Above-threshold ionization with highly-charged ions in super-strong laser fields: I. Coulomb-corrected strong field approximation

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by  Michael Klaiber

Above-thresholdionizationwithhighly-chargedionsinsuper-strong
laserfields: I.Coulomb-correctedstrongfieldapproximation
Michael Klaiber, Enderalp Yakaboylu, and Karen Z. Hatsagortsyan
Max-Planck-Institut fu¨r Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany∗
(Dated:January25,2013)
Aimingattheinvestigationofabove-thresholdionizationinsuper-stronglaserfieldswithhighlychargedions,
wedevelopaCoulomb-correctedstrongfieldapproximation(SFA).TheinfluenceoftheCoulombpotential
oftheatomiccoreontheionizedelectrondynamicsinthecontinuumistakenintoaccountviatheeikonal
approximation,treatingtheCoulombpotentialperturbativelyinthephaseofthequasi-classicalwavefunction
ofthecontinuumelectron. InthispapertheformalismoftheCoulomb-correctedSFAforthenonrelativistic
regimeisdiscussedemployingvelocityandlengthgauge. Directionizationofahydrogen-likesystemina
3
strong linearly polarized laser field is considered. The relation of the results in the different gauges to the
1
Perelomov-Popov-Terent’evimaginary-timemethodisdiscussed.
0
2
PACSnumbers:32.80.Rm,42.65.-k
n
a
J I. INTRODUCTION ThestandardSFAtechniquehasalsobeenmodifiedtoinclude

4 Coulombfieldeffectsoftheatomiccore. Thesimplestheuris-
2
ticapproachisthe,so-called,Coulomb-Volkovansatzinwhich
Due to advances in laser technology strong near-infrared

] laser fields nowadays are available up to intensities of theVolkovwavefunctionintheSFAmatrixelementisreplaced
h 1022W/cm2[1]andmuchstrongerlaserfieldsareenvisagedin byanheuristicCoulomb-Volkovwavefunction[39–49].Inthe
p lattertheCoulombfieldistakenintoaccountviaanincorpora-
nearfuture[2]stimulatingtheinvestigationoftherelativistic
- tionoftheasymptoticphaseoftheexactCoulomb-continuum
m regimeoflaser-atominteractioninultra-strongfields. Thepio-
wavefunctionintothephaseoftheheuristicCoulomb-Volkov
neeringexperimentinthisfieldwascarriedoutbyMooreetal.
o wavefunction[50]. Consequently,thecouplingbetweenthe
[3]. Theyhaveinvestigatedtheionizationbehaviorofatoms
t
a andionsinastronglaserfieldatanintensityof3×1018W/cm2. CoulombandlaserfieldisneglectedintheCoulomb-Volkov
. ansatzandtheapproachfailswhentheelectronappearsinthe
s Severalfurtherexperimentshavebeendevotedtorelativistic
c continuumaftertunnelingclosetotheatomiccore[51].
laser-inducedionization[4–10].
i
s Numericalinvestigationofthedynamicsofhighly-charged Followingamorerigorousapproach,theeikonalapproxima-
y ionsinsuper-strongfieldshasbeencarriedoutin[11–21]. The tion[52]hasbeenproposedtoapplyforstrongfieldproblems
h
standardanalyticalapproachesinthefieldofnonperturbative [53]. In the latter, nonrelativistic free-free transitions in the
p
[ laser-atominteractionarethestrongfieldapproximation(SFA) laserandtheCoulombfieldhavebeenconsideredemploying
[22–24]andtheimaginarytimemethod(ITM)[25,26]. For aneikonalwavefunctionforthecontinuumelectron. Herethe
1 atheoreticaltreatmentoftherelativisticeffects,theSFAhas laser field is taken into account exactly, while the Coulomb
v been generalized into the relativistic regime in [27, 28] and fieldisviatheeikonalapproximation. Theeikonalapproxima-
1
the ITM in [29–32], respectively. In the standard SFA the tionhasbeengeneralizedin[54]toincludequantumrecoilef-
6
7 influenceoftheCoulombfieldoftheatomiccoreisneglected fectsatphotonemissionandabsorption. ACoulomb-corrected
5 intheelectroncontinuumdynamicsandthelatterisdescribed SFAfornonrelativisticionizationemployingtheeikonalwave-
. bytheVolkovwavefunction[33]. Accordingly,thepredictive functionhasbeenfirstproposedin[55]. Similarapproaches
1
0 poweroftheSFAisthebestfornegativeionswherenolong havebeenconsideredin[56–61]. Recently,thenonrelativistic
3 rangeforcesoftheparentsystemactontheionizedelectron. Coulomb-corrected SFA based on the eikonal-Volkov wave
1 ForatomsormoleculeswithlongrangeCoulombforcesthe functionforthecontinuumelectronhasbeenfurtherelaborated
: performanceoftheSFAdowngradestoaqualitativelevel[34]. in[62,63]andappliedformolecularstrongfieldionizationand
v
i Thisistrueespeciallyforhighly-chargedions. high-orderharmonicgeneration. TheCoulomb-correctedSFA
X Inthenon-relativisticregimetheITMhasbeensuccessfully hasalsobeenextendedtoincluderescatteringeffects[64,65].
