PHYSICAL REVIEW A JANUARY 1999VOLUME 59, NUMBER 1

d ionization: Stratified symmetrical electron emissionand resonantly structured ionization continuum

P. L. Shkolnikov, A. E. Kaplan, and S. F. Straub*Electrical and Computer Engineering Department, The Johns Hopkins University, Baltimore, Maryland 21218

~Received 18 February 1998!

We have discovered theoretically that ionization of a one-dimensional quantum well by an ultrashort sub-cycle ~d-like! electromagnetic pulse results in photoelectrons propagating both along the field and in theopposite direction in well-separated shells whose velocities reflect quasiresonances in the ionization con-tinuum. For both a quantum well and a three-dimensional hydrogenlike atom, a unipolar, i.e., strongly asym-metrical,d ionization produces an approximately symmetrical, stratified photoelectron cloud.@S1050-2947~99!09701-2#

PACS number~s!: 32.80.Fb, 32.80.Rm, 42.50.Hz, 42.65.Re

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The theory of photoionization, one of the most fundametal and rich phenomena in atomic physics, was substantrevised for high laser intensities. In particular, Einstein’s fmula for the photoelectron kinetic energy,Ek5\v2Eion,while valid for relatively weak monochromatic fields, breadown in laser fields strong enough to cause ionization evethe photon energy\v is much smaller than the ionizatioenergyEion . Almost all strong-field effects discovered so f~above-threshold ionization, barrier-suppression ionizatand atomic stabilization, to name a few@1#! are, however,caused by quasimonochromaticoscillating electic fields oflasers. Recently@2#, we theoretically demonstrated the feability of intense ultrashort~near-femtosecond or subfemtosecond! subcyclepulses which, we believe, call for yet another important extension of the theory of interactionelectromagnetic radiation with matter. Indeed, in contrasthe most commonly studied optical fields, no particular fquency could be ascribed to a subcycle pulse; the notionpulse envelope becomes equally useless. Together withfeasible ultrashort time duration, this renders irrelevant scommon tools of the theory as the rotation-wave and slovarying-envelope approximations. Moreover, for a subcypulse shorter than typical times of a quantum system~such asthe ‘‘ionization time’’ t ion5h/Eion , whereEion is the ioniza-tion potential of the system!, the pulse amplitude and duration are no longer meaningful if taken separately. Instead,pulse’s effect on the system is solely determined by thepulsearea, Q[*dt8«(t8), «(t) being the pulse’s electric field;Q,in turn, is proportional to the average momentum imparby the pulse~by contrast,Q'0 for an oscillating pulse!.Indeed, such pulse may be modeled by thed function oftime. Then the wave functions just before (c0) and after~C!the pulse are related simply asC5exp@2(ieQ/\)(r j #c0 ,where the sum is taken over all the charged particles insystem@3#. It is easy to see now that the average momentof the system changes byDp5eQ, while the change in theaveragetotal energy looks like a classical relation betwethe changes in the momentum andkinetic energy: for a one-

*Permanent address: Abteilung fur Quantenphysik, UniversUlm, Ulm, Germany.

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particle system, for simplicity, DE[*dr C* H0C2*dr c0* H0c05(Dp)2/2m1p0Dp/m, wherep0 is the av-erage momentum of the system in the initial state, andH0 isthe system’s Hamiltonian without a subcycle pulse~field-freeHamiltonian!. ~This classical-like behavior is apparently duto the fact that the system does not have time to move duan ultrashort pulse.! One important consequence of this roof the pulse area is that the definition of the high-fieldgime, based on comparing the pulse field with the ‘‘atomifield ;AEion, is not applicable tod-like pulses.

Besides their apparent theoretical importance, ultrashsubcycle pulses of high intensity, being capable of stronexciting or ionizing many quantum systems almost instanneously on the systems’ time scale, may become an imtant new probing tool and a source of new effects in nonear optics and atomic and molecular physics. Sohowever,d ionization~andd excitation! remains largely un-explored. One important exception is ionization of Rydbeatoms by almost unipolar, ‘‘half-cycle’’ pulses of the durtion ~;0.5 ps! comparable to Rydberg orbital times, whicdiffers substantially from the ionization by lasers or blonger field pulses@4#. In this paper, we predict a qualitatively new effect, to our knowledge. We show, theoreticaand by computer simulation of a one-dimensional~1D! quan-tum well ~QW! and a 3D hydrogenlike atom, that the photelectron cloud produced byd ionization displays a stronglystratified spatiotemporal structure@5#: electrons are emittedin well-separated bunches both in the direction of the pufield and in the opposite direction. For a relatively smpulse area, the photoelectron cloud is approximately symetrical in regard to the pulse-field direction; this symmevanishes for higher amplitudes.

