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

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by  Michael Klaiber

Above-thresholdionizationwithhighly-chargedionsinsuper-stronglaserfields:
II.RelativisticCoulomb-correctedstrongfieldapproximation
Michael Klaiber,∗ Enderalp Yakaboylu, and Karen Z. Hatsagortsyan
Max-Planck-Institut fu¨r Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany
(Dated:January25,2013)
WedeveloparelativisticCoulomb-correctedstrongfieldapproximation(SFA)fortheinvestigationofspin
effectsatabove-thresholdionizationinrelativisticallystronglaserfieldswithhighlychargedhydrogen-like
ions. TheCoulomb-correctedSFAisbasedontherelativisticeikonal-Volkovwavefunctiondescribingthe
ionizedelectronlaser-drivencontinuumdynamicsdisturbedbytheCoulombfieldoftheioniccore.TheSFA
indifferentpartitionsofthetotalHamiltonianisconsidered.Theformalismisappliedfordirectionizationof
ahydrogen-likesysteminastronglinearlypolarizedlaserfield.Thedifferentialandtotalionizationratesare
3
calculatedanalytically.TherelativisticanalogueofthePerelomov-Popov-Terent’evionizationrateisretrieved
1
withintheSFAtechnique.ThephysicalrelevanceoftheSFAindifferentpartitionsisdiscussed.
0
2
PACSnumbers:32.80.Rm,42.65.-k
n
a
J I. INTRODUCTION exactly,whiletheCoulombfieldviatheeikonalapproxima-

4 tion. The nonrelativistic Coulomb-corrected SFA based on
2
theeikonal-Volkovwavefunctionhasbeenappliedrecently
SincethepioneeringexperimentbyMooreetal. [1]laser

] fieldsofrelativisticintensities(exceeding1018 W/cm2 atan formolecularhigh-orderharmonicgeneration[48–50]. The
h infraredwavelength)havebeenappliedfortheinvestigation eikonalapproximationhasbeengeneralizedtoincludequan-
p tumrecoileffects[51,52]. TherelativisticSFAbasedonthe
of strong field ionization dynamics of highly charged ions
m- [2–8], see also [9]. A lot of effort has been devoted to the generalizedeikonalwavefunctionhasbeenproposedin[53].
However,thefinalresultshavebeenobtainedonlyintheBorn
numericalinvestigationofthedynamicsofhighly-chargedions
o approximation.
inasuper-strongfields[10–20].
t
a Acommonanalyticalapproachforstrongfieldatomicpro- InthispaperwedeveloptheCoulombcorrectedSFAforthe
.
s cessesisthestrongfieldapproximation(SFA)[21–23]which relativisticregimebasedontheDiracequation,extendingthe
c hasbeenappliedforthetreatmentofrelativisticeffects[24,25]. correspondingnonrelativistictheory,seepaperI(thefirstpaper
i
s ThemaindeficiencyofthestandardSFAisthattheinfluence ofthissequel),intotherelativisticdomain. Thiswillallowus
y
oftheCoulombfieldoftheatomiccoreisneglectedforthe tocalculatespin-resolvedionizationprobabilitiestakinginto
h
electrondynamicsinthecontinuumandthelatterisdescribed accountaccuratelytheCoulombfieldeffectsoftheioniccore.
p
[ bytheVolkovwavefunction[26]. Whilethisiswell-justified IntheCoulombcorrectedSFA,theeikonal-Volkovwavefunc-
fortheionizationofanegativeion,foratomsand,moreover, tionisemployedforthedescriptionofthefinalstateinstead
1
for highly-charged ions it is not valid and the standard SFA oftheVolkovone. TheinfluenceoftheCoulombpotentialof
v
canprovideonlyqualitativelycorrectresults. Amodification theatomiccoreontheionizedelectroncontinuumdynamics
4
6 ofthetheoryisrequiredtotakeintoaccounttheinfluenceof is taken into account via the eikonal approximation. Direct
7 theatomiccoreonthefreeelectronmotion. ionizationofahydrogen-likesysteminastronglinearlypolar-
5 The Coulomb field effects during ionization have been izedlaserfieldisconsidered. Twoversionsoftherelativistic
. treatedinthePerelomov-Popov-Terent’ev(PPT)theorybased Coulomb corrected SFA are proposed that are based on the
1
0 ontheimaginarytimemethod [27,28]. InthePPTtheorythe usage of different partitions of the total Hamiltonian in the
3 barrierformedbythelaserandatomicfieldisassumedtobe SFAformalism. Thephysicalrelevanceofthetwoversionsis
1 quasi-staticandthetunnelingthroughitiscalculatedusingthe discussed.
: WKB-approximation. TherelativisticPPT-theory[29–32]has
v Theplanofthepaperisthefollowing: InSec.IItherela-
i providedthetotaltunnelingrate. However,spineffectsarenot tivisticCoulomb-correctedSFAisdeveloped. Thedifferential
X
investigatedthoroughly[33].
andtotalionizationratesforhydrogen-likesystemsarederived.
r ThestandardSFAtechniquehasbeenmodifiedtoinclude
a ThemodifiedSFAusingaspecificpartitionoftheHamiltonian
the Coulomb field effects of the atomic core. A heuristic
is considered in Sec. III and is used for the calculation of
Coulomb-Volkovansatzhasbeenusedforthispurpose, see
ionizationrates.
[34–45]. Inamorerigorousway,theSFAismodifiedbyre-
placingtheVolkovwave-functionintheSFAtransitionmatrix
elementbythe,so-called,eikonal-Volkovone[46,47].Thelat-
terdescribestheelectroncontinuumdynamicsinthelaserand
theCoulombfield. Herethelaserfieldistakenintoaccount
II. RELATIVISTICCOULOMB-CORRECTEDSFA
Weconsidertheinteractionofahighly-chargeionwitha
∗Correspondingauthor:[email protected] stronglaserfieldintherelativisticregimewhichisdescribed
2
bythetime-dependentDiracequation: The hydrogen-like system interacts with a strong linearly
polarizedlaserfield
(cid:32) (cid:33)
1 V(c)
iγµ∂ + γµA −γ0 −c ψ=0, (1)
µ c µ c
E(η)=−E cos(ωη), (8)
0
whereγµ aretheDiracmatrices,V(c) =−κ/ristheCoulomb
potential of the atomic core, κ the typical electron momen-
tumintheboun√dstate,determinedbytheboundstateenergy whereη=kµx /ω,andkµ =(ω/c,k)isthelaser4-wave-vector.
