tressef-atom or Floquet-

I.E. Bayfield, Excited atomic and molecular states in strongelectromagnetic fields

362

t(w’) Fig. 9.3. The phase-averaged induced transition probability as a function of time at the 5-photon resonance (v =

0.769 shown in fig. 9.2.

=

0.2945) for IdFl

Again after Maloney and Meath [199].

mode techniques) have been developed for a strong coupling, full quantum mechanical treatment of the two-level problem. These have been recently reviewed [2]. Most of the work has concentrated on low order (k0 ~ 1) resonances and the calculation of transition probabilities as functions of frequency and field strength, see fig. 9.2. At dF/w ~y~ I the antiresonant terms ignored in the rotating wave approximation are found to be important, producing large time-dependent corrections in the phase-averaged transition probability [199, 200]. These corrections are notably important at large k0, see fig. 9.3. Let us now turn to the calculation of multiphoton bound-bound transition rates in specific atoms. There are not as many of these calculations as for multiphoton ionization, the latter process being better studied experimentally, see section 11. Available theoretical excitation rates for real atoms have invariably be computed using perturbation theory [201—205]. Improved versions of perturbation theory involve (1) the use of perturbed states in corrections for level shifts [6], (2) new procedures for summing over the diagrams representing transitions through various intermediate states [203] and (3) in the case of hydrogen, application of the Coulomb Green’s function [190, 191, 206—209]. Two-photon rates for transitions from H(2s) higher been considered [209]. 1S to6’S) andstates Cs (6shave9d) have been calculated, Two-photon rates for the ruby laser transitions He (2 along with the 3 or 4 photon Nd-glass laser transitions K(4s -÷ 4f), Cs(6s 6f) and Na(3s 7s), [202]. Bound-bound multiphoton transitions such as these resonantly enhance ionization and are therefore discussed further in section 11. Microwave multiphoton transitions of order 3 ~ k 0 ~ 7 have recently been observed between neighboring highly excited levels of hydrogen in the region 40 ~ n ~ 60 [166, 210]. The observed rates are competitive with the rates for ionization in the same field. This means that the microwave ionization of Rydberg states is a multiphoton process, at least for hydrogen atoms and certain frequencies. See section 11 for further discussion of this subject also. -*

—~

—~

-+

10. Multipho ton electron detachment in negative ions As was the case for static applied fields, there is reason to believe that calculations using delta function atomic model potentials might usefully apply to nonresonant strong oscillating field phenomena in systems involving short range forces such as that of an electron outside a neutral atomic core. Although atomic negative ions often contain paired outer electrons, still one-electron calculations provide useful estimates [2111. At any rate, delta-function potential model calculations of field induced bound-free electronic transitions provide us with a qualitative feeling for what to expect both from numerical calculations and from experiments on strong field atomic multiphoton

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

363

ionization. Most of these model calculations utilize a semiclassical approximation at one stage or another. Those introducing the classical action and trajectory at the outset we shall call “classical trajectory” calculations [55—59],whereas those using only the saddle point method to estimate integrals we shall term “semiquantal” [212—215]. In many cases the two techniques are equivalent and give identical answers; however, sometimes this is not the case, because of other approximations made. Before discussing the semiclassical results we shall consider the recent full-quantum onedimensional model of Geltman [216, 284], keeping in mind that the semiclassical work finds some differences between the 1-D and 3-D delta function cases [56]. Let the potential energy be V(x)

=

—l/(2n2) having a wave function

which has one bound state with energy E ~x) = rr’12 eixI/~~, as well as continuum states with energies Ek ~k(x)

=

ei~+ Re,

k2/2 and barrier-reflection wave functions

x ~ x~0÷,

with R

ink)_i and T = ink (1 + ink)~.

—(1 +

=

We wish to solve the time-dependent Schroedinger equation using either the F r or A p forms for the interaction with a linearly polarized wave. In the former case

~

t)

at

1 1 a2 1 _[_.~.~—~+JT(x)+xFocoswtJtPis

t~0,

with the boundary condition ~i 1(x,0) = ~b(x). If we use instead (l/c)A p, let the wave function be called ~‘2• The reason for considering both forms for the interaction is that low-field perturbation theory is easiest using the F r form, while ultrahigh field perturbation theory (where V(x) perturbs an oscillating electron) is easiest usingA p. The relationship between wave functions is 2(t’) dt’} ). [xA(t) — A The x A part of this transformation is that of the momentum-translation transformation [183]; there is no difficulty however, if physical probabilities are calculated from rather than ~ [184, 185]. The ionization probability is =

~‘2 exp {!

~-

f

~i

1

P(t) =

f

~—

dk I(~kI’Pl)I = 1 —

1’~øbk~’l>I.

Numerical calculations are performed by integrating the Lippman—Schwinger integral form of Schroedinger’s equation for ~‘2’ after introducing the Green’s function for the oscillating electron G2

=

~-

f

{

dk exp i [k(x

—

x’) —

I

dr

(~ +

364

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields =

~J— e~1~(t t’)~’2exp

Ct

—

—

t’)~[(x — x’) —

(cos wt

—

cos

The numerical results are roughly consistent with the Keldysh parameter 7K w/nF determining the regions of “true” multiphoton absorption (7K ~ 1) and static-field-like tunneling (7K ‘~ 1) [213]. There are signs that another parameter is also relevant, in that 7K = 6.5 does not exhibit multiphoton power-law dependence when the frequency is low (w = 10-2). As seen in fig. 10.1 the numerical results for intermediate field strengths and intermediate values of 7K exhibit a transient “tunneling” rise in P(t) during the first field oscillation. The transient mechanism is relatively ineffective for further ionization at later times. The transient rate is proportional to F2. At still later times the usual ionization mechanism builds up a sizeable contribution to P(t) and finally dominates; if saturation has not occurred, the latter mechanism is a high order process with P(t) F21’0 when 7K> 1. The amount of ionization F, due to the transient mechanism is found to be Fi

=

~(n3F)2 f [(n2~)i]

where f 2o.,)~~ 2. The probability F, dominates F(t) for a total time 1 is given in table 10.1 for (n P,/W(k 0), where W(k0) is the rate for the late-time process. Although the physical nature of the transient mechanism is not yet certain,3Sit state is a possible forfield the of observed F2 dependence of He inexplanation a microwave frequency 1 OiOHz of the ionization probability the~ 40iO~and the atoms are exposed to less than 100 cycles of field [236], where (n2~)i ~ 200, of n3F oscillation. log t (cycles)

1 ps

I ns

o -2-IOi24345~6

-8

5OO’~ I.

I

-

It

lilill

Ii

Fig. 10.1. The ionization probability P(r) of a one-dimensionalmodel atom, piotted versus the time, after Geltman [216]. The various curves are for different values of the Keldysh parameter ‘~Kat early times, or the field strength F at late thnes. The plateau regions are discussed in the text.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

365

Table 10.1 The plateau function fi (x), 2w). x = i/(n (n2o)~

fi

2 2.5 3

5.76

4.39 2.08

4 8 i2

1.41 1.11 1.06

A second recent nonperturbative model quantum calculation is that of Manakov and Rapoport [2151. They considered a three-dimensional zero-range S-state atomic potential and a circularly polarized field. The latter stipulation greatly simplifies the theory, as a transformation to a rotating coordinate frame eliminates the time-dependence and leads directly to a quasienergy state problem; this was already seen in section 7 where perturbation theory was used to obtain the structure of the hydrogen atom in a circularly polarized field. In the present situation the delta function potential leads to nonperturbative results. After the transformation 4(r, t)

t),

~

the time-dependent Schroedinger equation with the interaction V(t) = eF (x cos

,it

+ y sin tot)

reduces to th acF/at = [H 0—

+ eFx]c1(r, t)

Q4.

If [Q(r)

—

~

0,

then (i ,(r,

t)

=

~gntm~, ~(r, t)

with + 27r/w) = Thus

~

is the quasienergy operator for this problem. To find the quasienergy states Ø,(r), one solves 2/2m — eFx g]Ø~(r) = (~2/2m)~(r), (Q — g ~ = [p with the boundary condition [64] Q

—

6, (r) rZ (hr A

—

1/a

0)/47r,

—

2/(2ma~).

E0= —~ solution from Green’s function theory is Ø~(r)~- [m/(2irifi)]ul2f r’I2dt exp~{e’t

+

~2

+ !~(sinwt

—

i)

2eFx ~~ri2

+

366

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

—

4. wt e2F2/ 2mw 21t—— \ wt —sm 2 —

A procedure is found for selecting out of the above i~, (r) only that part irregular at r imposing the above boundary condition. A formula for a complex energy E results: 2 (t)’12dt exp (_i {i — exp [i-y~2t](i — s~ (~)1/2 = 1 — (4iri)’/ 2~t))

5

=

0, and then

f-t~

7K is the Keldysh parameter, I —E~,is the atomic ionization potential and 6 —=lIw/21 is where related to the order k 0 of the ionization process by =

(1/11w + 1> =

(1/26 + 1>.

was 2 numerically evaluated and the real and imaginary parts taken to obtain the The integral for Ei~~ energy shift and width (ionization rate) results shown in figs. 10.2 and 10.3. A characteristic field strength for the problem is F

213/e11 0 8m and a characteristic frequency or energy is w 0 The shift is seen to deviate from that of second order perturbation theory for w 0.5w0 and F ~ 0. iF0. Thus it would seem that deviations from perturbation theory are large primarily when the parameter F/F0 is not small. On the other hand, deviations from the power law dependence of nonresonant perturbation theory are large when2. a different parameter is not small; at least for fewphoton ionization, this parameter is (ko’yK)

l02

lO~

10

0

10

10

F Fig. 10.2. Model calculations of a bound-level energy shift due to strong-field coupling to a continuum, after Manakov and Rapoport [215]. The strength of the circularly polarized field is F and the frequency w.

Fig. 10.3. Model calculations of the ionization rate asa funcdon of the inverse of the Keidysh parameter ‘~Kand the frequency ~ of a circularly polarized field, again after Manakov and Rapoport [2i5].

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

367

This result is important, as the transition region had been believed to be near ~ 1, whereas now it seems shifted to 7K ~ k~.This change can explain the recent Nd-glass laser multiphoton ionization results [217] that exhibit an expected power-law dependence for values of 7K perhaps as high as 0.3 when k~~ 0.1. Note that the reduction in effective multiplicity at high fields predicted here is also what was expected for bound-bound multiphoton excitation, see fig. 9.1. For low fields (F ~ F 0) and 2 using the saddle point method. high multiplicity k0, (-~y~ k0 ~ 1),one can evaluate the integral for Eu1 The corresponding semiquantal results agree with those of Popov, Kuznetsov and Perelomov [57] who earlier considered this very same problem using the complex classical trajectory technique to obtain E~E 0_~~~(F/Fo)2{l +~w2)

—i{(F/2F0)exp [—(2F0/3F)(l

—y~/l5)]}.