An important distinction of our approach is our focus othe photoelectron wave function in the coordinate represtation. This focus is productive because of the uniq‘‘trans-spectral’’ coherence of thed ionization, as opposed tothe ionization by a white-noise signal of a long timewidwith an equally broad spectrum. It is this coherency thmaps the energy-level structure of a quantum system intoordered spatiotemporal structure of the photoelectron clorather than into a random superposition of electron buncthat would result from a white-noise ionization. While thionization of Rydberg atoms by ad-like electrical field‘‘kick’’ has been considered before@6#, in particular when at

490 ©1999 The American Physical Society

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PRA 59 491d IONIZATION: STRATIFIED SYMMETRICAL . . .

spatially uniformd function of time was used to approximathe field of a fast incident electron@7#, no attempts have beemade to predict the spatiotemporal behavior of photoetrons. We are unaware of any research ond ionization of aQW.

The Schro¨dinger equation for an electron interacting withe pulse electric field«(t) is

i\]C~ t,r !/]t5@H0~r !2er•«~ t !#C~ t,r !, ~1!

H0 being the field-free Hamiltonian. We assume here thatpulse field is uniform across the quantum system~the dipoleapproximation!. One can show that this assumption is jusfied for d-like pulses ifEion!mc2, which coincides with thelimits of nonrelativistic quantum mechanics and is appently the case for any atom, and even more so for QWModeling «(t) by the Diracd function of time, «(t)→«d

5«pd(t/tp), «ptp[*2`` «(t)dt where«p is the pulse ampli-

tude, one obtains a simple relation between the wave fution before, c0 , and immediately after the pulse~att501) as ~see, e.g., Ref.@6#!

C~01 !5c0~r !exp~ ipd•r /\!,

pd5e«ptp5E2`

`

e«~ t !dt5eQ, ~2!

where pd is the classical~equal to the quantum averag!momentum transfer due to thed kick. Equation~2! is ob-tained by neglectingH0 in Eq. ~1! in the presence of thepulse field~the impulse approximation@3,6#!.

The evolution of the system att.0 is governed by thefield-free Schro¨dinger equation, withC~01! as the initialfunction. We express solutions to this equation throughfull orthonormal set of both bound@c j (r ), with Ej as theeigenvalue# and continuum@cE(r ), for a given energyE#eigenfunctions ofH0 as

C~ t,r !5( bjc j~r !e2 iE j t/\1C ion ,

~3!

bj5E drc j* ~r !eipd•r /\c0~r !,

where ( stands for the summation over the bound staNote that for weakd kicks, bj' ipdd1,j /\5 if j

(R) , whered1,j5*c0(r )rc j* (r )dr is the transition dipole moment between the initial state and thej th eigenstates, andf j

(R) is theRabi phase. Of primary interest to us is the ‘‘ionized part’’the total wave function,C ion5*dE b(E)cE(r )e2 iEt/\, thatcomprises the eigenfunctions of the continuum and, thefore, describes the motion of photoelectrons. We investigin more detail two fundamental quantum systems: aquantum well~QW! and a 3D hydrogenlike atom. Much othe symmetry of thed ionization, however, is not related tparticular potentials. It is easy to show, e.g., that for any rand inversion-symmetricalH0 , immediately after a highlyasymmetrical~unidirectional! d kick, the spatial distributionof photoelectrons is exactly symmetrical with regard tofield direction. Indeed, bound eigenfunctionsc j (r ) of suchH0 can always be chosen to be real and of a definite pa

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with the ground-state wave function being even; assumec0 is even. Furthermore, the amplitudesbj are real~imagi-nary! for even ~odd! c j , so that att501, Eq. ~3! can bewritten as

C ion~01,r ![M ~r !1 iN~r !,

M ~r !5@cos~pd•r /\!2b0#c02( bjevenc j

even, ~4!

N~r !5sin~pd•r /\!2( bjoddc j

odd,

where bothbjeven and bj

odd are real, as areM and N. SinceM (2r )5M (r ) and N(2r )52N(r ), uC(01,r )u25M2

1N2 is an even function ofr and, therefore, represents ainversely symmetrical photoelectron distribution. Intuitivelthis symmetry might be expected, as the initially symmecal electron distribution is instantaneously ionized by a uform field.

