How to understand the tunneling in attosecond experiment?

0.35 MB·

by  Ossama Kullie

How to understand the tunneling in attosecond experiment?
Bohr-EinsteinphotonboxGedanken experiment,tunnelingtimeandthewaveparticleduality
Ossama Kullie
Theoretical Physics, Institute for Physics,
Department of Mathematics and Natural Science, University of Kassel, Germany∗
7
1
0 The measurement of the tunneling time (T-time) intoday’s attosecond and strong field (low-frequency)
2
experiments,despiteitscontroversialdiscussion,offersafruitfulopportunitytounderstandtimemeasurement

n
and the time in quantum mechanics. In addition, as we will see in this work, a related controversial issue
a
J is the particulate nature of the radiation. Different models used to calculate the T-time will be discussed

8 in this work in relation to my model of real T-time, Phys. Rev. 92, 052118 (2015), where an intriguing
1
similaritytotheBohr-EinsteinphotonboxGedanken experimentwasfound. Thetunnelingprocessitselfis

] stillnotwell understood, but Iam arguingthatascattering mechanism (by thelaser wavepacket) offers a
h
p possibilitytounderstandthetunnelingprocessinthetunnelingregion. Thisisrelatedtothequestionabout
- thecorpuscularnatureoflightwhichiswidelydiscussedinmodernquantum opticsexperiments.
t
n
a Keywords: Attosecond physics, tunneling time and time measurement in attosecond experiments, time-
u
energy uncertainty relation, time and time-operator in quantum mechanics, Bohr-Einstein’s photon box
q
[ Gedanken experiment,multiphotonprocessing,Comptonscattering,wave-particleduality.

1
v
2 I. INTRODUCTION Newtonian time quantity, and hence does not obey an ordi-
1
0 nary TEUR. Nevertheless, Hilgevoord concluded in his work
5 There is no doubt that the advent of ’attophysics’ opens [13] that when looking to a time operator a distinction must
0
new perspectives inthe study of time resolved phenomena in bemadebetweentheuniversaltimecoordinate t,ac-number
.
1
atomicandmolecularphysics[1–5],thetunnelingprocessand likea space coordinate, and the dynamical timevariable of a
0
7 the tunneling time (T-time) in atoms and molecules [6–10]. physical system situated in space-time; i.e. clocks. Accord-
1
Attosecond science concerns primarily electronic motion and ingly in [10, 14] it was shown that the T-time is intrinsic,
:
v energytransportonatomicandmolecularscalesandisoffun- i.e. dynamicallyconnectedtothesystem(internalclock)after
i
X damental interest to physics ingeneral. The time-energy un- the classification of Busch [15, 16] and [17] (chap. 3). Fortu-
r certaintyrelation(TEUR)receivesanewbreathduetotheac- nately, Bauer’s introduction of a self-adjoint dynamical time
a
tual problems of quantum informationtheory and impressive operator in Diracs relativistic quantum theory [12, 18], sup-
progressoftheexperimentaltechniqueinquantumopticsand ports the results of [10]. In [18, 19] Bauer concluded that
atomicphysics[2,11]. Inmypreviouswork[10],Ishowedthat the dynamical time operator provides a straightforward ex-
usingtheTEUR(preciselythattimeandenergyareconjugate planation (within standard relativistic quantum mechanics)
variables) leads to a nice relation to determine the T-timein of the T-times, which is measured in the photoionization ex-
goodagreement withthe experimental findinginthe attosec- periments, compare the discussion in [14]. In this respect,
ondexperiment (forHeatom) [9],(1attosecond=10−18 sec- Bauer also rejects the claim of Dodonov [11] that no unam-
ond). TheT-timeandtimeitselfinquantummechanics(QM) biguousandgenerallyacceptedresultshavebeenobtainedfor
arecontroversial,andthereisstillcommonopinionthattime the time operator [18,20]. Moreover, Bauer showed [18] that
plays a role essentially different from the role of position in the Mandelstam-Tamm uncertainty associated with the ob-
quantum mechanics (although it is not in line with special servable Tˆ largely overestimates the internal time standard
relativity, [12]) and that time is a parameter, like a classical uncertainty asalreadydiscussedbyKullie[14].
A similar controversial issue to the time issue (and the
∗
Electronicmail: [email protected] T-timeandTEUR)isthewave-matterdualityandthepartic-
2
ulatenatureofthelight[21,22],sincetheEinsteinhypothesis
-0.1
-0.2
of the quanta as a carrierof hν based on the Planck hypoth- -0.3 -sqrt(Zeff F)
-0.4
-0.5
epthshieostcooonfrptwuhsaecsuqlguaiarvnehtnyizpbaoyttiohGnes.iosNfo.trhLigeeiwneainslelyrignsyt1eE9m2s6=fr[2oh3mν],.NaenTwdhteoinntd.eerAemds pot. energy in a.u.-------1000011-.......26987311 -2-sIqprtx(Zieff Fx)e,-xm(FhaB)=(xxma) xm(F)Ion. potential -Ip xe,+ xe,c
we will see in this work, the two issues closely appear in to- --11..45 AElteocmtriicc pfioetledn otifa lt,h -eZ leafsf/exr , -x F (F=0.07 a.u.)
day’s attosecond experiments (ASEs). Indeed, since the ap- --11..67 Veff=-Zeff/x-x F (F=0.07 a.u.)
--11..98 -2Ip Veff for F=Fa
pearanceofQMtimewascontroversial,thefamousexampleis 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.010.011.012.013.014.015.0
x axes in a.u.
the Bohr-Einstein weighing photon box Gedanken experiment
FIG. 1.
(BE-photon-box-GE).In[10]Ishowedwithasimpletunneling (Color online) Graphic display of the potential and the
effective potential curves, the two inner and outer points xe,± =
model that the tunneling in the attosecond experiment is in-
(Ip±δz)/2F and the barrier width dB = xe,+−xe,−, the “classi-
triguinglysimilartotheBE-photon-box-GE,wheretheformer calexit”pointxe,c=Ip/F andxm(F)=(Zeff/F)1/2thepositionat
can be seen as a realization to the later, with the electron as themaximumofthebarrierheighthB(xm),(notexa=xm(F =Fa)),
seetext.
a particle (instead of the photon) and an uncertainty in the
γ <1(field-)ionization happens by atunneling process (for
energybeingdeterminedfromthe(Coulomb)atomicpotential K
due to the electron beingdisturbed bythe fieldF,instead of F <Fa),whichmeansthattheelectrondoesnothaveenough
energy to ionize directly, and therefore it tunnels (or tunnel-
(the photon) being disturbed by the weighting process and,
ionizes)throughabarriermadebytheCoulombpotentialand
as a result, an uncertainty inthe gravitational potential [24],
theelectricfieldofthelaserpulseandescapesattheexitpoint
as shown by the famous proof of Bohr (see for example [25]
tothecontinuum, asshowninfig1,seethefollowingsection.
p. 132) to the uncertainty (or indeterminacy) of time in the
We pay attention to one important case study in attosecond
BE-photon-box-GE[15,16,24].
physics,theT-timemeasurementinASEperformedbyKeller
The T-time and the tunneling process itself in the ASEs
[7–9] and we will refer to it as the Keller ASE (KASE). In
arehotdebated,andthelaterisstillratherunresolvedpuzzle.