As w -÷ 0 the second term has the expected static field tunneling behavior discussed in section 2. If we assume F expanded ~ F0 in theinManakov—Rapoport model, theterm oscillating inside the 12 only can be a power series and integrated by termexponential to get a perturbation integral for E’ expansion in F2: ~hI2 (EF_—_E~’12= 1 \ I /

—

~

~ n1

(1~2fl

‘7k6/

R

R “

(—1 )~(~ +

1

‘~

2n + 1

~ m—n

(n

—

2m6Y2 + ~ m)!(n + m)!

where EF is the energy of the electron freely-oscillating in the field. The coefficient of F” has terms involving e ±2n that correspond to n-photon processes, although processes of lower order m also contribute to this term; this was recognized early by Kovarski and Perel’man [164] in their general development of atomic multiphoton ionization using Green’s function techniques based upon the oscillating electron propagator. Ionization only comes from those terms in the above expansion that have R~complex, i.e. with e + 2m6 0. Thus the ionization has contributions from all orders m such that ‘~

m~k~—(I+E~)/(Ilw), where k~is the shifted multiplicity that accounts for the additional absorption from the field needed to leave the final-state electron oscillating in the field. To utilize the model results of fig. 10.3 in estimates of electron detachment rates for negative ions, one must scale according to quantum defect theory. If the wave function at large r for a negative ion differs from that for the delta function by a factor i~,then the matrix elements of eF r scale as ~2 assuming they are dominated by contributions at large r. For the H ion in the ground ls2 state, u~~ 2.8 [215]. Thus the vertical axis in fig. 10.3 should actually be r(H-)n—2. Taking this scaling into account, we might expect that a CO 2 will multiphoton 2 laser flux about 10” detach an electron from H with an observable efficiency inof a typical TEAwatt/cm laser pulse time. Whether this is true or not remains to be seen. Much earlier de!ta function calculations have been carried through for both linear and circular polarizations, using both semiquantal [212, 214] and classical trajectory techniques [55—59]. According to these calculations, in the high order limit (k 07~’ 1) the ionization rate for linear polarization is larger by a factor ~‘

(3/7?.)i/2

exp [+(2F0/3F)’y~/30].

368

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

The reason is believed to be a result of the added energy absorption needed in the circular polarization case to give the electron the requisite angular momentum. Let us now consider why the complex (time) trajectory technique illustrated in section 2 for static field ionization is still useful when the field is oscillatory. In one dimension, Newton’s equation and the complex boundary conditions become =

F cos wt,

~(t0)

x(t0)

0,

We remember that under the barrier (t0 equations is =

Pxo

+~-sinwt,

~

iv,,.

=

t

~ 0) the time is imaginary. The solution of the above

i; +-~sinwt0,

PxO

~ = p~0(t—

t~)—

4

[cos wt

—

cos wt0],

where the classical turning point t0 is determined from 2(t 2 or fl 0) [Pxo + (F/w) sin wt~] iwto

=

i

sinh~[7K(~

—

For Px~~ 0, to is shifted from the imaginary axis into the complex plane. The wave function ~1’(P2’ t 2) for the electron to be free with momentum P2 at time t2 is then given, according to the complex trajectory method, approximately by 3I2(_i)e_iE0t2

dtifdxi e

f2

~i(P2, t2)~ (2~)_

2t2x1,t~)V(x,)

Ø0(x,),

where V(x,) is a short range potential, Ø0(x,) is an approximate initial state wave function 2(vnry’ — ~ e_L~n~~ ~m(~)

and W

S

c,(vn)~’ — P2 x

2 is a reduced action in the mixed (p2, x~)representation. Using the saddle point method,

~‘~:

l~’(p ~ where

2w

a

t)

=

2~V

a

—w c~

1[t’n

sinh2r 0

] [1

~-i

1/2

(~~j) I

~m(~o)

exp i[l~/(p2, t2xç°),t(0))

—

E0t21,

(

tanh r0 + ~2 ~0 it~,F[ The nature of =the time development of the classical action under the barrier is shown in fig. 10.4 for different values of w; at large w much of the action develops near the emergence point x(0) where the particle “velocity” is lowest. The increased time under the barrier results in an increasing effective width of the barrier with increasing w determined by x(0) = ~n 7K = 2(2n2F)’ [1 + (1 + w2/n2F2)”2]. 27K

2

—

~2

2

wu,, l+(1+7K)’1 The final results for the ionization probability for arbitrary (F, w, 1,) are quite complicated [156—

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

369

o7

t/to Fig. iO.4. Typical evolution of the reduced action S during the development of subbarrier motion in complex time, for values ofthe 7K = 0, 30, 100 for curves i, 2, 3, respectively. After Perelomov et ai. [57]. Keldysh adiabaticity parameter

159], especially when the procedure is generalized to a three-dimensional delta function and arbitrary elliptical polarization. A three-dimensional Fourier decomposition in momentum space leads to a final rate W expressed as a sum over all energetically allowed multiphoton processes. For the case of linear polarization and an initial state having quantum numbers n, 1, m, the time averaged rate is F, m)

=

I,, IC,,,I2(6/7r)’12 (21+i)(i + 1:1) [F( 1

+72)i/2}

A(w, 7K’ m) exp — [~.-~-~(7K)}~

mI +

where

A(w,

7K’ m)

4(3”2(ImI!Y~(1

f eY2(x~ y2)Imldy

a(7K)~ 2[sinh~7K_

—

—

3 f/ L7K

1 \ + —j-, sunh 7K

(1

-1

~‘7K

—

exp [—a(7K)(k

+7~K)2 k

+7~)h/2

L7K

1

(1

_~n /

keff =—

j,

—

keff)]

[(2~YK(k_k~f))’I2J

~

+72)u/2]

~l +

1

For m = 0 the function *‘ 0(x) is plotted in fig. 10.5. The function g(7K) is shown in fig. 10.6 and the frequency dependence of the prefactor before the exponential shown in fig. 10.7. A series of thresholds is observed as higher order processes can occur at lower frequencies. Among the limitations of these semiclassical results is the assumption that the electron is only incident once on the barrier, i.e. there is no interference due to multiple incidence. This possibility

o

~

‘

0

20

X Fig. 10.5.

40

60

~

~5~1

W

Fig. 10.6.

Fig. 10.7.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

370

has been investigated only at 7K = 100 and w,,/w = 10, where interference is found to be unimportant [59]. Generally speaking, the results of the semiclassical approach are qualitatively consistent with the behavior shown in fig. 10.3, for all polarizations. Quantitatively, however, there are sizeable polarization effects. The polarization C is defmed by A(t)

[— ~1sin wt,~’cos wt,

o]

and we consider a state with m = 1 = 0, in the high order limit 7K W~ ç~I~(I,,/2)[ (I

3

]

F 3/2 i/2

~(~_)

2F

~ 1

72

exp {—-~~[1 _-j~- (1 _~-)+..

except near e = ±1.So the rate decreases as C increases, because the electron trajectory is curved more, making traversal through the barrier more difficult. In the other limit 7K ~ 1, the most probable number k of photons absorbed is k~~

+ (I~/w)[e2/(1 — e2)] ln [27K1(’ — C2)112]

and the width of the distribution of multiplicities is ~ {[~2/(l —

C2)]

I,,/w(ln 7K)3} 1/2,

both smallest for linear polarization (C = 0). In this 7K ~ 1 limit, the rate again decreases as C increases. When z~k~ 1, the threshold effects characteristic of linear polarization disappear. Kotova and Terentiev [218] have begun with the classical trajectory formulation based upon the oscillating free-electron Green’s function and have derived a formula for rate enhancement due to a resonant intermediate state. Again a short range potential was assumed. They restricted themselves to a m = 0 -÷ m = 0, N-photon first transition into a deep nondegenerate intermediate state, Nw = w,, — w,,’ ‘~ w,,. Far from resonance the non-resonant rate WN dominates; near the intermediate state resonance the rate increases according to N

—

2~2l,,++ 1l\/ C.,, C,,,

2

(w,,

—

w,,’ 1’~,14 Nw)2 + l’~/4(27i) 2NWN’ —

where 1’N is a composite “power broadening” matrix element. We postpone a detailed discussion of resonance effects until the next section, where they are discussed carefully in terms of field-modified perturbation theory for the case of real atoms rather than for short range potentials. We close this section on model calculations of possible use to future negative ion strong-field experiments by mentioning the recent development by Cohen-Tannoudji and Avan of a strong-field version of the theory of the interaction of a discrete state embedded in a continuum [2191. Even negative ions such as W do have resonance states in the continuum, some of them metastable against autoionization because of selection rules, e.g. W(2p2 3P°),[220]. The theory of Fano [193] when extended to strong fields leads to satelite lines split by twice the Rabi frequency; if one of these energetically lies below the field-free ionization limit, its width can be small, see fig. 10.8. The ionization rates were obtained as a function of the field strength F.

J.E. Bayfield, Excited atomic and molecular states in strongelectromagnetic fields

371

Amplitude

t2~—Xt2

E

11> 1

0

Wo

E

E13~=Xfl 1

Energy Fig. iO.8. The energy spectrum of an initia “discrete” level of energy W0 > 0 embedded in a continuum, 2~ as is narrow, modifiedasby it alies strong below E = 0. After cohen-Tannoudji and Avan [2i9]. external oscillating electric field with a Rabi frequency Xiii. Note that the dressed-state with energy R~

11. Multiphoton ionization of excited atoms 11.1. Semiclassical calculations We now turn to the field-induced ionization of atoms, using a one-electron approximation. We are concerned with efforts where the Coulomb force is considered simultaneously with the external field. First we will discuss semiquantal and classical trajectory attempts to treat the Coulomb force problem. Second will be the development of usual perturbation theory using resolvent operator techniques to account for field-induced changes in atomic structure. We then mention some aspects of the time dependence of ionization phenomena, before considering the multiphoton ionization of excited hydrogen and other specific cases. This section ends with the question of ionization in the ultrastrong field limit, where the external field strength exceeds the mean Coulomb field strength in the normal atom. The use of oscillating electron orbitals was actually introduced by Keldysh [213] in his pioneering semiquantal discussion of multiphoton ionization. Without formal justification, he used perturbation theory to calculate the amplitude for a transition beginning with a field-free atom ground state and ending on an oscillating free electron state. No Coulomb interaction was included in the latter. Assuming linear polarization, the final state is ~.

~

1/

eF

.

1 ~-~p+--~-smwt~ 1 / eF .

dT

Resonant enhancement was also considered, via intermediate oscillating bound states

~,1i~(r, t) = Ø,,fr) exp (i/1l[1,t

—

(d,,F/w) sin wt]},

where the Stark energy in a static field is taken as The rigorous development of such QES procedures has been discussed in section 3. One major source of error in the Keldysh approach turns out to be the omission of the Coulomb interaction in the fmal state [214]. Let us consider the electric dipole matrix element between field-free initial and final states, in momentum representation =

ff5 ~

“~i~ e(F cos wt) r en/a0 dr/(ira~)’/2= 8(ira~12e#1F V~(l+ p2a~/~l2)2.

372

I.E. Bayfleld, Excited atomic and molecular states in strong electromagnetic fields

The Keldysh procedure is then to let the momentum become the generalized momentum p + (eF/w) sin Wi’ in an application of Fermi’s golden rule 1. W=—~ hm Re

ff5 dptfdt’ co:wt cos wi” ~

eF

xexpj_f~J+-_-_tp+—sinw7-~ 2m~ w

dTJJ.

A portion of the integrand is expanded in a Fourier series in

v0~(p+

eF~

+— sin cot~v01,(p +—sin wt’)

~j

t 2]dT)

—

sin wt) exp

2

~

J1,,+L(p +~-~sin or) +4mw2 _~~)t~

to obtain

wff5~f~IL(p)I2~

~

‘~),

~2m~4rn~

where L(p)

mi—1 +du

F

v

0,,

F2

u

(~+ a— u~exp {~—f

[j,

+ ~—(p + ~—~) 1(1

d —

V2)1~’2)

and the contour encloses the segment (—1,1). The exponential in the contour integral is rapidly oscillating, suggesting an evaluation using the saddle point method, the saddle points u3 being determined as usual by 2 0. = (1/2m)[p + (eF/w)u3] Each u 5 then contributes a portion L3 to L(p): L(p) ~

~2(7ra~)h/2

..IL.