Furthermore, the photoelectronpropagation is approxi-mately symmetrical with regard to the field direction. Thsymmetry is purely quantum. Indeed, a classical particleby the unidirectional field of a sufficiently strongd kickwould move in the direction of the force. Quantumd ioniza-tion, on the other hand, is similar to hitting water in a shlow bucket: a kick would spill water all over the rim, unlesthe kick is very strong, in which case almost all the wawill move along the kick.~Similarly, the scattering of a particle off a power center is isotropic for small incident mmenta @3#.! This symmetry holds for any inversionsymmetrical Hamiltonian, but is easier to see in odimension. The continuous spectrum of such a 1D Hamtonian is doubly degenerate. We may choose the eigenfutions c6(x) for a given energyE to mirror each other:c2(x)5c1(2x). Then

C ion~ t,x!5E0

`

dE~b1c11b2c2!e2 iEt/\,

~5!

b65E dx C~01 !c6* .

For weak kicks, i.e., whenapd /\!1 ~a being a characteristic size of the system!, we have

b1~E!'2b2~E!' i ~pd /\!dj ,E ,~6!

ub1~E!u2'ub2~E!u2,

so that the photoelectron spectra are almost identical tospectrum of the dipole matrix elements of the field-frHamiltonian, while photoelectrons are almost equally liketo propagate in both directions. To avoid misunderstandinit is important to keep in mind that even a weak kick, i.e.d pulse of a small area, may create photoelectrons with vhigh energy and momentum, although rarely. Indeed, theergy spectrum of photoelectrons, according to Eq.~6!, isclose to the dipole momentum spectrum of the QW, whspreads, strictly speaking, to infinitely high energy~note alsothat the energy spectrum of ad function is flat!. For example,under the conditions of Fig. 2 below, the kinetic energyabout 4.6% of emitted electrons is larger than the depththe quantum well.

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492 PRA 59P. L. SHKOLNIKOV, A. E. KAPLAN, AND S. F. STRAUB

This approximate ‘‘forward-backward’’spectralsymme-try holds for strong kicks as well, although becomes leexact. Combined with the above choice of the eigenfutions, it leads to thespatialsymmetry of the expanding photoelectron cloud. For detailed consideration, we now turntwo particular quantum systems.

(i) One-dimensional quantum well (QW).We consider aquantum well in the idealization whereby no solid-statefects are taken into account; we believe, however, thatqualitative picture we predict here will largely hold for a resemiconductor structures. A 1D quantum well in thex axis isdefined by the potentialU52U05const,0 at uxu<a, and50 elsewhere. We choosec1 as a plane wave that propagates from left to right, with both incident and reflectewaves on the left of the well, and only one~transmitted! rightto it @8#. Hence

c1~x,a!5eixkE1Ae2 ixkE,

c1~ uxu<a!5B1eixkU1B2e2 ixkU,~7!

c1~x.a!5CeikEx,

kE5A2mE/\, kU5A2m~E1U0!/\.

Similarly, c2(x)5c1(2x) propagates from right to leftThe constantsA, B6 , andC are found from the continuity oc6 anddc6 /dx at x56a as

A5k02~gU* 2gU!gE

2D21,

B652kE~kU6kE!gEgU61D21,

~8!C54kEkUgE

2D21,

D5~kU1kE!2gE22~kU2kE!2gE

22k05A2mU0/\,

gE~U ![e2 iakE~U !,

and uAu21uCu251. The QW is fully ‘‘transparent,’’uCu2

51, if

2akU5np, n integer; ~9!

in this case, the amplitude of the wave function insidewell is maximal,uc6umax

2 5@11k04 sin2(2akU)/4kU

2 kE2 #21. The

respective resonant energies

En5~n/S!22U0 , n>S[2ak0 /p ~10!

coincide with the positive eigenenergies of an infinitely hiwell of the width 2a, whose walls begin at the energy2U0 .~Scharacterizes the binding strength of the QW, as@S#11 isthe number of the bound states in the well;@S# is the integerpart of S.! Maxima of the QW transparency, in effect ‘‘continuum shadows’’ of the QW, are at the core of the bunchlionization of the QW. This effect, however, is not simplyconsequence of the transparency resonances@Eq. ~10!#. In-deed, while even and oddn in Eq. ~10! both correspond tothe zero reflectivity of a QW, they play drastically differerole for d ionization, due to the following easily verifiablselection rule:ud1,En

u2 zeros out for evenn and reaches loca

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maxima at approximatelyEn odd ~again, in full analogy withthe dipole selection rules in the infinite QW!. „For a QWwith @S#11 even, an extra maximum appears in the dipspectrum just above the bottom of the ionization continuubelow the lowest resonance allowed by Eq.~10!, as a resultof the fact that the continuous functionud1,Eu2 is equal tozero at bothE50 and at the lowest ‘‘regular’’ resonance.…