µ
via c2 − Ip = c4−c2κ2, Ip the ionization potential, Aµ is [thecoordinateandthemomentumprojectionsaredefinedas
the4-vectorpotentialofthelaserfield(atomicunitsareused r ≡r·eˆ,r ≡r·kˆ, p ≡p·eˆ, p ≡p·kˆ,and p ≡p·(kˆ×eˆ),
E k E k B
throughout). The transition amplitude for the laser induced withtheunitvectorskˆ andeˆ inthelaserpropagationandthe
ionizationprocessfromtheinitialboundstateφ(c)intoanexact polarizationdirection, respectively]. Thevector-potentialis
i
continuum state ψ with an asymptotic momentum p can be chosenintheGo¨ppert-Mayergauge:
written[24]
(cid:90)
i
M(c) =− d4xψ(r,t)A/φ(c)(r,t), (2) Aµ ≡(ϕ,cA)=(−r·E,−kˆ(r·E)). (9)
fi c i Generally, the SFA in different gauges do not coincide and
correspondtodifferentphysicalapproximations[55–57]. In
withA/ ≡γµA . Equation(2)isstillexactwiththeexactwave
µ paper I we have seen that in the nonrelativistic regime the
functionψ(r,t)describingthedynamicsoftheelectroninthe
Coulomb-corrected SFA in the length gauge leads, first, to
ionic and the laser field. Here, we have used the standard
rathersimpleexpressionsfortheCoulomb-correctedionization
partitionofthetotalHamiltonian:
amplitude,seeEq. (I.28)[referstoEq. (28)ofpaperI],which
H = H +H , (3) isduetothecancellationofther·EinteractionHamiltonian
0 int
inthematrixelementbytheCoulomb-correctionfactor;and,
H = cα·pˆ +βc2+V(c)(r) (4)
0 second, to results coinciding with the PPT-ionization rates.
Hint = βA/, (5) Asthelatterprovidesagoodapproximationtoexperimental
observations, in the relativistic regime we choose a gauge
whereα=γ0γ,β=γ0aretheDiracmatrices,andcthespeed
whichgeneralizesthelengthgaugeintotherelativisticdomain
oflight. Weconsiderahydrogen-likehighlychargedionin
[58],thatistheGo¨ppert-MayergaugedefinedbyEq. (9).
the relativistic parameter regime, i.e., the wave function of
theinitialboundstateφ(c)fulfillstheDiracequationwiththe In the conventional SFA the exact continuum state is ap-
i
HamiltonianH andisgivenby[54] proximatedbytheVolkov-wavefunctionwhichisidenticalto
0
thefirstorderWKB-approximationofthecontinuumelectron
φ(ic)(r,t) = κ√3/π2 (cid:118)(cid:117)(cid:117)(cid:116)Γ(cid:16)23−−cI2p2Ip(cid:17)(2κr)−cIp2 imbnyatthtihoeenlarisseeparlcafihceieelmdve.edAntinsoytfhstetehmreealeatxitciavciitmstwipcraoCvveoeumfluoenmncttbio-ocfnotrhψrie(srca,tept)dprwSoFixtAhi-
c2
(cid:104) (cid:105) therelativisticeikonal-Volkovwavefunction. Similartothe
×exp −κr−i(c2−Ip)t vi, (6) non-relativisticcase,seepaperI,firstweapplytheWKBap-
proximationtothesolutionoftheDiracequation(1),thenthe
wherethebispinor resultingequationswillbesolvedviaaperturbativeexpansion
(cid:32) (cid:33) intheCoulombpotentialV(c).
χ
v = i (7)
i iIp σ·rχ Due to the gauge covariance of the Dirac equation, we
cκ r i switch to the velocity gauge with its vector potential A˜µ =
describesthespin-upand-downstates(i=±),withthetwo- A˜µ(η) = (0,cA˜), A˜ = −(cid:82)η dη(cid:48)E(η(cid:48)) and after solving the
−∞
componentspinorsχ+ = (1,0)andχ− = (0,1),respectively, wave-equation go back to the Go¨ppert-Mayer gauge. The
andthePauli-matricesσ. quadraticDiracequationinvelocitygaugebecomes
(cid:32) (cid:33)(cid:32) (cid:33)
1 V(c) 1 V(c)
i(cid:126)γµ∂ + γµA˜ −γ0 +c i(cid:126)γµ∂ + γµA˜ −γ0 −c ψ=0, (10)
µ c µ c µ c µ c
 
−(cid:126)2∂2+ ic(cid:126)(cid:16)k/A/˜(cid:48)+2A˜·∂+α·∇V(c)−2V(c)∂0(cid:17)+ Ac˜22 + Vc(c2)2 −c2ψ=0. (11)
whereA˜(cid:48) ≡ 1 dA˜,andwehaveinserted(cid:126)toindicatetheWKB-expansion. Letusassumethatthesolutionhastheformψ=eiS/(cid:126),
ωdη
3
thenthecorrespondingequationforS is
(cid:32) A˜ V(c)(cid:33)(cid:32) A˜µ V(c)(cid:33) (cid:126)(cid:32) k/A/˜(cid:48) α·∇V(c)(cid:33)
∂ S − µ +g ∂µS − +gµ0 + ∂2S − − =c2.
