Microwave multiphoton processes in highly excited atoms

Prog. Quant. Electr., Vol. 6, pp. 219 243. Pergamon Press Ltd. 1980. I'rintcd in Great Britain

0079-6727/79/1001 0219505.00/0

U

MICROWAVE

MULTIPHOTON

IN HIGHLY

EXCITED

PROCESSES ATOMS

J. E. BAYFIELD Department of Physics and Astronomy, University of Pittsburgh, Pa 15260, U.S.A.

1. I N T R O D U C T I O N Novel experimental techniques employing atomic beams excited by laser beams now make possible new studies in the physics of atoms strongly perturbed by external electromagnetic fields. An example of this is the recent work with static electric fields, known for its familiar Stark splitting of excited atomic energy levels, and for field ionization via quantum mechanical tunneling. These two effects enter into the external field alteration of the real and imaginary parts of the complex eigenenergies for the atom "dressed" by the external field. The thermal beam experiments on sodium, m and the fast beam experiments on hydrogen, rE)have paralleled new approaches to the theory of strong static field effects. The latter include complex scaling, t3) semiclassical, t4) Pade approximate tS) and other approaches. These often begin with a picture that includes in the zeroth order Hamiltonian a part of the interaction with the external fieldJ ~ol For weak static electric fields it is known that the parabolic or Stark quantum numbers n, nl, n2, m characterize the eigenstates of the system. It is also known that the usual perturbation theory based upon field-free basis states and an expansion in powers of the field strength results in an asymptotic expansion, divergent after a term of some order in the field is reached. One must cut off the series at a point, which now is known to depend strongly upon the quantum numbers. The accuracy of the cut-off series therefore depends upon the quantum numbers; for established n~, n2, and m, the expansion parameter now seem not to be the characteristic field F, = < r 2 > ~ n - -~ atomic units, but rather a smaller characteristic field varying as n-4. f2~ For the more general problem of an atom in a strong oscillating field, the frequency e~ becomes an added physical parameter in the problem, greatly enriching the variety of observable phenomena. In particular, one expects that in the low-frequency limit the photon character of the electromagnetic radiation field should produce no dramatic effects, while in the high frequency limit, where the photon frequency can match atomic energy-level separations, quantum effects should dominate, except perhaps in some very strong field limit. The search is on for a list of useful parameters that characterize the response of the atom to the external field; parameters like the n - 4 F field ratio for the static-field case. As we shall see below, at least two such parameters have been tentatively identified, namely a field-induced dipole moment strength/~ and the Keldysh parameter 7. The square of~, is the ratio of the freeatom electron Coulomb binding energy to the free-electron oscillation energy in the external field. Nonhydrogenic highly-excited states do differ from hydrogenic ones, because of core effects arising from the other inactive electronsF 'a) However, the complications arising from the core are expected to matter less when the external field is strong enough for its interaction with the active electron to dominate over the core interactions. Thus, hydrogen is the prototype atom in strong-field atomic physics; for simplicity, the topics of this paper will be primarily discussed in terms of the work on hydrogen. Only occasional reference to work on multielectron atoms will be made. Optical and static-field phenomena in multielectron systems are discussed further in several recent reviews, tg'~°,a x) Perturbation theory calculations can be carried through for the oscillating field case, and are appropriate, for instance, for the recent three-photon ionization experiments on H(1 s).tl 2) Parallel experiments on the lower excited states (n = 7, 8, 9, 10) at the lower frequencies of the i.r. region have been pursued; t~3'~4) here, perturbation theory results for two- and threephoton ionization at 10/~m are already available. (~5) As co is further reduced, the use of perturbation theory is more questionable, especially for field strengths and atomic level separations where high-order photon absorption is required for significant field-induced 219 aPQE 6/4 - A

220

J.E. BAYFIELD

atomic excitation and ionization probabilities. The present case of microwave frequencies and highly excited states with 40 < n < 70 is well within the region of high-order photon absorption. The experiments that have been carried out to probe this region are three-fold in nature, in that they can primarily investigate: (1) the quasienergy levels of the dressed atom while in the field ;~16) (2) the multiphoton excitation probability for an induced transition between field-free states of different n ;tl 7) or (3) the multiphoton ionization probability, tl 8) For ~o between 5 and 10GHz, and for the above range in n, multiphoton excitation energetically involves photon absorption of order 2-20, while multiphoton ionization involves processes of apparent order 50-500. Dressed-atom studies, excitation and ionization are, of course, all linked together as different attributes of the atom's response to the field. Section 2 of this paper discusses the experimental techniques used heretofore to study microwave-induced strong-field phenomena in highly-excited atoms. Section 3 describes the results of the experiments, and Section 4 outlines the status of the various theoretical approaches to this new subject. 2. FAST A T O M I C

BEAM T E C H N I Q U E S

Past experiments on microwave multiphoton phenomena in Rydberg states have utilized laser excited fast atom beams. ~19'2°'21) Typical atom kinetic energies are in the keV region, corresponding to atom velocities of order 108 cmsec-1. The beam is passed through a microwave waveguide or cavity having beam entrance and exit holes. In principle, the range of atom velocities available covers 107-10 l° cm sec 1, while a practical range of microwave field region lengths is 1 m m - 1 m. Thus, exposure times of the atom to the field can be varied f r o m 10 - 5 t o 10 - 1 1 sec, the latter time being shorter than the field oscillation period. In practice, signal-to-background limitations have set a lower limit to observable transition probabilities of 10-4 ; combining this with the range of available exposure times gives a range of observable transition rates of 10-10tlsec - t. The high-rate region is most likely to exhibit unusual nonlinear effects such as physical saturation, and is therefore of special interest. There are other possible experimental approaches. Laser-excited atomic beams at thermal energies should in principle make studies possible at lower transition rates. Alternatively, laser-excitation of atoms in a low-pressure glass bulb can be used if the microwave field is pulsed to set the exposure time. The fast beam approach enjoys several advantages. One is relatively unimportant complications arising from transitions induced in the highly excited atom by the 300 K background thermal radiation. Such transitions are easily seen in experiments using thermal atoms, tz2'23) and can be troublesome. The speed of fast atoms can reduce the entire history of the individual atom under study to a time interval of 10-6 sec or less, which is short enough for thermal transition probabilities to be less than 1~, even for n as high as 50. Collisional disturbances in fast-atom experiments are also relatively weak. These are of two kinds, as a highly excited fast atom can collide either with a slow residual atom or molecule left within the vacuum system, or with another fast beam atom (excited or unexcited) traveling with a slightly different velocity. The latter "merged fast beams" type of collision has been studied, t24) Although low energy cross-sections for ionization, electron transfer and excitation transfer can have geometrical magnitudes n+~za2 approaching 10-8 cm 2, the low density p of fast beams keeps these collisional probabilities below 1~0, as long as beam particle fluxes are less than 1012 atoms/sec/cm 2 (p < 104 atoms cm-3). The collisions involving the residual gas molecules are of a high-energy type, in that the relative velocity of the colliding partners is much higher than the orbital velocity of the active electron in the highly excited atom. For fixed collision energy, the cross-sections for these processes scale roughly as 0.1 nZTta2, ~25) producing collisional transition probabilities at n = 60 of 0.3 ",, for a 10 8 Torr vacuum in a 3m-long atom flight-path. In deliberations upon possible collisions effects in strong field experiments, one should also keep in mind the possibility of "radiative collisions", i.e. collision processes that arise from photon absorption during the collision itself. ~9)