(1 _u2)h/2 exp

{~—j~’ [, + ~—(p

+~v)2](h

_V2)1/2}

For a multiphoton process of reasonable order k0> 1, we expect perturbation to apply 2 C theory 2m1,,. As an approxiwhen the integral for W is dominated by small contributions in the region.p mation, therefore, Keldysh put p = 0 in the prefactor of L(p), expanded the exponential to order p2 and evaluated the resulting integrals. The results are qualitatively similar to the model results of the last section, see for instance fig. 10.3. Nikishov and Ritus [214] used the semiquantal approach to show that the Coulomb force in the final state cannot be ignored, if one wants theoretical results to be at all comparable to the results of experiments on atoms. They consider an initial S-state and a k 0 ~‘ 1 process, and introduced the effects of the Coulomb field through a Coulomb form factor that modifies the matrix elements obtained assuming a short range interaction. The continuum final state wave function that they used contains only a perturbation correction for the Coulomb force. Their approximate results included

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

373

the modification of the Keldysh rates by the Coulomb factors [F,/F(l + 7K)2] ~n, circular polarization; [4F,,/F](210n)_

3/2,

linear polarization, ~

> (F/F)’/2 ~ 1.

No result was obtained for small orders k 0, or forpolarization, linear polarization in the low field,inhigh 7K 1. It was concluded that for circular Coulomb corrections the frequency initial and region final states were equally important, while for linear polarization the final state effect is most important. Since the region of possible validity of all the above works is F,,/F ~ 1, we see that the above expressions for the Coulomb corrections are very large, especially for high-order processes in weakly bound states. This situation is similar to that for static field tunneling, where the corresponding Coulomb correction is (2F~/F)(2/°n) [161. A parallel effort has been made to include the Coulomb force in the final state when applying the classical trajectory technique [222]. The trajectory equation becomes =

—l/x2 + F cos wt.

In general the Coulomb force affects the action both through the trajectory and directly through the Hamiltonian. The region of applicability of a Coulomb perturbation approach is determined by a comparison of the Coulomb and external field forces at the point x(0) of emergence from under the barrier. The ratio of forces is —2(fl\ 1 r —2 2 Fl + + 2\l/212 ~ X_‘~‘~_~LI 7K ~ =I.±’_\L ‘. 7K~ ~ n4F[l+(l+ 2)1/2]2 F x(O) F F co 11 + (1 + 7?)”2J ‘~vn7KJ F K ~

For large 7K this ratio becomes unity when ~ (n’F)’12 = 4(F/F)112, where F~ 1 / 1 6n4 is the field strength for classical field ionization. At small 7K’ we must have instead F ~ 1 6F~ independent of ‘y,~.At any rate Perelomov, Popov and Kuznetsov numerically integrated the classical trajectory equation and otherwise modified the action to include the Coulomb force via classical perturbation theory. They found for 7K ~ 1 that the exponential factor in W is modified, the relative values f(7K’ a) of the exponent being listed as a function of 7K in table 11.1 for the two cases of no Coulomb correction (a 0) and for the limit of a region of applicability given approximately by n2 w2/F = 0.3 or a 0max• Only recently have laser experiments at optical frequencies reached high enough values ofF to satisfy this criterion. Comparison with the semiclassical predictions would require a further assessment of the Coulomb effect on the prefactor for the caseyK~i

11.2. Perturbation theory Considerably greater progress in the calculation of accurate multiphoton ionization rates has been made in the area of standard perturbation theory using unperturbed atom basis states, the theory being modified in a self-consistent way to include induced spectral shifts and widths [6]. The question of applicability limits for perturbation theory has not yet been answered by high-power laser experiments in the optical region. The behavior of the microwave ionization of highly excited

I.E. Bayfield, Excited atomic 4nd molecular states in strong electromagnetic fields

374

Table 11.1 The effects of the Coulomb interaction in the final state upon the exponential factor multiphoton ionization in theThe near7K ~ for 1, after Perelomov et al. [222]. tunneling regime exponent with no Coulomb interaction is ~~K’ a = 0), while that for n2~’.’2/F’~ 0.3 iSft’YK, a = omax) K’~0~ 3 5 7 10 15 20 25 32.7 40 100

fl7K,a=amax)

1.365 1.832 2.156 2.510 2.909 3.194 3.416 3.683 3.883 4.799

0.972 1.44 1.78 2.15 2.586 2.895 3.156 3.423 3.638 4.640

hydrogen atoms does seem, however, to be in a non-perturbative region involving combined highorder photon absorption and tunneling; this will be discussed below. Application of the semiclassical approach to theory might in the near future concentrate on atomic ionization of excited states by fields in the infrared and microwave regions, while field-free state perturbation theory calculations should be made for all cases accessible to experiment. The results of the modified perturbation theory [6, 192] are particularly interesting in that they reveal the nature of induced atomic structure effects on the ionization rate for resonant processes. Let us consider atomic two-photon ionization from state 1 to a free-electron state with momentum K via an intermediate resonant state 2. If 1(w) is the spectral profile for the radiation of polarization e, the rate is an integral over final state electron momenta and w, as well as a sum over all possible intermediate atomic states m: =

8ir3a2 n’~~ dcZjc~ +

w’—w

f dwfdw’ww’I(w)I(w’)~ [

—

W~,+S~ +

KmTml

+s~l~ in +lFk in

When only one intermediate state is included, the spectral shift and width functions for that state are —

=

e 2

‘

27ra1r211 0J

w(w — w (w — 2, W2l

— S2) 1(w) dw 2 2 — ~2) + 72

e 2 J~ 72W 1(w) dw and ~‘a = 27ro~Ir21I 2 2~ ~ (w — w21 — s2) + 72 —

The induced atomic structure shift ~2 and width 62 for the intermediate state have components due to the coupling between all the important states: S2~52(u)+S2(~),

where

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

52(1)

2~

2,a~I

/

wI(w)dw

375

wI(w)

w

2~ra~

52~tC)

21 — w ‘ 2 d~ mKIr~ 8ur311 12 dczK,

f dw j dwK1r 2

~ — (wK

+ w,

21

2)’

and jr~2l also 72~72(o)

+

72(1)

+72~c),

where

/

2, 72(1) 27r2aIr~I2w 2c~ wI(w) lr~ 2dw. 1Ir~2I 21 1(w21), and 72orc) 2ir 2I The important point to notice here is that the effective “widths” and “shifts” in the energy denominator for the rate W2) are not the functions ~2 and 72 but rather the resonant functions Sa and [‘s. In the low field limit Sa becomes ~2 and Fa becomes 72. Near resonance, as the field intensity I is increased the width ITa becomes larger than the detuning w — w 21 and the intensity dependence changes from P to I. However, at still larger fields Fa saturates at the width of the radiation and the intensity dependence can return to P. The values of Fa and ‘~aabove were obtained assuming no shifts and widths in the continuum states; even so, they already show that strong field resonance effects can be quite varied and complicated. Line shapes are no longer Lorentzian and can be sensitive to details, see fig. 11.1. This has also been emphasized recently by Faisal [223] who obtains similar results for four photon ionization via a set of nearly resonant intermediate states. His Tmatrix exhibits the continued-fraction character also inherent in Lambropoulos’ resolvent operator perturbation theory as well as in the two-level treatments discussed in section 6: 72(0)

K’

~(a/c2)w~

[[

~O3

~LJ~

02

~

211IIL~ VK3 V32 V21 V10 ~~23~ ~O3 + ~j173KJ[ 01

~ ~02 —

1V 2 121 ~O3 +~173K 1V 2/(fi 231

Here the

i~,,, are

frequency detunings 11 z~,,= nllw — 1E

0 — E,,j. At high intensities, as the frequency w is varied the effective multiplicity can vary from j4 off-resonance to 10 when ~ = 0 and A03 ~73x~Here 7~is the induced width of the uppermost bound level due to coupling to the continuum; it is proportional to I.

0

I

2

3

4

Fig. 11.1. A frequency dependence of the transition probability for two-photon ionization at frequencies near that for resonant bound-bound transitions to an intermediate state, after Lambropoulos [192]. The quantity ~ is the frequency detuning from resonance. Curve B is the result omitting all level-shift effects, while the curve A includes them.

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

376

As one might expect, the saturation of intermediate state transitions not only affects resonance lines and effective multiplicities but also the very time dependence of the ionization probability itself. Kazakov, Makarov and Fedorov have considered this question of time dependence, employing the Fano method [193] to the case of resonant quasienergy states [224, 225]. Their approach improves upon the work already mentioned in this section in that the entire continuum is treated as such, rather than as a single level. Again for the case of resonant two-photon ionization, a resonance interaction at frequency w’ between states i~,0 and i~i1 is accounted for by using the QES ~01(t), see section 4: ‘Po, 1(t)

exp (— ‘o, 1t)

4~o, 1(t),

where ~1/2

c~o,i(t)~7~[(l

+ F?/4,

~/~2

The full

~

±-~~-) e~tlIio±ri(1

where the detuning is ~ ~

~1/2

=

E1

—

E0

—

w’ and

41W’l, ~ W’/IW’l,

Ff

‘o,.~~(E0+E1+w’~&~2),

~

W’

—~(lP1Id F’II1~.

wave function is expanded in terms of the two QES and the usual continuum: C0(t) exp (—i 50t) 40(t) + C1(t) exp (—i 51t) ‘11(t) + 5 dE CE(t) e_iEt~/IE.

‘I’(t)

Introducing as usual the Fourier transforms ~, 1(t)

—f d.Ea0, 1(E + So~+ w)e~t,

CE’(t)_ IdE bE’(E + E’)e_~Et,

a system of algebraic equations is obtained ~1/2 (E — ‘o 1 — w)a0, 1(E) = ;ç~(I ~ 5 WE’ bE’(E) dE’,

~)

(E

—

E’)bE’(E) ~J~WE’

where WE

=

~‘I’Ek1

[_ (1

1/2ao(E) + (1 +-~j)ai(E)] _ft)

F~i~>.

For now consider the problem of two discrete states (here QES with quasienergies S 0 + w and g~+ w) interacting with a continuum background. The results give the continuum QES e

~E(t)

where L~

~( —

i(E

~

‘~+ ga), from which the amplitude for ionization can be obtained

W’WE çexp {—i(E5

A(E~t)_E~j

—

E5—E

E)t}

—

exp {—i(Eb

1 -

—

Eb—E

where the poles Ea, Eb of A (E, t) are determined 2 by F21~ ~ A. iF 1 iF +...4~.]

E)t}

—

11

J.E. Bayfleld, Excited atomic and molecular states in strong electromagnetic fields

and ~,

377

F1 are level-shift and width functions due to ionization taken as the values at E = 0 of

~i(E)+~(E)f~n~fdE1[IWE,I2/(E+Eo+w_EP+ie)]. At really strong fields, one should not set i~ = A.(O) and F1 = F1(0), [219], see the model potential discussion at the end of the last section. The final ionization probability is ImEb —1 —41m I[expfi(E~* E’—E~, F1F~ 2(1exp(2 ImEa ImEat)— 1 +e~P(2ImEbt) —Eb)t} —1 32IEa_Eb1 Thus three constants characterize resonance ionization; the two quantities Im Ea and Im Eb are damping constants, while Re (Ea — Eb) is the oscillation frequency of P(t). If one of these three constants is unusually large, then a linear time-dependence occurs for the time interval r’ ~ t ‘~ r, where min[(Im E 5Y~,(ImEbY’ ,‘{Re(Ea

—

Eb)}~], r

max[(Im Ea)’, (Im Eb)’].

If on the other hand all three constants have comparable magnitudes, then the time r’ to reach linear behavior is comparable to the time r for total ionization of the atom, and no linear region exists. Several interesting limits for the above Fedorov probability P(t) should be mentioned. (1) If the ionization widths are narrow, F1 l~I~ ~l, then Ea, b ~ 41 + w and dP(E t) dE

2~F.F~t[&(E— 1 =(A(E,t)f

0—w)+6(E— 51—w)],

(2) If the field widths are narrow instead, Ff ~

+ E — iF1/21, then Ea ~

K+

w — ~/2, Eb

~‘~‘+

A1

+ w + ~/2 — iF,/2 and 2t cIP(E t) F.F dE Llo~2[6(E

0+w+w’—E)+6(E1—w—E)].