µ c µ0 c c c i c c
(cid:126)
TheWKBexpansionS =S + S +...yieldsfollowingequationsupthefirstorderin(cid:126)/i
0 i 1
(cid:32)(cid:126)(cid:33)0 2A˜ ∂µS A˜2 2V(c)∂ S V(c)2
: ∂ S ∂µS − µ 0 + −c2 =− 0 0 − , (12)
i µ 0 0 c c2 c c2
(cid:32)(cid:126)(cid:33)1 2A˜ ∂µS k/A/˜(cid:48) 2V(c)∂ S α·∇V(c)
: 2∂ S ∂µS − µ 1 =−∂2S + − 0 1 + . (13)
i µ 0 1 c 0 c c c
(cid:112)
Theseequationscanbesolvedperturbativelywithrespectto withε = c c2+p2 andtheconstantofmotionΛ ≡ k· p/ω
thepotentialtermV(c). Ifwedefine Further, Eqs.(17)and(19)canbesolvedviathemethodof
characteristicsas
S = S(0)+S(1), (14)
0 0 0 dS(1) dxµ V(c)
S1 = S1(0)+S1(1), (15) dσ0 = ∂µS0(1)dσ =− c π0, (24)
thefollowingequationscanbederived dS(1) dxµ
1 = ∂ S(1) (25)
dσ µ 1 dσ
∂µS0(0)∂µS0(0)− 2A˜µ∂cµS0(0) + Ac˜22 −c2 =0, (16) = −α·∇V(c) + ∂2S0(1) +∂ S(1)∂µS(0)+ V(c)∂tS1(0),
V(c) 2c 2 µ 0 1 c2
∂ S(1)πµ =− π0, (17)
µ 0 c wherethetrajectoryisgivenby
2∂ S(0)πµ =∂2S(0)−k/A/˜(cid:48), (18)
µ 1 0 dxµ
α·(∇V(c)) =πµ. (26)
∂ S(1)πµ =− dσ
µ 1 2c
Thisleadstodσ=dη/Λ,withtherelativistickineticmomen-
∂2S(1) V(c)∂S(0)
+ 0 +∂ S(1)∂µS(0)+ t 1 , (19) tuminthelaserfield
2 µ 0 1 c2
π(η) ≡ p(η)=∇S(0)+A˜(η)
A˜µ 0
whereπµ = −∂µS0(0) + c istherelativisticmomentum. Eq. = p+A˜(η)+kˆ(p+A˜(η)/2)·A˜(η), (27)
(16)givestheVolkov-action,i.e., cΛ
ω (cid:90) ∞(cid:32) A˜2(cid:33) andthecorrespondingrelativistickineticenergy
S(0) =−p·x− A˜·p+ dη(cid:48). (20)
0 c(k·p) 2c
η (p+A˜(η)/2)·A˜(η)
cπ0(η)≡ε(η)=−∂S(0) =ε+ . (28)
Ontheotherhand,thesolutionofEq. (18)is t 0 Λ
k/A/˜ ThenthesolutionsofEqs. (24)and(25)read
S(0) =− . (21)
1 2c(k·p) S(1)(r,η) = (cid:90) ∞dη(cid:48)ε(η(cid:48))V(c)(cid:0)r(η(cid:48))(cid:1), (29)
HereweshouldnotethatS(0)andS(0)generatethefullVolkov 0 η c2Λ
solution. Thecorrespondin0gequati1onsintheGo¨ppert-Mayer (cid:90) ∞ dη(cid:48) (cid:34)α·∇V(c)(r(η(cid:48)))
S(1)(r,η) = (30)
gaugecanbefoundviaagaugetransformationwithangener- 1 Λ 2c
atingfunctionχ=A˜(η)·r. Afteracoordinatetransformation η 
from(t,r)to(η,r)theybecome − V(c)(r(ηc(cid:48)))∂0S1(0) −∂µS0(1)∂µS1(0)− ∂2S20(1)
(cid:18) ε (cid:19)
S(0)(r,η) = p+A˜(η)− kˆ ·r (22)
0 c withtherelativistictrajectoryoftheelectroninthelaserfield
+(cid:90) ∞η(cid:48)(cid:32)ε+ (p+A˜(η(cid:48)Λ)/2)·A˜(η(cid:48))(cid:33), r(η(cid:48))=r+(cid:82)ηη(cid:48)dη(cid:48)(cid:48)p(η(cid:48)(cid:48))/Λ,startingattheionizationphaseη
η withcoordinater.
(1+kˆ ·α)(α·A˜) We can evaluate the explicit parameters for the applied
S1(0)(η) = 2cΛ , (23) eikonalapproximationestimatingtheimposedconditionsfor
4
the expansion of Eqs. (14) and (15): S(1) (cid:28) S(0), and Thus,therelativisticeikonal-Volkovwavefunctionreads
0 0
S(0) (cid:28) S(1). For this purpose we use order of magnitude
1 0 ψ (r,t)
estimations: f
S0(0) ∼ Ipτc ∼ EEa, (31) = 1+ (1+kˆ ·α)α·A˜(η)+2c(cid:82)Λη∞dη(cid:48)α·∇V(c)(r(η(cid:48)))
0
(cid:114)  cu
S0(1) ∼ log EEa∼1 (32) ×(cid:112)(2πf)3εexp[iS0(0)(r,η)+iS0(1)(r,η)], (42)
0
(cid:114)
S(0) ∼ A˜ ∼ E0τc ∼ Ip (33) with the bispinor uf for the spin-up and -down final states
1 cΛ c c2 (f =±)
(cid:90) (cid:114) (cid:114)
S1(1) ∼ dtcrr˙EE((tt))2 ∼ cr1Ec ∼ cIp2 EE0a. (34) uf = (cid:112)(1+√(σε·p/)cχ2f)/2χf , (43)
2(c2+ε)
Here, the first two equations were already derived in paper
I,seeEq. (I.22), Ea ≡ (2Ip)3/2. Further,takingintoaccount where χ+ = (1,0) and χ− = (0,1). The last term in the pre-
thetypicaltimeduringtunnelingτc ∼ γ/ω = κ/E0,withthe exponential[∝∇V(c)]describesthespin-orbitcouplingduring
Keldyshparameterγ = ωκ/E0 [21], thevalueofthet(cid:112)ypical ionization(under-the-barriermotion). Viaa p/c-expansionof
velocityoftheelectronduringtunnelingr˙Ec ∼κ[κ≈ 2Ip], theDiracequationintheatomicandthelaserfielditcanbe
the interaction lengt√h rEc ∼ r˙Ecδτc, the uncertainty of the shownthatthespin-orbitcoupling,givenintheHamiltonian
initialtimeδτc ∼1/ κE0 [fromthesaddle-pointintegration bytheterm
δτ2 ∼ 1/S¨˜(t )], ε(η) ∼ c2, Λ ∼ 1 [the electron is at rest at
c s c (cid:104) (cid:105)
the tunnel exit], and A˜ ∼ E τ , we come to the following σ· p(η)×∇V(c)(r(η))
0 c H = , (44)
conditionsfortheapplicabilityoftheeikonalapproximation: SO 4c2
(cid:114)
E I canbeevaluatedalongthemostprobabletrajectoryasfollows
p
(cid:28)1 and (cid:28)1. (35)
Ea c2 p ∂V(c) p ∂V(c) (cid:32)I (cid:33)3/2
H τ ∼ k τ , E τ ∼ p , (45)
TheactionfunctionS(1)canbeapproximatedbythefirstterm SO c c2 ∂rE c c2 ∂rk c c2
1
inEq. (30):
wherethetypicalvaluesfor p ∼ κ, p ∼ κ2/c, r ∼ r κ/c
E k k Ec
(cid:90) ∞ α·∇V(c)(r(η(cid:48))) havebeenused. Itisofhigherordersmallnessthantheterms
S(1)(r,η)= dη(cid:48) . (36) keptintheeikonalwavefunctionandisneglectedinthefurther
1 2cΛ
η calculation.