r usedtotreatCoulombfieldeffectsduringtheionizationinthe Here the Coulomb field is taken into account exactly in the
a
quasi-staticregimeandthewell-knownquantitativelycorrect quasi-classical electron continuum trajectories that are later
Perelomov-Popov-Terent’ev (PPT) ionization rate has been pluggedintothephaseofthequasi-classicalwavefunction.
derived[35,36]. ThePPTtheoryusesthequasi-classicalwave Intherelativisticregime,similartothenonrelativisticcase,
functionforthedescriptionofthetunnelingpartoftheelec- thestandardSFAisonlyexponentiallyexactsincetheCoulomb
tron wave packet through the quasi-static barrier formed by field is neglected during ionization, whereas the ITM [29–
thelaserandatomicfield,withmatchingofthequasi-classical 32]canprovidealsocorrectpreexponentialfactors. Canthe
wavefunctiontotheexactboundstatewavefunction[37,38]. quantitatively correct relativistic ionization probabilities be
derivedviatheSFAtechniqueaccountingCoulombfieldeffects
accurately? Therelativisticgeneralizedeikonal-Volkovwave
function(takingalsointoaccountquantumrecoil)hasbeen
∗Electronicaddress:[email protected] derived in [66]. The Coulomb corrected SFA based on this
2
wavefunctionhasbeenproposedin[67].However,finalresults A. ThestandardSFA
have been obtained only in Born approximation, i.e. via an
expansion of the eikonal wave function with respect to the Weconsiderahighly-chargedhydrogen-likeioninteracting
Coulomb field, which, in fact, reduces the transition matrix withalaserfield. ThedynamicsisgovernedbytheHamilto-
elementtotheoneinthestandardsecondorderSFA. nian
With this paper we begin a sequel of papers in which
H = H +H , (1)
wedeveloptherelativisticCoulomb-correctedSFAbasedon 0 int
theDiracequation,generalizingthenonrelativistictheoryof
whereH istheHamiltonianoftheatomicsystem
0
[55,63]andapplyitforthecalculationofspin-resolvedquanti-
tativelycorrectionizationprobabilities. RatherthantheVolkov H =pˆ2/2+V(r), (2)
0
wavefunction,theeikonal-Volkovwavefunctionisemployed
asfinalstateoftheCoulombcorrectedSFA.Theinfluenceof withtheatomicpotentialV(r),themomentumoperatorpˆ and
Coulombpotentialoftheatomiccoreontheionizedelectron coordinatevectorr(atomicunitsareusedthroughout). The
continuumdynamicsistakenintoaccountviatheeikonalap- interactionHamiltonianduetothelaserfieldinlengthgaugeis
proximation. Thelattermeansthatthequasi-classical(WKB)
H (t)=r·E(t), (3)
approximationisappliedfortheelectroncontinuumdynamics int
and, additionally, the Coulomb potential is treated perturba- withthelaserelectricfieldE(t). Thetimeevolutionoperator
tivelyinthephaseofthequasi-classicalwavefunction. The U(t,t )oftheatominthelaserfieldcanbeformulatedviathe
0
formalismisappliedfordirectionizationofahydrogen-like Dysonequation
systeminastronglinearlypolarizedlaserfield.