Weak d kicks simply map this structure into the photoelectron energy spectrum. As the kick area grows, the sptral maxima and minima shift from Eq.~10! to higher~lower!energies for forward-~backward-! moving electrons—the ef-fect that could be viewed as ad-ionization analog of theStark shift. Figure 1 shows that the momentakU(6)min forwhich ub6(E)u25min, depend almost linearly on the~nor-malized! momentum transferq5pd a/\:

akUmin~6 !'6q1np/2,

~11!n5nmin ,nmin12, . . . ,

wherenmin is the smallest evenn allowed by Eq.~10!. Themaxima ~except for the above-mentioned near-bottom o!follow that pattern as well, with more pronounced but dimiishing oscillations. Physically, this means that the velocitof the photoelectron bunches are determined, for a giQW, by the pulse area, so that Eq.~11! might be used tomeasure areas of ultrashort pulses.~The oscillations in Fig. 1are reminiscent of the behavior of quasienergies in a sudressed two-level system@9#.! While Fig. 1 ~and Fig. 2 be-low! has been obtained for a QW with@S#57, we haveobserved qualitatively similar pattern for 1<@S#<30. Equa-tion ~11! becomes almost obvious for kicks so strong thmost of C ion comes from the high-energy eigenfunctiowhich can be approximated by plane waves; thenb6(kU);c0(6kU2q), wherec0(k) is the initial wave function inthe momentum representation. As an important consequethe photoelectron spectrum after a strongd ionization pro-vides a ‘‘snapshot’’ ofc0 .

Thed pulse translates the structured ionization continuinto a spatial stratification of the photoelectron cloud. Thphenomenon may be calledcoherent ionization, because ofthe trans-spectral coherency of all the photoelectron specomponents, which have the same initial phase; the resu

FIG. 1. The momentakU for which the spectrum of the forwardpropagating photoelectrons for a 1D quantum well reachesmaxima~solid lines! and minima~dotted lines!, vs normalized mo-mentum transferq5pd a/\; @S#57 is assumed.

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PRA 59 493d IONIZATION: STRATIFIED SYMMETRICAL . . .

a well-ordered stratification of the photoelectron clouBased on Eq.~5!, Fig. 2 displays a sequence of weseparated moving peaks inuC ion(t,x)u2. The velocities of therespective electron bunches almost coincide withA2Emax/m,whereEmax are the energies of the local maxima of the phtoelectron energy spectraub6(E)u2, while the peak dividespropagate with the velocities that correspond to minimathose spectra. Moreover, the relative heights of the peakuC ion(t,x)u2 reflect the relative spectral amplitudes at thespective maxima. One can show that for weak kicks theyproportional to (uE0u1E)24E21/2, E0 being the ground-leveenergy.

Figure 2 also reflects the approximate backward-forwsymmetry of the photoelectron cloud. The visual similar

FIG. 2. Symmetry and stratification of the photoelectron cloin the d-kick ionization of a QW:~a! evolution from t50 to t5400t0 ; ~b! at t5400t0 . t0[h/U0 , @S#57, andq51; the ioniza-tion probability is;1025.

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of the emission parallel and antiparallel to the field is obous. Quantitatively, the ratio of the electron emission alothe force and in the opposite direction is just 1.2, with ttotal ionization probability forq51 used quite substantial—;1025. Even for q510 ~50% short of the kick that fullyionizes the well!, for which the ionization probability is;7% and the ground level is almost totally depopulated,forward-backward ratio is still as small as;3. As a result,the qualitative similarity remains, although one peak begdominating the picture.

(ii) 3D hydrogen and hydrogenlike atom (HA).The am-plitudesb(k) of the d ionization from the ground state cabe extracted from the calculations on atomic collisio@11,12#; for an arbitrary initial state, see Ref.@13#. Based onRefs.@12,6#, we writeC ion for the HA as follows~in atomicunits!:

C ion~ t,r !5E dk x~k!c2~k,r !exp~2 ik2t/2!, ~12!

c2~k,r !5~2p!23/2G~11 i /k!ep/2k

3eik•r1F1~2 i r21,1,2 ik•r2 ikr !, ~13!

b~k!516

~2p!3/2 ep/2kGS 12i

kD3F 11~q2k!2

11q22k222ik G i /k

3q22q•k~11 ik21!

@11~q2k!2#2~11q22k222ik !. ~14!