Finally,theionizationamplitudeintheCoulomb-corrected
Infact,theorderofmagnitudeofthetermsinr.h.s. ofEq.(30)
SFAwillhavethefollowingformintherelativisticregime:
are
T ∼ (cid:90) dηV(c) ∼ (cid:114)E0 (cid:114)Ip, (37) M(fci) = −i(cid:90) ∞dη(cid:104)p(η)r,m|Hintexp[−iS0(1)(r,η))]|0r,i(cid:105)
1 cr E c2 −∞
Ec a ×exp[−iS˜(η)]. (46)
E (cid:90) E (cid:32)I (cid:33)3/2
T ∼ 0 dηV(c) ∼ 0 p , (38)
2 c3 Ea c2 withthespatialpartsoftheinitial|0r,i(cid:105)andthefinal|p(η)r, f(cid:105)
(cid:90) spinorsandthecontractedaction
E τ E I
T ∼ 0 dηV(c) c ∼ 0 p, (39)
3 c2 rEc Eac2 (cid:90) ∞ (cid:40) [p+A˜(η(cid:48))/2]·A˜(η(cid:48))(cid:41)
S˜(η)= dη(cid:48) ε−c2+I + . (47)
p Λ
andT isvanishingbecauseinthetunnelingregion∆V(c) =0. η
4
Fromthelatter,weestimatetheratios:
In the adiabatic regime (ω (cid:28) I ,U , with the pondero-
p p
(cid:114) motive potential U = E2/4ω2) the time integration in the
T I E p 0
2 ∼ p (cid:28)1, (40) amplitudeofEq.(46)iscalculatedwithSPM.Asinthenon-
T1 c2 Ea relativisticcase,herealsothedisturbanceofthesaddle-point
(cid:114) (cid:114)
T I E conditionsduetotheCoulombfieldisneglected. Thisisinac-
3 ∼ p (cid:28)1, (41)
cordancewithourapproachtotakeintoaccounttheCoulomb
T c2 E
1 a
fieldperturbativelyinthephaseoftheWKBwavefunction:
therefore, only the first integrand term may be maintained S(1)isalreadyproportionaltotheCoulombfieldV(c)(r),there-
0
in Eq. (30). Further, one can see from Eqs. (33) and (34) fore, in the trajectory r(η) it should be neglected. Mathe-
that S(0) (cid:28) 1 and S(1) (cid:28) 1, which allows us to expand the maticallythisisjustifiedasonecanseefromtheestimation
1 1 √
correspondingexponents. ∂ S(1) ∼ I E /E [similartoestimationsinEqs. (31)-(34)].
η 0 p 0 a
5
Itshowsthat∂ S(1) isnegligiblysmallcomparedto I , and, approximatedby:
η p
consequently, the saddle-point condition ∂ηS˜(ηs) = 0 is not (cid:90) η0 iκ
disturbed. Thelatterleadstothefollowingtwosaddlepoints −iS(1)(r,η )= dη(cid:48)
s Λc2
percycle: ηs
ε(η )+∂ ε(η )(η(cid:48)−η )+ ∂2η,ηε(ηs)(η(cid:48)−η )2
 (cid:115)  × s(cid:12) η s s 2 (cid:12) s
ωη1 = arcsin−Ep0/Eω +i 2Λ(ε−(Ec02/+ωI)2p)−p2E (48) −(cid:90) η(cid:12)(cid:12)0xd+η(cid:48)piEκΛ(ηs)(η(cid:48)−ηs)− E2(rΛηks()η(η(cid:48))(cid:48)2−ηs)2(cid:12)(cid:12) ,(51)
ωη2 = π−arcsin−Ep/Eω −i(cid:115)2Λ(ε−(Ec2/+ωI)2p)−p2E. where ηs 2 (cid:12)(cid:12)(cid:12)x+ pEΛ(ηs)(η(cid:48)−ηs)− E2(Ληs)(η(cid:48)−ηs)2(cid:12)(cid:12)(cid:12)3
0 0 (cid:112) (cid:112)
i 2I r 2I iE(η ) 2I
r (η) = p E − p(η−η )+ 0 p(η−η )2
k 3c 3c s 2c s
The amplitude is now evaluated at these phases. First, we
considertheactionS˜(ηs)intheexponent: −E(6ηc0)2(η−ηs)3, (52)
(cid:113) (cid:113) (cid:16) (cid:17)
(cid:104)2Λ(ε−c2+I )−p2(cid:105)3/2 pE(ηs) = i 2Λ(ε−c2+Ip)−p2E ≈i 2Ip 1−5Ip/36c2 .
Im{S˜(η )}= p E (49) (53)
s 3|E(η )|Λ
0 ε(η ) = c2−I (54)
s p
with|E(η )|= E (cid:112)1−(p /(E /ω))2. Sincetherealpartgives ∂ηε(ηs) = −pE(ηs)E(η0)/Λ (55)
0 0 E 0 ∂2 ε(η ) = E(η )2/Λ (56)
anunimportantphaseintheresultingamplitude,onlytheimag- η,η s 0
inarypartisgiven. Thisfunctionintheexponentdominates Otherparametersareωη =arcsin(cid:2)−p /E /ω(cid:3),κ≈ (cid:112)2I (1−
0 E 0 p
themomentumdistributionanddeterminesthemaximumof
I /4c2), Λ ≈ 1 − I /3c2, r (η) ≈ 0, E(η ) ≈ E(η ). To
themomentumdistributionwhichislocatedataparabolawith p p B s 0
have compact expressions, expansions on the small param-
p = I /3c+p2/2c(1+I /3c2), p =0. Inthefollowing,we,
k p E p B eterI /c2 havebeenusedabove[I /c2 ≈ 0.25forhydrogen-
therefore, evaluate the less important pre-exponential factor p p
like uranium]. The starting coordinate of the trajectory
onlyatthisparabolaandneglectdeviationsfromit. Inevalu-
r is found from the saddle-point condition of the integral
atingS0(1) atthesaddlepointη=ηs,notethattheintegration (cid:82) d3rexp[−ip(ηs)·r−κr], that is rs/rs = p(ηs)/(iκ). With
inS0(1) startsatthesaddlepointηs,whentheelectronenters theseparameterstheintegralinEq.(51)canbecalculated:
thebarrierduetotheeffectivepotential,andgoestoinfinity (cid:104) (cid:105)
when the electron is far away from the core. However, the Qr ≡ exp −iS(1)(r,ηs)
upperintegrationlimitcouldbesetatthemomentη0,when (cid:34)2I (cid:35)1− Ip
the electron leaves the barrier since further integration over = exp p 6c2
thecontinuumleadseventuallyonlytoanunimportantphase c2 λ1−cIp2
othfethbeararmieprliistusdheo.rTthceomtimpaerewdhwicihththteheelleacsterronpesrpieonddasnudndtheer = exp(cid:34)2cI2p(cid:35)(cid:32)1− 6Icp2(cid:33)Q1nr−cIp2 (57)
integrandcanbeexpandedaroundthesaddlepointη :
s
withthesmallparameterλ=−r·E(η )/4I . Intheexpression
s p
above only the leading order term in 1/λ are retained. For
−iS(1)(r,η )=(cid:90) η0dη(cid:48) iκ (50) justificationofourapproximations,wecompareinFig.1our
s ηs Λc2 analytically derived Coulomb-correction factor Qr with the
ε(η )+∂ ε(η )(η(cid:48)−η )+ ∂2η,ηε(ηs)(η(cid:48)−η )2 exactresultofanumericalcalculationofEq.(50),andwith
×(cid:12) s η s s 2 s (cid:12), theCoulomb-correctionfactoroftherelativisticPPT[31]. De-
(cid:12)(cid:12)r+ p(ηs)(η(cid:48)−η )+ FL(ηs)(η(cid:48)−η )2+ kˆE(ηs)2(η−η )3(cid:12)(cid:12) viation of our approximate Coulomb-correction factor from
(cid:12) Λ s 2Λ s 6cΛ2 s (cid:12)
the exact one occurs mostly due to a linearization of the λ-
dependenceintheanalyticalexpressionswhich,however,is
whereF (η)=−E(η)−kˆ(p(η)·E(η))/cΛistheLorentz-force neededforthefurtheranalyticalintegration.