(cid:90) t
Inthisfirstpaperofthesequel,webeginwiththenonrel- U(t,t )=U (t,t )−i dtU(t,t(cid:48))H (t(cid:48))U (t(cid:48),t ) (4)
0 0 0 int 0 0
ativisticCoulomb-correctedSFAtoshowinthemostsimple t0
case the scheme of the Coulomb-corrected SFA. The SFA
whereU isthetimeevolutionoperatoroftheatomicsystem
formalismisappliedtotreattheCoulombfieldeffectofthe 0
withoutthelaserfield. Thematrixelementforalaserinduced
atomic core during ionization systematically and to obtain transitionfromtheinitialatomicgroundstate|φ(t)(cid:105)=|0(cid:105)eiIpt,
quantitativelycorrectresultswhich,inparticular,forthetotal
withthegroundstateenergy−I ,andtheionizationpotential
ionizationratecoincidewiththePPTresult. Twoversionsof p
I ≡ κ2/2, into a continuum eigenstate of the total system
thetheorybasedonthevelocityandlengthgauge,respectively, p
|ψ (t)(cid:105)withanasymptoticmomentumpisthengivenby
areconsidered. ComparisonwiththePPTtheoryiscarriedout p
andthephysicalrelevanceofthetwoversionsisdiscussed. A (cid:90) ∞
M =−i dt(cid:104)ψ (t)|H (t)|φ(t)(cid:105). (5)
conclusionisdrawnconcerningtheschemeoftherelativistic p p int
−∞
generalizationoftheCoulomb-correctedSFA.Inthesecondpa-
perofthesequel,therelativisticCoulomb-correctedSFAwill In the SFA, the final continuum state is approximated by a
bedeveloped,andthenextpaperinthesequelwillbedevoted Volkovstate|ψV(t)(cid:105),i.e. aneigenstateofaHamiltonian,where
p
tospineffectsinrelativisticabove-thresholdionization. the electron is only interacting with the laser field [33]. In
coordinatespaceitisgivenby
The plan of the paper is the following: In section II the
nonrelativistic Coulomb-corrected SFA in the length gauge (cid:104)r|ψV(t)(cid:105)=exp[iS(0)(r,t)]/(2π)3/2. (6)
is considered and differential and total ionization rates for p 0
hydrogen-likesystemsarederived. Thenextsectionisdedi- The function in the exponent S(0)(r,t) = (p + A(t)) · r +
catedtotheCoulomb-correctedSFAinvelocitygauge. The (cid:82)∞dt(cid:48)(p+A(t(cid:48)))2/2istheclassi0calactionofanelectronin
comparisonofthedifferentversionsoftheCoulomb-corrected t
alaserfieldinthelengthgauge. NotethattheVolkovwave
SFAiscarriedoutinSec.IV,andtheconclusionisgivenin
functioncoincidesexactlywiththewavefunctioninthezeroth-
Sec.V.
order WKB approximation for the system. The ionization
matrixelementintheSFAyields:
(cid:90) ∞
M =−i dt(cid:104)p+A(t)|H (t)|0(cid:105)exp[−iS˜(t)] (7)
p int
−∞
II. NONRELATIVISTICCOULOMB-CORRECTEDSFAIN
THELENGTHGAUGE with S˜(t) = (cid:82)∞dt(cid:48)[(p+A(t(cid:48)))2/2+κ2/2]. In the adiabatic
t
regime,whenthelaserfrequencyωissmallerthantheground
stateenergyI andtheponderomotivepotentialU = E2/4ω2,
InthissectionweshowhowthenonrelativisticCoulomb- p p 0
withthelaserfieldamplitude E , thetimeintegrationinEq.
correctedSFAinthelengthgaugeisdeveloped. Ratherthan 0
(7)canbecarriedoutingoodaccuracyviathesaddlepoint
theusualVolkovwavefunction,itemploystheeikonal-Volkov
method(SPM),see,e.g.,[68]. Thisyields
wave function to describe the electron continuum dynamics
accurately,takingintoaccounttheCoulombfieldeffectofthe (cid:115)
(cid:88) 2π
atomic core. As we will see in this way the PPT ionization M =−i (cid:104)p+A(t )|H (t )|0(cid:105)exp[−iS˜(t )], (8)
ratescanberecoveredwithintheSFAformalism. p iS¨˜(t ) s int s s
s s
3
wheret aretheso-calledsaddlepointsoftheintegrablefunc- andthetotalionizationrate:
s
tiondefinedbyS˙˜(ts)=0. AfterapartialintegrationinEq. (7), (cid:114)3 E3/2 (cid:34) 2κ3(cid:35)
thetransitionoperatorinthematrixelementcanbetransformed w(z) = 0 exp − . (15)
fromHint toV(r)[69]: π2κ5/2 3E0
(cid:115) The SFA ionization rates of Eqs. (14) and (15) for a short
(cid:88) 2π
M =−i (cid:104)p+A(t )|V(r)|0(cid:105)exp[−iS˜(t )]. (9) rangepotentialcoincidewiththeITMresult[71].Thephysical
p iS¨˜(t ) s s reasonisthatneglectingtheatomicpotentialaftertheelectron
s s
istransferredintothecontinuum,isjustifiedfornegativeions.
Inthecaseofalonglaserpulsethedifferentialionizationrate
isexpressedviathematrixelementasfollows[68]:
C. SFAforahydrogen-likesystem
dw ω
= |M |2, (10)
d3p 2π p
Inthecaseofatomicionization,theCoulombpotentialof
theioniccorecannotbeneglectedintheelectroncontinuum
where the summation in Eq. (8) is carried out only over the
dynamics. Therefore, to obtain an accurate ionization rate,
saddlepointsofonelaserperiod.
the wave function of the continuum state |ψ (t)(cid:105) in the SFA
p
ionization amplitude is approximated by the eikonal wave
function(insteadoftheusualVolkovfunction)whichaccounts
B. SFAforanegativeion
fortheCoulombfieldeffectoftheioniccore[55,63].