this experiment an elliptically polarized laser pulse is used
In the (low-frequency) ASEs the idea is to control the elec-
tronic motion by laser fields that are comparable in strength with ω0 = 0.0619au(λ = 736nm), the ellipticity parameter
ǫ = 0.87, while the electric field strengths are in the range
totheelectricfieldintheatom. Intoday’sexperimentsusual
intensities are ∼ 1014Wcm−2, for more details we refer to F =0.04−0.11andforHeatom Ip=0.90357au.
the tutorial [4–6, 26, 27]. In the majority of phenomena in
attosecond physics, one can separate the dynamics into ado-
II. THE TUNNELING TIME
main “inside” the atom, where atomic forces dominate, and
“outside”, where the laser force dominates, a two-step semi-
classical model, pioneered by Corkum [28]. Ionization as the Usually the tunneling process in the ASEs is explained
transition from“inside”to “outside” of the atom plays akey by a simple picture, like the one shown in fig 1 for the He
roleforattosecondphenomena. AkeyquantityistheKeldysh atom. It is based on the strong field approximation (SFA)
parameter[29], or Keldysh-Faisal-Reiss approximation [29–31]; for an intro-
ductory review see [32]. This simple picture is very useful in
γK = pF2Ipω0=τKω0, (1) explaining the experiment, although itisstrictly trueonly in
lengthgauge[33–35]. Physicalquantitiesareindependentofa
whereIp denotestheionizationpotentialofthesystem(atom gauge transformationaslongasexact equations (orthesame
or molecule), ω0 is the central circular frequency of the laser ordersofapproximation[36])areemployed. Indeed,thelength
pulseorthelaserwavepacket(LWP)andF,throughout this gaugeorthedipoleapproximationoftheinteractionHamilto-
work,stands forthepeakelectricfieldstrengthatmaximum, nian due to the G¨oppert-Mayer gauge-transformation (when
andτK denotestheKeldyshtime. Hereafterinthiswork(un- long-wavelength approximation applied) also has the advan-
less itis clear), atomic units areused, where ~=m=e=1, tage that it leads to an expression for the interaction energy
the Planck constant, the electron mass and the unit charge involving mathematical quantities, each of which has a ready
areallsetto1. Atvalues γ >1one expects predominantly physicalinterpretation[37].
K
photo-ionization or multiphoton ionization (MPI), while at InthetunnelingprocessintheASEs,accordingtotheSFA
3
(seealso[38]),theelectrontunnelsandescapesthebarrierre- with the real T-time point of view, with the real time of the
gionattheexitpointxe,+,seefig1,withapproximatelyzero FPI description of [9](although Sokolovski [44], [17] chap. 7,
kinetic energy, more precisely the electron velocity along the argued that T-time described by the FPI is imaginary), and
fielddirectioniszeroandnegligibleintheotherdirections. In theentropicformulation(orthestatisticalapproach)of(real)
[10](hereafterKulliemodel)weshowedthattheuncertaintyin T-time of Demir [46], the relation in eq 2 presents a real T-
theenergy,whichisrelatedtothehightofthebarrierhB(xm), time model which explains the T-time in KASE in a good
canbequantitatively discernedfromtheatomicpotential en- agreementwiththeexperimentalfinding[10,14],comparefig
ergyattheexitpoint∆E∼|V(xe)|=|−Zeff/xe|forarbitrary 2, 3. Although the model is simple, it is important in the
strengths F ≤Fa, where Zeff is the nuclear effective charge quest to answer the question: how to understand the T-time
andFa=Ip2/(4Zeff)istheatomicfieldstrength[39,40][41]. andtunnelingprocess,thetimemeasurementandthetimein
WiththeTEUR,∆E·∆T ≤1/2,oneobtainsthesymmetrical quantum mechanics [14].
(ortotal)T-time[10]: Secondly,thetreatmentin[10,14]benefitsfromtheinter-
nal(intrinsicordynamical)timepointofview(internalclock,
1 1 1
τT,sym=τT,i+τT,d = 2(cid:18)(Ip+δz) + (Ip−δz)(cid:19), (2) [17] chap. 3), this requires one to choose a reference point
[2, 13, 24], which can be at best determined by the (natural)
whereδz = Ip2−4ZeffF. Therelationineq 2, besides the internal properties of quantum mechanical systems (e.g. ion-
q
mathematical simplicity, aids a conceptual reasoning [10, 14]
ization potential Ip). This is important as it enables one to
and the discussion further below. The physical reasoning of
identify or map the internal delay time (a time interval) as a
thisrelationisthefollowing: thebarrieritselfcausesadelaying
delaytimeinanexternalclockwithout,certainly,contraction
timeτ ,whichisthetimedelaywithrespecttotheionization
T,d ordilationofthetime(scaleunity)intervaloftheclocks (i.e.
at atomic field strength Fa, where the barrier is absent (i.e. no effects like in the relativistic theory.) We also note that
thebarrierheight,thebarrierwidthandδz arezero),itisthe theinternalclockortheintrinsictimepointofviewissimilar
timedurationto pass the barrierregion(between xe,−,xe,+) to what occurs in special relativity, where a moving particle
andescape atthe exitpointxe,+ tothe continuum, formore has its own time in its inertial frame, which differs from the
detailssee[10]. Thefirsttermτ ineq2isthetimeneeded
T,i timefromtheviewpointofother inertialframes,asdiscussed
to reach the entrance point xe,− from the initial point xi, in[14]. Finallywementionthatsomeauthors[42,43]usethe
compare fig 1. At the limit F → Fa, δz → 0 and the total notation ts =tR+itT, wherets refersto the solution ofthe
timebecomes the ionization timeτT,sym = I1p at the atomic saddlepointequation,tR ortherealpartofts denotedasthe
fieldstrengthFa. ionizationtime(aftertunneling)andtheimaginaryparttT as
TheT-timeisacontroversialissuefromdifferentpointsof
the T-time. The partition of ts this way, in a real part for
view,aswewilldiscussinthefollowingsubsections.
ionization andanimaginarypartforthe tunneling, lets some
questionsbeopened. WediscusthisinthenextsubsecIIB.It
isworthwhileto mention (seediscussionin[10])that ineq 2,
A. A real or an imaginary quantity
δz becomes imaginary δim,z =iδz (i the imaginary number)
forF ≥Fa andthatistheabove-thresholdregion.
First, the most reasonable argument is that that T-time
isarealquantity,asclarifiedbySteinberg[17](chap. 11)and
Bu¨ttiker [17] (chap. 9), although many authors claim, and it
B. Many different approaches
is widely accepted, that it is an imaginary quantity [42, 43],
[44](and[17]chap. 7). Theimaginarytunnelingtimepointof
viewreliesonthefactthat thetunnelingisclassicallyforbid- The scattering theory concept iswidely used to calculate
den. Although that is important, it is not instructive, in the the T-time from the scattering time (for example Bu¨ttiker-
case it obscures an insight and otherwise accessible concep- Landauer or Pollak-Millertime, for details see[49,50]. How-
tual understanding [45]. One has to mention that a complex ever, following Collins [49] this is not justified. Recently
time point of view (i.e. real and imaginary parts) would not Landsman [9] showed that these time approximations are in
change the conclusionofour works,because thecrucialpoint disagreementwiththeexperimentalfindingofKASE,andthat
in our discussion is a real part of the T-time. In agreement the FPI with a coarse graining procedure fits well with the
4
120 120
110 110
100 Eτxp Zdeaftfa 100 EExxpp LBaongdes man
Time (as) 5678900000 τττTTTT,,,,dn nduu,mm Z ,Z eZfeKeffCfuffllKleCimuelleelmineteinti Time (as) 5678900000 τττnTTu,,dnm,u ,mZ ,Te ZfDfeCSflfeECmQleenm tie nti
40 40
30 30
20 20
10 10
00.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0 1 2 3 4 5 6 7 8
Field strength (au) Intensity (au)
FIG. 2. FIG. 3.