L
due to the laser field. As the correction factor is evaluated Using the Coulomb-correction factor Qr of Eq. (57), the
at the parabola with p = I /3c + p2/2c(1 + I /3c2) and spatialintegrationinEq.(46)canbecarriedout.TheCoulomb-
k p E p
p = 0, we derive the trajectories at these special values of correctedpre-exponentialmatrixelementreads
B
momentum.Duringthemotionunderthebarrierthemagnitude m˜ (η )=(cid:104)p(η ) , f|r·E(η )(1−kˆ ·α)Q |0 ,i(cid:105). (58)
of the coordinate variation in the kˆ-direction (imaginary) is fi s s r s r r
significantly smaller than in eˆ-direction [r (η)/r (η) ∼ κ/c] ItisremarkablethattheCoulomb-correctionfactorinEq. (58)
k E
andtheintegrandcanbeexpandedbyr (η). Itistakeninto cancels the dependence of the matrix element on the elec-
k
account also that the ionization event happens close to the tric dipole operator r · E and, approximately, also the pre-
laser polarization axis: rk (cid:28) rE. The integral is, therefore, exponentialtermr−Ip/c2 oftheinitialstatewavefunction[when
6
(cid:112)
withβ= 2Ip/candm−+ =m+− =0. Eqs. (61)and(62)are
200 exactin p /c,the p /ctermcanbeimportantatξ (cid:29)1. Asin
E E
100 thenon-relativisticcase,seepaperI,thematrixelementhasa
singularityatthesaddlepointη andthemodifiedSPMhasto
50 s
beappliedwhichinducesanadditionalpre-exponentialfactor
Qr
20
1 1
= . (63)
10 [p(ηs)2+κ2]2 4(p(ηs)·p˙(ηs))2(η−ηs)2
5
The total pre-exponential factor after the η-integration in
0.05 0.10 0.15 Eq.(46)becomes:
FγI=G.01.0:1TahnedCIpo/ucl2o=mb0-.c2o5r(reeqctuiiovnalfeancΛttotorZQr=v9s1t)h:e(bplaarcakm,estoelridλ)tfhoer mˆfi(ηs)= √2π32−(cid:113)2cI2Γp((cid:16)23Ip−)322c−I2p23cI(cid:17)p2|Ee(xηps(cid:104))|2c32I2−p(cid:105)cIp2 mˆfi, (64)
numericalcalculationviaEq. (50)withoutapproximations, (blue,
short-dashed)theanalyticalresultofEq.(57),(red,long-dashed)the with
Coulomb-correctionfactoroftherelativisticPPT[31]. √ (cid:16) (cid:17)
2c(2c+ip ) 1+ β
mˆ++ = (cid:113) E 2 (65)
8c4+6c2p2 +p4
E E
theapproximationr≈rE,i.e.,ionizationhappensatthelaser β2c(cid:16)20c4+2ic3p +13c2p2 +icp3 +2p4(cid:17)
pploileadriGzao¨tpiopnerat-xMisa,yiserugsaeudg].eTwhhiischissiagncoifincsaenqtulyenscimepolfifitheestahpe- + 3√2(2c−ip )(cid:16)E2c2+p2(cid:17)E3/2 (cid:113)4cE2+p2E ,
calculationoftheionizationmatrixelement. E E E
√ (cid:16) (cid:17)
In this paper we are concerned with spin-unresolved ion- 2c(2c−ip ) 1− β
ization probabilities. Therefore, we are free to choose the mˆ−− = (cid:113) E 2 (66)
orientation of the quantization axis. In the following we as- 8c4+6c2p2 +p4
E E
sumethattheinitialspinoroftheboundstateisalignedalong (cid:16) (cid:17)
tehleemlaesnetrimnaEgqn.e(t5ic8)fiienldthdisirceacstieoyni.eTldhse:spin-dependentmatrix +β23c√220(c24c−+2ipic3)p(cid:16)E2c+21+3cp22p(cid:17)2E3/2−(cid:113)ic4pc3E2++2pp24E ,
m˜fi(ηs) = (cid:90) d3r2πc2κ√3/22Iεp (cid:118)(cid:117)(cid:116)Γ(23−−cI2p2Ip)(cid:32)|E8(κηIsp)|(cid:33)−cIp2 aisnudsmeˆd−+ =mˆ+− =0. InthelEaststeptheEfollowingexpEansion
c2
×exp(cid:34)2cI2p(cid:35)exp(cid:2)−ip(ηs)·r−κr(cid:3) (59) −(cid:113)2πiS¨˜0(ηs) ≈ √2π (cid:32)1+ 41Ip(cid:33). (67)
×(cid:32)1− Ip (cid:33)u+(cid:16)1−kˆ ·α(cid:17)v 4(p(ηs)·p˙(ηs))2 4(2Ip)3/4|E(ηs)|3/2 24c2
6c2 f i Wenotethatanarbitraryconfigurationofthequantization
= 23−π2cI(cid:113)2p(Γ2I(cid:16)p3)−49−223cIIp2p|(cid:17)E(cid:2)(pη(sη)|)cIp22e+xκp2(cid:104)(cid:3)22cI2p(cid:105)mfi, (60) ”athuxepis”incaaintnidabl”edaonawcdcntoh”m,eip.fielni.s,ahmlesdjt=avtie±ah1aa/rv2oetfaottwrioojn=.inE1dx/ep2pl,eictnhidteleyrn,otstasinoteclduetsbitooantthes
c2 s isgivenby
wherethefactorm fordifferentspintransitionsis (cid:88)
fi |1/2,m(cid:105)(cid:48) = |1/2,m(cid:48)(cid:105)D (R) (68)
√ (cid:16) (cid:17) m(cid:48),m
m++ = (cid:113)2c(2c+ipE) 1+ β2 (61) where D (R)isthe j =m1(cid:48)/2representationoftherotation
8c4+6c2p2 +p4 m(cid:48),m
E E matrix. Since the initial quantization axis is along the laser
(cid:16) (cid:17)
β2c 168c4−16ic3p +142c2p2 −8icp3 +25p4 magneticfield,anyquantizationaxiscanbealignedwithtwo
− 24√2(2c−ip )E(cid:16)2c2+p2(cid:17)E3/2 (cid:113)4c2E+p2 E , angles,seeFig. 2,withtherotationmatrix
m−− = √(cid:113)2c(2c−ipE)(cid:16)1−E β2(cid:17) E E (62) Dm(cid:48)m =ee−iiζζ11//22scions((ζζ22//22)) −eei−ζ1i/ζ21/c2ossin(ζ(2ζ/22/2)). (69)
8c4+6c2p2 +p4
(cid:16) E E (cid:17) Therefore the transition matrix element mˆ(cid:48) for arbitrarily
β2c 168c4+16ic3p +142c2p2 +8icp3 +25p4 fi
− 24√2(2c+ipE)E(cid:16)2c2+p2E(cid:17)E3/2 (cid:113)4c2E+p2E E , rotatedspinorsisgivenmˆb(cid:48)fyi = D∗fjmˆjkDki (70)
7
FIG.2:Anyquantizationaxiscanbealignedwithtwoangles.
