Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218
Received September 12, 1989; accepted November 11, 1989
Above-threshold ionization (ATI) by intense, short-pulse lasers is studied numerically, using the stretched hydro-gen atom Hamiltonian. Within our model system, we isolate several mechanisms that contribute to the ATIprocess. These mechanisms, which involve both excited bound states and continuum states, all invoke intermedi-ate, off-energy shell transitions. In particular, the importance of excited bound states and off-energy shell bound-free processes to the ionization mechanism are shown to relate to a simple physical criterion. These processes pointto importance differences in the interpretation of ionization characteristics for short pulses from that for longerpulses. Our analysis concludes that although components of ATI admit of simple, few-state modeling, the ultimatesynthesis points to a highly complex mechanism.
INTRODUCTION
The phenomenon of above-threshold ionization (ATI) hasreceived considerable attention over the past few years.",2
The inherent complexity of the process has been revealed byexperiments that have characterized the electron spectra ingreat detail.3 Theoretical analysis, on the other hand, hasbeen hampered by the apparent breakdown of conventionaltechniques in low-field theories of multiphoton physics.The development of new, in particular nonperturbative,techniques has been the primary thrust of recent theoreticaleffort.
The past year has also seen a proliferation of simple nu-merical models designed to aid in understanding the basicnature of ATI.4 In large part, the impetus has come fromexperiments on time scales that are numerically tractable.Several models of varying complexity have been consideredthat incorporate some, but not all, aspects of the real atomicsystem. Each of these has specific advantages and draw-backs, but all have contributed to added insight into theproblem. In particular, these models have proved useful forevaluation of the relative efficacy of analytic approxima-tions attempted in describing ATI. However, the most sur-prising result of these numerical experiments has been thatmost qualitative features of the observed ATI spectra arereproduced with almost any simple model. The differencesamong the various models have been primarily in the de-scription of details of the atomic structure, and a simpleinference might be the absence of any dependence of ATI onspecific atomic structure. This conclusion concurs with the-ories that involve only free-electron dynamics in intensefields.5 The alternative view is that ATI is, in fact, a highlycomplex process and that each simple model describes only apart of the entire mechanism. This is the viewpoint that thepresent paper attempts to illustrate.
Departures from strictly perturbative behavior in multi-photon ionization have been reported for a wide range ofatoms, frequencies, and field strengths. ATI is one of themanifestations that range from enhanced ionization in themicrowave region6'7 to the phenomena of multiple ionizationat higher frequencies. 8 These diverse phenomena exhibit
some common features that include, aside from the commonmultiphoton character, (a) that they are all strong-field ef-fects with respect to the considered systems and (b) that thequalitative features, and some of the quantitative character,may be recovered by use of simple classical or statisticalmodels. 7 ,
9 -"1
We infer, therefore, that this behavior is a reflection of themechanism involved in these diverse processes and presentsthe opportunity to relate ideas in different frequency re-gimes. For example, in the microwave frequency regime theexistence of a diffusive mechanism for ionization has beenattributed' to strong distortion of the higher excited statesby the external field. Here we present arguments and nu-merical evidence for analogous behavior at laser frequenciesand show that several aspects of the ATI phenomenon canbe addressed from the perspective of a model on these argu-ments.
The bulk of theoretical research addressing the questionof high-field ATI has focused on the effects of intense laserfields on the continuum states of the atom.2 An implicitassumption of most of these studies has been that interac-tions within the bound states of the atom are well represent-ed by perturbative estimates. This may, in fact, be a reason-able assumption when one is considering long interactiontimes, in which case, as is elegantly shown by experiment, 2
free-electron effects are dominant and mask the basic atom-ic process. For shorter pulses, the fact that the strength ofthe laser field used in a typical ATI experiment is strongerthan that of the binding field for most of the excited states ofthe atom being irradiated must be considered. Using suchconditions, one can expect that the part of the multiphotonprocess that occurs within these highly distorted excitedstates will not be simply described by a perturbative calcula-tion.
In a recent paper 3 we presented a physical picture ofshort-pulse, intense-field ATI that was motivated by a mod-el of multiphoton ionization that focused attention on therole of the excited bound states in the ionization process.Contained in the study were numerical calculations thatwere used to test the predictions of the heuristic model. Asimple criterion was also constructed to gauge the impor-
tance of the bound-state mechanism to the multiphotonprocess. Here we summarize the arguments contained thereand further extend the analysis to isolate other contribu-tions to the ATI process. As will be seen below, our numeri-cal scheme, though simple, has the advantage of permittingthe study of the role of specific couplings (bound-bound,bound-free, and free-free) in ATI. We use this to dismantleand subsequently rebuild the features reported in ATI.
PRELIMINARIES
We begin by considering a physical picture of the multipho-ton process involved in ATI that incorporates the field-induced distortion of the excited states. The first questionthat arises is: When should an excited state be consideredas being distorted by the external field? The simplest mea-sure is to require the field-induced width of the state begreater than the spacing to an adjacent level. When levelsare sufficiently broadened that they overlap, we shall speakof these levels as being strongly mixed. A justification forthis nomenclature can be found in the fact that any excita-tion of this region of overlapping states by absorption of aphoton will produce a wave function that is an admixture ofall the overlapping levels. Obviously, mixing starts with thehigher levels of the atom, where field-free states are closesttogether, and moves lower with increasing intensity of thephoton field.
When intermediate steps of a multiphoton processes, suchas ATI, pass through the mixing regime, strong deviationsfrom simple perturbative behavior can be expected. This isin large part due to the importance of higher-order shift andwidth processes in the mixing regime. A simple rule ofthumb to determine the importance of mixing to ATI can beobtained by considering whether the mixing regime extendsbelow the threshold for one-photon ionization (multiphotonthreshold, in general). If it does not, one could imagine thatthe excitation process would simply jump over the mixingregime and that a perturbative calculation would provide areasonable description of the process. On the other hand, ifthe mixing regime extends more than the energy of onephoton below the ionization threshold, a simple picture ofthe excitation process would show it passing through themixing regime. This would naturally lead to a situation thatwould not be well described by perturbation theory.