In these first two cases we have the existence of an ionization rate. However, if (3) we consider early times t ~ r, then 3 P3(t) ~ (F~F~/48)t and the “rate” is proportional to t2, [285]. Another case of unusual time-dependence is (4) when + L~E 0 and F 1 Ff, where Ea = = — ~/2 — iF1/4 and P4(t)

=

1

—

exp (—F1t/2)[l +

~1I’1t

+

More recently [226a, 266b, 286] , two additional sources of experimental ionization widths were considered, namely (a) 100% ionization “hole-burning” near the resonance center, (b) quasienergy level splitting during a finite-time slow switching on of the field. The earlier work had assumed instantaneous switching. For the case of a smoothly switched-on field, the ionization width experimentally observed would be roughly 2/3),F~(F~t)V2], F = max [lit, F~, Ff, (aF~)r(

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

378

where the last two possibilities arise from (a) and (b) respectively. Some experimental situations that are concerned with I’ will be discussed below. Of course, some experiments also exhibit additional effects due to a finite field bandwidth [287]. 12. Multiphoton ionization of excited hydrogen and helium atoms In many respects the hydrogen-like and helium-like atoms are as different from each other as two kinds of atoms could be. In hydrogen one has a single quite active valence electron and nearly degenerate states highly susceptible to external perturbations. In helium, however, the “valence” electrons are paired and minimal in number, leading to a tightly correlated closed electron shell and fractionally large singlet-triplet splittings. The lack of degeneracy is maximal in the excited states of He; the high 1 states are nearly hydrogenic and therefore form the exception to this (see section 2). Thus He is the stereotype for “normal” atoms without degeneracy; yet in really strong fields the interaction can be larger than lower level fine structure separations or Rydberg level separations, making states effectively degenerate [7]. So perhaps the hydrogen atom is not so abnormal when one is concerned with strong external oscillating fields. Field-free atomic structure is well understood for both H and He; it is in calculating matrix elements, Green’s functions and the like where hydrogen makes theory easier and more exact, since the wave functions are accurately known.

12.1. Excited hydrogen in ionizing oscillating fields The general role of the hydrogen atom’s spectrum in its multiphoton ionization has been discussed specifically by Kovarski and Perel’man [227]. The electric dipole approximation was assumed in making a diagram expansion of the S-matrix. Hydrogen-atom oscillating Stark orbitals were used as basis states, thus including those virtual photon processes responsible for adiabatic energy absorption exactly, rather than as a perturbation. The 1 and m degeneracy in hydrogen enhances this absorption, the effect being linear in the field; this was discussed in section 7, where we were concerned with the photon-replica atomic structure lines that reflect these virtual processes. The inequality for radial matrix elements (R~’~ i)_2

~

(R~/

1)2<1

n’ * n

implies that electric dipole coupling between states with the same n dominates over that to all other n; this underlies the notion that it is useful to choose a QES basis that takes the coupling with fixed n into account. Then the Green’s function or propagator G0 used to partially describe photon absorption is that which develops the oscillating Stark orbitals, and vertices in diagrams correspond to interactions with the field that mix quasienergy levels of different n. The S-matrix expansion corresponds to the unusual diagrams

~

)~.):

~:

a... L~..

The first or left-hand diagram is meant to describe a direct coupling of the initial state and the fieldperturbed continuum, no photons being absorbed other than those necessary to quasi-adiabatically

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

379

distort atomic structure. Quantum tunneling is the ionization process that involves a horizontal electron transition through a barrier in the field-distorted potential, i.e. it proceeds within this QES picture via “virtual” processes only. Thus the first diagram is that for pure quantum tunneling; it dominates at low frequencies and high field strengths. The terms of higher order in G0 correspond to the “true” absorption of one or more photons in the presence of strong “virtual” absorption; these then correspond to combined multiphoton and tunneling processes. In the low field limit diagrams of order M> k0 are unimportant, and indeed away from resonances the diagram of order k0 makes the largest contribution. It can give the usual power law behavior I~~0 only when the uppermost highly excited level n* involved is not strongly mixed with levels of other n. the Thus for such 2, i.e. photon replica behavior there an upper limit onsignificantly the field strength roughly F< w(n*) spectrum. of theis level ~ does not extendofinto that of neighboring n. The usual Fourier expansion of oscillating orbital phases exp {—ip~sin wt}

=

J,,,(p~)

exp (imcot)

leads to products of Bessel functions in the summation over diagrams that generates the ionization probability. For the Mth order diagram involving QES of order N 1,. . Ne,, . . . NM 1: 26[EK— E ~ IAQK(N!,N2,N3,. . .NM l)1 0 —M’fiw + (N1 +. . +N~ 1)~w]

.,

.

N5, K

where

~ {i~}

V~1~,2 ...

J~M

1,KR~l,ll,Nl).. .R(1M_ilM_1NM_l)

~

s=i

and R(j5, 15,N5)

=

~

15(pf)J~(p1~)}/{~,5

—

—

(l~ s)IIw}. —

Here {j~}= {n5, a5, iç} is the set of quantum numbers labeling the Q~S,both discrete and continuous, obtained in the discrete case by field-mixing of the degenerate states of a given value of n = n5. The p)’s are the roots of the QES secular equation A + FZ I(~~ + TI 1 = TI = I A • Lb \. ue~ t,tW ~ ua ~ r nsll3J ‘j~ rail — ~a5~ , the symbols ;, j3~run over the field-free quantum numbers for fixed n. The 6-function assures that the change in energy from the initial value E0 to the continuum state value EK is equal to the sum of energies of all the absorbed photons, both those involved in the development of the photon-replica aspects of the oscillating basis states and those involved in the transitions between n-levels. Now, as the field F increases, diagrams with M 1 increases. This shift in importance of diagrams arises because of the corresponding factors ~N(P)’~)’ N * 0, are not only no longer small, but they saturate. The major terms in EM P(M) are those containing the maximum number of saturated Bessel functions. Suppose that for such a term the indices of the Bessel functions having N5 p,”s are N5~,A~2- . Ps~.,. Then the number r of absorbed “virtual” photons is ~ 1N5, and k0-photon ionization is described by the diagram of order k0 r, When the freely oscillating electron energy EF ~ ~w, the tunneling —

—

380

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

diagram M = 1 becomes important while the M> 1 diagrams decrease with F according to the asymptotic behavior of the Bessel functions. An important aspect of the above Kovarski perturbative QES picture of strong field ionization in hydrogen is that a natural inclusion of the atomic distortion (due to the field driving the electron) leads to an expected saturation of the corresponding “virtual” processes at high field strengths. Consequently there is a lowering of the effective multiplicity of the ionization probability as well as a smooth transition in the high field limit to the pure quantum tunneling regime where all photon absorption is “virtual” in the dressed atom representation. Our discussion has ignored intermediate state resonance effects that are certainly present and may often play a major role. In the Kovarski picture they are present through the energy denominators in the R (f3, i~,N~)and can be pictured as occurring when photon replica levels for states with different quantum numbers match energetically. One might wonder under what conditions can the strong field phenomena predicted by QES theory be observable in experiments. In the visible wavelength region, k0> 1 in H requires that n ~ 2. If the field strength parameter pj’s is to be at least of order unity for n5 = n, then 2F/w n2F/(k 2)’ 2n4Fk~= (F/F~)(k 1~ n 02n 0/8). Since static field ionization occurs at fields of F 2F,~,we see that for small k0 the edge of the strong field region coincides with the edge of the region of rapid ionization via tunneling. As k0 is increased we expect the nonionizing part of the strong field region to grow. Thus for H, multiphoton experiments involving n = 2 that use Nd-glass laser photons of 1 eV energy will be strongly limited by the competition with tunneling. Experiments with the 0.1 eV photons of the CO2 laser should, however, prove very interesting for H(n = 2) and fields between 4 X 10~to 4 X l0~volt/cm, i.e. laser power fluxes of 4 X 1010 to 4 X 1012 watt/cm. At CO2 laser wavelengths, ionization is multiphoton for n < 10. As the wavelength is further reduced through the far infrared and into the microwave region, the range of interesting n increases and values for the field strength parameter p much greater than one can be investigated. For the case of transitions beginning from an s state in helium, a parallel development of the perturbative QES strong field picture would lead us to expect that higher field strengths are needed in general to saturate the M > 0 diagrams. Because of their large quantum defects, the s and p states are nondegenerate and absorb energy only in second order. Multiphoton ionization of states with 1 ~ 2 would, however, behave in a more hydrogenlike manner. Recently calculations of 2- and 3-photon ionization of the nS states of hydrogen have been carried through using regular perturbation theory accurate to first nonvanishing order in the electric dipole approximation [228]. The Coulomb Green’s function was used, and expanded in terms of Sturmian functions [190]. Numerical results are given only for n 1, 2 and 3, for a comparison with several earlier perturbation theory calculations. Similar calculations have been made for the CO2 laser 2photon ionization of then = 8,9 and 10 levels of hydrogen [229],see fig. 12.1. No experiments on excited hydrogen have been completed in the infrared except for the hydrogen discharge studies of Dubreuil and Chapeile [100, 168], already mentioned in 7. of Although observed laser 2, section a number possiblethey collisional ionization ionization of the n = 6, make 7, 8 and 9 levels at 0.5 in MW/cm effects in the discharge an interpretation terms of multiphoton ionization uncertain. For n = 8, the required fields for ionization are low compared to that expected on the basis of fig. 12.1. Let us now turn to the microwave ionization of highly excited hydrogen atoms [166, 174]. At a frcquency of 10 GHz (w = 1.2 X l0~a.u.) and for n = 48, the Bessel function J 1(O) is experimentally observed to reach saturation at a field strength F1 = 60 ±30 volt/cm. This was a result of the photon replica experiments already mentioned in section 7. On the other hand, the field strengths required

381

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

// \

H (n~48)

\

2

I

1626

1 I

9.6

I

I

10.0

I

10.59 il

10.4

Wavelength Fig. 12.1. Two-photon ionization of then hydrogen in the C0

‘10

,~

(EL) =

8 levels of

2-laser wavelength region, as theoretically determined by Justum and Maquet [229].The solid curve is the result of a perturbation calculation using a Sturmian expansion of the Coulomb Green’s function, while the other curves were obtained using the average-frequency technique of Bebb and Gold [201].

~ 00

~

~ I

9.5

I 0.5

I

11.5

Frequency (GHz) Fig. 12.2. A comparison of microwave multiphoton excitation (right) and ionization (left) signals for n = 48. The open circles are the backgrounds in the excitation experiment.

1 are experimentally observed to be about 15 times for an ionization probability of order 108theory sec we thus expect a high degree of “virtual” photon larger. On the basis of the Kovarski QES absorption at the field strengths required for ionization, with diagrams of order M ‘~ = 78 making large contributions. This has been experimentally confirmed in studies of the field strength dependence of the ionization probability that determine the effective multiplicity to be k~f= 12 ± 3 [230]. The question of whether the ionization is purely tunneling (M = 0) or involves true photon absorption processes (M> 0 diagrams) has also been recently answered by experiments that investigate the frequency dependence of the ionization probability [210]. Resonances in the ionization were observed for frequencies between 9.4 and 11.6 GHz, eliminating pure tunneling as the ionization mechanism at least near the resonances. As expected, the locations of resonances were observed to slowly shift to lower frequencies as the principal quantum number n of the atom was slowly decreased. The existence of the microwave ionization resonances implies that multiphoton transitions between states of different n should also proceed at observable, indeed rapid rates. This expectation has been confirmed by direct measurements of bound-bound transitions involving an increase in n [210]. A comparison of the observed excitation and ionization resonances is shown in fig. 12.2 for n = 48. The field strength is the same for both curves to within a few percent. Marked with vertical arrows are the frequencies fork = 5, 6 (An = +1) and k = 10, 11 (An = +2) multiphoton boundbound transitions between field-free states, the lower-energy one being the initially-prepared state n = 48. The excitation experiments did not discriminate between strongly Stark-shifted k = 5 (~n + 1) transitions and Ic = 10 (An = +2) transitions. What is clear is a marked 10% or 1 GHz “red shift” of the ionization resonance frequency away from the ~n = +1 field-free atom resonance frequency. This is to be expected from the multilevel theory of resonance enhancement of transitions to a continuum [223, 231]. Since the spacing of field-free levels is i~E~ = n~3, the local anharmonicity near n = 48 is b~E~/E~ = 4/n or 8%, close to the observed red shift.