in terms of the initial one mˆ . For instance, in the case the
fi
quantizationaxisisalongthelaserelectricfieldthematrixis
1−i −1−i
Dm(cid:48)m =12+i 1+2i , (71)
2 2
while in the case the quantization axis lays along the laser
propagationdirection,ityields
Dfi =√√112 −√1√12, (72)
2 2
andthecalculationofthecorrespondingtransitionamplitudes
isstraightforward.
Thedifferentialionizationrateaveragedbytheinitialspin
polarization and summed up over the final polarizations is
definedas FIG.3:ThedifferentialionizationratefortheparametersI /c2=0.25,
p
andE /E =1/25:(a)relativisticcalculationviaEq.(74);(b)non-
dw ω(cid:16) 0 a √
d3p = 2π |mˆ++(ηs)|2+|mˆ+−(ηs)|2+|mˆ−+(ηs)|2 relativistic calculation via Eq. (I.31), ∆(cid:107) ≡ Ea/E0(E0/ω) is the
longitudinalmomentumwidth.
(cid:17)
+|mˆ (η )|2 exp{−2Im[S0(η )]}, (73)
−− s s
whichwiththehelpofEq. (64)willread
dw 23−4cI2p (cid:18)1+ 2pc2E2 − 45cIp2 − 1924pc2E4Ip(cid:19)(2Ip)3(cid:16)1−cIp2(cid:17)ωexp(cid:16)4cI2p(cid:17)
=
d3p π2Γ(cid:16)3− 2cI2p(cid:17)|E(ηs)|3−2cI2p (cid:18)1+ 2pc2E2(cid:19)2
×(cid:32)1+ 41Ip(cid:33)exp−2(cid:104)2Λ(ε−c2+Ip)−p2E(cid:105)3/2.
12c2  3|E(ηs)|Λ  respecttothenonrelativisticone. Whilethenonrelativisticdis-
tributionpeakisat p =0, p =0,intherelativisticcaseitis
E k
(74) shiftedto p =0, p = I /3c. Aswehaveshownin[64],this
E k p
momentumshiftisduetotheunder-thebarrierdynamicsand
Notethatthereisanadditionalfactoroftwo,thatarisesdue
arisesalreadyattheionizationtunnelexitbut,notwithstanding
tosummingupoverthetwosaddlepointsperlasercycle. The
of the large momentum transfer from the laser field during
momentumdistributionationizationintherelativisticregime
theelectronexcursionaftertheionization,thecharacteristic
isplottedinFig.3. Asitisalreadymentioned,thedistribution
momentumshiftofthepeakofthedistributionisdetectable
islocatedaroundaparabola,seealso[59–63].
farawayfromtheinteractionzoneatthedetector.
Characteristicfeaturesofthemomentumdistributioninthe
relativisticregimearethelargeparabolicwingscorresponding
tothelargelongitudinalmomentum(alongthelaserpropaga-
tiondirection)whichariseduringthefreeelectronmotionin
thelaserfield. However,amoreinterestingfeatureofthisdis- Whentheexponentinthedifferentialionizationrateisex-
tributionisthesmallshiftofthepeakofthedistributionwith pandedquadraticallyaroundtheparabola p = p , p =δp ,
E E B B
8
p = I /3c+p2/2c(1+I /3c2)+δp ,therateexpressionreads 10
k p E p k
dw 23−4cI2p (cid:18)1+ 2pc2E2 − 45cIp2 − 1924pc2E4Ip(cid:19)exp(cid:16)4cI2p(cid:17)(2Ip)3(cid:16)1−cIp2(cid:17)ω u. 8 (cid:68)
=
d3p π2Γ(cid:16)3− 2cI2p(cid:17)|E(ηs)|3−2cI2p (cid:18)1+ 2pc2E2(cid:19)2 a. 6 (cid:64)
0 4
×(cid:32)1+ 41Ip(cid:33)exp(cid:40)− 2Ea (cid:32)1− Ip (cid:33) (75) w1 (cid:144)
12c2 3|E(η )| 12c2 2
s
 
−|E(cid:112)(2ηIsp)|(cid:16)1−δp7722BIcp2(cid:17)2 + (cid:18)1+ 8Icp2 + 2δpc2Ep22k(cid:16)1+ 2141cIp2(cid:17)(cid:19)2. 00.00 0.05 0I.p10c² 0.15 0.20
InEq. (74)anexpansionontheIp/c2parameterhasbeenused FIG.4:ThetotalionizationratevsionizationpotentialforE0/Ea=
toclearlyindicatetheIp/c2-scaling. WithoutIp/c2-expansion 1/32:(blue,short-dashed)intherelativisticCoulomb-correctedSFA
theexponentialfactorofthedifferentialionizationratereads viaEq.(78),(red,long-dashed)inthe(cid:144)relativisticCoulomb-corrected
dd3wp ∝ exp−(1+2√Ξ32Ξ)|E3c(3ηs)| − E(cid:112)(ηIsp) (cid:113)√13+Ξ(√δΞp2−2B1) mReofd.i[fi3e2d].SFAviaEq.(98),and(black,solid)relativisticPPTfrom
1+Ξ2
− E(cid:112)(2ηIsp)(cid:114)83Ξ(cid:16)Ξ2+3(cid:17)δDp22k, (76) afrnedetehveowlvaivnegfautnocmtiiocnsyosfttehme,infuitlifialllisntagtethφe(ice)q(ru,att)iodnescribesthe
(cid:113) √ (i∂t−H0)φ(ic)(r,t)=0, (80)
whereΞ= 1−Υ/2( Υ2+8−Υ),Υ=1−I /c2,and
p
withH = H−H = H = cα·pˆ +βc2+V(c)(r). However,
0 int 0
(cid:115)
(cid:16) (cid:17) √ 2 in general, as it is pointed out in [55, 56], the SFA can be
D ≡ Ξ2+2 4 Ξ2+1− √ +1 (77) developed employing different partitions of the total Hamil-
Ξ2+1 tonianwhichwillresultindifferentphysicalapproximations.