To elaborate, the nonresonant perturbative picture ofmultiphoton processes has the electron moving from theinitial bound state to a final continuum state in a series ofvirtual steps. The virtual nature of these intermediate stepsimplies that little actual population appears in the excitedlevels that the electron transits on its way to the continuum.When one or more of the intermediate steps pass throughthe mixing region, field-induced widths compensate for theenergy defects, and the excitations into this region becomereal rather than virtual, resulting in an increase in the popu-lation of the states in the mixed region; then a two- (ormore-) step process becomes important. By this we meanthat population will be moved into the mixed states, where itwill be temporarily trapped and subsequently ionized by aprocess that will exhibit, naturally, a weaker dependence onthe intensity I. In many cases, the initial step from theground state into the mixing region will be saturated, andthe intensity dependence of the total ionization will vary as
IP, where p is the number of photons required to ionize fromthe mixing regime. The photoelectron spectrum can also beexpected to change with participation of the mixed states.As was pointed out recently, 4 the formation of Volkov-likebound states is more likely with increased principal quan-tum number n. These states enhance the formation of high-energy photoelectron peaks and, thus, the continuum elec-tron probability should be shifted to higher energy peakswith increased involvement of highly excited states. Final-ly, the saturation of the initial step, from the ground state tothe mixed states, would mean that the energy-conservingpoints in the continuum are now decided from the interme-diate excited states, possibly leading to the observed sub-structure' 5 in the ATI spectrum.
Having constructed a picture of ATI that includes many ofthe effects of strong-field physics, we now consider a simplesystem with which to test these ideas numerically. We con-sider the stretched hydrogen atom' 6 (SHA) interacting witha spatially homogeneous field F(t)cos(t + Xp). The SHAmodel is based on the realization that the dipole matrixelements coupling highly excited states in three-dimensionalhydrogen display a simple pattern. This is most clearlydemonstrated and interpreted in terms of the Stark states. 7
In terms of the parabolic quantum numbers n, ni, n2, and m,transition matrix elements from a state (n, 0, n - 1, 0) to (n',nl'= k, n'-k-1,0) are well approximated by C(An, k)n2 .Thus there is a marked propensity for highly excited statesto undergo k = 0 transitions and thus to remain on the ladderof extreme states (n, 0, n - 1, 0). It is the extended nature ofthese extreme states that permits a quasi-one-dimensionaldescription. The largest coefficients for the on-ladder tran-sitions C(1, 0), C(+2, 0), and C(+3, 0) are 0.32, 0.11, and0.059, respectively. For the lower n states this simple be-havior breaks down, but the SHA still provides a useful low-dimensional model Hamiltonian. In particular for theground state, the SHA includes only a single channel forexcitation and ionization. However, it is in keeping with ourmotivation that the results from this model, though qualita-tive, provide insight for a more realistic theory.
Before proceeding to a description of the numerical analy-sis, let us first define a quantity that is a measure of theimportance of the mixing regime to the multiphoton process.For simplicity, we consider the first multiphoton threshold,namely, the one-photon threshold. It is reasonable to do so,as the one-photon threshold reflects the lowest external per-turbation for which mixing effects are relevant. We labelthe one-photon and mixing thresholds by their associatedprincipal quantum numbers, represented by n, and n2, re-spectively. For our model Hamiltonian, for a frequency wand field strength F,
n-()-1/2,
n2 = (2.7/F)-1/5. (1)
The mixing threshold n2 is a simple two-level estimate and isobtained by requiring that the one-photon Rabi width beequal to the level spacing. More-realistic generalizationswould involve the inclusion of level shifts in estimating n,and n2 or using the nonresonant Rabi width. With thesechanges the two thresholds would become functions of boththe field strength and the frequency, but, for our purpose,the simple estimates suffice.
B. Sundaram and L. Armstrong, Jr.
416 J. Opt. Soc. Am. B/Vol. 7, No. 4/April 1990
A
An = n2 I
n 2
An,
A
AR>1
I
(n)
A
AR<1
(b)
Fig. 1. Schematic of the hydrogen atom for the parametric regions(a) R > 1 and (b) R S 1. n is the one-photon threshold and n2 themixing threshold.
The ratio of the two thresholds R = n2/nl is an usefulindicator of the role of mixing in the ionization process.Consider the schematic of the energy levels shown in Fig. 1,where four photons nominally ionize the atom. Figure 1(a)corresponds to R > 1, where the mixing sets in above the one-photon threshold. As shown, the intermediate steps bypassthe mixed states, and a perturbative calculation would re-flect the dominant contributions. In real terms, this wouldcorrespond to very low fields or to low fields and high fre-quencies. This is to be contrasted with Fig. l(b), for whichR< 1. Here the mixed states are levels that are important tothe ionization process, and nonresonant, perturbative esti-mates would not reflect their contributions. This situationwould occur for extremely strong fields or moderate fieldsand low frequencies. In the case of Fig. 1(b) the possibleconsequences are the following:
(a) Population is driven into a large number of the high-er excited states,
(b) Over some time scale, there is a likelihood of popula-tion's being trapped in these states,
(c) The subsequent ionization of these states providesanother mechanism for gaining access to the continuum.The ionization process, through either absorption of a pho-ton or tunneling, would alter the final electron spectrum.