382

I.E. Bayfield, Excitedatomic and molecular states in strong electromagnetic fields

In summary, the microwave experiments on highly excited hydrogen establish the existence of a strong-field regime at field strengths below that for very rapid ionization. The picture presented by the Kovarskii QES theory for hydrogen is basically correct, and its implications for higher frequencies (especially in the infrared) should be taken seriously. A remaining interesting question about the resonances shown in fig. 12.2 concerns their widths. Confining ourselves to the case of excitation first, experimental values for the individual widths discussed earlier in this section are the bandwidth t1 = 100 MHz, ionization width 50 MHz, field width Ff = 100 MHz and quasienergy switching width F~(I’ 1I2= 70 MHz. These do not seem to quite 1t) experimental field was that along the electric explain the 500 MHz width in the figure. However, the field direction of a TE 10 waveguide, i.e. the highly excited atom beam passed through the narrow dimension of the waveguide field. Thus the atoms saw a Gaussian-like pulse envelope; in this case the effective interaction time teff is less than the time t determined by the geometrical length of the waveguide, because high-order (k> 1) transitions occur mainly near the field maximum. This reduces t to taft ~ t/k [224], making the bandwidth contribution 500 MHz and the switching contribution smaller than before. Thus the observed widths can be understood. Under these short pulse conditions, it is interesting3, to that the the analysis regime where probability may be expected as note mentioned in ionization section 11.is Ainfull of the the ionization resonance widths cannot to vary with t yet be completed, since the effective multiplicity near resonance has not yet been measured (the above-mentioned value Of keff = 12 was at 9.9 GHz only, well off the resonance peak in fig. 12.2. As we shall see below, in the resonance frequency region we may expect the further complication that keff will be strongly frequency dependent. 12.2. Ionization of excited helium atoms In the helium atom some experiments have investigated the five-photon ionization of the 23S metastable state using a Nd-glass laser [232] and the three-photon ionization of the 21S metastable state using a ruby laser [234]. Both these ionization processes are resonantly enhanced. The 2’S—6’S resonance is shown in fig. 12.3 at two different laser intensities. The optical Stark shift is clearly observed and is proportional to the intensity. The 23S experiments were the first to demonstrate a frequency dependence in keff near a resonance, with keff reaching values larger than k 0 because of the complexity of resonant shift-width functions and the interference of amplitudes of several contributing intermediate states, see fig. 12.4. The same behavior has now been also seen in a number of laser ionization experiments on atomic ground states [4, 235], see fig. 12.5 for example. The fast atomic beam techniques used in the highly excited hydrogen experiments discussed above can also be used for studies 4He(ls on highly excited helium atoms [30, 31, 236]. The 4ls)3S at 9.92 GHzand hasother an observed effective multiplicity microwave ionization of the state keff = 2 [236], a value much less than those observed in H. This highly excited triplet state has a sizeable energy defect and is effectively nondegenerate up to quite high microwave field strengths. This reduces the effects of photon replica formation, which are now quadratic in the field strength. As a result, at the field strength required for an ionization rate of 108 sec’, resonant enhancement by multiphoton transitions to intermediate states can be less effective than in hydrogen, with low order-diagrams M ~ 0 dominating the ionization of helium S and P states at least at some frequencies. More experiments are required before much more can be conjectured about helium. We close this section with mention of a recent exploration of the hypothetical question, what happens to the ionization probability as the strength of an instantaneously-switched field or an

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

~‘

IJ~~ 28844

I 28840 2’S-6S

...4

___ I

1~

1

37700

2 x Frequency (cm~ )

4

Fig. 12.3. The 5-photon Nd-glass laser ionization ofHe (2’S) 1S—6’S intermediate state resonance, after Delone et al.the [234]. near 2 Note the position of the field-free resonant frequency at the left; the resonance curves are for two different laser field intensities.

k

383

X Frequency

37900 (cm~

Fig. 12.4. The variation L~.kin multiplicity away from k

for the ruby-laser ionization of He 23S, after Bakos et al. 0 [232]. The locations of possible n = 12, 13 and 14 intermediate state resonances is indicated.

5

2 _______

9400

9450

9400

9450

Frequency (cm~

(a)

(b)

Fig. 12.5. The 11-photon ionization of the xenon atom, showing (a) the frequency dependence of keff and (b) the frequency dependence of the ionization probability [235].

adiabatically-switched field is increased indefinitely? Maybe the probability does not increase indefinitely, if we define ionization to be the removal of the electron from the locale of the atomic nucleus [122, 237]. Consider the conditions achieved in some highly excited state microwave ionization experiments, namely 7K ~ I and F ~ F~.The latter means that the electron is close to or in the region of classical escape from the nucleus, and is relatively free to respond to the external oscillating field. The former condition implies that the classical oscillation energy ~F is greater than the field-free atom ionization energy 1~, =

(~/v)2= (nF/w)2

=

l/’y~~’ 1.

Since the field strength parameter (in hydrogen) for the strong virtual photon absorption needed to sustain a large field-driven oscillatory motion of the electron is then large: p~n2F/wn(nF/oi) l/’y~’ l,

384

J.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

it seems that the transition to a highly oscillating yet localized-electron “ultrastrong field” regime might occur at fields strengths less than that required for essentially instantaneous ionization, i.e. ionization within one electron orbit period. Such a regime would be characterized by equally spaced oscillator-type energy levels for the system atom-plus-field. The system would become somewhat stabilized against ionization; with a further increase in field strength the ionization probability in some short fixed time would actually decrease as predicted by “inverse” perturbation theory where the Coulomb force perturbs the oscillating electron via momentum transfer scattering to produce ionization [237—239]. The existence of such an ultrastrong field regime remains to be established, with the question of field-switching effects critical to both theory and experiment. 13. Excited states of molecules in strong fields The straight-forward extension to molecules of the concepts discussed in the previous sections leads to the prediction of many interesting strong-field induced molecular phenomena. Let us consider the diatomic molecule case, where the adiabatic (oscillating) orbitals and quasienergy (fieldperturbed) levels are functions of the internuclear separation R. Again nonresonant fields can create level shifts and splittings, as well as photon replicas that because of saturated “virtual” photon absorption are as “real” as the field-free levels .E(R) are in the absence of the field. Since molecular structure in the intense field can be quite different from that with no field, so can be molecular processes such as predissociation [240, 241] and inelastic slow atom—atom collisions [242—252] - Not only can new “radiative” processes occur that involve photon absorption between field-free levels at resonant values of R where 11w E1(R) — E2(R) [253—256], but strong fields can under suitable circumstances “quench” normal field-free processes through the splitting of normal near-degeneracies. Of course, static fields can also alter predissociation [257, 258] and collision cross-sections [259, 260]. But the additional variables of frequency and polarization greatly increase the richness of oscillating field phenomena. The quasienergy states of molecules have been theoretically considered, and the Born—Oppenheimer approximation for quasienergy states developed [246, 261]. Fig. 13.1 shows the rotational quasienergies of a,’ E diatomic-molecule ground vibrational state as a function of the field strength parameter. There are many regimes where the angle between the electric field and the molecular axis is an important additional parameter. In fig. 13.1 the field is assumed to resonantly couple the first and second vibrational states, and is to be linearly polarized in a direction not far from the molecular axis, where the coupling is weak. Fig. 13.2 shows schematically how a q = +1 lower-state photonreplica level can intersect a q = —1 upper-state replica to produce a molecular energy level crossing at some value of internuclear separation R. The splitting of adiabatic curves at the adiabatic photonreplica crossing is in general due to both the Coulomb and external fields, while the nonadiabatic coupling that produces transitions between adiabatic QES can be due to both molecular internuclear motion and to the field. When the order of the photon replicas involved is not small, the fieldinduced molecular processes have rates that are strongly dependent upon field strength. An example of this is shown in fig. 13.3 where the cross-section for the energetically forbidden process H(ls) + H(ls)-+H(ls) + H(2p),

KE

9.0 eV,

is shown as a function of field strength at the Nd-glass laser and at the CO2 laser frequencies [246] A collisional kinetic energy of 9.0 eV is assumed, making the absorption of one 1 eV photon or ten 0.1 eV photons during the collision a requisite for a nonzero cross-section.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

385

The observation of isotope-selective CO2 laser dissociation of polyatomic molecules has created considerable interest in molecular multiphoton excitation processes. Fig. 13.4 shows some perturbation theory rates for multiphoton excitation of i~v= + 1 vibrational transitions in CO. The fields required for rapid laser-induced chains of nearly-resonant one-photon transitions up the molecules vibrational ladder can be lower. Fig. 13.5 shows some typical results for this case. The reader is refered to the reviews on infrared laser isotope separation and infrared laser-induced chemistry for further discussions of this last topic [265—267].

~

—6

~

Internuclear Separation

Fig. 13.1. Dependence of the values of vibrational 1Z groundquasienergy state of a upon the interaction energy ~ of a diatomic molecule with a linearly polarized oscillating field, after Makarov and Fedorov [262]. The quantity ~ is in units of the anharmonic energy defect of the molecule; the levels of the field-free molecule are = 0) = —p(p — 1). The applied field is assumed to resonantly couple the levels p = 0 andp = 1.

IQ~

1010

Fig. [~ ,, 13.2. The field-free [E,2(R)]

and two photon-replica 2(R)] potential energy curves of a molecule in a strong resonant oscillating electric field.

1011

1Q12

2

Loser Power Flux (W/cm Fig. 13.3. The laser-induced collision cross-section for [H(ls) ÷H(ls)J (b~~) + 9.0 eV -÷ [H(ls)

+

H(2p 0)] (a~E~) at CO2 and Nd-

glass laser frequencies, after Kroll and Watson [246].

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

386

E 5 ulO-

S

0

0

~J~o

~ ~

1Q00

1100

Q_ 4

-

121\A

900 Frequency (cm ‘)

1000

(b)

(a)

Fig. 13.4. Some multiphoton absorption coefficients for nonresonant CO

2 laser excitation of one vibrational quantum 2, the of energy HCI curves in

for three-photon transitions at 1011 and watt/cm2. (a) CO and (b) HC1, after Merchant Isenor [263]. The CO curves are for two-photon transitions at io~watt/cm

940

950

Frequency (cm~) Fig. 13.5. A calculated frequency dependence of the steady-state power absorption of a model SF 2, after Hodgkinson and Briggs [264]. All the vibrational modes of6the molecule molecule in awere CO2intercoupled laser field to each and to v with other a strength of the ~ laser-excited watt/cm 3 mode, resulting in the broad aspect of the curve. The narrow peaks are due to resonant enhancement of the upper i.’3 levels by multiphoton processes. An anharmonic oscillator model was used for each mode, with parameters chosen to fit what is known about SF6.