Aswenotedintheprevioussection,theVolkovwavefunc-
Thecalculationofionizationratesisstraightforwardinthe
tionisidenticaltotheelectronwavefunctioninthelaserfield
caseofionizationofanegativeion. Thelattercanbemodeled
byazero-rangepotentialV(z)(r) = −(2π/κ)δ(r)∂ r, withthe in the zeroth order WKB-approximation. A systematic im-
matrixelement(cid:104)p|V(z)|0(z)(cid:105)=−√κ/(2π)[70]. Inarsinusoidal provementofthisstatecomparedtotheexactcontinuumstate
laser field A(t) = (E /ω)sin(ωt) the saddle point equation canbeachievedemployingtheWKB-approximationforthe
0 wavefunctionofanelectronexposedtothesimultaneousac-
yields:
tionofthelaserandtheCoulombfield. FromtheSchro¨dinger
(cid:115) equationforanelectroninaCoulombpotentialV(c)(r)=−κ/r
sin(ωt )=− pE +i γ2+(cid:32) p⊥ (cid:33)2 (11) andalaserfieldE(t)
s E /ω E /ω
0 0 (cid:126)2
i(cid:126)∂ψ=− ∆ψ+V(c)ψ+r·E(t)ψ, (16)
with the Keldysh parameter γ ≡ κω/E0, pE ≡ p·eˆ, p⊥ ≡ t 2
|p−(p·eˆ)eˆ|,eˆ ≡E /|E |. Inthetunnelingregime(γ(cid:28)1)the
0 0 theansatzψ=eiS/(cid:126)yieldsthefollowingequation
saddlepointsinonelasercyclecanbegivenapproximatelyvia
aperturbativesolutionofEq.(11)withrespecttoγ:
(∇S)2 (cid:126)∆S
−S˙ = +V(c)+r·E+ . (17)
(cid:113) 2 i 2
(cid:34) (cid:35) κ2+p2
ωts1 = −arcsin Ep/Eω +i |E(t )|/ω⊥ UsingtheWKB-expansionS =S0+ (cid:126)iS1+...,weobtainthe
0 0 equation
(cid:113)
ωts2 = π+arcsin(cid:34)Ep0/Eω(cid:35)+i |Eκ(2t0+)|/pω2⊥. (12) (cid:32)(cid:126)i(cid:33)0 : −S˙0 = (∇S20)2 +V(c)+r·E(t), (18)
(cid:113)
with|E(t )|= E 1−(ωp /E )2. Insertingthesaddlepoints S0istheclassicalactionofanelectroninthelaserfieldandthe
0 0 E 0
intoEq.(10),yieldsthedifferentialionizationprobabilityofa atomicpotential. Intheeikonalapproximationthepartialdif-
ferentialequationforS issolvedperturbativelyintheatomic
negativeion 0
potential V(c). The zeroth order solution gives the Volkov-
ddw3p(z) = 2π2|ωE(t0)|exp−2(cid:16)κ32|E+(tp02⊥)|(cid:17)3/2 (13) actioSn0(0)(r,t)=(p+A(t))·r+ 12(cid:90) ∞dt(cid:48)(cid:0)p+A(t(cid:48))(cid:1)2, (19)
t
Since the ratios pE/(E0/ω) and p⊥/(E0/ω) are smaller than with A(t) = −(cid:82)t dt(cid:48)E(t(cid:48)), whereas the first order solution
oneinthecaseoftunnelionization,wecanexpandthefunc- −∞
reads
tionintheexponentquadraticallyintermsofmomentumand
(cid:90) ∞
neglectthedependenceinthepreexponentialfactor. Withthis S(1)(r,t)= dt(cid:48)V(c)(cid:0)r(t(cid:48))(cid:1), (20)
wearriveatthedifferentialionizationrate: 0
t
 
ddw3p(z) = 2πω2E0 exp−32Eκ30 − Eκ0p2⊥− κ33Eω032p2E, (14) w(cid:82)tti(cid:48)thdtt(cid:48)h(cid:48)pe(ttr(cid:48)(cid:48)a)jeacntdorpy(to)f≡thpe+elAec(ttr)o.nTihnetthimeelatsecranfiebledirn(tte(cid:48)r)p=retred+
4
asthetimeandrasthecoordinateoftheionizationevent.Thus, verymomentofionization. Thus,itisjustifiedtoexpandthe
theapproximatewavefunctionoftheelectroncontinuumstate argumentinS(1)describingthetrajectoryoftheelectron,upto
0
inthelaserandCoulombfield,whichistermedastheeikonal- secondorderaroundthesaddlepointt ,i.e. aroundtheinstant
s
Volkovwavefunction,inthenonrelativisticregimeis ofionization:
1 r(t(cid:48))=r+p(t )(t(cid:48)−t )−E(t )(t(cid:48)−t )2/2. (24)
ψ(c)(r,t)= exp{iS(0)(r,t)+iS(1)(r,t)}. (21) s s s s
p (2π)3/2 0 0
Further,themomentumdistributionoftheamplitudeisdom-
inatedbytheexponentialfunctionthatislocatedaroundthe
Ittakesintoaccounttheinfluenceoftheatomicpotentialquasi-
laser polarization direction, i.e. we can assume in the pre-
classicallyuptofirstorderandwillbeusedintheSFAampli-
exponential function p = p eˆ and p(t ) = iκeˆ. Addition-
tudeofEq. (5). E s
ally,itcanbearguedthatthetunnelionizationstartsmainly
Letusestimatetheapplicabilityoftheeikonalapproxima-
tion givenby the condition S(1) (cid:28) S˜. The perturbedaction in the area around the laser polarization axis r = rEeˆ, i.e.