As was stated above, the variable R and the demarcator R= 1 merely indicate the onset of the effects described. Fur-ther characterization of the mixing would be necessary if onewished to assign a more quantitative role to this parameter.Such characterization is complicated by the fact that thepulse profile introduces other factors relevant in determin-ing fully the effects of the two thresholds.
Our numerical calculations employ the following one-di-mensional Hamiltonian (in atomic units), in the dipole ap-proximation:
H = P2 -- 1+ F(t)cos(wot + p)x, x > .2 x
The time-dependent solution 4) is expressed in terms of theunperturbed basis to be
(t) = E an(t)*I exp(-iEnt) + J dkak(t)qfk exp(-iEkt),n=1
(2)
where *i are the unperturbed wave functions and ai repre-sent the expansion coefficients. Using Eq. (2) in the Schr6-dinger equation leads to a set of coupled equations for theexpansion coefficients given by
idn(t) = E Znm(t)am(t) + J dkZnk(t)ak(t),
m
iak(t) = E Zkm(t)am(t) + J dKZkK(t)aK(t),
m
(3a)
(3b)
where Zij denote the coupling matrix elements. The prima-ry difficulty in solving Eqs. (3) lies in the choice of an appro-priate representation of the continuum. We resort to adiscretized continuum, also referred to as pseudodiscretestates.' 8"19 This allows the terms involving the continuumto be simplified as
I dkZkak J dkZnkak - ZnkjakjA,Jo j~=0 jk-Aj/2 j=0
(4)
where k and A, are the center and the width of the jthinterval, respectively, and n, denotes the number of pseudo-discrete states considered. To obtain relation (4), onemakes the implicit assumption that the integrand is invari-ant over the interval Ai. Using relation (4) in Eqs. (3) for thebound-free and free-free terms leads to an equivalent treat-ment of both bound and continuum states and a set ofcoupled differential equations
nlb+n(.
i$'j(t) = E exp(iwj1 t)Zi1(t)y1(t),1=1
(4)
where yi = ai for bound states and y = Ai'1 2aki for continuumstates. nb is the number of bound states included, and wjl isthe energy difference between levels j and 1. Choosing allthe widths Ai to be equal to, say, A is equivalent to puttingthe system in a box of length (A)-'. The matrix elements forthis system are known,' 9 and the coupled equations weresolved numerically. For reasons of numerical tractability,however, it is easier to work with the p * A form of theinteraction, retaining the A2 term as well. This is due to therelative sizes of the bound-bound, bound-free, and free-freematrix elements. With r -E, the free-free matrix elements,for k near k', are much larger than the others, leading topossible stiff differential equations. This likelihood is mini-mized in the p -A construct. In most of the cases presented,we consider a shaped pulse such that E(t) and A(t) aresimultaneously zero, both at time t = 0 and at the time ofmeasurement. This consideration permits the interpreta-tion of the expansion coefficients as probability ampli-tudes.2 0 For cases involving square pulses the conditionthat A(t) be zero at both the initial and the final times isemployed. It should be mentioned that, from our calcula-tions, the features are not seen to be gauge dependent.
At this juncture, the limitations of this simple treatmentshould be clearly stated. They are the following: (a) The
:
B. Sundaram and L. Armstrong, Jr.
Vol. 7, No. 4/April 1990/J. Opt. Soc. Am. B 417
results are affected by the basis size as well as by the densityof pseudodiscrete continuum states; (b) as expected, theprocedure fails for long times, as the method for dealing withthe continuum amounts to putting the system in a box.Errors appear on time scales that correspond to the ionizedelectron reaching the sides of the box. However, errors canbe controlled by the choice of A, the spacing of states in thecontinuum. For the calculations reported, the choices ofbasis size and total interaction time are such that theselimitations do not affect the conclusions. As a further check,we confirmed that the appropriate low-field limits are repro-duced by our calculations. This was done primarily by con-sidering the effective order of nonlinearity (defined below),which is a useful indicator of low-field, perturbative behav-ior. Comparison of total ionization rates by using the SHAmodel with those from a more realistic, three-dimensional,treatments show a consistent discrepancy of less than anorder of magnitude. This comparison is cited more to putour calculations in perspective than for any quantitativeassessment.
R
1.3 1.2 1.1 I
0.1
0.01
.0
D 0.001
0.
0.000 1
RESULTS AND ANALYSIS
Our computations considered a wide range of frequenciesand field strengths, but, in the interest of clarity, we illus-trate our arguments through a few representative cases. Toproceed systematically, first we consider processes withinthe bound states and isolate the role of mixing. We thenshow that additional mechanisms arise on including thebound-free transitions in our deliberations. These compet-ing processes are further complicated by free-free effects,where no simple picture emerges.
Figure 2 shows the intensity dependence of ionized andtrapped fractions for a typical case. The trapped fraction isdefined as the population in states labeled by quantum num-bers n 2 n2 , where n2 is given by Eqs. (1). The intensityrange considered is 10-4-3 X 10-3 a.u., corresponding toroughly 7 X 1012-4 X 1014 W/cm2 , and the photon frequencyis 0.09 a.u. (nominally, six-photon ionization). The pulseduration t was considered to be 50 fsec, and an envelopefunction sin2 (7rt/tp) was used. Convergence was achievedwith a basis of 30 bound states and 240 continuum states.