Microwave multiphoton processes in highly excited atoms

221 II

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FIG. 1. An apparatus employingcollinear fast atom and laser beamsin studies of highly-excitedatoms in strong external fields. Both 30keV and 400keV particle beams are available, along with both pulsed and c.w.laser beams. Various parts of the main beam line are electricallyisolated ("I")for fastion kinetic energylabeling(seetext). D.P. meansa diffusionpump.Johnston MMI and Bendix M-306 particle multipliers are among the fast-ion and fast-atom detectors used. In the fast-beam experiments performed to date, the observed particles have been the fast ions produced by microwave ionization of the highly excited fast atoms, rather than the electrons. The ions have so much momentum that their trajectories are not significantly altered by the microwave field ; calculated proton deflection angles are bounded by 0.03 mrad for a 40 V c m - 1, 1.5 G H z field, an angle 100 times less than the beam collimation angle. A major disadvantage of fast beams is the relatively large motional electric field F = v x B arising from the atom's motion through residual magnetic fields within the apparatus. Cancellation of the Earth's magnetic field is important for most experiments. As will become clear below, the fast-beams approach is unusually flexible, with laserexcited states of almost any kind of atom being accessible to study. To date, all strong external field experiments with laser-excited hydrogen or helium atoms have utilized fast beams. However, pulsed laser techniques for exciting thermal hydrogen atoms are being developed. (~2) Figure 1 is a schematic of the fast-beam laboratory apparatus at the University of Pittsburgh. Let us begin by tracing through the time history of a highly excited atom in this apparatus, concentrating on the atom's production and detection. Then we shall discuss those special features that correspond to the different types of specific experiments that can be carried out. The production of nearly monoenergetic fast neutral particle beams begins with an ion beam that is then charge-neutralized by ion-atom electron-capture collisions. At Pittsburgh, two sources of fast ions are available : one a 20 keV test stand, and the other a 400 keV Van de Graaff accelerator. Both units start with ion sources utilizing radio-frequency discharges. The best beams one can expect from such arrangements have 10-6 A intensities through two 1 cm-diameter collimation apertures, 3 m apart. Laboratory reference-frame kinetic-energy spreads are typically between 10 and 100eV, depending primarily upon ion-source design and mode of operation. An analyzing magnet is used to select a beam with the desired charge and mass from the multicomponent accelerator output beam. Let us take a 10 keV proton beam as an example. The protons are passed through a gas cell, possessing beam holes in its ends ; here the electron capture collisions occur, that convert about 30% of the proton beam into a neutral hydrogen-atom beam. An essential aspect of this conversion is that not all the hydrogen atoms are in the ground state ; there is a distribution of final states varying roughly as n-3 for n > 2. (26)

222

J . E . BAYFIELD

The laser-excited fast atom technique favored today utilizes c.w. COz laser excitation at wavelengths near 10/,tm of n = 10 atoms initially in the beam. (19) C.w.u.v. argon-ion laser excitation of the 2s state has also been used. (18.2o) For heavy gas-cell target particles (Ar, Xe, C 6 F 6 ) , about 5~o of the atoms are in the 2s state, whereas only 0.1 ~o have n = 10. However, the transition probabilities for exciting a level with a given high n are 100 times higher for n = 10 than for n = 2. Also, 200 W of single-line CO2 laser power is available, whereas only 0.5 W of 361 nm radiation can be obtained from large u.v. argon-ion lasers. Thus, signals can be as much as 1000 times more favorable in those CO2 laser experiments that utilize n = 10 atoms, independent of their other parabolic quantum numbers. I f a static electric field is used to split the n = 10 energy levels in Stark-state selected experiments, the advantage of the COz laser arrangement over the u.v. laser one is reduced by one or two orders of magnitude, depending upon which Stark sublevels are involved in the laser excitation process. We have noted one essential aspect of the fast-beam technique, namely that electron capture collisions are used to obtain an initial level of atomic excitation ; a far-u.v, laser is not needed. A second essential point is that the high atom-velocity produces a significant firstorder Doppler effect when the exciting laser beam is made collinear with the atom beam, as is done in Fig. 1. Doppler fine-tuning of the laser frequency in the rest frame of the atom is a key to laser excitation using fixed wavelength lasers. (2°) A static electric field for Stark-tuning the atomic energy levels into resonance with the laser frequency is an alternative approach. (1'2) Figure 2a shows a Doppler scan of 2s to high-n transitions for the u.v. laser case. Figure 2b shows observed ion beam energies for the excitation of hydrogen to states with various high values of n, compared with values E(n) calculated from the formula

v / \Av(.))

e(n) = 2 - ~ 1

Here v is the laser frequency, and h(Av(n)) is the difference in accurate ionization potentials for the two excited states involved in the laser transition. Table l lists experimental design parameters for the CO2 laser excitation technique. (~9) When the grating-tuned COa laser is used, the various wavelengths available make possible a continuous tuning of the effective laser frequency, since the fractional laser line-spacing is at most 0.3~o, and the Doppler tuning range is more than 0.5~o. The atom energy can be varied either by changing the ion acceleration voltage, or by varying a voltage applied to the electron-capture gas cell. The former option is awkward, in that the magnetic analyzer must be changed also. The gas cell in Fig. I can be operated at voltages up to at least 30 kV, resulting in large adjustable beam energy shifts, that occur because the incoming ion is accelerated by the applied voltage, while the outgoing neutral atom suffers no change in kinetic energy. The electric field that the outgoing n = 10 atoms pass through is strong enough to mix states with differing I and m, but not different n. Varying the gas-cell voltage does vary the mixing effects, but not enough to be significant in experiments carried out to date. All ions in the beam are removed after the gas cell by a strong abrupt lateral electrostatic deflection. TABLE 1. P a r a m e t e r s for p r e s e n t a n d possible f u t u r e s o u r c e s o f fast h i g h l y excited a t o m s H ° beam emittance Useful H " b e a m intensity H* beam energy spread T y p e o f ion s o u r c e C h a r g e e x c h a n g e gas H + - t o l l ° c o n v e r s i o n efficiency H (n - 10) f r a c t i o n in n e u t r a l b e a m H (n - 10) b e a m flux C O z laser p o w e r o u t p u t C O 2 laser b e a m l o s s - f a c t o r C O 2 laser " ' l i n e w i d t h " R e s o n a n c e D o p p l e r linewidth C02 laser n = 10 ~ n - 44 p u m p i n g efficiency H (n = 44) b e a m flux P r o t o n - t o - H (n - 44) c o n v e r s i o n efficiency

1.2 t n r a d c m 0.6 l~A < +20eV r.f. d i s c h a r g e xenon 0.07

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Microwave multiphoton processes in highly excited atoms

223

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Calculated beamenergy, keV FIG. 2. (a) Portions of a Doppler-tuning scan of the u.v. laser excitation of hydrogen atoms initially in the metastable 2s state. (b) A comparison of expected and measured hydrogen atom energies needed for Doppler tuning into resonance the laser excitation of high-lying levels. The data for CO2 laser excitation is for different laser lines: n = 44, P-22; n = 46, P-16; n = 47, P-14; n = 48, P-I 2; n - 49, P-10; and n = 50, P-8.