+ p2E(cid:113)4 (cid:0)Ξ2+1(cid:1)3(cid:16)(cid:16)√Ξ2+1+1(cid:17)Ξ2− √Ξ2+1+1(cid:17). TthheesmtaanidnadrdefipcairetintcioynoifntdhiecaCteodulaobmovbe-c,oisrrtehcattedthSeFiAnflbuaesnecdeoonf
c2
the laser field on the atomic bound dynamics is completely
UsingtheexpressionEq. (76),theintegrationovermomentum neglected. In this section we propose a modified version of
spacecanbedoneanalytically. Ityieldsforthetotalionization theSFAemployinganotherpartitionoftheHamiltonianinthe
rate: SFAformalism. Themainmotivationistousesuchaparti-
w = (cid:16)23−4cI2p (cid:17)(cid:114)3(cid:32)1+ 161Ip(cid:33)exp(cid:32)4Ip(cid:33) teivoonluwtihoenr.eItnhepalartsiecrufilaerl,dwheasusaectohnetrfioblulotiwoinngtoptahretibtioounn:dstate
Γ 3− 2Ip π 72c2 c2
c2 H = H˜ +H˜ , (81)
0 int
×(2Ip)47−3cI2p exp(cid:34)−2Ea (cid:32)1− Ip (cid:33)(cid:35). (78) H˜0 = cα·(cid:104)pˆ −kˆ(r·E(η))/c(cid:105)+βc2+V(c)(r) (82)
E12−2cI2p 3E0 12c2 H˜int = r·E(η). (83)
0
InFig. 4thetotalionizationrateofEq. (78)iscomparedwith Inthispartitionthetransitionmatrixelementreads
theITMresultofRef.[32]. TheCoulomb-correctedrelativistic (cid:90)
SFAandtherelativisticPPTyieldsresultsforthetotalioniza- M˜(c) =−i dtd3rψ+(r,t)H˜ φ˜(c)(r,t), (84)
fi f int i
tionratewhichareclose,thoughtheyarenotidentical. Inthe
nextsectionwemodifytheCoulomb-correctedSFAwiththe
wheretheboundstatedynamicsisdeterminedbythefollowing
aimtoobtainionizationprobabilitiesclosertotherelativistic
time-dependentDiracequation:
PPTresult.
(cid:110) (cid:104) (cid:105) (cid:111)
i∂φ˜(c) = cα· pˆ −kˆ(r·E(η))/c +βc2+V(c)(r) φ˜(c). (85)
t i i
III. MODIFIEDCOULOMB-CORRECTEDRELATIVISTIC Inthisspecificpartitiontheinitialboundstatewavefunction
STRONG-FIELDAPPROXIMATION φ˜(c)istheeigenstateoftheinstantaneousenergyoperator
i
InthestandardSFAintheGo¨ppert-Mayergauge,thetransi- Eˆ =cα(p+A(η))+βc2 (86)
tionmatrixelementisgivenbyEq. (2)where
[65, 66], with A(η) = −kˆ(r·E(η))/c in the Go¨ppert-Mayer
H =r·E(1−kˆ ·α) (79) gauge, seeEq. (9). Notethatinthenonrelativisticlimitthe
int
9
standardSFAinthelengthgaugecorrespondstothepartition andthetransitionstoexcitedstatesaresuppressedbyafactor
inwhichtheinitialboundstateistheeigenstateoftheenergy ofE /E .
0 a
operator[55]. Thispointprovidesanotherargumentinfavor With the wave functions of the bound state Eq. (90), the
oftheappliedpartitionintherelativisticcase. differentialandthetotalionizationratescanbecalculatedin
ThetermincorporatingthelaserelectricfieldinEq. (85) thesamewayasintheprevioussectionbefore. Thestructure
describesthespindynamicsoftheboundelectronbeforeion- oftheionizationmatrixelementremainsthesame
izationandhastobeconsideredinthefurthercalculation. We
athreedcyonnacmerinceadlZbeyetmheanspsipnlidttyinngamoficthseinbothuendbostuantedesntaetregiaensddbuye M(c) =−im˜ (η )exp−(cid:104)2Λ(ε−c2+Ip)−p2E(cid:105)3/2 (92)
tothespininteractionwiththelasermagneticfield. Therefore, fi fi s  3|E(η0)|Λ 
in solving Eq. (85) we will restrict the bound-bound transi-
tions only to the transitions between different spin-states at
wherem˜ (η )isgivenbyEq.(59),inwhichthefollowingre-
fi s
fixedquantumnumbers{n, j,mj}. Thedipoleapproximation placementshouldbedoneinordertoaccountforthedifference
for the laser field will be adopted for the description of the
in the spinor operator of the interaction Hamiltonian in the
boundstateevolution: E(η) ≈ E(t), sincethetypicallength-
modifiedSFA:
scaleforthebounddynamicsismuchsmallerthanthelaser
wave length: rb/λ ∼ ω/κc ∼ γ(E0/Ea)(κ/c) (cid:28) 1. When (cid:16) (cid:17) (cid:34) A(η )·α(1+kˆ ·α)(cid:35)
theinitialspin-polarizationisalongthelasermagneticfield u+f 1−kˆ ·α vi →u+f 1+ s 2cΛ v˜i, (93)
|φ(c)(cid:105)=|φ(c) (cid:105),thennospintransitionsoccurbecausetheinter-
B,±
actiontermVˆint(t)≡−(α·kˆ)(r·E(t))doesnotcausespin-flip wherev˜ =v exp[±iA˜(η)(1−2I /3c2)]isthespinorpartofthe
(cid:104)φ(c) |Vˆ (t)|φ(c) (cid:105)=0.Accordingly,inthiscasewearelooking i i p
B,∓ int B,± statesgiveninEq.(90). ThespinoroperatorinbracketsinEq.
forthesolutionofEq.(85)intheform:
(93)comesfromthespinorpartoftheVolkovwavefunction.