We characterize the intensity dependence in terms of theeffective order of nonlinearity k, usually defined in terms ofthe total or partial ionization rate P and intensity I as
dInPd ln I
Lowest-order perturbation theory (LOPT) would predict avalue of 6 for k. The ionization curve in Fig. 2 shows threedistinct regimes characterized by differing slopes. At thelower intensities (less than 7 X 10-4), the slope is 2.5,changing with increasing intensity to the LOPT value -6and, finally, -1.1. The discrepancy, at low intensities, be-tween the measured slope and the LOPT prediction is aconsequence of the short interaction time considered in ourcalculations. For these short times, the rate regime is notattained, and the ionized fraction shows rapid oscillations22
in time. This portion of the curve has been included pri-marily for completeness rather than for any predictive value.With increasing intensity, the rate regime is achieved withinthe time considered, and the LOPT result is seen. Further
10-5 10 .00 1 0.00 1
Intensity(a.u.)Fig. 2. Fractional ionized (solid curve) and trapped (dashed curve)population as a function of intensity at a frequency of 0.09 a.u.Note that a shaped pulse with a total duration of 50 fsec was used forthe calculations.
increase in intensity saturates this transition rate. The in-tensity at which k becomes approximately linear (1.1 in thiscase) is referred to as the saturation intensity.
As can be seen from the upper axis in Fig. 2, increasingintensity corresponds to R - 1 from above, and, at an inten-sity of 1.6 X 0-3 a.u., R is close to 1. The relationshipbetween decreasing values of R and the change in slope isconsistent with earlier predictions, as is the correlation be-tween the change to quasi-linear slope and R 1.
The high-intensity regime exhibits the striking featurethat significant population remains in the bound states atintensities above the saturation value. This is surprising, asthe usual interpretation associates the transition to linearslope with the total depletion of neutral atoms in the interac-tion volume. The linear slope is then seen as a consequenceof expansion of the phase volume, that is, the ionization ofatoms not in the focal spot. The presence of the trappedpopulation, however, suggests a need to alter this definitionfor the case of short pulses.
Let us now examine these features more closely by consid-ering the six-photon case in detail. Figures 3(b) and 3(c)show the electron spectra for two values of the intensity for X= 0.09 a.u. The intensities for Figs. 3(b) and 3(c) are 9.0 X10-4 and 1.6 X 10-3 a.u., respectively. In Fig. 3(a) the popu-lation in the first 30 bound states is shown, as a function ofthe principal quantum number n, for the two intensity val-ues. These intensities correspond to R > 1 and R < 1,respectively. The correlation between increased excitationof the high n states at the higher intensity and the appear-
i I . I
\ A/V
B. Sundaram and L. Armstrong, Jr.
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III
I /
I I I I I .I .. II
418 J. Opt. Soc. Am. B/Vol. 7, No. 4/April 1990
(b) -
Lk~~~~.kr~~~~~ii~_0 0.2 0.4 0.6 0.8 1
0.02
0.015
0.01
0.005
0
,E(a.u.)0 0.2 0.4 0.6 0.8 1
0 10 20 30
Quantum Number n
(a)Fig. 3. Excited-state population and electron spectra at a frequency of 0.09 a.u. with a shaped pulse. (a) The solid curve is at an intensity of1.6 X 10-3 a.u.; the dashed curve corresponds to an intensity of 9.0 X 10-4 a.u. (b), (c) Continuum population as a function of energy (a.u.) at (b)9.0 X 10-4 a.u. and (c) 1.6 X 10-3 a.u.
ance of high-energy ATI peaks is quite clear. Further, theuniform excitation of a large number of bound states is alsoconsistent with the physical picture constructed in terms ofthe relative positions of the two thresholds.
The role of R is emphasized in Table 1, where a compari-son of the ionized and trapped fractions for several frequen-cies is shown. The intensity, in all cases, is 1.6 X 10-3 a.u.,and the total interaction time is also held fixed. Recall thatour simple estimate for parameter R is a qualitative measureof the relative importance of processes involving the mixedstates to ones that bypass them. As was emphasized above,this model does not include the field-profile characteristicsthat are relevant for determining the ionized and trappedfractions. Further, excitation of the mixed states proceedsthrough the first excited state. For fixed field strength, theeffective rate for this first step drops off with decreasing w.Therefore, given the constant interaction time, the depletionof the ground state is strongly frequency dependent. Theratio Pti of trapped to ionized population normalizes the
Table 1. Trapped and Ionized Fractions As aFunction of Frequency for Fixed Field F0 = 0.04 a.u.a
X (a.u.) R N Pi (ionized) Pt (trapped) Pt; = PtPi
0.3 1.8 2 2.39 X 10-1 7.87 X 10-8 2.3 X 10-70.2 1.47 3 7.96 X 10-1 3.94 X 10-8 0.5 X 10-70.148 1.26 4 1.22 X 10-1 1.87 X 10-4 1.5 X 10-30.09 0.94 6 4.29 X 10-1 1.05 X 10-2 2.4 X 10-20.046 0.70 11 1.16 X 10-1 6.31 X 10-3 5.4 X 10-2
a N is the minimum number of photons required for ionization.
results at different frequencies and is a better measure of thecompeting processes. Thus the efficacy of R is better judgedby this ratio. The equivalence of the two quantities is clear-ly seen from Table 1.
At a frequency co = 0.2 we see a deviation from the patternestablished by the other frequency values. It is useful toexplore the reasons for this anomaly, as it is exploited for asubsequent result. At this frequency there is a near reso-nance between the states n = 1 and n = 2. The mixingregime starts from n = 3, and the one-photon threshold is atn = 2. Thus the near resonance provides a mechanism forjumping over the mixed states and diminishing their contri-bution to the ionization process. This mechanism is shownbelow to decide the role of the mixed states in producingsubstructure in the ATI peaks.
Having established a correlation between the parameter Rand the excitation and ionization processes, we focus onthree different aspects of the problem to explore this rela-tionship further:
(1) The time dependence of the excited and ionized frac-tions,
(2) The role of intermediate reasonances in creating thepeak substructure,
(3) The electron spectrum produced from excited statesabove and below the mixing theshold.