(cid:82) 0 (cid:82) at the outskirts of the atom in direction of the laser elec-
canbeestimatedS(1) ∼ V(c)dτ∼ dτx˙/x∼log(r /r )∼
√ 0 Ee Ei tric field. This typical value for the initial coordinate of
log( Ea/E0)∼1,usingthepotentialV(c) ∼κ/rE (rE ≡r·eˆ), the trajectory r is justified via the saddle point condition
(cid:82)
the initial coordinate before tunneling rEi ∼ vcδτc, the ve- for the integral d3rexp[−ip(ts) · r − κr] which leads to
locity vc ∼ κ, the uncertainty of the initial time δτc [in rs/rs = p(ts)/(iκ). Thus, the integrand in the expression of
thelatter, weusethetime-widthofthesaddle-pointintegra- theCoulomb-correctionfactorofEq. (20)canbesimplified:
(cid:113) √
tion δτc ∼ 1/ S¨˜(ts) ∼ 1/ κE0] and the tunnel exit coor- 1 = (cid:12) 1 (cid:12). (25)
dinate rEe ∼ κ2/E0. While the Volkov-action is estimated r(t(cid:48)) (cid:12)(cid:12)rE +pE(ts)(t(cid:48)−ts)−E(ts)(t(cid:48)−ts)2/2(cid:12)(cid:12)
S˜ ∼ p(τ )2τ +I τ ∼ E2τ3+I τ ∼ E /E withthetunneling
timeτ ∼c γ/cω=pκc/E d0etcermipnecdbyathe0Keldyshparameter Furthermore,themotionaftertheelectronhasleftthebarrier,
c 0
andtheatomicfieldE =κ3,theeikonalapproximationforthe contributesonlyasanunimportantphaseinthepreexponential
a
nonrelativisticionizationproblemisvalidwhen factor in Eq. (23) and the integration limit can be set at the
tunnelexit: ωt = −arcsin(cid:2)p /(E /ω)(cid:3). Withthesesimplifi-
0 E 0
E0 (cid:28)1. (22) cationstheintegralinEq. (20)canbeevaluated:
NotethatE0/Ea <1/16iEnathetunnelingionizationregimefor exp(cid:104)−iS0(1)(r,t)(cid:105) = −11++√√11++44λλ√11+4λ
ahydrogen-likeion.
To be able to handle the additional term S(1) in the ≈ 1 +O(λ), (26)
0 λ
SFA transition amplitude, we have to make simplifica-
t−ioVn(cs).(cid:16)rT+h(cid:82)e∞tidmt(cid:48)e(cid:48)pd(et(cid:48)r(cid:48)i)v(cid:17),atciovrereospfoSn0(d1s)tgoitvheenpbotyen∂titSal0(1e)n(rer,gt)yo≈f woridthertohfe√smEa0l/lEqau(cid:28)ant1it,yseλeE=q.√−(r22·)E.(Itns)f/a2cκt,2ownehiccahniesstoimf tahtee
t λ ∼ x E /κ2 ∼ v τ E /κ2 ∼ E /E . We underline that in
theionizedelectronintheremotefuture,afterithasescaped c 0 a c 0 0 a
allexpressionsafterEq.(22)expansionsinthisparameterare
fromtheboundstate. Sincetheelectronlefttheatomicsystem
employed.