Let us now complete the tracing of the history of a fast beam particle in the apparatus of Fig. 1. The n = 10 atoms can be laser-excited at all points along the main beam axis when the Doppler-tuning is on-resonance and no atom-destroying external fields are present. In the different kinds of microwave experiments, the atoms pass through either one or two nonzero microwave field regions. The last of these is always adjusted to ionize the laser-excited atoms, producing "signal" fast protons in our example case. These protons are selected out of the beam by a parallel-plate electric field ion analyzer, and detected with an electron multiplier. There are, however, other fast protons made in the beam by ionizing collisions with residual gas molecules. Normally these collisions would occur along the entire 3 m neutral beam path; all the fast atoms contribute, not just the excited ones. This source of"background" fast protons is considerably reduced by a technique called atom kinetic-energy labeling. A positive voltage of 100 V or so is applied to the ionizing microwave field structure. Neutral atoms enter the structure, while protons come out. Thus the "signal" protons receive an increment in kinetic energy that protons produced outside the microwave field region do not receive. The parallel-plate ion energy analyzer (27) and a following retarding field "filter lens" energy analyze{ TM combine to separate out energy-labeled protons from all the others, reducing the background signal by two orders of magnitude. The signal is further enhanced by three orders of magnitude or more, by amplitude modulating the laser beam and selecting the signal by means of phase-sensitive detection. This detection method may be either analogue or digital, depending upon whether the electron multiplier is coupled to analogue

224

J.E. BA'CFIELD

electronics or fast-pulse electronics. Signal-to-noise ratios of 100 or 1000 : 1 are achievable. An easily observable and unavoidable background signal arises from CO2 laser photoionization of beam atoms with n > 11 while within the energy-labeled region. Using 5 W of COz power, c.w. signal count rates as high as 105 ions sec 1 have been achieved. The past microwave experiments can be divided into three categories" (1) Dressed-atom spectroscopy. (2) Measurement of multiphoton excitation probabilities (MPE). (3) Measurement of microwave ionization probabilities (MPI). The first two categories use two microwave field regions in tandem ; photon absorption in the first field does not necessarily lead to atom ionization in these cases. The third category of experiment uses one field only. The ionization experiments involve measurements of the relative intensity of the fast proton signal as a function of microwave field strength F, microwave frequency ~o, and atom principal quantum number n. Other quantities such as the angle between the microwave field and laser field polarization vectors, as well as the remaining parabolic quantum numbers for the atom, are variables not yet investigated. In an ionization (MPI) experiment, the signal is presumed to be dominated by spatially separated laser excitation and microwave ionization steps ; this stepwise situation also holds true, by definition, for category (2) M P E studies, but not for the spectroscopic experiments. Typically, M PI experiments have involved 2 m of laser excitation region, followed by 5 cm of microwave field region. In an excitation experiment, atoms are first laser-excited to a level with a selected high value of n, then further excited (or deexcited) by a first microwave field to a field-free level having some adjacent value of n, and finally ionized by passage through a second microwave field. The field strength in the first microwave region is somewhat less than or equal to that required for observable ionization rates, while the field strength in the second region is well above that needed for complete ionization of atoms having the relevant values of n. In the spectroscopy experiments the beam energy is not set to that for resonant one-photon laser excitation from n = 10 up to the desired high-lying state. Rather, the Doppler tuning is adjusted for a multiphoton resonance involving the simultaneous absorption of one laser photon and some number, k, of microwave photons. Such processes can, of course, only occur within a microwave field region; under these tuning conditions, laser excitation outside does not occur. After leaving the microwave region the beam will contain highly excited atoms in one or more levels, to be subsequently ionized in the second microwave field region. The next section of this paper discusses in turn the results of each type of experiment. Before starting this, we should mention some known experimental problems that make these experiments challenging. The principal difficulties arise from the extreme sensitivity of highorder photon absorption processes (as well as quantum tunneling phenomena) to the strength of the applied field. If the microwave field strength varies over the transverse spatial profile of the atom beam, then different atoms in the beam undergo transitions at a given rate for different values of input microwave power. The net result of this will be a slower variation with power of the overall observed transition probability than that for the uniform field case. The best past ionization and excitation experiments have used TMom o axial cavity microwave fields, with central transverse spatial field strength variations of about 0.5~o over the atom beam profile. A typical cavity would be 10.4 cm long, 2.3 cm in diameter, with 0.8 cm diameter entrance and exit holes for a 0.2 cm diameter atom beam. (Careful alignment of the apparatus is important !) Such a cavity provides a reasonable rectangular axial field profile (along the beam direction). The 1 m-long first microwave field region shown in Fig. 1 has an even better axial field profile, with the beam entrance and exit apertures both 1 cm in diameter. For frequency tunability, this particular field structure is a TElo mode waveguide, with a traveling microwave propagating along the atom beam direction ; it has been used primarily in the spectroscopy experiments. Great care is required in the development of such tunable-frequency microwave structures, so that they have no internal resonances of their own that make the strength of the field seen by the atoms vary nonlinearly with the apparent field strength measured elsewhere in the microwave system.

Microwave multiphoton processesin highly excitedatoms

225

An interesting question concerns the origin(s) of the widths of the Doppler-tuning resonances shown in Fig. 2a. These Doppler ion-energy widths of 50-100 eV correspond to apparent photon-energy widths of about 0.003 c m - 1. Estimates indicate that three effects may be making important contributions to the width. First, since all n = 10 sublevels can contribute to the resonance signal, the n = 10 fine structure separations (z9) amounting to a total of 158 MHz enter into the line width. Second, the spread in Doppler shifts due to the ion beam and laser beam angular spreads (nominally about 1 mrad each) is not insignificant. A third signal contribution may come from unresolved Stark level splitting, due to remaining stray and motional electric fields, believed at present to have strengths perhaps as large as 0.1 V cm. All three of these partial width factors can be reduced. In particular, the n = 10 fine structure problem can be greatly reduced by exciting the atoms while in a strong static field with resolved Stark-state a-esonances/2) One can then hope to adiabatically reduce the static field strength to low values, if desired. In fast beam spectroscopy the inherent resolution limit due to ion-source energy spreads can be surprisingly good; this is because the ion acceleration process compresses the axial velocity spread that enters into the parallel-beam Doppler-spread term for the collinear double-beam type of experiment33°'3 ~,3z) 3. E X P E R I M E N T A L

OBSERVATIONS

3.1 Studies of Microwave Photon Absorption Let us consider a highly excited hydrogen atom in a state of principal quantum number n near 50, exposed to a linearly-polarized microwave field. Let the microwave frequency 09 be considerably less than the energy splitting A E , - - n -3 atomic units, between levels of adjacent n. As the microwave field peak-strength F is turned on and increased, a dipole moment # is dynamically induced by field mixing of the nearly degenerate sublevels of different nx, nz and m. The interaction of the first-order moment p with the oscillating field F cos (090 produces a nonstationary atomic state, that is expected in the low-frequency limit to adiabatically-oscillate periodically in time with a phase factor q~oso - exp [ - i#F/(hco) sin(~ot)] that satisfies the partial Schrodinger equation [#" F cos (¢ot)]qJoso = ih?~oso/~t. Insofar as nonadiabatic effects may occur when the field strength oscillates through zero, the actual atomic wavefunction may involve a time-dependent linear combination of q~osoS, having different values for the parameter #. For each contributing value of #, an experiment measuring the spectrum of the atom while dressed in the field will yield the squares of the amplitudes within the Fourier decomposition of qJoso : +oso

=-

.