Here G is the gamma function, and1F1 is the confluenthypergeometrical function. Substituting Eqs.~13! and ~14!into Eq. ~12!, we deriveC ion as

C ion~ t,r !5E dk4 exp~2 ik2t/2!exp~2 ig/k!

p2k@12exp~22p/k!#exp$2k21tan21@2k/~11q22k2!#%

3@q22q•k~11 ik21!#

@11~q2k!2#2~11q22k222ik !exp~ ikr !1F1~11 ik21,1,ik•r1 ikr !,

~15!g[ 1

2 ln@11~q2k!2#@11~q1k!2#2 ln@11~q2k!2#;

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the Kummer transformation of the confluent hypergeomefunction was used in deriving Eqs.~15! to reduce the dependency on the direction ofr . Integrated over energy, the angular distribution of photoelectrons based on Eq.~14! is pre-sented in Ref.@7#, and displays an apparent approxima‘‘forward-backward’’ symmetry for a kick as strong asq50.2. The photoelectron spatial distribution based on~15! for a weak kick~Fig. 3! shows an apparent similaritwith Fig. 2: an approximate symmetry with regard to tfield direction and pronounced maxima~shells!. Now, how-ever, a shell does not necessarily correspond to elect

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with close velocities, as is the case for a QW. Moreover,spectra ofub(k)u2 @Eq. ~14!# do not contain any peaks foweak kicks. We believe that the shells in Fig. 2 are in farelated to the 3D nature of the process, and reflect the strspatiotemporal diffraction of the after-kick photoelectrocloud, which may result in a time-derivative behavior similto that of initially nonoscillating electromagnetic~EM!pulses@14#. We will address this problem elsewhere.

The feasibility of the predicted effects depends onavailability of ultrashort subcycle pulses. In this regard, ocould argue that propagating subcycle EM pulses of wh

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494 PRA 59P. L. SHKOLNIKOV, A. E. KAPLAN, AND S. F. STRAUB

ever length are unphysical, as they contain a dc componThis argument, however, applies strictly only to the far-fiearea, while many experiments are feasible in the near-fiarea. Moreover, although known half-cycle pulses~HCP’s!@4# do consist of two parts of the opposite signs, practicaall the ionization is due to the first, strong and short, paOur computer simulations for a pulse fitted@10,14# to anexperimental HCP confirm that we can safely neglectHCP ‘‘tail.’’ Whether such pulses ared like depends furtheron the ratio of the pulse duration to the time scale ofaffected quantum system. For ionization, this time scaledetermined byEion : t ion5h/Eion . d ionization of the ground-state hydrogen atom, therefore, would require a pulse dtion of !;10216 sec, which is beyond reach~but could be-come feasible in the future with the methods proposedRef. @2#!. The effects considered here for the HA could, hoever, be observed in the ionization of Rydberg atoms, w

FIG. 3. Symmetry and stratification of the photoelectron cloin the d-kick ionization of a ground-state hydrogenlike atom at510 a.u. after the kick;q50.2 a.u., and the ionization probabilit'0.01.x andz are in atomic units of length.

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t ion in picoseconds, by existing subpicosecond HCP’s, use.g., the visualization technique of Ref.@15# in combinationwith time gating. Similarly, relatively long pulses can emlate ad kick for the ground-state ionization of a shallow QWMoreover, our simulations forfinite-width pulse ionizationshow many features of thed ionization even for the pulses along ash/U0 . Therefore, with, e.g.,U0510 meV, ;200-fspulses will be short enough to observe the characteristratification; this duration is close to that of already avaable HCP’s. An almostd-ionization-like picture would re-quire pulses about four times shorter; even shorter pulses@2#will be needed for deeper QW’s. The evolving stratified phtoelectron distribution might be observed using a combition of near-field microscopy with ultrafast time-resolvespectroscopy proposed recently for studying other spatiotporally localized electronic phenomena in semiconducQW’s ~see, e.g., Ref.@16#!. Since a particularly spectaculasymmetry is predicted for a relatively weak kick and, consquently, a relatively low degree of ionization, one may eploy additional techniques to improve the signal-to-noisetio by, e.g., cooling the QW structure.

In conclusion, we have discovered theoretically thatdionization of a one-dimensional QW by an ultrashort~d-like!electromagnetic pulse would result in an ordered symmecal stratification of the photoelectron cloud, as the traspectral coherence of thed-like pulse maps the structure othe QW ionization continuum into photoelectron spatioteporal distribution. A similarly ordered, symmetrical photelectron cloud would result fromd ionization of a 3D hydro-genlike atom. The predicted effects may initially be observexperimentally in Rydberg atoms or in QW’s with a lobinding potential.

This work was supported by AFOSR. The authors appciate useful discussions with I. Bialynicki-Birula.

ev.

d,

is,

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