afterionizationandrecollisionisnotconsideredhere,itspo-
Wecometotheconclusionthatinthenonrelativisticregime
tentialenergyisvanishingforasymptoticallylargetimesand
thereforeitisjustifiedtouseS˙(1)(r,t)≈0. Consequently,the theCoulomb-correctedSFAamplitudediffersfromtheonein
0 thestandardSFAbythefollowingCoulomb-correctionfactor:
additionaltermS(1) intheexponentoftheamplitudehasno
influenceonthes0addlepointequationandleavesthesaddle Q =− 4Ip . (27)
pointsunchanged[73],however,itcanchangethepreexponen- nr r·E(t )
s
tialbyafactorexp[−iS(1)(r,t )]. Thus,theCoulomb-corrected
0 s Thetransitionamplitudecanthenbeexpressedinaverysimple
SFAamplitudeofionizationreads:
form:
(cid:90) ∞ (cid:90) ∞ (cid:110) (cid:111)
Mp(c) = −i −∞dt(cid:104)p+A(t)|Hint(t)exp[−iS0(1)(r,t)]|0(c)(cid:105) Mp(c) =4iIp −∞dt(cid:104)p+A(t)|0(c)(cid:105)exp −iS˜(t) . (28)
(cid:104) (cid:105)
×exp −iS˜(t) , (23) Thissimpleformfortheionizationamplitudeinlength-gauge
Coulomb-corrected SFA is achieved because the Coulomb-
where|0(c)(cid:105)istheelectronboundstateintheCoulombpoten- correctionfactorQnr cancelsthedipoleinteractionfactorr·E
tial. Thenexttaskistofindananalyticexpressionforthenew inthelength-gaugematrixelement.
preexponential factor for times t = t . Physically, S(1)(r,t) Theoccurringmatrixelementissingularatthesaddlepoint:
s 0 √
correspondstothesumofpotentialenergiestheelectronpos-
1 2 2κ5/2
sessesonitstrajectory. Whentheelectronhasleftthevicinity (cid:104)p+A(t)|0(c)(cid:105) =
oftheatomiccore,thepotentialenergyissmallandthereare π(cid:2)κ2+(p+A(t))2(cid:3)2
(cid:114)
no further contributions to S(1)(r,t). Since we consider the κ 1
0 = − , (29)
tunnelingregimewhereE0/ω (cid:29) κ,thissituationsetsinata 2πE(ts)2(t−ts)2
5
whereinthelaststeponlytheleadingorderterminE /E is ThecorrespondingVolkovwavefunctiondescribingthefree
0 a
retained, and the integral in Eq. (28)must be calculated via electroninthelaserfieldinthisgaugeis
the modified SPM [68], taking into account the pole during
theintegration. Comparedtothecaseofazero-rangepotential ψV(r,t)= √1 exp[ip·r+iS˜(t)]. (34)
thisyieldsacorrectionfactorintheamplitudeof 2π3
M(c) 23/2E Inthecaseofionizationofanegativeion,theionizationampli-
= a. (30)
M(z) |E(t )| tudeinthestandardSFAinvelocitygaugeisgivenbyEq. (9)
0
wherethepreexponentialmatrixelementisreplaced:
ThiscorrectionfactorisknownfromITM[35]butappearsto
bereproduciblealsowiththeSFAtechnique. Thedifferential (cid:104)p+A(t)|V|0(cid:105) →(cid:104)p|V|0(cid:105). (35)
ionizationrateinthecaseofaCoulombpotentialoftheatomic
coreis Since the matrix element (cid:104)p|V|0(cid:105) is constant and does not
dependonmomentuminthecaseofashort-rangepotential,it
 
ddw3(pc) = π42ωEκ036 exp−23EEa0 − Eκ0p2⊥− κ33Eω032p2E, (31) iitshoenidiszetanattniiocdanalrdatomStphFleAito.undeeinfothrealneneggtahtigvaeuigoen.Tishgeareufgoer-ei,ntvhaeroiavnetrailnl
InthecaseofaCoulomb-potentialasioniccorethesituation
andthetotalionizationrateyields
is different. Here the preexponential matrix-element is not
(cid:114) (cid:34) (cid:35) a constant and the different momentum dependencies could
3 κ7/2 2κ3
w(c) = 4 exp − . (32) lead to a gauge dependence. The Coulomb corrected SFA
πE1/2 3E0 basedontheeikonal-Volkovsolutioncanbedevelopedforthe
0
velocitygaugesimilartothatintheprevioussection.Thesame
These rates are identical to the PPT-ionization rate [35, 36]. stepsleadtothefollowingfinalexpressionfortheionization
The momentum distribution of the ionized electrons in the amplitude,cf. Eq.(28),
non-relativisticregimeindicatesthattheemissionofelectrons