Figures 4(a) and 4(b) show the excited-state and ionizedpopulations as a function of time, measured in cycles of theexternal field. A frequency of 0.09 (six-photon ionization)and a shaped pulse of peak field strength 0.05 a.u. are consid-
-I0.002
0.0015
0.00 1
.0005
0
_4
.20.
0
10.1
0.01
0.00 1
0.000 110 -6
10 -a1 -7
lw w w w w s w w w w w w w w . s s w
....................
B. Sundaram and L. Armstrong, Jr.
Vol. 7, No. 4/April 1990/J. Opt. Soc. Am. B 419
ered. The pulse duration is 24 cycles (50 fsec), and thepopulations are sampled when A(t) is zero, in accordancewith the discussion above and with Ref. 20. Ionization setsin at a critical field, which, in keeping with earlier analysis,2 3
is found to be independent of the pulse duration. Withrising field there is simultaneous increase in both ionizedand excited population. The excited-state population ex-hibits several peaks, which reflect the competing processesof depletion through ionization and excitation from theground state. We have verified that the oscillation in excit-ed-state population is not related to fluctuations of theground-state population but is a consequence of ionization.Figure 4(c) shows the electron spectrum at the end of thepulse. The spectrum is seen to be noisy, particularly for lowenergies. Indications of the origins of this behavior can befound by a comparison with Fig. 5, which shows analogousplots for a square pulse. In this case initial transients areseen to drive population into the excited states, which decayrapidly into the continuum. The electron spectrum showsbetter-defined ATI peaks, without much of the substructureseen in the case of the shaped pulse. A possible explanationof the differences is based on the role of dynamical shifts andwidths in the case of the shaped pulse, which would bringdifferent higher excited states into resonance as a function oftime. The subsequent decay of these states would producesatellite ATI peaks, which would constitute the substruc-
0.015
0.01
0.005
0
.e0
-*. 0.8
0 .2
0.15
0.1
0.05
0 0.2 0.4 0.6 0.8 I
E(au.)
0 5 10 15 20 25
0.01
0.005
0
r.._
L0r
0.4
0.3
0.2
0.1
0
0.1
0.05
0
0 0.2 0.4 0.6e (a.u.)
0.8 1
0 5 10 15 20 25
Time(in cycles)Fig. 4. (a) Excited-state fraction and (b) ionized fraction as afunction of time. = 0.09; Fo = 0.05; sin 2(t/tS) pulse. (c) Electronspectrum as a function of energy at the end of the pulse (50 fsec).
Time(in cycles)Fig. 5. Same as Fig. 4 but for a square pulse.
ture. With a square pulse, the quasistatic atomic spectrumwould discount such a mechanism. If this explanation werecorrect, the field profile would play an important role in theelectron spectrum, at least for short pulses. For longerpulses the role of the field profile would not be a factor,especially if all the atoms were fully ionized before the peakfield were attained. The role of the higher excited states inproducing satellite peaks also suggests that tuning the fre-quency to be resonant with one of the lower excited stateswould inhibit the substructure.
This picture can easily be verified through a frequencyscan that is tuned through a resonance with a lower excited,relatively undistorted, state. Figure 6 shows the fractionalpopulation as a function of energy at four values of thefrequency. The intensity is 2.5 X 10-4 a.u., and the pulseenvelope sin2(7rt/2t,) is used; all four frequencies correspondto nominal four-photon ionization. In Fig. 6(a) the consid-ered frequency is near resonant for a three-photon transitionbetween n = 1 and n = 3; Figs. 6(b)-6(d) are increasinglydetuned from this resonance. The appearance of substruc-ture within the ATI peaks increases with the increasingdetuning. A simple two-state diagonalization, involving n =2 and n = 3, is inadequate to resolve the structure. Asanticipated, the nonresonant cases appear to involve severalstates that are dynamically drawn into resonance and con-tribute to the ionization. In the near-resonant case, thedominant mechanism is through the n = 3 state. In Fig. 6(b)a prominent peak is seen within the negative energy states.
B. Sundaram. and L. Armstrong, Jr.
420 J. Opt. Soc. Am. B/Vol. 7, No. 4/April 1990
0.10
0.08
0.06C:0
o
._
0.04
0.02
0.000.10
0.08
0.06
0.04-
0.02
0.00 -
0.0 0.2 0.4 0.6 0.0
Energy e(a.
Fig. 6. Fractional population as a function of energy (a.u.) for (a) w = 0.148, (b) w = 0.142, (c)in a.u. For all cases the intensity is 2.5 X 10-4 a.u.
This corresponds to population centered around the n = 5state and is not a subthreshold peak.24
The multilevel nature of the process is further stressed inFig. 7, where electron spectra arising from different approxi-mations are compared. Figure 7(a) retains all the boundand continuum couplings, while Fig. 7(b) excludes the con-tinuum-continuum coupling. The overall ionized fractionis nearly the same for the two cases, whereas the electronspectra are considerably different. Figure 7(c) considers asingle bound state coupled to the continuum, a situationreminiscent of the Volkov state approximation.2 5 Given theshort pulse times (50-100 fsec), the spectra do not developfully in this approximation, but the rates are seen to beseveral orders of magnitude lower. From our perspective,the short-pulse, intense-field regime is characterized bycompeting rates and associated time scales that require in-clusion of all possible couplings. It is, therefore, highlyunlikely that approximations that neglect part of the spec-trum would be successful.
We have nearly exhausted our treatment of the bound-bound part of the ATI mechanism, but, before proceeding tobound-free and free-free considerations, let us look at theelectron spectra arising from ionization of the excited states,which are relevant not only to the origins of substructure butto other contributing mechanisms in ATI. Figures 8 and 9show the population distribution in both bound and contin-uum states for a range of initial states. The field parametersare a frequency of 0.148 and a field strength of 0.05, whichputs the mixing threshold at n = 3 and R 1.2. Thus initialstates n 2 2 are also within the one-photon threshold. Thehighly localized bound-state distributions, for states abovethe mixing threshold, are distinctive, as is the similarity intheir corresponding electron spectra. Although all the fea-tures are not fully understood, we suggest that they reflect
6x0 '
4x10
2x10
00.015
0
co
.