(Color online) Graphic display of tunneling time verti- (Color online)Graphicdisplayof tunnelingtimevertical
cal axes (in attosecond) vs field strength F in au. τT,d eq 2 (red axes (in attosecond) vs the intensity in au. τT,d,τnum are plotted
ZeffKullie,blueZeffClemanti,andτnumeq3(orangeZeffKullie, at the experimental value of [9]. τT,d (upper, blue) eq 2, and τnum
brownZeffClemanti). τT,d,τnum areplottedwithF asafreevari- (lower, brown) eq3withZeffClemanti, seetext. Theblacksquares
able. Experimentalvaluesfrom[9]. arenumericalintegrationofTDSEQform[47],datakindlysentbyI.A.
Ivanov. Thepointswitherrorbarsareexperimentaldata,green(large
circles) from Landsman [9] and the light blue (small circles) from R.
Boge[48],datakindlysentbyA.S.LandsmanandR.Boge.
KASE measurement data. Collins’s most critical point [49],
eq2presentsadelaytimerelativetothelimitatatomicfield
which addresses the question: how long does it take for an
electron to tunnel through a barrier, was the conclusion that strength τT,d(F = Fa) = 1/(2Ip) ≡ τT,d,Fa, by subtracting
thelatterfromformerweget
thescattering time,defined throughtheinverseofthetransi-
tpioorntpbreohbaavbioilritoyfmthaetreilxe,ctisronno’tsrteulnanteedlintgo.tIhteisdyanssaomciiacateldtrwanitsh- τT,d−τT,d,Fa = 2(Ip1−δz) − 21Ip = 2(I(pδI−pz)δz) ≡τnum(3)
thefinite lifetimeanddecay ofmetastable states, inthiscase δ δk 1 δ k
= +··· ···=
2Ip2 2Ipk+1 2Ip (cid:18)Ip(cid:19)
beingthe tunneling electrons treated asquasi-particles which kX=1
aredecaying froma state on one sideof the potential barrier
whereinthesecondlineanexpansionoftheform(1−x)−1=
into another state on the opposite side of the barrier. Since
1+x+x2+..., x = (δ/Ip), is used, and 1/(2Ip) is the zero
thisquantifiesthedecayofmetastablestatesandnotthetran-
order term (k = 0)). In fig 2 the T-time τT,d of eq 2 and
sittimeofanelectronacrossabarrier,thistimecanbequite
τnum ofeq3arecomparedwithexperimental dataof[9],the
large and is a steady-state picture which does not reflect the
field strength F is the free variable and runs between 0.04−
dynamical nature ofthe tunnelingparticle(s). Andaccording
0.12, the small dark square (at the right lower corner) marks
toCollins,itcanbeseenonthebasisofthetime-independent
Fa, where τnum becomes zero and τT,d = 1/2Ip as already
pictureandbyanalogywithaparticledecay,thatthescatter-
mentioned. The crucial difference between τnum and τT,d is
ingtimerepresents ameantimeinwhichacertainlikelihood
thattheformertendstozeroatatomicfieldstrength(δz(Fa)=
of a tunneling event may take place. This does not reflect
0), limF→Faτnum = 0, which can happen only numerically,
theactual timeofatunnelingprocess[49]. Asimilarpointof whereas quantum mechanically limF→FaτT,d = 2I1p is the
viewwasconsideredbyFockandKrylov[51],inregardstothe
second part of the total ionization time, eq 2, at atomic field
lifetimeandTEUR,seediscussionbyAharonov[52]. strength (τT,sysm = 2I1p + 2I1p = I1p), as discussed in [10].
Collinsthenshowed(usingaGaussianwavepacketinthe This can be seen infig2 forZeff =1.6875 of Clementi [53],
simulation),thatthephase-timeisoverallthebestonetouse where Fa ≈ 0.12au, which has to be chosen in this region
whenmomentumskewingoftheinitialwavepacketisnotsig- and matches the experiment. Zeff = 1.375 of Kullie (Fa ≈
nificant. Unfortunately,in[14]weshowedfirstthatthephase 0.14 au) [54] is the better choice in the region for small field
timeinattosecond experiment leads tothe KeldyshtimeτK, strengths,fordetaileddiscussionsee[10].
eq 1 (an approximation that neglects the effect of the core In his work [19], titted the problem of time in quantum
potential, and could be important for the evolution of the mechanics, Bauer mentioned that time interred in the time
wave packet), which is far from the experimental finding [9]. dependentSchro¨dingerequation(TDSEQ)hastobeidentified
Andsecondthatatime-delayrequiresustochooseareference with a parameter (parametric time tparm) that corresponds
(point);delaysinnumericalsimulationcanreferinprincipleto tothetimecoordinateofthelaboratoryframeofreferenceas
anyarbitrarilychosenreference[2,14]. Indeed,becauseτT,din claimed by Briggs [55]. Although τnum differs qualitatively
5
from a τparm (defined below), because it is constructed form matrixtheory)tocalculatetheT-timeintheASEs(theatto-
the dynamical (internal) time of the systems, it looks to be clock),arguedthattheARMrequiressufficientlythicktunnel-
identicalwiththeparametrictimeτparm,i.e. onecanwrite ingbarriers[42,57,58]. TorlinaelalcalculatedtheT-timeof
the(field-)ionizedelectronfromthegroundstateoftheHydro-
1
τparm=τnum=tT,d− 2Ip, (4) gen atom, which isexposed to an attosecond laser pulse [42].