withavanishingfinalmomentumismostprobable. Thelon- (cid:90) ∞ (cid:104) (cid:105)
M(c) = −i dt(cid:104)p|Q p·A(t)+A(t)2/2 |0(c)(cid:105)
gitudinal and the transversal widths of the distribution are p nr
√ √ −∞
∆ = E /E E /ωand∆ = E /E κ,respectively. (cid:110) (cid:111)
(cid:107) 0 a 0 ⊥ 0 a ×exp −iS˜(t) . (36)
Concludingthissection,withintheSFAS-matrixformalism
andemployingtheeikonal-Volkovwavefunctionforthede-
In contrast to the length gauge calculation, the saddle point
scriptionofthelaser-drivenelectroncontinuumdynamicsdis-
ofS˜ laysnotonthesingularityofthepreexponentialmatrix
turbedbytheatomicCoulombpotential,aswellasneglecting
elementandthestandardsaddlepointapproximationcanbe
recollisions,onecanderivequantitativelycorrectdifferentialas
applied. Ityieldsfortheamplitude
wellastotalionizationratesthatcoincidewiththeexpressions
obtainedwithinthePPTquasi-statictheory. Inthenextsection (cid:104) (cid:16) (cid:17)(cid:105)
2E p (p −2iκ)−2(p −iκ)κarctan pE
weapplytheCoulomb-correctedSFAformalisminvelocity M(c) = a E E √ E κ
gauge. p πp2|E(t )|3/2
E 0
III. NONRELATIVISTICCOULOMB-CORRECTEDSFA ×exp−(cid:16)κ23+|E(pt2⊥0)(cid:17)|3/2. (37)
INVELOCITYGAUGE
TheionizationdifferentialrateinthevelocitygaugeCoulomb-
It is well known that the SFA is, in general, not gauge- correctSFAreads
invariant and the SFA in different gauges correspond to dif-  
ftheereinotnpizhaytisoicnarlaatepporfoaxnimhyadtiroongse.n-IlniktehiisonseucstiinogntwheeCcoalucluolmatbe- ddw3(pc) = 4πκ26Eω03 exp−32Eκ30 − Eκ0p2⊥− κ33Eω032p2E
cthoerrreecstuedltsSFoAftihnevPePloTcitthyegoaruygaen.dLathteer,lewnegwthi-lglacuogmepCaroeuiltowmitbh- ×1+ 4pκ22 − p4κ arctan(cid:18)pκE(cid:19)1+ 2pκ22
corrected SFA to answer the question: in which gauge the E E E
Cofotuhleomiobn-iczoartrieocnterdateSFoAf aisnmhyodrerorgeelenv-lainktefioornt?heWcealcwuillaltuiosne + 4pκ22 arctan(cid:18)pκE(cid:19)21+ pκ22. (38)
E E
thisinformationinthenextpaperforthedevelopmentofthe
relativisticCoulomb-correctedSFA. The ionization differential rate in the velocity gauge differs
InvelocitygaugetheHamiltonianisgivenbyEq. (1)with fromthatinthelengthgauge,seeEq.(31),bytheexpression
theinteractionHamiltonian inthecurlybracketsinEq.(38). Toobtainthetotalionization
rate, the p -integration can be carried out analytically, but
⊥
H (t)=p·A(t)+A(t)2/2. (33) p -integrationhastobeaccomplishednumerically.
int E
6
1.2 10
a
1.0 1
0.8 0.1
wPPT0.6 (cid:72) (cid:76) wPPT 0.01
wi0.4 (cid:144) wi0.001(cid:144)
10(cid:45)4
0.2
10(cid:45)5
0.0 0.02 0.03 0.04 0.05 0.06
0 20 40 60 80 100
E E
∆ 0 a
60 b FIG.2:TheratioofthetotalionizationratederivedintheSFAwith
50 respecttothePPTratevstheparamete(cid:144)rE0/Ea:(black,solid)inthe
Coulomb-correctedSFAinlengthgauge,(green,dash-dotted)inthe
40 (cid:72) (cid:76) Coulomb-correctedSFAinvelocitygaugeforγ=0.1,(blue,dashed)
∆ 30 standardSFAwithaCoulomb-potential,(red,dotted)standardSFA
withazero-rangepotential.
20
10
0
fieldusingtheCoulomb-correctedSFAinlengthandvelocity
5 10 15 20
gauge. InFig.2wecomparethetotalionizationratesinthese
Z approximationswiththePPTionizationratefordifferentval-
uesof E /E . Forcomparisonalsotheratesinthestandard
0 a
FIG. 1: (a) The ratio of the total ionization rate derived in the
SFAarepresentedusingashort-rangepotentialandaCoulomb
Coulomb-correctedSFAwithrespecttothePPTratevstheparam-
√ potential. Allapproximationsshowthesamequalitativebe-
eterδ = E /E /γ: SFAinlengthgauge(black, solid), andSFA
0 a havior,buttheabsolutevaluesoftheratesdiffersignificantly.