Acknowledgement This work was partially supported by the U.S. National Science Foundation. References [1] [2] [3] [4] [5] [6] [7] [8]

[9] (10] [11] [12] [13] [14]

[151

H.W. Drawin and F. Emard, Zeit. f. Physik 270 (1974) 59. D.R. Dion and J.O. Hirschfelder, Adv. in Chem. Physics 35 (1976) 265. J.S. Bakos, Advan. Electron. Electron Phys. 36 (1974) 57. N.B. Delone, Usp. Fiz. Nauk 115 (1975) 361 [Soy. Phys. — Usp. 18 (1975) 169]. C. Grey Morgan, Rep. Prog. Phys. 38 (1975) 621. P. Lambropoulos, Advan. Atom. Molec. Phys. 12 (1976) 87. H.A. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Academic Press, 1957). E. Schroedinger, Ann. Phys. (Leipzig) 80 (1926) 437. P.S. Epstein, Phys. Rev. 28 (1926) 695. V. Rojanski, Phys. Rev. 33 (1929) 1. G. Luders, Ann. Physik 68 (1951) 301. J.D. Bekenstein and J.B. Krieger, Phys. Rev. 188 (1969) 130. G. Wentzel, Zeit. 1. Physik 38 (1926) 518. C. Lanczos,Zeit. f. Physik62(1930)518;65 (1930)431;68 (1931) 204. M.H. Rice and R.H. Good, J. Opt. Soc. Am. 52 (1962) 239.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

387

[16] B.M. Smirnov and M.I. Chibisov, Zh. Eksp. Teor-Fiz. 49 (1965) 841. [17] S.P. Alliluev and l.A. Malkin, Zh. Eksp. Teor. Fiz. 66 (1974) 1283 [Soy.Phys.

— JETP 39 (1974) 627]. M.H. Alexander, Phys. Rev. 178 (1969) 34. [19] J.O. Hirschfelder and L.A. Curtiss, J. Chem. Phys. 55 (1971) 1395. [20] K. Helfrich, Theoret. Chem. Acta (Berlin) 24 (1972) 271. [21] N.A. Gushina and V.K. Nikulin, Chem. Phys. 10(1975) 23. [22] R.J. Damburg and V.S. Kolosov, J. Phys. B9 (1976) 3149. [23] G.P. Drukarev and A.P. Scherbakov, in: Problems in theoretical physics, 1: Quantum mechanics, eds. M.G. Veselov, Y.V. Novozhiorand, P.P. Pavinsky (Leningrad University, 1974) pp. 54—62. [24] T. Yamabe, A. Tachibana and H.J. Silverstone, Phys. Rev. 16 (1977) 877. [25] J.B. Delos and W.R. Thorson, Phys. Rev. Lett. 28(1972)647. [26] L.l. Ponomarev and T.P. Puzynina, Zh. Eksp. Teor. Fir. 52 (1967) 1273 [Soy. Phys. — JETP 25 (1967) 846]. [27] L.I. Ponomarev, Zh. Eksp. Teor. Fiz. 55 (1968) 1836 [Soy. Phys.-JETP 28(1969) 971]. [28] G.J. Hatton, WL. Lichten and N. Ostrove, Chem. phys. Lett. 40(1976) 437. [29] G.J. Hatton, Phys. Rev. A16 (1977) 1347. [30] J.E. Bayfield, in: The physics of electronic and atomic collisions, eds. J.S. Risley and R. Geballe (Univ. of Wash. Press, Seatle, 1976) p. 726.

[18]

[31] J.E. Bayfleld, Rev. Sci. Instr. 47 (1976) 1450. [32]

R. Loudon, Am. J. Phys. 27 (1959) 649.

[33] R.F. O’Connell, Phys. Lett. 60A (1977) 481. [34] [35]

[36a] [36bJ [37a] [37b] [38] [39] [40] (41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

A.R.P. Rau, Phys. Rev. A16 (1977) 613. N. Zsi-Ze and W.W. Po, J. Phys. Rad. 7(1936)193. S. Haroche, M. Gross and M.P. Silverman, Phys. Rev. Lett. 33 (1974) 1063. C. Fabre and S. Haroche, Opt. Commun. 15 (1975) 254. T.W. Ducas, M.G. Littman, R.R. Freeman and D. Kleppner, Phys. Rev. Lett. 35 (1975) 366. M.G. Littman, M.L. Zimmerman, T.W. Ducas, R.R. Freeman and D. Kleppner, Phys. Rev. Lett. 36 (1976) 788. D. Kleppner, in: Atomic Physics 5 (Plenum Press, 1977). R.T. Hawkins, W.T. Hifi, F.V. Kowalski, A.L. Schawlow and S. Svanberg, Phys. Rev. A15 (1977) 967. T.W. Ducas and M.L. Zimmerman, Phys. Rev. A15 (1977) 1523. H.C. Bryant, B.D. Dieterle, J. Donahue, H. Sharifian, H. Tootoonchi, D.M. Wolfe, P.A.M. Gram and M.A. Yates-Williams, Phys. Rev. Lett. 38 (1977) 228; also to be published. R.R. Freeman and G.C. Bjorklund, Phys. Rev. Lett. 40(1978)118. M. Hehenberger, H.V. McIntosh and E. Brandas,, Phys. Rev. AlO (1974) 1494. D.R. Herrick, J. Chem. Phys. 65 (1976) 3529. D.S. Bailey, J.R. Hiskes and A.C. Riyiere, Nuc. Fusion 5 (1965) 41. A.C. Riviere and D.R. Sweetman, Proc. Sixth mt. Conf. on Ionization phenomena in gases, Paris (1963) p. 105. N.Y. Fedorenko, V.A. Ankudinov and R.N. Il’in, Zh. Tekh. Fiz. 35 (1965) 585 [Soy. Phys.-Tech. Phys. 10 (1965)461]. R. LeDoucen and J. Guidini, C.R. Acad. Sc. (Paris) B268 (1969) 918. R.N. Il’in, in: Atomic Physics 3, eds. S.J. Smith and G.K. Walters (Plenum Press, 1973) pp. 309—326. i.E. Bayfield, G.A. Khayrallah and P.M. Koch, Phys. Rev. A9 (1974) 209.

[51a) R.P. Feynman, Rev. Mod. Phys. 20 (1948) 267. [Sib] [52] [53] [54]

R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). C. Garrod, Rev. Mod. Phys. 38 (1966) 483, 315. I.C. Percival, in: Atomic Physics 2, ed. P.G.H. Sandars (Plenum Press, 1971) p. 345. LD. Landau and E.M. Lifshitz, Kvantovaya mekhanika (Nauka, Moscow, 1974) [Quantum mechanics (Pergamon, London,

1965)]. L.P. Kotova, A.M. Peralomov and V.S. Popov, Zh. Eksp. Teor. Fir. 54 (1968) 1151 [Soy.Phys.-JETP 27 (1968) 616]. [56] AM. Perelomov, V.S. Popov and M.V. Terent’ev, Zh. Eksp. Teor. Fir. 50 (1966) 1393 [Soy. Phys.-JETP 23 (1966) 924]. [55] [57] [58) [59] [60] [61]

A.M. Perelomov, V.S. Popov and M.V. Terent’ey, Zh. Eksp. Teor. Fir. 51(1966) 309 [Soy. Phys.-JETP 24 (1967) 207]. A.M. Perelomov and V.S. Popov, Zh. Eksp. Teor. Fir. 52 (1967) 514 [Soy. Phys.-JETP 25 (1967) 336. A.M. Perelomov, V.P. Kuznetsov and V.S. Popov, Zh. Eksp. Teor. Fir. 53 (1967) 331 [Soy. Phys.-JETP 26 (1968) 222]. S.N. Kaplan, G.A. Paulikas and R.V. Pyle, Phys. Rev. 131 (1963) 2547. R.N. Il’in, V.A. Oparin, I.T. Serenkov, E. Soloy’ev and N.V. Fedorenko, Zh. Eksp. Teor. Fir. 59 (1970) 103 [Soy. Phys.-

JETP32 (1971) 59]. V.A. Oparin, R.N. Il’in, I.T. Serenkov and E.S. Solov’ev, Pisma Zh. Eksp. Teor. Fir. 12 (1970) 237 [JETP Lett. 12 (1970) 162]. [63] V.A. Oparin, R.N. Il’in, I.T. Serenkov, E.S. Solov’ev and N.Y. Fedorenko, Pisma Zh. Eksp. Teor. Fir. 13 (1971) 351 [JETP Lett. 13(1971)249]. [64] Yu.N. Demkov and G.F. Drukarev, Zh. Eksp. Teor. Fir. 47 (1964) 918 [Soy.Phys.-JETP 20(1965) 614]. [65] ‘.A. Cahill, J. Richardson and J.W. Yerba, Nucl. Instr. Meth. 39 (1966) 278. [62]

388 [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76]

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields T.W. Ducas, M.G. Littman, R.R. Freeman and D. Kleppner, Phys. Rev. Lett. 35 (1975) 366. R.V. Ainbartsumyan, G.I. Bekov, V.S. Letokhov and V.!. Mishkin, Pis’ma Zh. Eksp. Teor. Fir. 21(1975)295 [JETP Lett. 21(1975) 279]. M.G. Littman, M.L. Zimmerman and D. Kleppner, Phys. Rev. Lett. 37 (1976) 486. T.F. Gallagher, L.M. Humphrey, R.M. Hill and S.A. Edelstein, Phys. Rev. Lett. 37 (1976) 1465. T.F. Gallagher, L.M. Humphrey, W.E. Cooke, R.M. Hill and S.A. Edelstein, Phys. Rev. A15 (1977) 1937.

P. Jacquinot, Invited Paper, CNRS Int. Colloquim on Atomic and molecular states coupled to a continuum, Ausois, France, to be pub!, in Jour. de Physique Colloque, 1978. A.F.J. Van Raan, G. Baum and W. Raith, J. Phys. B9 (1976) L173. M.L. Zimmerman, T.W. Ducas, M.G. Littman and D. Kleppner, J. Phys. Bli (1978) Lll. C.F. Barnett, J.A. Ray and A. Russek, Phys. Rev. AS (1972) 2110. C.F. Barnett and J.A. Ray, Phys. Rev. AS (1972) 2120. L.R. Carison, J.A. Paisner, E.F. Worden, S.A. Johnson, C.A. May and R.W. Solaxz, Lawrence Livermore Lab. Report No.

UCRL-78034, 1976. [77] V. Pankonin, P. Thomasson and J. Barsuhn, Astron. Astrophys. 54 (1977) 335 and references therein. [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88]

W.B. Somerville, Repts. on Prog. in Phys. 40 (1977) 483. T.H. Troland and C. Heiles, Astrophys. Jour. 214 (1977) 214. M. Ruderman, in: IAU symp. No. 53, Physics ofdense matter, ed. C. Hansen, Boston (Reidel, 1974) p. 117. R. Gajewski, Physica 47 (1970) 575. A.F. Starace, J. Phys. B6, (1973) 585. J.H. Van Vleck, in: The theory ofelectric and magnetic susceptibilities (Oxford Press, 1932) p. 178. L.I. Schiff and H. Snyder, Phys. Rev. 55 (1939) 59. A.R.P. Rau and L. Spruch, Astrophys. Jour. 207 (1976) 671. R. Garstang, Repts. on Prog. in Physics 40 (1977) 105. R.F. O’Connell, Astrophys. Jour. 187 (1974) 275. G.L. Surmeian and R.F. O’Connell, Astrophys..Jour. 193 (1974) 705. A.R.P. Rau, Phys. Rev. A16 (1977) 613.

[89] [90] A.R. Edmonds, J. Phys. B6 (1973) 1603.

F.A. Jenkins and E. Segre, Phys. Rev. 55 (1939) 52. C.D. Harper and M.D. Levinson, Opt. Commun. 20 (1977) 107. W.R.S. Garton and F.S. Tomkins, Astrophys. Jour. 158 (1969) 839. R.O. Mueller and V.W. Hughes, Proc. Nat. Aced. Sciences (USA) 71(1974) 3287. [95] Yu.N. Demkov, B.S. Monozon and V.N. Ostrovskii, Zh. Eksp. Teor. Fir. 57 (1969) 1431 [Soy. Phys.-JETP 30 (1970) 775]. [96] L.A. Burkova, I.E. Dzyaloshinskii, G.F. Drukarev and B.S. Monozon, Zh. Eksp. Teor. Fir. 71(1976) 526 [Soy.Phys.-JETP 44(1976)276]. [97] G.F. Drukarev and B.S. Monozon, Zh. Eksp. Teor. Fir. 61(1971)956 [Soy. Phys.-JETP 34 (1972) 509]. [98] W.L. Lichten, Phys. Rev. A3 (1971) 594. [99] R.H. Kagann and T.C. English, Phys. Rev. Al3 (1976) 1451. [100] M. Born, Vorlesungen Uber Atommechanik (Springer, Berlin, 1925). [101] J. Avron, I. Herbst and B. Simon, Phys. Rev. Lett. 39 (1977) 1068. [91] [92] [93] [94]

[102] [103] [104] [105] [106]

N.B. Delone, B.A. Zon, V.P. Krainoy and V.A. Khodovoi, Usp. Fir. Nauk 120 (1976) 3 [Soy.Phys.-Usp. 19 (1976)711]. W. Pauli, Handbuch der Physik 24(1933)164. M. Born and V. Fock, Z. Physik 51(1928)165.