TheyalsoclaimthatT-timeisimaginary(seeIIA)andnoreal
whereτparmdenotesatimeintervalsuchthatτT,d =τparm+ tunneling delay time is associated with the tunnel-ionization
1/(2Ip) and 1/(2Ip) is the ionization time (at atomic field [42], i.e. for F < Fa, and optical tunneling is instantaneous.
strength Fa). Thus, the identification of the timein the TD- Instantaneous heremeansthatarealT-timeiszero,although
SEQasaparametrictimeindicatesthelackofapropermap- itshould, infact, representthe timeof areal dynamical pro-
ping of the time from the internal (time-frame) to an exter- cess and that at the tunnel exit the electron is far from the
nal time (time-frame) and not an inherent property of the nucleus,i.e. thetunneldistanceisnotnegligibleasconfirmed
timeinTDSEQi.e. inquantummechanics,inagreementwith experimentally by [59]. However, it can be easily shown that
HilgevoordpointofviewandBushclassificationasmentioned therequestofsufficientlythicktunnelingbarrierscorresponds
in sec.I. Thus; so τnum is identical with the time interring totheregionofsmallfieldstrength, forwhich(at opticalfre-
the TDSEQ, when mapping the dynamical (internal) timeto quencies) most likely γK ∼ F1 > 1, see fig 1 and eq 1, and
an external (laboratory) time by ignoring the reference point that is the MPI region, where tunneling is not probable. In
(i.e. neglecting1/(2Ip)),asclearlyseenineq4(andinfigs2, addition, it could be that the imaginary T-time of Torilina
3, see below.) Whereas τT,d counts the internal time (inter- et al is related to the phase time of Collins [49] as discussed
val)andtransformsittotheexternaltime(interval,measure- above, see also [14]. In this case one expects that the phase
ment data) when the electron moves from inner to the outer timeoftheelectronicwavepacket,likethephasevelocity,has
region (the tunneling or ionization process) due to the inter- no real physical significance, although it can be viewed as a
actionwiththelaserfield. Contrarytothe(parametric)time characteristicoftheevolutionofthewavepacketinthesense
of measurement thought by Aharonov [52], that the time of describedbyMessiah[60],seediscussionin[14].
measurement belongs to the observing apparatus, which has
been corrected in his recent work [24]. Note a time interval
1. In depth discussion
refers to a clock, which measures the time quantity with re-
spect to a reference point, whereas a time variable denotes a Apoint,whichIthinkisimportanthere,isthat thecalculations
ofTorlinaelalshouldbecomparedwiththetime-of-arrival([17]chap
time quantity of any type, see subsec IIA 2nd part. And as
3, [61, 62]), [63]. The later concerns the subsequent propagation of
discussedin[14]perhapsonlytheclassicalNewtoniantimeis thetunnel-ionizedelectron (seebelow). Onenotes thatinattosecond
a parametric (external non-dynamical) type of time. In fig 3 andstrong-fieldsciencethe(tunnel-)ionizationprocesscannotbeunder-
stoodinitsconventionalformknowninquantumoptics[64]. Important
the experimental data of [48] (light blue) and [9] (green) are
here(compare[42]): FirstthecalculationoftheionizationtimeofTor-
displayed, they are from the same set of experimental data, lina et al (hereafter Torlina model) relies on the dynamic interaction
but arenewed calibrationprocedure of the field strength was oftheoutgoingelectronwiththeremainingcore,i.e. theeffectsofthe
long-rangepotential(startingattheexitpoint),andthattheionization
used in the later. One sees that the T-time points result-
isnotyetcompletedatthemomenttheelectronexitsthetunnelingbar-
ing from numerical integration of the TDSEQ of Ivanov [47] rier(notethedifferentdefinitionsofthebarrier,seebelow). Thislooks
(black squares), lie somehow below the experimental data of similartotheconceptofthetime-of-arrivalmethodandtheobservable-
typeoftimeaftertheclassificationofBusch([17]chap3,whereinthis
[48],aftershiftingtheexperimentaldata(donebyLandsman)
casetheatomactsasasource(confiningtheelectronicwavepacketup
one sees that ourτnum eq 3, where the insertedF values are toRe(t)=0)[17]chap. 1,andwheretherealtimecounting,asTorlina
etalalsoargued,startsatthetunnelexitoftheirmodel(which,how-
taken from the experiment, lies below the experimental data
ever,definestheborderbetweeninnerandouterregions.) Indeed,the
of Landsman [9] inasimilarway the points of Ivanov [47] lie
potentialattheexitpointxexit=xe,+definestheenergyuncertainty
belowtheexperimental dataofBoge[48][56]. inthemodelofKullie[10],hereweindicateacommon point(consid-
eration)betweenthetwomodels,andthatthepotentialenergyatthe
Nevertheless,Collinsconclusionshowsthattheusualscat-
exit point is the central quantity, although the exit points xe,+ and
teringconceptswhenusedfortheASEsarebestsuitedtothe xe,c,andtheassignmentoftheinnerandouterregionsinthetunneling
MPIregion,wheremomentum skewingoftheelectronicwave process aredifferentinthetwo models, seefig1. Wearguethat, the
electronescapesthetunnelexitxe,+ (wheretheenergyuncertaintyis
packet is small [49], compare subsec IIC. Therefore, it is not
defined),xe,+>xe,−>xi (xiistheinitialposition,seefig1)isreal
surprisingthat Torlinael al, when usingARM (analytical R- and henceit isreached by areal time, and in theouter region (after
6
xe,+,whereaccordingtheSFAtheelectronisfree)theeffectofthelong assumes that xe,c = Ip/F is roughly corresponds to the exit of the
rangepotentialdecreasesrapidly,i.e. themeasurementdataof[9]cor- tunnelingbarrier. Hence,thetunnelingprocesshasadifferentmeaning
respondstothecrossingofthebarrierregion(betweenxe,−andxe,+) betweenthetwomodels,whereTorlinaetaldefineitbyanimaginary
whenthetunnelingprocessoccurs[9,10],inthelengthgaugepicture. T-time,andbyrealandimaginarytrajectoriesintheinsideandoutside
OurrealT-timeτT,d representsadelaytimewithrespecttothenat- regions,where“tunnel-exit”isacomplexintegrablesingularitypointof
ural reference point (i.e. ionization at the atomic field strength Fa), thepotential. Thiscan becompatiblewiththevelocitygauge, where
whichiscertainlyreal,sincetheexitpointxe,+≥xe,−>xi(equality thebarrierisabsent,ornotaphysicalone(thebarrierregioniscrossed
forFa)isreal,andquantummechanicallysuchatunneling/ionization inanimaginarytimeelapse). Indeed,asmentionedabovethedefinition
dynamicaltimecannotbezeroorpurelyimaginary. of thebarrier and thebarrier regionsin attosecond physicsaregauge
Andsecond,theauthorsof[42]claimthatalthoughthemeasured dependent[33]. Thissuggests(butitneedsascrutinytobeaccepted)
quantity(theelectronmomentum)isreal,thetrajectoriesintheARM thattheTorlinamodel(whenappliedtothetunnelingregionγK <1)
method are not classical, in thesense that thetrajectories have both isquantitativelyequivalenttothemodelofKullie(providedbothmod-
realandimaginarycomponentsallthewaytothedetector,wherethey elshavethesameordersofapproximation,seediscussionbelowabout
claimthattherealpartofthetrajectorystartsneartheoriginwithout theinitialtime),althoughthetwomodelsarecompletelydifferentcon-
an explicitdefinition[65]. This, inturn, shows that inthe tunneling cepts,andthelengthgauge(usedinKulliemodel)hastheadvantageof
process,realandimaginarycomponentsofatrajectoryand(hence)real presentingaclearphysicalpicture[14,18],becausenophysicalquantity
andimaginarycomponentsoftimecanexist(quantummechanically)in correspondstothevectorpotentialinthevelocitygauge. Note,Torlina
both the insideand outside regions (under the barrier and after tun- etalalsoconcludedthatthetotalionizationratedependsonlyonwhat
neling,despitethedifferencesinthedefinitionoftheseregions). Thus happenstotheelectronwhileitistunneling[65],hencetheconfusionis
thebarrierregionisnotnecessarycapturedsolelybyanimaginarytime mainlyduetothedifferentdefinitionsofthetunnelingprocessandthe
componentalthoughitisclassicallyforbidden. barrierregionortheinsideandoutsideregions.