After adding in the phase factor for the field-free atomic electron's binding energy E, - 1/(2n 2) atomic units, we see that the photon absorption is expected to result in dressed atom spectral lines with "quasienergy components ''(9,33,34) having values E. +_kho~, - c o < k < ~ . The relative intensity of a line is given by

Ik" = ~ j2 (#i(n)F'~ ,

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Figure 3 shows such a dressed hydrogen atom spectrum measured for 09 = 7.829 GHz, for CO2 laser excitation from n = 10 using the P-22 line at 1045.022 c m - x, and for a microwave field strength F = 13 V c m - 1.,6) The lines are labeled by their values of k and n, calculated assuming no large resonance frequency shifts. As explained in the last section, this experimental spectrum is obtained by fast atom-beam Doppler-tuning; laser frequency in the rest frame of the atom is varied by changing the atom kinetic energy. Seen in the Figure are the expected series of satellite lines of different orders k, located at dressed atom energy values

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number n exposed to a relatively weak microwave field. The quantity k is the photon replica order number.

E, +__khco. These satellites are often called photon replicas. Shown are some replicas with I k I < 5 for n = 43, 44, and 45. At the chosen value of the field strength F, the replica intensities decrease with increasing k. The chosen microwave frequency co can be termed nonresonant, in that the k = - 5 replica of n = 44 does not coincide with the k = 5 replica of n = 45 ; similarly for possibly overlapping replicas ofn = 44 and 43. If two replicas of adjacent n were to coincide, then there would be a resonance condition for a multiphoton excitation or deexcitation process, involving the field-free levels having those values of n. The details of dressed atom spectra can be studied as a function of the primary parameters of the problem, namely n, co and F. Important features are the replica intensities, center frequencies and widths. For the data shown, the widths are experimental, as discussed in Section 2. Within experimental error, the center frequencies for the data in Fig. 3 lie at values expected in the absence of microwave-induced level shifts. Thus, here the intensities remain as the most easily studied quantities. Figures 4 and 5 show the variation of intensity with microwave power for k = 1 replicas, for two different values of n and co. In both cases, the intensity is seen to initially rise, peak, then fall, and finally remain fairly constant. The peaking is also observed for higher values of k, but at higher field values. This is what is predicted on the basis ofJkz (#F/hco) relative intensity expectations, assuming # to be dependent on n and co, but not upon F or k. Figure 6 essentially shows the field values Fk for some peaks plotted as a function of k, for uJ constant and n almost constant.Good quantitative agreement with expectations is seen. Figures 7 and 8 essentially show the variations of Fk with n and with o) respectively, k now being held constant. From these curves one infers that/~ is proportional to R z, and varies weakly with u~, or about as o~ 1/2 for the narrow range of frequencies 6-8 GHz. The intensity plateaus observed above the maxima in Figs 4 and 5 are not yet understood. The squared Bessel functions oscillate with increasing F, and have zeros not observed in the data. It appears that a linear combination of oscillating Stark orbitals is needed to explain the plateau regions. o c (./')

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Microwave m u l t i p h o t o n processes in highly excited a t o m s

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Figure 9 shows a dressed atom spectrum at a much higher field strength than was used for Fig. 3. The value F = 45 V c m - 1 now used corresponds to plateau region conditions for all replicas with ] k ] < 3. What is observed in Fig. 9 are many new replicas, having higher order k up to 15. A number of apparent replica-frequency power shifts are present. These have been studied as a function of microwave field strength. Some k = 2 and k = 3 shifts appear to be real microwave power shifts, in that no strong overlapping higher-order replicas are expected to interfere. This is not the case for the higher-order replicas. Experimental work with higherfrequency resolution is needed for replica-shift data to be easily interpreted. At this point let us digress briefly for a glimpse at the nature of future, more sophisticated, spectroscopy experiments than those just described. The approach in the past towards dealing with the atomic quantum numbers n l , nz, m has been to minimize stray electric fields i

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Microwave multiphoton processes in highlyexcited atoms

229

so as to work with collections of degenerate states. Indeed, it has been found that the phenomena observed in the spectroscopy experiments are not significantly altered by deliberately increased motional static fields, as long as their strength is less than 10~ of the peak microwave field strength. ~6) Yet it would be valuable if one knew the values of nl, n 2, m for the n = 10 atoms in the beam one is working with. It seems feasible to carry out multiphoton experiments with selected static-field Stark states; several laboratories are working in this direction. Figure 10 shows a calculated Stark-split n = 10 to n = 43 Doppler excitation spectrum, constructed assuming equally populated n = 10 sublevels for hydrogen. ~35) A 0.5 V c m - 1 static field is assumed, along with near perfect experimental resolution. Exact matrix elements coupling each possible (n = 10, n = 43) pair of sublevels were numerically computed ; the final spectrum sums over all possible transitions. Many of the lines in Fig. 10 have contributions from more than one pair of states. An enormous number of lines are predicted to have significant intensities. It is unlikely that resolution will be improved enough to directly observe such a spectrum. An ingenious optical double resonance (ODR) Stark-state technique has recently been devised to deal with the state-selection problem336) T w o CO2-1aser transitions are stepwise induced. The first is done in a strong static electric field of tens ofkV c m - 1, between one Stark sublevel ofn = 7 and one ofn -- 10. The static field strength is then reduced to a low value (by the atoms moving from the first field region to another one), and a second transition induced between the populated n = 10 sublevel and a desired high-n sublevel. The same laser beam can be used for both transitions, if the strengths of the two electric fields are appropriately adjusted, along with proper choices of laser-line and atom-beam energy. Nonadiabatic transitions between a laser-pumped level and another level in the same Stark manifold can occur as the static field is reduced; in practice, this is not too serious a problem, although nonadiabatic repopulation of the level with opposite value of n~ - n2 has been observed. (2) Figure 11 presents a theoretical state-selected photon replica spectrum for a microwave dressed-atom spectroscopy experiment. ~35~The transitions that are selected are:

n, nl, nz, m = 7, 0, 6, 0--* lO, O, 9, 0 ~ 4 8 ,

nx, n2, 0.