0.01
0.005
00.02
0.2 0.4 0.6
u.)w = 0.138, (d) w = 0.130, where w is the frequency
0.015
0.01LF
0.005
0 0.2 0.4 0.6 0.8 1
E (a.u.)
Fig. 7. Distribution of electrons as a function of energy . w = 0.09a.u., F = 0.04, and no = 1. (a) 30 bound states; (b) 30 bound states,no continuum-continuum coupling; (c) 1 bound state. (a) and (b)use a shaped pulse, (c) is for a square pulse.
F. l X 1
B. Sundaram. and L. Armstrong, Jr.
Vol. 7, No. 4/April 1990/J. Opt. Soc. Am. B 421
0.001
0 10 20 30
0.5
0
0.5
0
0.5
0
0.05
0
0 10 20 30Quantum Number n
Fig. 8. Population distribution within the bound states X = 0.148,F = 0.05 for different initial states no for a shaped pulse of duration20 cycles (25 fsec).
the increasing importance of diagonal couplings within boththe bound and the continuum states. These couplings couldbe introduced by dressing the bare atomic states, a proce-dure that isolates the diagonal couplings and reflects theimportance of off-energy shell transitions. This dressinghas to be done consistently for both bound14 and continu-um26 states. For example, simply dressing the bound stateswould give the probability of populating the sth peak Ps,from within the one-photon threshold, to be
P. 8 Js(Xb)IVbEl,
where Xb = 3/2n(n - 1)(F/1), VbE is a bound-free matrixelement, and J, is a Bessel function of the first kind. Includ-ing dressed continuum states2 6 yields, to first order, a combi-nation of Bessel functions that does not fully reproduce theresults obtained. However, our analysis indicates that thismay account for at least some of the processes involved inATI.
That a simple dressed-atom picture is insufficient to de-scribe ATI is suggested in Fig. 10, where the electron spectrafor a range of frequencies are shown, at fixed field strength.The frequency range shown changes the minimum numberof photons for ionization from six to seven, although we haveconsidered a range from four to nine photons. In large part,the behavior is as expected, with the spectra evolvingsmoothly with frequency and the first peak position movingto lower energies. Let us now focus on just two frequencies,
0.005
0.002
0.02
0.01
0
0.02
0.01
0
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
Electron energy E (a.u.)
Fig. 9. Electron spectra for the same conditions as for Fig. 8.
0
0.005C:
0
CL)03.4
0
0.005
0
0.005
0
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
Electron energy (a.u.)Fig. 10. Electron spectra for field strength F = 0.05 a.u. startingfrom the ground state for a range of frequencies. A square pulse ofduration 5T is considered. The frequency w is (in a.u.) (a) 0.080, (b)0.082, (c) 0.084, (d) 0.086, (e) 0.088, (f) 0.090, (g) 0.092, (h) 0.094.The arrows indicate missing peaks.
no=15
0
0.0005
0
0.01
00.2
0.1
0
. I T lII I 1
no=
0.10
0
0.05
0
0.10
0
0.02
0
-- I ------f----- - -- _-1 II I I i
no=5
------- --------------
B. Sundaram and L. Armstrong, Jr.
I
422 J. Opt. Soc. Am. B/Vol. 7, No. 4/April 1990
w = 0.086 and 0.088 a.u. These are representative of cases inwhich a small change in frequency results in significantdifferences in both the electron spectra and the ionized frac-tion. We see 21% ionization at w = 0.086, which is substan-tially lower than the 58% ionization obtained at 0.088. Byanalyzing these two cases in detail, we illustrate anotherclass of processes important to ATI. In both cases, sixphotons are needed for the nominal ionization process, and n= 3 is the one-photon, field-free ionization threshold. Fig-ure 11 shows the corresponding electron spectra at the twofrequencies, and it is clear that the cases are qualitativelydissimilar. The glaring difference is contained among thelow-energy peaks, where, for w 0.088, an expected peak(indicated by the arrow) is missing. The high-energy peaksare also better resolved in this case, which can be understoodin terms of the increased ionization. For longer interactiontimes, the envelopes of the higher-energy peaks are seen tobe similar in the two cases.
As demonstrated above, we can isolate any near reso-nances in the excitation process by considering the popula-tion in the individual bound states as a function of time.This treatment shows that, for w = 0.088, an intermediatestep exhibits a near-resonant transition with the n = 3 state.In contrast, the n = 5 state mediates the transition to thecontinuum at the lower frequency. The fact that n = 3 is theone-photon threshold and hence is strongly coupled to thecontinuum is consistent with the enhanced ionization seenat the higher frequency. At w = 0.086, the trapping mecha-nism suggested above comes into play. How does this resultin the missing peak?
0.008
0.006
0.004
.002
r.0Z 0co
Z 0.030
0.02
0.01
0
Fig. 11. E(b) w = 0.0O
0.03 (b)
0.02
0.01 ' 40
0
0.02
0.01
0 0.2 0.4 0.6 0.8 l
Electron energy (a.u.)Fig. 12. Electron spectra for F = 0.05 and co = 0.088 a.u. with nocontinuum-continuum coupling. The initial state is (a) the groundstate and (b) n = 3. Missing peaks are indicated by the arrows.