in velocity gauge (red, dashed); (b) The parameter δ for different
nuclearchargesZ atafixedangularfrequencyω = 0.05a.u. and TheCoulomb-correctedSFAincreasestheionizationrateby
E /E =1/25.ThelaserintensityisI=5.6×1019×(Z/10)6W/cm2. severalordersofmagnitude. Thisisinaccordancewiththein-
0 a
tuitivepicturethattheCoulomb-potentiallowersthetunneling
barrierandthereforefacilitatestunneling. Further,itshouldbe
mentionedthattheCoulomb-correctionisonlydependingon
In Fig. 1 (a) we compare the total ionization rate calcu- E0/Ea,butnot,e.g.,onIporω.
latedwithintheCoulomb-correctedSFAinthelengthorve- Thus,fromtheresultsofthisandtheprevioussectionsone
locitygaugewiththePPTratefordifferentvaluesofthepa- canconcludethattheCoulomb-correctedSFAshowsagood
√
rameter δ = E /E /γ = (E /E )3/2(I /ω). This parame- agreementwiththePPTtheoryonlyinlengthgauge. Thisisa
0 a 0 a p
ter arises since the deviation in the two gauges depends on messagethatshouldbetakenintoaccountinthegeneralization
the curly bracket that is a function of p /κ with the typi- oftheCoulomb-correctedSFAintotherelativisticdomain.
E
cal value for the momentum in laser polarization direction
√
p ∼∆ = E /E E /ω. Whilethelength-gaugeresultcoin-
E (cid:107) 0 a 0
cideswiththePPTone,thevelocitygaugeresultstendstothe V. CONCLUSION
PPT-rateonlyinthelimitsδ→0[55]andδ→∞,deviating
fromthelatteratintermediatevaluesofδ. Thisisevidentfrom
WehaveappliedtheCoulomb-correctedSFAforionization
Eq. (38),sincethecurlybracketgoestooneinbothlimits. For
ofhydrogen-likesystemsinastronglinearlypolarizedlaser
intermediatevaluesoftheparameterδ,thedeviationcanbe
field.Thenonrelativisticregimeisconsideredtoshowhowthis
largerthanafactorof2. Notethatinthetunnelingregimethe
approximationworksandhowtousethedevelopedprocedure
parameterδcanvaryinthetotalrangeof(0,∞). InFig.1(b)
forafurthergeneralizationoftheapproximationintotherela-
we show the value of δ for different nuclear charges Z and
tivisticdomain. TheappliedCoulomb-correctedstrong-field
a suboptical angular frequency. It can be seen that for this
approximationincorporatestheeikonal-Volkovwavefunction
parametersetthevalueofδlaysinanareawheretheresultsin
forthedescriptionoftheelectroncontinuumdynamics. The
thetwogaugesdiffersignificantly.
latter is derived in the WKB approximation taking into ac-
counttheCoulombfieldoftheatomiccoreperturbativelyin
thephaseoftheWKBwavefunction,i.e.,inphysicalterms,
IV. COMPARISONOFDIFFERENTAPPROXIMATIONS thedisturbanceoftheelectronenergybytheCoulombfieldis
assumedtobesmallerwithrespecttotheelectronenergyin
Intheprevioussectionswehavecalculatedtheionization thelaserfield. Wehavederivedananalyticalexpressionfor
ofahydrogen-likesysteminastronglinearlypolarizedlaser theionizationamplitudewithintheCoulomb-correctedSFA
7
in length and velocity gauge. A simple expression for the gauge. The SFA in different gauges, in fact, corresponds to
amplitudeisobtainedwhenusingthelengthgaugewhichis different partitions of the total Hamiltonian used to develop
due to the fact that the Coulomb correction factor (ratio of theSFA[72]. Therefore,onecanconcludethattherelativistic
theCoulombcorrectedamplitudetothestandardSFAone)in generalizationoftheCoulomb-correctedSFA,whichwillbe
thisgaugecancelsthefactoroftheelectric-dipoleinteraction carriedoutinthenextpaperofthissequel,shouldbebasedon
Hamiltonian in the matrix element. Moreover, a Coulomb thepartitionofthetotalHamiltonianthatinthenonrelativistic
correctionfactorcoincidingwiththatderivedwithinthePPT limitcorrespondstothepartitionofthelengthgaugeSFA.
theoryisobtained. Thedifferentialandtotalionizationrates
arecalculatedanalytically. Thecalculatedtotalionizationrate
in length gauge is identical to the PPT-rate, while in the ve-
Acknowledgments
locitygaugeitcandeviatefromthePPTresultuptoafactor
of2. TakingintoaccountthatthePPT-rateprovidesagood
approximationforexperimentalresults,wecanconcludethat Valuable discussions with C. H. Keitel and C. Mu¨ller are
theCoulomb-correctedSFAworkssuccessfullyinthelength acknowledged.
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