P.W. Langhoff, S.T. Epstein and M. Karplus, Rev. Mod. Phys. 44 (1972) 602. J. Musher, Annals of Physics (N.Y.) 27 (1964) 167. [107] H. Sambe, Phys. Rev. Al (1973) 2203. [108] Ya.B. Zeldovich, Usp. Fir. Nauk 110 (1973) 139 [Soy.Phys.-Usp. 16 (1974) 427]. [109] Ya.B. Zeldovich, N.L. Manakov and L.P. Rapoport, Usp. Fir. Nauk 117 (1975) 563. [110] J.M. Okuniewicz, J. Math. Phys. 15 (1974) 1587. [111] R.H. Young, W.J. Deal Jr. and N.R. Kestner, I. Mol. Phys. 17 (1969) 369. [112] E.A. Power and T. Thirunamachandran, J. Phys. B8 (1975) L167; 8 (1975) L170. [113] F. Bassani, J.J. Forney and A. Quattropani, Phys. Rev. Lett. 39 (1977) 1070. [114] P. Lambropoulos, G. Doolen and S.P. Rountree, Phys. Rev. Lett. 34 (1975) 636. [115] K.H. Yang, Annals ofPhys. (N.Y.) 101 (1976) 62; 101 (1976) 97. [116] B.A. Zon and E.I. Sholokho, Zh. Ekps. Teor. Fir. 70 (1976) 887 [Soy. Phys-JETP 43 (1977)461]. [117] C.H. Townes and A.L. Schawlow, Microwave spectroscopy (McGraw-Hill, New York, 1955). [118] N.L. Manakoy, V.D. Ovsyannikov and L.P. Rapoport, Zh. Eksp. Teor. Fir. 70(1976) 1697 [Soy. Phys.-JETP43 (1976) 885]. [119] C.H. Townes and F.R. Merritt, Phys. Rev. 72 (1947) 1266. [120] D. Prosnitz and E.V. George, Phys. Rev. Lett. 32 (1974) 1282. [121] D. Prosnitz, D.W. Wildman and E.V. George, Phys. Rev. A16 (1977) 1165.

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields

389

[122] J.E. Bayfield, Invited Papers, European Study Conf. on Multiphoton processes, Seillac, France, April 1975. [123] P.M. Koch, post deadline paper presented at the Intern. Conf. on Multiphoton processes, Rochester, New York, June 1977. [124]

J.E. Bayfield, Invited Paper, CNRS Intern. Colloquim on Atomic and molecular states coupled to a continuum, Aussois, France, June 1977, to be published in Jour. de Physique Colloque, 1978. [125] D. Andrick and L. Langhans, J. Phys. B9 (1976) Ll. [126] A. Weingartshofer, J.K. Holmes, G. Caudle, E.M. Clarke and H. Kruger, Phys. Rev. Lett. 39 (1977) 269. [127] L.S. Brown and R.L. Goble, Phys. Rev. 173 (1968) 1505. [128] N.M. Kroll and K.M. Watson, Phys. Rev. A8 (1973) 804. [129] W. Happer Jr., Phys. Rev. 136 (1964) A35. [130] M. Allegrini and E. Arimondo, I. Phys. B4 (1971) 1008. [131] C. Cohen-Tannoudji, in: Laser spectroscopy, eds. R.C. Brewer and A. Mooradian (Plenum Press, 1976). [132] P.R. Fontana and P. Thonman,Phys. Rev. A13 (1976) 1512. [133] W.E. Lamb Jr., Phys. Rev. 85 (1952) 259, see Appendix IV. [134] V.1. Ritus, Zh. Eksp. Teor. Fir. 51(1966)1544 [Soy. Phys.-JETP 24 (1967) 1041]. [135] S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. [136] J.H. Shirley, Phys. Rev. 138 (1965) B979.

[137] B.R. Mollow, Phys. Rev. A12 (1975) 1919. [138] C. Cohen-Tannoudji and S. Reynaud, J. Phys. BlO (1977) 345. [139]

S.R. Barone, M.A. Narcowich and F.J. Narcowich, Phys. Rev. A15 (1977) 1109.

[140] C.R. Stroud, Invited Paper, Fourth Rochester Conf. on Coherence and quantum optics, June 1977. [141] S.-!. Chu and W.P. Reinhardt, Phys. Rev. Lett. 39 (1977) 1195. [142] S.S. Hassan and R.K. Bullough, J. Phys. B8 (1975) L147. [143] F.T. Hioe and J.H. Eberly, Phys. Rev. All (1975) 1358. [l44a] J. Von Neumann and E. Wigner, Phys. Z. 30 (1929) 467. [144b] G.J. Hatton, Phys. Rev. A14 (1976) 901. [145] F. Schuda, C.R. Stroud and M. Hercher, J. Phys. B7 (1974) L198. [146] F.Y. Wu, R.E. Grove and S. Ezekiel, Phys. Rev. Lett. 35 (1975) 1426; Phys. Rev. A15 (1977) 227. [147] J.L. Picque and J. Pinard, J. Phys. B9 (1976) L77. [148] A. Schabert, R. Keil and P.E. Toschek, Opt. Commun. 13 (1975) 265. [149] P. Cahuzac and R. Vetter, Phys. Rev. A14 (1976) 270. [150] 5. Feneuille and M.G. Schweighofer, J. Physique (Paris) 36 (1975) 781. [151] W.A. McClean and S. Swain, J. Phys. BlO (1977) L143. [152] 5. Yeh and P. Stehle, Phys. Rev. A15 (1977) 213. [153] A.M.F. Lau, Phys. Rev. Al4 (1976) 279. [154] N.F. Perel’man, Zh. Eksp. Teor. Fir. 68 (1975) 1460 [Soy. Phys.-JETP 41(1976) 822]. [155] B.A. Zon and B.G. Katsnel’son, Zh. Eksp. Teor. Fir. 65 (1973) 947 [Soy. Phys.-JETP 38 (1974) 470]. [156] M.V. Fedorov and V.P. Makarov, Opt. Commun. 14 (1975) 421. [157] B.A. Zon and B.G. Katsuelson, Opt. Spektrosk. 40 (1976) 952 [Opt. Spectrosc. 40 (1976) 544]. [158] B.A. Zon and T.T. Urazbaev, Zh. Eksp. Teor. Fir. 68 (1975) 2010 [Soy. Phys.-JETP 41(1976)1006]. [159] A. Maquet, Phys. Lett. A48 (1974) 199. [160] B. Dubreull and J. Chapelle, Phys. Lett. A46 (1974) 451. [161] B.A. Zon, N.L. Manakov and L.P. Rapoport, Opt. Spektrosk. 38 (1975) 13 [Opt. Spectrosc. 38 (1975) 6]. [162] B.A. Zon, Opt. Spektrosk. 42 (1977) 13 [Opt. Spectrosc. 42 (1977) 6]. [163] V.A. Kovarskii, Zh. Eksp. Teor. Fir. 57 (1969) 1217 [Soy. Phys.-JETP 30(1970) 663]. [164] V.A. Kovarskii and N.F. Perel’man, Zh. Eksp. Teor. Fir. 60 (1971) 509 [Soy. Phys.-JETP 33 (1971) 274]. [165] V.A. Kovarskll, N.F. Perel’man and S.S. Todirashku, Kvant. Elektron. 1(1974) 2417 [Soy. J. Quantum Electronics 4(1975)

1344]. [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178]

J.E. Bayfield, in: Multiphoton processes, eds. P. Lambropoulos and J. Eberly (J. Wiley, 1978). W.R. Sairman, Chem. Phys. Lett. 25 (1974) 302. B. Dubreuil and J. Chapelle, Jour. de Physique 34 (1973) C2-105. B. Dubreuil, H.W. Drawin and J. Chapelle, Z. fur Physik B20 (1975) 117. A.O. Barut and H. Kleinert, Phys. Rev. 156 (1967) 1541. J.E. Bayfield and P.M. Koch, Phys. Rev. Lett. 33 (1974) 258. W.W. Hicks, R.A. Hess and W.S. Cooper, Phys. Rev. AS (1972) 490. B. Dubreuil and J. Chapelle, Phys. Lett. 60A (1977) 115. J.I. Gersten and M.H. Mittleman, Phys. Rev. All (1975) 1103. D.H. Kobe, Phys. Rev. Lett. 40 (1978) 538. N.H. Burnett, H.A. Baldis, M.C. Richardson and G.D. Enright, AppI. Phys. Lett. 31(1977)172. E.A. McLean, J.A. Stamper, B.H. Ripin, H.R. Griem, J.M. McMahon and S.E. Bodner, Appl. Phys. Lett. 31(1977) 825. C.C. Choi, W.C. Henneberger and F.C. Sanders, Phys. Rev. A9 (1974) 1895.

390 [179] [180] [181] [182] [1831

[184] [185] [186] [187] [188] [189] [190] [191] (192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208]

[209] [210] [211]

[212] [213] [214]

fE.