AnotherpointisthattheworkofTorlinaetalisanoperativecon-
This leads to the conclusion, that the partition ts = tR+itT
cept(comparebelowsubsecIIC)itdoesnottouchtheconceptoftime
mentionedaboveinsubsecIIA,i.e. animaginaryparttT fortunneling
(anditscontroversialdiscussion)inquantummechanics. Indeed,itas-
and a real part tR for ionization after the tunneling, is at least not
sumesimplicitlyaparametrictime,pointofview,compareeq4. Itis
uniqueifacceptable, apart form thefact thattimedelayrequiresone
worthwhiletomentionthatthereexistsclassicalprocedures,whichare
to choose a reference system [14, 27]. This partition is only in line
withtheclassicalpointofviewthatthedynamicsunderthebarrieris quantitativelyequivalenttothetheTorlinamodel,e.g. thepropagating
ofclassicaltrajectorieswithinatwo-stepmodelor(CCSFA)whichin-
classicallyforbidden,hencenorealT-timecanexistforthedynamicsin
volvesclassicalMonteCarlo-typesimulations[65]. IntheKulliemodel
thisregion,i.e. thetunnelingprocess,althoughquantummechanically
thereisnocomplicatedcalculationwithawave function,whichis,no
tunneling is a possible physical process. The partition goes back to
doubt,anadvantageoftheTorlinamodel. Alsonotrajectories(realor
Perelomov et al [38, 66, 67] (hereafter PPT model), it was based on
imaginary)areused,thecalculationsareachievedintime-energyspace
theargumentthatthe(semi-)classicaltrajectoryisdeterminedbythe
initialconditionsx(t0)=0,v(t0)=ik,k =(2Ip)1/2, wheret0 isthe basedontheTEUR.However,thegeometryofthebarrierisneeded. It
coversthetimeandT-timeconceptsinquantummechanics,andmakes
timewhenthefieldreachesitsmaximum.Thetimewaschosensuchthat
useof,oroffersaqualitativeorquantitativeconnectionto,thedifferent
theoriginfortherealtimeistheinstantwhenvexit=v(tR=0)=0,
fundamentalissuesofthequantummechanics,i.e. theBE-photon-box-
i.e. tR = 0 when the particle emerges from the barrier, but at the
same time [67] the exit point is real, xexit = x(tR = 0) = xc 6= 0, GE,thedoubleslitexperimentandtheTEUR.
Finally,thesituationlookssimilartotheequivalencebetweenmy
wherexc=Ip/F >>xiisusuallycalledtheclassicalexitpoint. This
model and the FPI description of [9] as mentioned in subsec IIA. At
makestheassumptionthattheT-timeisapurelyimaginaryquantity
thispoint,onehastomentionthatamodelcalledanentropicformu-
questionable. Tunnelingisaquantummechanicaleffectandtomybest
lationofT-time[46]exists,themodelisalsobasedontheTEURand
knowledgethereexistsnophysicalrestriction,whichforcesustoassume
theauthorsclaimthattheirmodeldescribesarealT-time,whichisin
that the dynamical time to overcome the barrier region, should be a
goodagreementwithKulliemodelandtheFPImodelof[9]. However,
purelyimaginaryquantity,whichisaclassicalstandpointanddidnot
crucialpointsstillhavetobeclarified,e.g. choosingthereferencesys-
reflectthequantumnatureoftheelectron’smotioninthebarrierregion,
temtocalculatethedelaytime,wherepossiblyitisoneofthesources
wherecertainlytheenergyconservationshouldnotbeviolated,itturns
out that afullunderstandingofthe tunnelingprocess in theASEsis of thenegative timeintheTorlinamodel[42] for F > Fa (Fa isthe
atomicfieldstrength). Indeed,onecouldconsideritasanartifactef-
stillwaiting.
fect,becauseTorlinaetalevaluatetheirtimestartingatthepeakofthe
We note that in the PPT model, used by Torlina et al [42], one
spectrum, regardlessthenatural parametersofthesystem undercon-
definestheinnerregionthroughtheassumptionthattheapproximate
sideration(theHydrogenatom),i.e. theionizationpotential(possibly
wave function ψin can besubstituted by the(field-free) ground-state
with a Stark shift) and the atomic field strength, where the Above-
wavefunction[68].Fromthisoneseesthattheborderbetweentheinner
threshold-ionizationprocessstarts. Theinitialrealtimezeroassump-
andouterregionsinthismodeliscomparablewiththeentrancepoint
xe,−=(Ip−δz)/(2F)≈Ip/(2F)=x˜eofthemodelofKullie[10,14] tiont0=0atthepeakintensity,intheattoclockoftheTorlinamodel
was criticizedbyZimmermanet al,supplementalinformation[69]. In
asseeninfig1. ForF =Fa,xa=xe,±=Ip/2Fa,thiscase(ionization
asδz=0)matchestotherealparttimeasdefinedbyTorlina,however myunderstanding,addingthetermτFa =1/(2Ip),seeeqs3,4,tothe
Torlinamodel,canremedytheinitialtimeproblem(andthenegative
xadiffersfromtheclassicalexitpointxe,c=Ip/F =2xaassumedin
time),whichenablesonetotakethesystemparametersinaccountand
Torlinamodel.Asasidenote,thismeansthatτT,i=1/(2Ip+δ)canbe
chosenasaninitialtimet0=τT,ifortheTorlinamodel,seediscussion countsthedelay(therealtimeparttR ofts)relativetotheionization
below. x˜eisalsocompatiblewiththeirchoiceoftheradiusofthesphere timeτFa atatomicfiledstrength. Usingtheinitialtimet0=0atthe
peakofthespectrumisequivalenttoeq4(comparefig2,3),thusthe
Tsehpaatramtienagntshtehiantnetrheansdo-ocuatlleerdreugniodners-1th/κe-b=ar(r2iIepr)r<eg<iona<as<giIvpe/nFin[57fi]g. realpartofthecalculatedtimebyTorlinatTRorlina=τnumorinother
words,oneobtainsforthetimeoftheTorlinamodel
1(betweenxe,− andxe,+ oftheKulliemodel)belongsroughlytothe
continuum(oroutsideregion)oftheTorlinamodel,althoughthemodel
Re(τs)≡tTRorlina+t0=τnum+t0= (t0=τFa)
7
C. The different regimes
witht0 =τd,Fa =1/(2Ip). Re(τs)isthencorrespondstotheactual
T-timeτT,d,seeeq2. Thisbecomesmoreapparent,whenonecompares
theresult of thenumerical integration of theTDSEQ (blacksquares) Despite the different views concerning the tunneling pro-
withτT,d,τnum infig3.