The atom energy is 12 keV, the laser frequency 1053.9253 cm - 1, the microwave frequency 09 = 7 GHz, and microwave field strength 15 V c m - ~. Only 3 W of laser power is assumed. The spectrum is presented as a function of the strength of the second static electric field. Reducing the atom beam energy by 1 keV lowers by about 2 V c m - ~ the strength of the second field required for a given spectral line; the strength of the first field would then also need readjustment. Stark-state-selected experiments may make possible high-resolution studies of microwave photon replica formation in the presence of an additional static electric field. The reduction in state degeneracy so achieved may lead to quite different results than those observed for entirely-degenerate excited state manifolds. 3.2 Multiphoton Bound-Bound Transitions The k = 1 photon replica for n = 48 and ~o = 8 G H z reaches its maximum for a relative field strength F/F, = 0.03, where F, ~ 1/(16n 4) is a characteristic field strength threshold for classical field ionization at the top of the peak instantaneous barrier. ~9) The region of field strengths F < 0.05Fc may be termed a perturbed-level region; the levels with different n, however, are weakly coupled, in that multiphoton excitation (MPE) has not been observed at these field strengths. Stated more carefully, even on resonance the M P E rates are less than 105 sec 1 for F < 0.1Ft. It is only when the field is further increased to values F/Fc > 0.2 that one observes signs of significant M P E effects in the two-field experiments about to be described. The region of observable M P E corresponds roughly to the region of on-resonance overlapping of field-saturated photon replicas; the latter region is roughly determined by the condition pF/ho9 = 0.5k', where k' is the order for An = 1 MPE. Strong M P E transitions occur in the field region for rapid ionization at F -~ 0.5F c.

230

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Excitation is observed in two-field experiments by varying the strength of the second field (the final ionizing one), with the strength and frequency of the first field also being stepped as parameters. (Ignoring the frequency of the second field is an oversimplification that has been made so far.) Thus, the quantity observed is the ionization threshold(s) for the atoms with different values of n p r o d u c e d by M P E in the first field. Doppler-tuning to a p h o t o n replica with k = 0 is helpful, here, as laser excitation outside the first field than makes no contribution to the signal. Figure 12 shows several threshold curves for the production of atoms with n near 44. (37) The curve marked k = 0 is for n = 44 Doppler-tuned directly into resonance ; for this curve we k n o w that the atoms being ionized have n = 44. The other two threshold curves are for D o p p l e r tuning to the k = 1 p h o t o n replica of n = 44, with the microwave frequency tuned for resonant M P E from n = 45. M P E from n = 44 to n = 43 is then nonresonant. The Figure shows that at the lower field strength the atoms produced have an ionization curve characteristic of n = 44, within experimental error. This is because the microwave power level of 0.05 W corresponds to a field strength far below that for the M P E region. A power of 1 W is, however, sufficient for M P E , as verified by the observed shift in ionization threshold to lower field values. The fractional shift in F to be expected can be estimated as 4/n ,.~ 10~o, if we take n - 4 to be the n-dependence of the ionization threshold; the observed shift in power is close to the expected value of 20%, and is in the right direction. It is interesting that the 1 W curve in Fig. 12 corresponds to the M P E population o f n = 45, with little signs of n = 44. Ionization curves that exhibit significant fractions of both n = 44 and n = 45 have also been obtained. It is possible to observe resonances in M P E transition rates, by varying the frequency of the first microwave field while holding all other variables constant. In particular, the strength of the first field must not vary with frequency by more than a percent or so. Figure 13 shows some experimental resonance data. (x7) The observation of such resonances depends u p o n

n=46

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I

232

J.E. BAYFIELD

setting the strengths of both fields to appropriate values within _ 5-10~o ; the n = 47 curve shown was probably taken with the first field a little too strong, thereby experimentally saturating the M P E transition both on and off resonance. It must be remembered that a resonance curve with fields set to detect An -- 1 transitions will also contain contributions from An > 1 transitions. The frequencies and widths of the M P E resonances provide important information, as do the magnitudes for the M P E transition rates. If the resonances in Fig. 13 are interpreted as arising from An = + 1 transitions (narrowing the "window" between the strengths of the two fields did not uncover other resonance frequencies), then the M PE resonant frequencies have apparent microwave power shifts of about 50k' MHz, where k' is the order of the M P E transition. A number of factors contribute to the widths of the observed M P E resonances. One, of course, is microwave "power broadening". The M P E (and MPI) transition rates can be determined from atom velocity, field region length, and observations of experimental saturation of signals at high enough field strengths. These rates were both about 100 M H z for the data of Fig. 13. Thus, the lifetime of the laser-excited atoms while in the microwave field was limited to the order of 10 s sec, giving a 100 M H z component to the resonance width. However, the data shown was taken with the first field produced within a short field region ; the excited atoms were exposed to a Gaussian-like field strength variation with time. Under these conditions the actual transition region is not the geometrical length of the waveguide along the beam direction, but a shorter length near the top of the Gaussian curve. 138'39) Thus, only the central 20 field oscillations are calculated to be important, rather than the full 100 oscillations; this leads to expected resonance widths of 500MHz, rather than 100MHz. Values near 500 M H z are those observed. New studies with the long waveguide region shown in Fig. 1 should not be limited by the effects of Gaussian-like pulse envelopes. 3.3. Multiphoton Ionization of Hi(4hly Excited Atoms The two extremely different field ionization mechanisms of nonresonant multiphoton ionization (M PI) and quantal tunneling are described schematically in terms of their effective potentials in Fig. 14. Usual M P I is discussed in terms of the field-free atom electronic binding potential, while tunneling views the potential as distorted by the external field interaction. The microwave ionization of highly excited hydrogen (45 < n < 57) for frequencies between 9.4 and l l . 6 G H z is very high order, 88 < ko < 173, when viewed as MPI. Certainly, the possibility of quantal tunneling effects or other "distorted potential" effects cannot be disregarded when the field oscillation frequency is not very different from atomic electron orbiting frequencies. We now review the results of ionization experiments to see what can be said about these matters. If the microwave ionization were MPI, and if the possibility of saturated M P E effects were ignored, then a field strength dependence F 2k° for the ionization rate W~(F)would produce an ionization probability curve very much like a step-function indeed. Figure 15a shows W~(F) for CO2 laser-excited n = 46 and n -- 47 states ; a TMolo cavity was used for ionization, with Energy

states

Multi photon ionizotion

I

Quontol tunneling

FIG. 14. Schematic pictures for non-resonant M P I and for static-field quantal tunneling, In MPI the external field produces photon absorption, whereas in tunneling the field distorts the electron's potential energy curve.