Bayfield, Excited atomic and molecular states in strong electromagnetic fields

R.F. O’Connell, Phys. Rev. A12 (1975) 1132. P. Lambropoulos, E.A. Power and T. Thirunamachandran, Phys. Rev. A14 (1976) 1910. J.I. Gersten and M.H. Mittleman, J. Phys. B9 (1976) 2561. T. Welton, Phys, Rev. 74(1948)1157. H.R. Reiss, Phys. Rev. D4 (1971) 3533. A. Decoster, Phys. Rev. A9 (1974) 1446. C. Cohen-Tannoudji, J. Dupont-Roc, C. Fabre and G. Grynberg, Phys. Rev. A8 (1973) 2747. H.A. Kramers, Rapport et Discussions, Solvay Congress Stoops, Brussels, 1948. W.C. Henneberger, Phys. Rev. Lett. 21(1968) 838. C.K. Choi, W.C. Henneberger, S.N. Mian and F.C. Sanders, Phys. Rev. A12 (1975) 719. Y. Gontier and N.K. Rahman, Lettre al Nuovo Cimento 9 (1974) 537. E. Karule, J. Phys. B4 (1971) L67. E. Karule, Atomic processes, Report of the Latvian Academy of Sciences (Iznatni, Riga, 1975) pp. 5—24. P. Lambropoulos, Phys. Rev. A9 (1974) 1992. U. Fano, Phys. Rev. 124 (1961) 1866. U. Fano and J.W. Cooper, Phys. Rev. 137 (1965) A1364. S. Rautian, in: Invited Papers of the Second Int. Conf. on Electrons in strong electromagnetic fields, Budapest, Hungary, 1975 (CRIP, Budapest) p. 150. I.M. Beterov and V.P. Chebotaey, Prog. in Quantum Electron. 3 Part 1 (Pergamon, Oxford, 1974). D.F. Zaretskil and V.P. Krainov, Zh. Eksp. Teor. Fir. 66(1974)537 [Soy. Phys.-JETP 39 (1974) 257]. V.P. Krainov, Zh. Eksp. Teor. Fir. 70 (1976) 1197 [Soy. Phys.-JETP 43 (1977)622]. J.V. Maloney and W.J. Meath, MoL Phys. 31(1976)1537. P.W. Atkins and R.C. Gurd, Chem. Phys. Lett. 16 (1972) 265. H.B. Bebb and A. Gold, Phys. Rev. 143 (1966) 1. B.A. Zon, N.L. Manakov and L.P. Rapoport, Zh. Eksp. Teor. Fir. 60(1971)1264 [Soy. Phys.-JETP 33(1971)683]. Y. Gontier and M. Trahin, Phys. Rev. A4 (1971) 1896; A14 (1976) 1935. E.J. Delsey and J. Macek, 1. Math. Phys. 17 (1976) 1183. Y. Heno, A. Maquet and R. Schwarz, Phys. Rev. A14 (1976) 1931. J. Schwinger, J. Math. Phys. 5 (1964) 606. S. Klarsfeld, Lett. Nuovo Ciinento 2 (1969) 548; 3 (1970) 359. A. Maquet, J. Phys. B7 (1974) L228; Phys. Rev. A15 (1977) 1008. A.!. Ignatiev, Zh. Eksp. Teor. Fir. 70 (1976) 484 [Soy. Phys.-JETP 43 (1976) 250]. J.E. Bayfield, L.D. Gardner and P.M. Koch, Phys. Rev. Lett. 39 (1977) 76. J.S. Risley, in: Atomic Physics 4, eds. G. zu Putlitz, E.W. Weber and A. Winnacker (Plenum Press, 1975) p. 488. A.!. Nlldshov and V.!. Ritus, Zh. Eksp. Teor. Fir. 50 (1966) 255 [Soy. Phys.-JETP 23 (1966) 1681. L.V. Keldysh, Zh. Eksp. Teor. Fir. 47 (1964) 1945 [Soy. Phys.-JETP 20 (1965) 1307]. A.I. Nikishov and V.1. Ritus, Zh. Eksp. Teor. Fir. 52 (1967) 223 [Soy.Phys.-JETP 25 (1967) 145]. N.L. Manakov and L.P. Rapoport, Zh. Eksp. Teor. Fir. 69 (1975) 842 [Soy. Phys.-JETP 42 (1976) 430].

[215a] [215b] I.J. Berson, J. Phys. B8 (1975) 3078. [216]

S. Geltman, J. Phys. BlO (1977) 831.

[217] L.A. Lompre, G. Mainfray, C. Manus, S. Repoux and J. Thebault, Phys. Rev. Lett. 36 (1976) 949; Phys. Rev. A15 (1977) 1604.

[218] L.P. Kotova nad M.V. Terentiev, Zh. Eksp. Teor. Fir. 52 (1967) 732 [Soy.Phys.-JE’FP 25 (1967)4811. [219] C. Cohen-Tannoudji and P. Avan, Invited Paper, CNRS Int. Colloquium on Atomic and molec. states coupled to a continuum, to be published in J. de Physique Colloque 1978. [220] G.W.F. Drake, Astrophys. Jour. 184 (1973) 145. [221] Nils Ryde, Atoms and molecules in electric fields (Almquist and Wiksell, Stockholm, 1976). [222] A.M. Perelomov, V.S. Popov and V.P. Kuznetsov, Zh. Eksp. Teor. Fir. 54 (1968) 841 [Soy. Phys.-JETP 27 (1968) 451]. [223] F.H.M. Faisal, J. Phys. B9 (1976) 3009. [224] A.E. Kazakov, V.P. Makarov and M.V. Fedorov, Zh. Eksp. Teor. Fir. 70 (1976) 39 [Soy. Phys.-JETP 43 (1976) 20]. [225] M.V. Fedorov, J. Phys. BlO (1977) 2573. [226a] M.V. Fedorov and A.E. Kazakov, Opt. Commun. 22 (1977) 42. [226b] B.L. Beers and L. Armstrong Jr., Phys. Rev. A12 (1975) 2447. [227] V.A. Kovarskll and N.F. Perel’man, Zh. Eksp. Teor. Fir. 61(1971)1389 [Soy. Phys.-JETP 34 (1972) 738]. [228] S.V. Khristenko and S.L. Vetchinkin, Opt. Spektrosk.40 (1976) 417 [Opt. Spectrosc. 40(1976)239].

[229] Y. Jastum and A. Maquet, J. Phys. BlO (1977) L287. P.M. Koch, L.D. Gardner and J.E. Bayfield, in: Beam-foil spectroscopy, eds. D.J. Pegg and l.A. Sellin (Plenum, New York, 1976) Vol. 2. [231] D.M. Larsen and N. Bloembergen, Opt. Commun. 17 (1976) 254. [232] J.S. Bakos, A. Kiss, L. Szabo and M. Tendler, Phys. Lett. A39 (1972) 283.

1230]

I.E. Bayfield, Excited atomic and molecular states in strong electromagnetic fields [233] [234] [235] [236] [237] [238] [239] [240] [241] [242]

[2431 [244] [245] [246] [247] [248] [2491 [250] [251] [252] [2531 [254]

[255] [256] [257] [258]

[259] [260] [261] [262] [263] [264] [265] [266] [267] [268] [269] [270] [271] [272] [273] [274] [275] [276]

[277] [278] [279] [280] [281] [282] [283] [284] [285] [286] [287] [288]

391

J.J. Wendoloski and W.P. Reinhardt, Phys. Rev. A17 (1978) 195. J.S. Bakos, A. Kiss, M~L.Nagaeva and V.G. Ovchihnikov, Fiika Plasma 1 (1975) 693. D.T. Alimov and N.B. Delone, Zh. Eksp. Teor. Fiz. 70 (1976) 29 [Soy. Phys.-JETP 43 (1976) 15]. P.M. Koch, Abstracts Int. Conf. on Multiphoton processes, Rochester, N.Y., June 1977, pp. 72—73. G.J. Pert, J. Phys. B8 (1975) L173. S. Geltman and M. Teague, J. Phys. B7 (1974) 422. J.I. Gersten and M. Mittleman, Phys. Rev. AlO (1974) 74. A.I. Voronin and A.A. Samokhin, Zh. Eksp. Teor. Fiz. 70 (1976) 9 [Soy.Phys.-JETP 43 (1976) 4]. F.I. Dalidchik and V.Z. Slonim, Zh. Eksp. Teor. Fiz. 70 (1976) 47 [Soy. Phys.-JETP 43 (1976) 25]. L.I. Gudzenko and S.!. Yakoylenko, Zh. Eksp. Teor. Fiz. 62 (1972) 1686 [Soy. Phys.-JETP 35 (1972) 877]. S.I. Yakovlenko, Zh. Eksp. Teor. Fir. 64 (1973) 2020 [Soy. Phys.-JETP 37 (1974) 1019]. N.F. Perel’man and V.A. Kovarskii, Zh. Eksp. Teor. Fiz. 63 (1972) 831 [Soy. Phys.-JETP 36 (1973) 436]. R.Z. Vitlina, A.V. Chaplik and M.V. Entin, Zh. Eksp. Teor. Fir. 67 (1974) 1667 [Soy. Phys.-JETP4O (1975) 829]. N.M. Kroll and K.M. Watson, Phys. Rev. A13 (1976) 1018. A.M.F. Lau, Phys. Rev. A13 (1976) 139. J.I. Gersten and M.H. Mittleman, J. Phys. B9 (1976) 383. M.H. Mittleman, Phys. Rev. A14 (1976) 586. A.M.F. Lau and C.K. Rhodes, Phys. Rev. A15 (1977) 1570. J.M. Yuan, T.F. George and F.J. McLafferty, Chem. Phys. Lett. 40 (1976) 163. J.C. Bellum and T.F. George, J. Chem. Phys. 68 (1978) 134. R.W. Falcone, W.R. Green, J.C. White, J.F. Young and S.E. Harris, Phys. Rev. A15 (1977) 1333. J.L. Carsten, A. Szdke and M.G. Raymer, Phys. Rev. A15 (1977) 1029. D.A. Copeland and C.L. Tang, J. Chem. Phys. 66 (1977) 5126. N.B. Delone and V.S. Livitsa, Soy.-J. Plasma Phys. 2(2) (1976) 198. M. Broyer, J. Vique and J.C. Lehmann, J. Chem. Phys. 64(1976)4793 and references therein. A.M.F. Lau and C.K. Rhodes, Phys. Rev. A15 (1977) 1570. J.-C. Gay,W. Schneider and A. Omont, Fourth mt. Cont. on Atomic physics, Abstr. of Contrib. Papers, eds. J. Kowalski and H.G. Weber (Univ. of Heidelberg, 1974) pp.546—548. A.I. Gurevich, Zh. Eksp. Teor. Fiz. 64 (1973) 1199 [Soy. Phys.-JETP 37 (1973)610]. M.V. Fedorov, O.V. Kudrevatova, V.P. Makarov and A.A. Somokhin, Opt. Commun. 13 (1975) 299. V.P. Makaroy and M.V. Fedoroy, Zh. Eksp. Teor. Fiz. 70 (1976) 1185 [Soy. Phys.-JETP 43 (1977) 615]. V.E. Merchant and N.R. Isenor, IEEE. J. Quantum Electron. QE-12 (1976) 603. D.P. Hodgkinson and J.S. Briggs, J. Phys. BlO (1977) 258. V.S. Letokhov and C.B. Moore, Soy. J. Quant. Electron. 6 (1976) 259. J.P. Aldridge, J.H. Birely, C.D. Cantrell and D.C. Cartwright, in: Laser photochemistry. Tunable lasers and other topics, eds. S.F. Jacobs, M. Sargent III, M.O. Scully and C.T. Walker (Addison-Wesley, New York, 1976). N. Bloembergen, C.D. Cantrell and D.M. Larsen, Proc. Nordfjord Conf. (Springer-Verlag, Berlin 1976). D. Banks and J.G. Leopold, J. Phys. Bll (1978) L5. D. Banks and J.G. Leopold, J. Phys. B1l (1978) 37. E.A. Power and S. Zienau, Phil. Trans. Roy. Soc. A251 (1959) 427. E.A. Power and T. Thjrunamachandran, Amer. J. Phys. (1978), to be published E.A. Power, in: Multiphoton processes, eds. P. Lambropoulos and J. Eberly (Wiley, 1978) p. 11. M.P. Silverman and F.M. Pipkin, J. Phys. B5 (1972) 1844. E.T. Jaynes and F.W. Cummings, Proc. IRE 51(1963) 89. C.R. Stroud, Phys. Rev. A3 (1972) 1044. E. Courtens and A. Szoke, Phys. Rev. A15 (1977) 1588. S. Ezekiel and F.Y. Wu, in: Multiphoton processes, eds. P. Lambropoulos and J. Eberly (Wiley, 1978) p. 145. H. Walter, in: Multiphoton processes, eds. P. Lambropoulos and J. Eberly (Wiley, 1978) p. 129. J.H. Eberly, Prog. in Optics 7 (1969) 359. J.R. Mowat, C.E. Johnson, H.A. Shugart and V.J. Ehlers, Phys. Rev. D3 (1971) 43. T.W.B. Kibble, A Salam and I. Strathdee, Nucl. Phys. B96 (1975) 255. P.L. Knight, Phys. Lett. 60A (1977) 182. R.J. Fonck, D.H. Tracy, D.C. Wright and F.S. Tomkins, Phys. Rev. Letters 40 (1978) 1366. S. Geltman, Ionization dynamics of a model atom in an electrostatic field, Preprint, 1978. D.L. Andrews, J. Phys. BlO (1977) L659. P.L. Knight, Opt. Comm. 22 (1977) 173. S. Wong and J.H. Eberly, Opt. Lett. 1(1977) 211. J. Brossel, in: Quantum optics and electronics, 1964 Les Houches lectures, eds. C. DeWitt, A. Blandin and C. CohenTannoudji (Gordon and Breach, 1965) pp. 284—5.