cess,itisstillpuzzlingandrathernotwellunderstood. Inthe
Anotherreasonablepossibility,toadjustthechoosingoftheinitial
time, isto consider xe,− as theinitialpositionwhere theprocess (of SFA oneusually assumes aphotoelectric effect mechanism or
tunneling or ionization) starts (which differs from the interaction in- multiphotonabsorption(multiphotonprocessing),itisimpor-
stant,thefirststepseediscussionin[10])andthustoconsiderthetime
toreachxe,− astheinitialrealtimeatthepeakintensity,andhence tantintheregimeoflargeKeldyshparameterγk >1andalso
τT,i =1/(2(Ip+δz))=t0, eq2, serves asareal initialtimefor the isthe usual process inquantum optics experiments, although
countingoftherealtimebytheTorilnamodel. Theimportanceofthis in quantum optics (where the field strength is weak) usually
followsfromthefactthatonlyaftertheinteraction[10],thepropaga-
an operational point of view is used [71, 72], based on the
tionvectoroftheelectronwavefunctionisidenticaltothepropagation
vectoroftheopticalvectorpotential. Inaddition,itiscompatible,as second quantization formalism,for thetheoretical treatments
discussedabove,withthedivisioninside(ψin)andoutsideofthebarrier and understanding of the experimental findings. It is initi-
regionatthepointxe,−inconnectiontotheTorlinaapproximation. In
atedbyRonyGlauber[72][Aphotoniswhataphotodetector
otherwordsτT,i=t0canbeconsideredastherealpartoftheT-time
thatcorrespondtheimaginarypartoftheT-timeofTorlinaetalinthe detects], which is in line with the operational time-of-arrival
insideregion,ortheabovementionedexpressionoftsshouldbewritten concept mentionedabove.
intheform
Taking into consideration the view of Collins [49] as dis-
τs=tTRorlina+t0+itT, witht0=τT,i=1/(2(Ip+δz)) cussed above, we prefer, see sec III, another point of view in
the region of small Keldysh parameter γk < 1 (the tunnel-
i.e.atthestart(afterthefirststep[10],theinteractionstep)thepointer
ingregime),andarguethatascatteringmechanism(orelastic
oftheattoclockpointsatt0=τT,i.AtF =Fathisbecomet0=τFa =
1/(2Ip),δz=0,wherethedifferencetoτT,iissmallbecauseδzissmall collision) is involved in the tunneling process in attosecond
inthetunnelingregion. experiments suchKASE,wherealargenumberofphotonare
Moreover,theimaginarypartofthetimeisthenthecharacteristic
involved,whichmeansthattheelectronrecoilduetothescat-
timeofevolutionofthewavefunction,inthesensebroughtbyMessiah
[60](seediscussionin[14]),andalsocouldbepresentafterthetunneling tering with LWP, with a drift along the radiation direction
andnotonlyinsidethebarrier. [73, 74]. It is worthwhile to mention that even the Compton
In the PPT model, used by Torlina et al and others, the time
effect can be explained by a semiclassical nonrelativistic ap-
ts = tR+itT or the solution of the saddle point equation, defines
proximation, Schro¨dinger [75], see [76] p. 222-225, where the
thestartingpointofthe(semi-classical)trajectory[65]. Animaginary
timecomponenttodescribetheT-timeseemstobearesultoftheexter- electron is described by a wave function. Indeed, the exper-
naltimeframe. Adynamicaltimeshouldbeconnectedtothesystem,
imental investigation of [73, 74] showed a type of nonlinear
i.e. oneshoulduseaninternalclock. Thesolutionofthesaddlepoint
equationiscomplex(i.e. oftheform Re(t)+Im(t))butthisdidnot Comptonscatteringathighlaserintensities,wheremanypho-
means, that theT-timeistheimaginary part of thesolution and the tonsparticipateinasinglescattering. Theoretically,anearlier
realpartistheionizationtimeaftertunneling.Infact,Perelomovetal
workofEberly[77]proposedanexperimentwithhighphoton
statedintheirwork[38]thattheionizationtimeatatomicfiledstrength
Faoccursinatimecomparablewiththeatomictime,whichisrealand densityρL(Watt/cm3)intheopticalfrequencyrangeω,where
should serve as an initial point or the reference point (τFa or τT,i) theeffecttobeobservedisthatthephotonsofthelaserbeam
of theattoclock, i.e. for thecounting oftheT-timeor theionization
interact (collectively) simultaneously with the electron, and
timeinattosecond experiment,becauseat thisfieldstrengththebar-
rierwidthequalszero(nobarrier),andthebarrierappearsforF <Fa, giveupmomentumandenergytotheelectrondependingona
asshowninourmodel,itincreasesgraduallyandbecomesinfinitefor nonlinearitydimensionlessparameter0<ε∼ρL/ω<1.
F → 0, wheretheT-timebecomesinfinite[10](tunnelingisnot pos-
sibleor thetunnelingprobabilityiszero) for theunperturbedsystem Thisisimportantbecauseitalsobringsuptheparticulate
(groundstate)asitshouldbe. natureoftheradiationinthediscussion,althoughLamb,ina
Finally,unfortunatelythereisstillnoexperimentaldataavailable nicepapertitled“anti-photon” [21],concluded andarguedto
for the Hydrogen (or a Hydrogen-like) atom [70], which is similar to giveupthephotonasaparticle[Itishightimetogiveupthe
theKASEforHe,thatmakesitpossibletocomparethedifferenttime useofword“photon”,andofabadconceptwhichwillshortly
approximationswithanexperimentalfindingandclarify(someof)the beacentury old. Radiationdoes notconsistofparticles,and
controversialissueoftheT-time,especiallytheabovementionedpoints, the classical, i.e. non-quantum, limit of quantum theory of
keepinginmindthatthetwoquantummechanicalconcepts(ofKullie radiation is described by Maxwell’s equations for the electro-
andTorlinaetal)couldbequantitativelyequivalentorbelongtotwo magnetic fields, which do not involve particles.] Ironically,
physically equivalent gauge pictures, similarlyto other quantitatively Einsteinhimselfdidnotacceptwave-particleduality(WPD).
equivalenttreatments,suchastheFPItreatmentof[9]ortheentropic He wrote: this interpretation... (WPD) .. appears to me as
realT-timeof[46].
8
onlyatemporarywayout[78]. Although,intheearlierdaysof ofpulsesisshowntoberelateddirectlywiththepropagation
quantummechanics,Duane[79]wasabletoexplaindiffraction velocityofpulsesinvacuum,whichisfoundtobesmallerthan
ofx-raysbyacrystalintermssolelyoftheirparticleaspect. thelightspeedinallcasesexceptaninfinitelyextendedplane
However, modern experiments of quantum optics have wave[81,83].