MicrOwave multiphoton processes in highly excited a t o m s

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the field inhomogeneity across the atom beam profile only 0.59/0. The curves are rapidly varying, rising two orders in magnitude for a 109/o increase in microwave field strength. The apparent order kerr(also called effective multiplicity) for these curves is 15 or 20, far less than ko. The saturated n = 47 signal is bigger than the n = 46 one, as a stronger CO2 laser line was used. The n = 47 atoms are ionized at weaker fields, as expected. Figure 15b shows ionization curves for two quite different values of n, namely 40 and 50. However, this data was taken with poor field nonuniformity ;(~8) the n = 50 data shown was worse in this respect than the n = 40 data. However, this Figure does demonstrate that the microwave ionization thresholds do vary strongly with n, more strongly than the n - 4 dependence of static field tunneling. If a dependence F ( n ) = n -~ is assumed, then ~ seems to be between 10 and 20, with the uncertainty arising from experimental difficulties in converting microwave-system inputpowers into field strengths seen by the atoms. All-in-all, the few studies of W~(F) done to date suggest that neither nonresonant M P I nor unmodified quantal tunneling is the mechanism underlying microwave field ionization. This is not surprising, in view of the nature of the basic photon absorption process uncovered in the spectroscopy experiments, and in view of the observation of large M P E rates at fields comparable to those for ionization. A definitive ruling-out of simple nonresonant ionization mechanisms comes from the direct observation of resonances in M P I transition rates. (~7) Figure 16 shows some of these resonances, as the frequency is varied from 9.6 to 11.4 GHz. The error bars on each data point set limits on signal variations arising from known resonances in the tunable microwave system ; these variations were measured by taking data over a much finer mesh of frequency values. The signal is expected to be exceedingly sensitive to frequency-dependent imperfections in the microwave system, in view of the field dependences observed in Fig. 15. There is one technique that establishes the resonances of Fig. 16 as really due to atomic structure enhancement of the ionization rate. What is done is to take data for adjacent values of n, while

234

J.E. BAYFIELD n:58

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F[(;. 16. Resonances in the microwave M PI of atoms excited from the 2s state. Similar data is obtained using CO 2 laser excitation from the n = 10 state. leaving the microwave system unaltered; this is possible for appropriate ionizing field strengths. The only variables changed in going from the n = 51 to n = 52 curves in Fig. 16 are the C O 2 laser line wavelength and the a t o m energy. It is very difficult to see how changing these variables slightly can produce the observed changes, except t h r o u g h actual changes in the value of n for the atoms. Figure 17 makes a comparison of M P E and M P I resonance curves, with ionization data for a given n presented above the corresponding excitation data. The resonances in the rates for the two processes do not lie at identical frequencies. Figure 18 shows extended, ratecalibrated data for n = 48, with ionization (b) now below excitation (a). M a r k e d in the excitation diagram are calculated field-free a t o m frequencies for An = 1 and An = 2 M PE. The ionization resonance frequency is half-way between the k = 5 and k = 6 M P E , An = 1, resonance frequencies. This is expected for ionization proceeding t h r o u g h a ladder of resonant intermediate states in an a n h a r m o n i c q u a n t u m system. (4°) F o r the present excited h y d r o g e n - a t o m case, the effective local a n h a r m o n i t y is A E , / E , = 4 / n ~ 8%, in close agreement with the observed fractional "red shift ''(41) in the ionization resonance frequency of 10%. A s u m m a r y of basic characteristics for the M P I process is given in Table 2. This table can be referred to when one reads Section 4, where we shall see how the various theoretical approaches to the M P I problem relate to the experimental data. An interesting picture of M P I emerges from the experiments performed to date. It is viewed as M P E resonance-enhanced p h o t o n absorption to near the top of the peak instantaneous barrier in the oscillating effective-potential for the tunneling problem (see the right hand side of Fig. 14). Actually, what we do is superimpose the effects of the two sides of Fig. 14, to obtain the picture of Fig. 19, taken at a time when the field oscillation is at its maximum. ~39) If such a picture is to possibly be valid, the orbiting atomic electron must have time to at least partially follow the field oscillations; this requires that (o f=

1 _v.

< 27r = T, a,

1 n3

= AE.

where v, and a, are the orbital velocity and radius, respectively. Thus, one must be in or near the M P E region of applied frequencies, which is the present situation. Crude estimates can be made for the height H of the p h o t o n absorption chain to the top of the barrier. In atomic units one has the barrier condition 1/2n 2 + Hco = 2 ~ F ~

Microwave multiphoton processes in highly excited atoms

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to be solved simultaneously with a photon absorption probability condition. If photon replicas near the top of the barrier are near their maximal intensities, then their relative intensity is approximately the maximal value for the square of the Hth-order Bessel function

This can be equated to the measured probability Pi for nearly-saturated ionization at the field value Fi, per cycle of field oscillation. The quantity Pi is typically 10 3. The two conditions are reasonably satisfied for H from 10 to 30. It is interesting that this picture predicts that the signal protons are actually produced in short bursts, repeating at the microwave frequency. The picture can clearly encompass resonance enhancement and the other observations listed in Table 2. As we shall see in the next Section, a theoretical basis for combined MP1 and potential distortion does exist, although no numerical calculations have been carried out yet. 4. T H E O R E T I C A L IONIZATION

APPROACHES OF HIGHLY

TO THE EXCITED

MICROWAVE ATOMS

4.1 Introduction Table 3 lists the different theoretical studies that bear upon one or more aspects of our microwave-field ionization problem. The three calculations employing a 6-function model of the atom pertain to those aspects of ionization that arise from the strong coupling of a single J P Q E 6 1 4 - E)

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bound electronic state with the electronic continuum states. Usual stationary-state perturbation theory may account for resonant intermediate state effects, at least for low field strengths. Alternative dressed-atom forms of perturbation theory begin not with field-free states, but rather oscillating Stark orbitals that are not stationary. The latter general approach is also often called quasienergy state theory; (9'33) its ideas were used in the interpretation of the dressed-atom spectroscopy data given in Section 3. Because of the large amounts of photon absorption involved, a quasienergy-state statistical model has recently been proposed ; this approach is based upon the notion of an essentially uniform distribution of (quasi) energies available for the electron to "diffuse" through while on its energyabsorbing way towards the continuum. Lastly, there are interesting classical Monte Carlo results. Table 4 lists some simple parameters for classical electron motion in the free-atom and oscillating electron limits to the general problem of an electron exposed to both Coulomb and external oscillating fields. Ratios can be made of the limiting values of each of the characteristic quantities, time, electron displacement, electron velocity or kinetic energy, and field strength. Most of the theories listed in Table 3 find at least some of these ratios to be important parameters. For n = 50 hydrogen in a 200 V cm ~, 1 GHz field, the ratios have TABLE 2. Characteristics of microwave ionization of highly excited hydrogen atoms (n ~ 4 0 , f = 10 G Hz) (1)

For a given transition probability, the peak field needed is a fraction of the static field value.

(2)

The dependence of the transition probability on field strength appears to be much weaker than that predicted by the perturbation theory for nonresonant multiphoton ionization.

(3)

Resonance structure exists in the frequency dependence of the ionization probability, with a dependence on atomic principal quantum number characteristic of the energy levels of the atom.

(4)

Multiphoton bound bound transitions are driven by the microwave field, with rates on resonance that are larger than the ionization rate at the same frequency.