As was stated above, the particular advantage of the close-
coupling method is the facility to exclude certain interac-tions. We employ this facility in treating the more interest-
(b) ing case of w = 0.088 and consider the electron spectrum inthe absence of continuum-continuum interactions. As canbe seen from Fig. 12(a), the removal of the free-free matrixelements results in enhancement of every alternate peak,and the higher-energy peaks are completely absent. How-ever, the profile of the low-energy peaks is similar to thatwhich occurs when all the couplings are retained [Fig. 11(a)].This similarity suggests that the low-energy end of the elec-tron spectrum results primarily from bound-bound andbound-free processes. Since we already know that the dom-
I l l I l l | | { ~~~~i i| ii i| nant bound-bound characteristic is the excitation through(a) the n = 3 state, let us consider the photoelectron spectrum
with n = 3 as the initial state. This spectrum is shown inFig. 12(b) and displays pronounced alternation in the lowerpeak heights, indicating that the n = 3 state, mediating theionization mechanism, is largely responsible for the profileof the lower peaks. In fact, a simple model provides a quali-tative explanation for this effect.
The model is most easily seen in a quantized field picture.Consider the schematic shown in Fig. 13 of a process involv-ing a bound state lb, p) and a continuum state le, m). We
I I _ . | , |ignore free-free effects in this model. With lb, N) as the0 0.2 0.4 0.6 0.8 1 initial state, a direct one-photon process populates le, N -
1). This is the low-field scenario in which only the lowestElectron energy (a.u.) peak will be seen. With increasing intensity, the ratio (b,
lectron spectra forF = 0.05 a.u. and 0.088 a.u. w=(a) and pIVIe, m)/ is large enough to allow the off-energy shell)6 a.u. The arrow indicates a missing peak. transition from le, N - 1) to lb, N - 2) to be significant. A
B. Sundaram and L. Armstrong, Jr.
Vol. 7, No. 4/April 1990/J. Opt. Soc. Am. B 423
subsequent one-photon process leads to a peak at le, N - 3),two photons away from the first peak. In the absence offree-free interactions, there is no mechanism to populate thestate le, N - 2) except through dressed bound states. Theabsence of the even peaks points to the reduced efficiency ofthis process, which is governed by the ratio of a bound-statediagonal matrix element and the external frequency. Themechanism outlined can be repeated several times, givingrise to the staggered peak structure observed.
In the case of w = 0.088, the ionization proceeds throughthe state with the largest bound-free matrix element, n = 3,of all the states within one photon of the continuum, and theabove mechanism is important. By contrast, the case of0.086 involves the n = 5 state, which has a bound-free matrixelement considerably smaller than that for the n = 3 state.Consequently, the ratio given above is too small for thestaggering of peaks to be seen at this intensity, although itshould appear at higher power levels.
The question of modifications to these processes when allthe interactions are included remains an open one. Thenumber of competing processes proliferates in the generalcase, and we have as yet not identified any clear pattern.What we have observed by considering the time develop-ment of the electron spectra is that the higher peaks areformed simultaneously. Peaks in well-demarcated energywindows are also seen to be impervious to changes in fre-quency and field strength. We speculate that, at low ener-gies, free-free couplings lead to population transfer from onepeak to another. At the high end of the spectrum, final stateeffects (modulation due to the external field) are important,leading to, perhaps, the formation of a Volkov-like state.2 5 26
By this we mean that the behavior of the envelope of high-energy peaks with intensity is suggestive of Volkov behavior,although the ratios of successive peak heights, that is, ratiosof Bessel functions, are not well described by the Volkovprescription. The intermediate regime remains a completemystery with the many possible competing processes.
The manifestations of several competing processes are notrestricted to the electron spectra but also affect the totalionization probability. For example, for the two frequencies
le,N-3>
leN-l> r
-------------- -----------------
r
b,N-2>
Ib,N>
Fig. 13. Schematic for model describing staggering of peaks in theelectron spectrum. r and c stand for rotating and counterrotating,respectively.
contrasted above, the effect of excluding free-free couplingsis considerably different. At w = 0.086, the relative changein ionization probability of 30% when continuum-continu-um interactions are dropped, whereas at 0.088 it is only 14%.This is not entirely surprising given the large number ofcompeting processes, leading to the possible rate bottle-necks. These considerations led to an earlier remark, in thecontext of dressed-state approaches, that both bound andcontinuum states have to treated consistently, that is, allstates have to be dressed by the field.
CONCLUSIONS
We have isolated, within a simple low-dimensional model,several classes of processes that contribute to the featuresobserved in ATI. The common feature shared by all theseprocesses is that they invoke off-energy shell transitions.The simplest component appears to be the excited boundstates, whose role is related to the smearing of part of thediscrete spectrum of an atom when interacting with a stronglaser field. It is demonstrated that physically motivatedmeasures can be constructed to gauge the importance of thisquasi-continuum, or mixed, regime to the final electronspectra. Inclusion of bound-free and free-free transitionsresults, unfortunately, in a proliferation of competing pro-cesses, making a complete synthesis elusive.
These mechanisms are not specific to particular field oratomic characteristics and are readily extended to otherfrequency and field regimes. For example, the large energygap between n = 1 and n = 2 for a hydrogen atom means anominal range of variance for a parameter such as R, at laserfrequencies and intensities. It is also clear that at extremefrequencies the parameter R is restricted only to one regime.Going to the Rydberg sequence and microwave radiationappears to be the other extreme of limited variability.There mixing is the dominant featured and is seen as thereason for the success of classical modeling. Infrared fre-quencies, perhaps, are ideal for exploring a wider range ofthe parameter R and merit further consideration.