beautifully confirmed the corpuscular character of the pho- The effective mass ML differs from the “sum” of all the
ton, Zeilinger [22,80]. Furthermore, Zeilinger concluded that single-photonmassesnph·mph=nωc20,wherenphisthemean
theresultssuggestthatonehastoabandonatleastthenotion numberofthephotoninthepulse,inthiscasemphisobtained
ofna¨ıverealismthatparticleshavecertainpropertiesthatare form the relation E = mphc2 according to the theory of rel-
independent of any observation (pre-existing properties). We ativity. mph =E/c2 =~ω0/c2 has been introduced formally
willrelyonthisconclusionofZeilingerandseethattheKASE, as an effective mass entering a formal Schro¨dinger equation
is a case (and generally in the ASEs) to realize the existence fora“photon-wave function”,thefieldamplitudeF [76]page
ofacorpuscular propertyof the(attosecond pulse)radiation, 153. Itisclearthatawavepacketisnotasingle-photonwave
preciselyofanattosecond LWP. andonlyinthelimitofaplanewave,isω0 theenergycarrier
of a single photon (where the photons density related to the
intensity IL ∼ F2 via IL/ω0.) We can consider the LWP as
III. THE TUNNELING PROCESS
a carrier of a finite number of energy quanta, the (average)
Ahallmarkofthe tunnelingregimeinstrongfieldioniza- numberofphotonsinthepulse. Similarly,ontheparticleside
tionistheKeldyshparameterγK,eq1. ForγK <1itfollows aplanmatter-wavecorrespondstoaparticlestreamandasin-
that gleparticleisdescribedbyamatter-wave packet (MWP). At
2Ip F this point I have to stress not to anticipate any possible mis-
γK = pF ω0<1⇒(ω0)>p2Ip=ke (5) interpretation of myclaiminthis section. Thereisnoreason
whereke isthewavevectororthemomentumoftheelectron. to interpret ML (or mph) as a mass of a small bullet (point-
Thelefthandsideofthesecondinequalityofthe relation(5)
particle) or a group of point-particles. However, light quanta
canbeinterpretedasanaveragemomentumofthelaserpulse
orthephotoncanbeconsideredasaparticleinthesensethat
pL=kL. Thisleadstothefollowing: it shows particle behavior as claimed by Grangier et al [84].
(ωF0)=kL = MLc, withML = (kcL)= (cFω0) (6) AclunddeasclcigohrdticnagntaolsCoocmonpstiosnto[8f5d]is(csreeete[7u6]n,itpsamgeo2v2in4g):inonceerctoanin-
directions,eachunithastheenergy~ωandtheimpulse~ω/c.
where ML is an ”effective” average mass (or simply effective
Hence for a radiation of LWP with an approximate intensity
mass),afeatureofthewholephotoniclaserpulseortheLWP,
IL≈ω0=ωph,nph≈1,oneobtains
cis the speedof lightand ω0 (inau) isthe photon energyor
thecentralcircularfrequencyoftheLWP.ML hasnothingto ILli≈mω0ML≈mph=ωph/c2⇒Fph=ωp2h/c=Ep2h/c (8)
dowiththerestmassof asingle-photon, whichhas anupper
Although it is difficult to interpret this relation, but a possi-
limitof mph ≤10−49g ≈10−17eV [81]. Indeed, different au-
ble simpleinterpretation wouldbe, Fph is a limitof the elec-
thorssuggestedamasscharacteristicforthelightpulse,where
tricfieldstrength(forafixedcircularfrequencyω0)forwhich
experimentallyitwasfoundthatthelightpulsespropagatein
a LWP can approximately be considered as a single-photon
thevacuumwithaspeedsomewhatsmallerthanthespeedof
pulse,inthesensethatitcanshowparticlebehavior. Asimi-
lightc[82]. Fedorov[83]claimsthatanonzeroinvariantmass
larsituationwhentheDeBrogliewavelengthbecomessmaller
of
mFedorov=N2πλw0a~cω20 (7) tohfatnhethaettgoesoemcoentrdyeoxfptehreimpeanrtt,iclien.tFhoerotphteiceaxlprearinmgee,nttahlisseitsupa
can be attributed to the light pulse, where wa >> λ0 is the very small value (F ∼ 10−5au), the typical field strengths
pulse width (waist) at z = 0, at the center where a Gaus- usedinthe ASEs∼0.01−0.11au. Inthecontext ofthefield
sianpulsebecomesapproximatelyaplanewave,ω0=ck0 the quantization, according to Purcell [86](see [76],page 153), it
central frequency of the laser pulse and N the photon num- makesnodifferencewhetherwethinkofρ=F2asasquareof
ber. mFedorov characterizes global features of the pulse as a theelectricfieldstrengthorthephotonprobabbilitydensityρ.
whole. It is astonishing to see that for λ0 = 2πwa one gets Nowwecanturnback toeq5keeping inmindtheabove
mFedorov/N = mph = E/c2 = ~ω0/c2. The invariant mass discussion, especially that our concern is the impact of the
9
LWPasacarrierofmomentum,whenitinteractswiththeelec- has no rest mass [22]. Clearly, the De Broglie wave relation
tron, whereLWP is agroup of lightquanta or photons prop- λ=1/p, withpthe momentum, isthe basis forthe wavena-
agating in the vacuum with a speed vc slightly smaller than ture of a particle and the basis for the photon hypothesis as
the speed of light c [82, 83], and showing aparticle-behavior. a carrier of a momentum k = mphc = ω/c [92] (De Broglie
Any further interpretation is beyond the scope of this work. Nobel-prizelecture), seeeq 10below,wheremph hereisusu-
Eq 5 has a simple interpretation in the language of QM, for ally interpolated as the “motional” mass of the photon [90].
kL <ke,(γK >1) the ionization happens through multipho- However,onehastoavoidamisinterpretationofmph because
tonabsorption(MPIregime). WhereasforkL>ke,(γK <1) therestmassofthephotoniszero(<10−49g)[81].
thetunnelingregime,onefiguresoutthat,astheaverageLWP IntableIthe effective massML forarange ofintensities
momentum is larger than the electron momentum, the tun- usedintheASEsisgiven. Theaveragenumberofphotonsnph
nel ionization happens by a heavy scattering of the (bound) iscalculatedwiththeassumptionthatthelaserpulsehitsthe
electron through the LWP. The LWP acts collectively and electron inits ground state orbital (1s for He atom, withthe
strongly and the electron is therefore, not able to form im- radiusre ≈1au), withthe cross-sectionAe =4πre2=4π,the
mediate metastable or virtual states, as it is the case in the areaofasphereofthegroundstateduringthetimeperiodof
MPI regime. The electron response happens on a fast time onecycleδte =1/ 2Ip (i.e. suchthattheprobabilityampli-
p
scale. Consequently, and unlike the MPI regime, the ioniza- tude offindingthe electrononasphericalshelloftheground
tiondependsmainlyonthefieldstrength,andnotsignificantly state during the time δt is ψ2 = 1.) The average number
onthe frequency (whichissignificant forthe MPI)[87] chap. of photons is then nph = (IL/ω0)Aeδte. Assuming a photon
1,andtheelectron ratherfollowsthevalueofthelaserinten- Compton scattering, see [90] and [91], the momentum trans-
sity(theintensityenvelopeofLWP)[87]chap. 2.3,seebelow, mitted to the electron is ∼ (ω0) [85], which is too small (as
c
thustheelectronundergoesadynamicaltransport[49],seesec expected) relativetothemomentumoftheelectronke, pkCe is
IIB. Conceivably, which is justified by Collins [49], the scat- giveninpercentintableI,wherepC =nph(ωc0). Notethata
tering time concept (the decay of metastable states, compare semiclassical approximation due to Schro¨dinger [75] explains
subsec IIB), when used to calculate the T-timein the ASEs, theComptoneffect. WhereasassumingacollectiveLWPscat-
hasnosuccessincasessuchasKASEasshownbyLandsman tering, the average momentum transmitted to the electron is
[9,50]andalreadymentionedinsubsecIIB. pW. Thequantity pkWe , given intable Iinpercent relative to
In accordance with the view of Colllins, subsec IIB, we the momentum of the electron, iscalculated as the following,
suggestascatteringmechanism(orelasticcollision)inthetun- seeA:
nelingregimeasmentionedabovesubsecIIC,itisnotthatof F 2
asinglephotonscattering(Comptonscattering),butthatthe pW = η α(cid:18)ω0(cid:19) , (9)
LWPcollectivelyscatterstheelectron. Thiscollectiveprocess whereα=1/cisthefinestructureconstant,whichisequalto
shows a particulate nature of the LWP, similar to the way thestrengthoftheinteractionofthephotonwiththeelectron.