Microwave multiphoton processes in highly excited atoms

237

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values T/T, = 10, fix/a. = 0.1, rio/v, = 1, a n d F/F. = 0.02. G e n e r a l l y speaking, the p h e n o m e n a being discussed in this p a p e r occur for F/F. < 10 -2, a n d for the o t h e r three p a r a m e t e r s , between 0.1 a n d 10. F o r c o m p a r i s o n , the g r o u n d state of h y d r o g e n , e x p o s e d to very high p o w e r 1.06 # m laser r a d i a t i o n ( F = 108 V c m - t), will have the following, similar, values for these p a r a m e t e r s : T/T,---20, fix/a, = 3, rio/v. = 2, a n d F/F, = 0.02. O n e a d d i t i o n a l p a r a m e t e r t h a t is quite different for the g r o u n d state a n d highly excited-state cases is the f r e q u e n c y - d e p e n d e n t low-field d y n a m i c polarizability,(51) which is small a n d second o r d e r for H(ls), while large a n d first o r d e r for the degenerate highly-excited levels. Before discussing specific theories, let us list in a qualitative w a y those features of o u r p r o b l e m that have been p r e d i c t e d by one t h e o r y or another.

TABLE3. Theoretical approaches to the microwave ionization of Rydberg atoms (1) One-dimensional 6-function atom, quantum mechanical: Geltman (1977)t42) (linear polarization). (2) Three-dimensional f-function model atom, quantum mechanical : Manakov and Rapoport (1975),t43~Berson (1975),(44~(circular polarization). (3) Three-dimensional fi-function model atom, semiclassical: Perelomov and Popov (1967),¢4~ Nikishov and Ritus (t967).(46) (4)

Stationary-state perturbation theory: Not yet done for large n.

(5)

Quasienergy-state perturbation theory: Kovarski and Perel'man (1971).t47) Quasienergy-state statistical model : Delone, Krainov and Zon (1978).t49)

(6)

(7) Classical model atom, Monte Carlo : Leopold and Percival (1978).~5°~

238

J. E. BAYFIELD TABLE 4. Classical parameters H(n) a t o m i c electron, n o field

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T~ = 3rcn3(2.4 × 10 -17 sec)

Displacement

a. = 3n2(5.3 x 10 -9 cm)

Free electron in field T

=

fix

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V~= n- l(2 x l0 s cm sec- x)

Field strength

F n = n - 3 ( 5 x 109 V cm -1)

Parameters

1

Z/Tn

eF mto2

fix~an

(/)

eF

fi~ -

me) F

fiv/v. ~ 7-1 F/F.

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atomic units ;

perhaps a better parameter is (Tko)- 1 = n F / ( k o o ) ) , where ko is the multiplicity threshold for MPI. (2) The region in (n, e), F)-space for observable nonresonant MPI is determined by y < 1. (3) A complete theory of nonresonant field ionization should have a quantum tunneling static field limit when ~ --* 0. (4) When y < 1 there is an effective lowering of the edge of the continuum, due to photonabsorption distortion of the continuum levels, imposition of the instantaneous barrier in the effective potential, or the like; different theoretical pictures view this in different ways. The apparent multiplicity for the single-bound-state problem becomes keff---ko - n ( 1 + c0, where c~ < 1 is a field-dependent correction. (5) The multiplicity threshold ko actually increases with field strength, because additional photon absorption is necessary to leave the final state electron oscillating in the field with a mean kinetic energy (1/2)fl~ : ko =

1+

+ 1

I, = 2 ~ a t o m i c units.

This effect produces steps in k o ( F ) as the threshold requirement for each additionally-needed absorbed photon is reached in turn. (6) As the field oscillation frequency co is increased from zero, the nonresonant MPI rate (for n and F fixed) increases, due to a narrowing of the effective semiclassical barrier. The barrier width Ar, in units of n a o , is na o

e)

1 +(1 +y2)1/2 •

For 7 << 1, this becomes F , / F as expected; in the other limit, 7 >> 1, this becomes 2 I , / ~ o , a quantity independent of F. (7) For a real atom with its infinity of excited states, the field is expected to distort boundstate wavefunctions and produce photon replicas, Stark shifts and a.c.-field powerbroadening. These effect the details of the ionization rate through intermediate state resonances. Such resonances can saturate at high field strengths, reducing keg. The introduction of the concept of an effective barrier in the oscillating field problem is justified through the development of time-dependent semiclassical approximations, working within the complex time plane. ~Sa)Time is imaginary under the instantaneous barrier. Then the electron's trajectory becomes complex, while still satisfying Newton's equations of motion at all complex times. This approach gives near-analytic results when the complex trajectory is that determined assuming the atomic binding potential is the radial b-function 6(r). ~45~Some numerical work solving the sub-barrier trajectory equations for V ( r ) = r - ~ has resulted in Coulomb corrections to the results of the b-function model, t54) For the simple one-dimensional b-function model, purely quantal calculations have been made t42'43) for

Microwavemultiphoton processesin highly excitedatoms

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comparison with the semiclassical results; there is a reasonable amount of qualitative agreement among thesd calculations. The semiclassical result for the ionization rate is a sum over all energetically-allowed multiphoton processes of increasing order above threshold:

W.,m= ~ w&(v, co,~.) k=k o

1 C z/'6"~1'2 (2l + 1)(/+ Im[)! ( F -2,

\,s,+3/2

"'

+

x As(co, 7)exp ( -- 2 F.

"~

The function g(7) is

(1 0 (7) =

1+

sinh- ~ 7 -

+

~,2)1/2~ 27

J

1 - 72/10, 7 << 1 -+ Z~(ln 27 - ~ ) , 7 >> 1

27

and is plotted in Fig. 20. The relative amplitudes A s are given by

A,,,(CO,7) =

4 1 ?2 (3rt)1/2 Iml! 1 + 72 k==k~oexp { - 2 [ s i n h - ~ 7 --(7/(1 + 72)1/2)~(k - k o ) } x cos

I(

27 (1 + 72) 1 , 2 ( k -

k°))'1/27J

where

co,,(x) -

xZt,,i +1 f£ exp ( - x a z ) z I"'1 2

.~

(1 7 r)l/~-

•

The rate W,~= varies strongly with the component m of angular momentum along the direction of the linearly polarized oscillating field. The term with m = 0 is largest. For unpolarized atoms, one averages over m to obtain 1

C

2//6"11/2/ F

1

lTV,,t=~n21 ,,,[ ~_~) ~(+y2)x,2)

\3/2

/

2F.

"~

A,,o(co,~;)expl__~ffO(,)}.

Figure 21 shows the frequency dependence of the factor before the exponential in this last expression, with the field strength held constant.

240

J.E. BAYFIELD

(I+), z)3/zAo (~,)~) /

OJn+ 2 COn+ I con

COn_ I

co

FIG. 21. Expected threshold effects in the nonresonant photon absorption required for multiphoton ionization of a bound state. As the frequency is lowered, the minimum number of photons required for ionization increases.

A p p r o x i m a t e results have been obtained for the "usual perturbation theory" region :

forT/T. >> 7 >> 1,

W., =

Ae)~)

exp(IJe))

,

k = k o - n(1 + c0;

and for the "nearly static field" region : for 7 << 1,

W.~ = Wsta,cexPL10\3 F J J ~ exp

- 5~(1

- y2/10) .