In summary, our primary conclusions are that
(a) A two-threshold characterization of the bound statesis effective in treating strong-field ATI;
(b) Many features of ATI require the excited levels toplay a dynamical role. This would suggest that the use ofeffective matrix elements is inadequate to represent theprocess;
(c) The populating of several excited levels owing tofield-induced shifts and widths could be the mechanism forpeak substructure as well as the anomalous effective ordersof nonlinearity observed3 for the electron spectrum. Fur-ther, assigning a dynamical role to the excited states wouldsuggest the importance of the field profile in describingshort-pulse ATI;
(d) The concept of saturation intensity may have to bemodified for short-pulse experiments;
(e) A large number of competing processes involving in-termediate, off-energy shell transitions contribute to thenature of the ATI spectrum.
Despite the limitations of models such as the one present-
B. Sundaram and L. Armstrong, Jr.
424 J. Opt. Soc. Am. B/Vol. 7, No. 4/April 1990
ed here, such models provide systems that can be character-ized in detail and subsequently provide insight for morerealistic analytic and numerical analysis. This may be par-ticularly useful given the shorter-pulse, more intense, laserexperiments projected for the future.
ACKNOWLEDGMENTS
This research was supported in part by a grant from theNational Science Foundation. The numerical calculationswere performed on a Cray-XMP computer at the PittsburghSupercomputing Center.
* Present address, Theoretical Division, MSJ569, LosAlamos National Laboratory, Los Alamos, New Mexico87545.
REFERENCES
1. See Feature on Multielectron Excitations in Atoms, W. E.Cooke and T. J. McIlrath, eds., J. Opt. Soc. Am. B 4, 705-862(1987).
2. N. K. Rahman, C. Guidotti, and M. Allegrini, eds., Photons andContinuum States of Atoms and Molecules, (Springer-Verlag,Berlin, 1987).
3. P. Agostini, F. Fabre, G. Mainfray, G. Petite and N. K. Rahman,Phys. Rev. Lett. 42, 1127 (1979); P. Kruit, J. Kimman, H. G.Muller and M. J. van der Wiel, Phys. Rev. A 28,248 (1983); P. H.Bucksbaum, M. Bashkansky, R. R. Freeman, T. J. McIlrath,and L. F. DiMauro, Phys. Rev. Lett. 56, 2590 (1986); R. R.Freeman, T. J. McIlrath, and M. Bashkansky, Phys. Rev. Lett.57, 3156 (1987); G. Petite, P. Agostini, and F. Yergeau, J. Opt.Soc. Am. B 4, 765 (1987); P. Agostini and G. Petite, Contemp.Phys. 29, 57 (1988).
4. C. Cerjan and R. Kosloff, J. Phys. B 20, 4441 (1987); J. Javan-ainen, Q. Su, and J. H. Eberly, Phys. Rev. A 38, 3430 (1988); L.Collins and A. L. Merts, Phys. Rev. A 37, 2415 (1988).
5. See the papers cited in Refs. 1 and 2 for theories involving free-electron dynamics.
6. J. E. Bayfield and L. A. Pinnaduwage, J. Phys. B 18, L49 (1985);Phys. Rev. Lett. 54, 313 (1985).
7. K. A. H. van Leeuwen, G. V. Oppen, S. Renwick, J. B. Bowlin, P.M. Koch, R. V. Jensen, 0. Rath, D. Richards, and J. G. Leopold,Phys. Rev. Lett. 55, 2231 (1985).
8. T. S. Luk, H. Pummer, K. Boyer, M. Shahidi, H. Egger, and C.K. Rhodes, Phys. Rev. Lett. 51, 110 (1983).
9. M. Crance, J. Phys. B 17, L355, 4333 (1984).10. Liwen Pan, J. Phys. B 18, L833 (1985).11. G. Casati, B. V. Chirikov, D. L. Shepelansky, and I. Guarneri,
Phys. Rep. 154,77 (1987).12. P. H. Bucksbaum, R. R. Freeman, M. Bashkansky, and T. J.
McIlrath, J. Opt. Soc. Am. B 4,760 (1987).13. B. Sundaram and L. Armstrong, Jr., Phys. Rev. A 38,152 (1988).14. Liwen Pan, B. Sundaram, and L. Armstrong, Jr., J. Opt. Soc.
Am. B 4,754 (1987).15. R. R. Freeman, P. H. Buchsbaum, H. Milchberg, S. Darack, D.
Schumacher, and M. E. Geusic, Phys. Rev. Lett. 59,1092 (1987).16. J. N. Bardsley and B. Sundaram, Phys. Rev. A 32, 689 (1985).17. H. A. Bethe and E. E. Saltpeter, Quantum Mechanics of One-
and Two-Electron Atoms (Springer-Verlag, Berlin, 1957).18. R. D. Cowan, The Theory of Atomic Structure (U. California
Press, Berkeley, Calif., 1981).19. S. Susskind and R. V. Jensen, Phys. Rev. A 38, 711 (1988).20. W. E. Lamb, Jr., R. R. Schlicher, and M. 0. Scully, Phys. Rev. A
36, 2763 (1987).21. S. I. Chu and J. Cooper, Phys. Rev. A 32, 2769 (1985).22. S. L. Haan and S. Geltman, J. Phys. B 15, 1229 (1982).23. P. Lambropoulos and X. Tang, J. Opt. Soc. Am. B 4,821 (1987).24. J. Javanainen and J. Eberly, in Multiphoton Processes, S. J.
Smith and P. L. Knight, eds. (Cambridge U. Press, Cambridge,1988), p. 88.
25. H. Reiss, Phys. Rev. A 22, 1786 (1980); see also Ref. 2, pp. 98-103.
26. L. Pan, "Multiphoton above-threshold ionization of atoms inintense laser fields," Ph.D. dissertation (The Johns HopkinsUniversity; Baltimore, Md., 1987); "Treatment of continuum-continuum coupling in the theoretical study of ATI," in Atomsin Strong Fields, C. Nicolaides, C. W. Clark, and M. Nahfeh,eds. (Plenum, New York, 1989).