thecollectiveprocessoftheparticlesshowsthewavebehavior Note that the interaction of anelectron with anintense laser
onthescreeninthedoubleslitexperiment(DSE).Thelatter fieldischaracterized by(F/ω0)2 [73](andref10insideit.)
happensevenwhenonesendstheelectronone-by-one[88],[89] In eq 9 the quantity pW determines the amount of an
(voted the most beautiful experiment by readers of Physics average momentum that is transmitted to the electron, and
Worldin2002,Sep1.) Fortheformersimilarexperimentscan is (depending on the unknown parameter η) much larger
be performed, like the single-photon double-slit experiment than p of Compton scattering as seen in table I. One notes
C
[84],ortheComptonscatteringexperiment[90,91]withsingle- that in the above consideration pW/pC ∼ nph−1(F2/ω03) =
photon LWP provided isexperimentally achievable. Zeilinger η(δtAeω02(1+ǫ2)−1 =η ·15.89(ω0,ǫaregiveninsec I). It
[22]concludedthattheresultsofthese(achieved)experiments isworthwhiletomentionthatinstrongfieldexperiments,the
confirm that the quantum state is not just astatistical prop- processofscatteringinthetunnelingandtheMPIregimesare
erty of an ensemble of particles. The conceptual questions complex and nonlinear [87]. As the process is nonlinear, it is
arising for photon interference are the same as those arising possible that η depends on the mean photon number nph in-
forinterference ofmassiveparticles, and inboth cases we see volvedintheinteraction(notthetotalnumberofthephotons
particle-likeandwave-likeproperties. Andthatinequivalence oftheLWP),whichdependsontheprobabilitydensitytofind
forcertaininterferenceexperimentsarisesbecausethephoton a photon in the interaction volume/area, compare mFederov
10
xa 1 2 3 4 5 6 7.5 the DSE with light (Young experiment) is used to prove the
In(au) 0.00285 0.0057 0.00855 0.0114 0.01425 0.0171 0.0214 wavenatureoflightandtheDSEwithmatterisusedtoprove
F(au) 0.040 0.057 0.07 0.081 0.090 0.099 0.11 the wave nature of particles. That the KASE (in the tunnel-
ML 0.0048 0.0067 0.008 0.0095 0.0106 0.0116 0.013 ingregime)isalsoaDSE,wasdiscussedin[14]inaccordance
nph 0.43 0.86 1.3 1.72 2.15 2.6 3.23 withtheuseoftheFeynmanpathintegralbyLandsman[9]to
pC% 0.0145 0.023 0.043 0.058 0.072 0.087 0.11 calculatetheT-time. Hence,thewave/particledualitynature
ke
pW% 0.23 0.46 0.69 0.92 1.15 1.38 1.72 of the radiation can be impressivelyshown in the ASEs such
ke
astheKASE,withthecautionthatthepresentworkpresents
TABLE I. a Intensity=x∗1014 inWcm−2,In=intensity, onlyastepinthisdirection,andcertainly,theexperimentwill
F = field strength, nph = mean photonic number, ML = enlighten and is decisively important to prove our approach.
α(ωF0),pC =nphαω0,pW =ηαF2/ω02,η=1,seetext. We discuss this further by drawing the following remarks to
ourattention.
eq7,whichdepends onN.
• Thewavepacketconceptrepresentsanunifyingmath-
Hencethedeterminationofaprecisevalueofη needsfur-
ematicaltoolthatcancopewithandembodynature’s
ther investigations, perhaps using wave packet dynamics. In-
particle-like behavior and also its wave-like behavior,
deed,itwillalsobefruitfultoachieveaCompton-typeexper-
Zettili[94],chap1.8.
iment using strong field attosecond isolated short pulses, e.g.
LWP in the visible light, UV or XUV range. Today the ex- • ML is an effective mass of the LWP, which acts col-
lectively (like a MWP) at a time scale much smaller
perimentally achievable pulsedurations aresufficiently short,
about67−130as[4,93]. Noteeq9isrelatedtoUp/c,where than1/ω0.
Up isthe wellknownPonderomotive energy andisdefined as • Thecorpuscularpropertycanbebestjudgedformthe
the cycle-averaged quiver energy of the electron in the laser momentum, pphoton,pparticle = ~k. For a stream of
field. For a free electron Up =χF2ω0−2, χ= 1/2 (for linear photons or a plane wave (the “motional” mass of a
polarizedlaserfield). photon mph = ω0/c2) mph ∼ nphmph = nphω0/c2,
InMPIregimetheaveragephotonmomentumisnotlarge wherenph istheaverage number ofthephotons. For
enough to destabilize the electron in its circular movement, aLWPML=kL/c= cFω0
andtheelectroniscapturedinametastablestate(byabsorb- • Fromeq5,kL>ke orcML>ke,whereML=kL/c,
ing a photon) with a new wave vector ke′ < ke and a new thecorpuscularnature(particle-behavior)oftheLWP
oscillation frequency (e.g.ωe′ = ωe−ω0 < ωe). In the case is a reasonable assumption, because the De Broglie
of virtual states the electron evolves successively through the relationisbasedonthesymmetrybetweenlight(wave-
virtual states to a final orbital or to the continuum) by ab- light)andparticles,asisclearinthefollowingrelation
sorbingportionsofenergy(manyphotons)fromtheradiation. [92]
Einstein’sinsighttothePlanckhypotheseswastoseethatab-
sorbingenergyfromradiationinaquantizedformleadstothe mphc= Ec = ωpch =p theparticlenature (10)
fact that light implies a quanta (photon), which carries the E=ck,λD = 1p thewavenature

energy ω0, although the MPI process, and the photoelectric
effectitself,isnottheregime,wherethecorpuscularproperty
emerges fromtheradiationorfromalaserpulse,i.e. noneed De Broglie wrote: I was guided by the aim to perform a real
forthequantization ofthe classicalelectromagnetic radiation physical synthesis,validforallparticles,ofthecoexistence of
(onlyquantizationofthematter)[37]. SimilarlytheCompton thewaveandofthecorpuscularaspectsthatEinsteinhadin-
effect can be explained by a semiclassical approximation as troducedforphotonsinhistheoryoflightquantain1905[95].
mentionedabove[75]. Intheattosecondexperimentweseethe Naturallythesymmetryineq10permitsbothdirections,wave
advantagethatinthesameexperiment,bothpropertiesofthe property forparticles or MWPandacorpuscular property of
radiation can emerge, the wave property (in MPI regime, no the radiation or LWP. Indeed, De Broglie insight was to see
photon-particleconceptneeded)andthecorpuscularproperty that for the light quanta from E = hν = hc/λ and E = cp
(having particle-behavior) in the tunneling regime. Roughly it follows that λ = h/p and the same holds for the particles,
the range of γK determines the different regimes. Likewise wherehisthePlanckconstant.