The semiclassical formula for ~sstatic obtained using parabolic coordinates is characteristic of quantal tunneling in the absence of interactions between different b o u n d states :(5 5)

Wsstatic =

(n2 + m)!nz!\F,J

exp

- ~].

The semiclassical results do not directly apply to the problem of microwave absorption within the ladder of degenerate atomic excited states ; they do, however, give one some feeling a b o u t the interaction with the c o n t i n u u m ; this feeling is summarized in points (1) through (6) above. A general formalism for M P I has been given that directly incorporates the degeneratestate p h o t o n a b s o r p t i o n m e c h a n i s m ) 47) Quasienergy basis states--here, hydrogen a t o m oscillating Stark o r b i t a l s - - a r e introduced at the outset. These O S O ' s are defined by

~:(r, t) =-~ CJ.i~O°i(r)exp( - ~i E,t)exp(- ipi sino~t) where the C~i are weighting coefficients characteristic of the coupling of spherical states ff°i(r) to form Stark states, and the P7 are solutions of the secular equation II ha~pT,~k -- VjT,II = 0 where

~k------ <~'°jl~'FI

o

As an example, the two possible n = 2 0 S O ' s would be

~E2t + ~ieF(2s [z [2p> sine~t ) .

1 o ( 1 Wzl'2(r, t) = ~ ( ~ 2 s -4- ~,°p)exp -

This O S O formalism then proceeds by introducing the O S O Green's function Go, divided into b o u n d - O S O and c o n t i n u u m - O S O parts: Go - Gon + Goc

Microwavemultiphoton processesin highly excitedatoms

241

where the bound state part is, for instance, i i* G~ = --i @ (t I -- t2)EtPn(rl, t l ) ~ , (r2. t2) n.i

---i(~)(tl

-

-

t2)~,ZC.j C.k ~.j(r o l )~b.k(r o 2) n,ij,k

× exp(~-~En(tl--ta)--ip'~(sincotl

- sincot2)).

At this point, the expected Fourier decomposition is made :

exp(--ipTsin~ot) =

~

Jm(pT)exp(imo~O.

m = - c~

A perturbation theory expansion is then made in powers of the coupling between OSO's induced by the oscillating'field; this divides the atom's interaction with the field into two parts. One type of photon absorption enters into the time-development of the OSO's, and does not lead to M P E or MPI ; the second type of photon absorption is that additionally needed for actual transitions to take place. At high field strengths there is a large amount of photon replica formation, i.e. photon absorption of the first type. Under these conditions, M P E and/or MPI may be of low order in photon absorption of the second type. The final M PI rate involves a sum of amplitudes over all possible orders, for both types of photon absorption. The important terms are those having highest order in the first type of absorption, while still having their Bessel functions close to saturation; these terms require lowest order absorption of the second type. Quantum tunneling is contained in the zeroth order term for photon absorption of the second type. Needless to say, numerical calculations based along the lines of this OSO analysis would be of great interest, but have not yet been carried out. An even greater emphasis on the degenerate-state photon absorption mechanism is made in the diffusion model for the microwave ionization process. ~49) In this model, no effective barrier is introduced in the electron's potential ; the edge of the continuum is left at its fieldfree energy value. It is assumed that the density and widths of the photon replicas contained in the dressed-atom spectrum are both so large that the replicas form a continuum of uniform density. Ionization is assumed to occur rapidly when the peak Stark splitting, #F ~ nZF, is comparable to the splitting between levels AE, = n 3 ; thus, the threshold field for diffusion would be For ~ n s. At, or above, this value the atomic electron is presumed to diffuse or drift around statistically among the quasi-continuum of quasienergy (photon replica) states, until it energetically reaches the continuum. The latter occurs only after a great deal of photon absorption, for the probabilities of photon absorption and induced radiation are almost the same. Once in the continuum region, a one-photon transition (of type two in the theory just discussed) leaves the electron permanently above the flee-atom ionization limit. Because the diffusion is statistical, intermediate-state resonance effects are predicted to be weak and quasistatistical. The diffusion time is calculated to be

z ~ 1/(nXlmzf 2) and for shorter times t << T the MPI transition probability is approximately Pi(t)

~

(t/z) U2 ~

coFn 11/2tl/2"

The principal problem with the diffusion model seems to be the weak F- or FE-dependence of the probability on field strength ; this arises basically from the assumption of effectivelysaturated photon absorption of type one dominating over processes involving two-or-more photon absorption of type two. The observed dependence of ionization rate on field strength is much stronger (kerrbetween 7 and 20) (see Fig. 15). Thus, the diffusion model seems to be an oversimplification. In addition, recent experiments on photon replica formation in hydrogen indicate only a weak interaction between OSO's of differing values of n. t16) Thus, the

242

J.E. BAYFIELD

threshold field for diffusion is better estimated (in hydrogen) by /~F ~ E, = 1/(2n2), or F ~ 0.5n 4 a factor of 25 higher than n-5 for n = 50. However, for nonhydrogenic systems, core effects are known to make Stark manifold much more interacting than in hydrogen ;~7) then diffusion may possibly set in at F ~ n - 5. This would explain the weak field dependence observed in the microwave field ionization of the (ls41s)3S state in heliumJ 56) Another theoretical approach based upon really-large amounts of photon absorption is a purely classical one. ~5°) Monte Carlo calculations are made of the possible electron trajectories for the electron in combined Coulomb and oscillating fields, using a classical microcanonical distribution of initial electron conditions that corresponds to a quantum distribution having equally populated Stark states (nl, he, m), for a given n. The linearly polarized microwave field is switched on and off according to F(t) = A(t)F cosogt

with exp [2(t -- tl)], A(t) =

1,

exp[--2(t--tf)],

0 < t <_ tl tl < t < t f ty < t < T.

After any given time T, the collected trajectories fall into four categories : (1) Trajectories on invariant tori, which probably never ionize. (2) Trajectories with rapid ionization, in a few field oscillations. (3) Trajectories passing through one or more very highly excited states (total energy corresponding to nI > 5nl), making relatively sudden transitions between those states before ionizing. (4) Trajectories which reach such very highly excited states, but not producing ionization at the final time T. That M P E and M P I are both significant and competing phenomena is consistent with the data. An interesting result of the Monte Carlo work is that the electron can remain "bound" (i.e. stay near the nucleus) for the time T, even though the field strength is so high that its total energy oscillates as a function of time by m a n y times the ionization energy. Under these conditions, a compensated electronic energy Ec is found to be a more meaningful measure of orbit stability than is the total energy. The quantity Ec is defined by 1 2 + vy2 + (vz - fl,,sin~t) 2] Ec = r 1 + ~[vx i.e. the oscillation velocity of the electron is removed from the total kinetic energy term in the expression for the total energy. It remains to superimpose such interesting classical effects with the quantum phenomena, including true M P E between the quantum energy levels of the real atom. This has been investigated for the anharmonic oscillator problem3 5~)

Acknowledgements The author wishes to thank R. Ove and P. M. Koch for helpful ideas on the development of experiments with atoms in individual Stark states.

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