φ˜(c) (r,t)=φ(c) (r,t)eiS(t), (87) NotethatthephaseintheexponentialinEq.(90)variesslowly
B,± B,± compared with the contracted action S˜. Therefore, it does
where φ(c) (r,t) = φ(c) (r)e−iε±t is the ground state before not modify the saddle point integration, but alters the pre-
B,± B,± exponentialterm.
switchingonthelaserfield, i.e., aneigenstateoftheatomic
unperturbedHamiltonian √
(cid:104)cα·pˆ +βc2+V(c)(r)(cid:105)φ(Bc,)±(r)=ε±φ(Bc,)±(r). (88) m˜++ = (cid:113)28cce4−+i2pcE6(c22cp2++ippE4) (94)
E E
Thespinquantizationaxisforφ(c) (r,t)statesisalongthelaser (cid:16) (cid:17)
magneticfielddirection. TakingB,±intoaccountthat β2e−i2pcE(2c+ipE) 8c5−40ic4pE +14c3p2E
(cid:90) η A˜(η)(cid:32) 2I (cid:33) − 12√2(cid:16)8c4+6c2p2 +p4(cid:17)3/2 ,
dt(cid:48)(cid:104)φ(c) |Vˆ (t(cid:48))|φ(c) (cid:105)= 1− p , (89) E E
whereA˜(−η∞)≡−(cid:82)B−η,∞± Ei(nηt(cid:48))dη(cid:48)B,,t±hewav2ecfunctio3nc2ofthebound −β2e−i2pcE(122c√+2ip(cid:16)8Ec)4(cid:16)−+268cic22pp23E++p34c(cid:17)p3/4E2−4ip5E(cid:17),
statewillread E E
(cid:34) (cid:32) (cid:33)(cid:35)
A˜(η) 2I
φ˜(c) (t) = φ(c) (t)exp i 1− p
φ˜(BBc,,)+−(t) = φ(BBc,,)+−(t)exp(cid:34)−i2A˜c2(cη)(cid:32)1−3c322cIp2(cid:33)(cid:35) (90) m˜−− = (cid:113)√82cc4e+i2pcE6(c22cp−2Ei+pEp)4E (95)
(cid:16) (cid:17)
TishoefZoeredmeranofspA˜l/itctin∼gEte0rτmc/icn∼theκ/pcha(cid:28)seo1f.tIhteiswasmveafllunwchtiiocnh −β2ei2pcE 8√c5+40ic4pE(cid:16)+14c3p2E(cid:17)+(cid:113)28ic2p3E +3cp4E +4ip5E
arisesfromthefactthattheperturbationtermVˆint(t)couples 12 2(2c+ipE) 2c2+p2E 8c4+6c2p2E +p4E
thelarge/smallspinorpartoftheinitialstatebispinorwiththe
small/largespinorpartsofthefinalstatebispinor. andm˜+− =m˜−+ =0. Withthisthedifferentialionizationrate
Theexplicitconditionjustifyingtheneglectoftransitionsto inthemodifiedSFAyields:
excited states can be given as follows. The transition prob-
aωbninl(cid:48)ity∼bIeptw(neeanndthne(cid:48)satraetetshenp→rincnip(cid:48)awl qituhanetnuemrgynudmiffbeerresn)cies dw = 23−4cI2p (cid:18)1+ 2pc2E2 − 1327cIp2 − 4124pc2E4Ip(cid:19)(2Ip)3(cid:16)1−cIp2(cid:17)ωexp(cid:16)4cI2p(cid:17)
tPrannn(cid:48)si∼tioEn0irnb/tωhenns(cid:48)ta∼teEn0irsb/PIsps,(cid:48)w∼hEil0erbthτeKp(rsoabnadbisl(cid:48)itayreofthaessppiinn- d3p π2Γ(cid:16)3− 2cI2p(cid:17)|E(ηs)|3−2cI2p (cid:18)1+ 2pc2E2(cid:19)2
quantumnumbers).PPTnhne(cid:48)r∼efoIre1τ, ∼ EE0, (91) × (cid:32)1+ 1421cIp2(cid:33)exp−2(cid:16)2Λ(ε3−|Ec(2η+s)|IΛp)−p2E(cid:17)3/2. (96)
ss(cid:48) p K a
10
andthetotalionizationrateis intotherelativisticregime. TheappliedCoulomb-corrected
strong-field approximation incorporates the eikonal-Volkov
w = Γ(cid:16)233−−4cI22pIp(cid:17)(cid:114)π3(cid:32)1− 772Icp2(cid:33)exp(cid:32)4cI2p(cid:33) (97) wdyanvaemfuicnsc.tiTonhefolarttthereidsedsecrriivpetidoninotfhteheWeKleBctraopnprcooxnitminautuiomn
c2 takingintoaccounttheCoulombfieldoftheatomiccoreper-
×(2Ip)47−3cI2p exp(cid:34)−2Ea (cid:32)1− Ip (cid:33)(cid:35). ttuerrmbast,ivtehleydinisttuhrebpahnacseeoofftthheeeWleKctBrownaevneefrugnycbtiyonth.eInCpohuylsoimcabl
E12−2cI2p 3E0 12c2 fieldisassumedtobesmallerwithrespecttotheelectronen-
0
ergyinthelaserfield. Theeikonal-Volkovwavefunctionis
InFig.4,thetotalionizationprobabilitycalculatedviathe applicablewhenthelaserfieldissmallerthantheatomicfield
modifiedCoulomb-correctedrelativisticSFAiscomparedwith E (cid:28) E andI (cid:46)c2.
0 a p
theresultofthestandardSFA,aswellaswiththatoftherela- Wehavederivedananalyticalexpressionfortheionization
tivisticPPT.Theconclusioncanbedrawnthatbothrelativistic amplitudeinalinearlypolarizedlaserfieldwithintheCoulomb-
Coulomb-corrected SFA give slightly different results com- correctedrelativisticSFAwhenadditionallysmallnessofthe
pared to the PPT for large Ip. The modified SFA is closer parametersofIp/c2 (cid:28)andγ(cid:28)1isused.Asimpleexpression
tothePPTresultthanthestandardSFA.Thetotalionization fortheamplitudeisobtainedwhenusingtheGo¨ppert-Mayer
probabilities in the relativistic Coulomb-corrected standard gauge. Moreover,inthisgaugeaCoulombcorrectionfactor
SFAislargerthantherelativisticPPTbyafactorsmallerthan (ratiooftheCoulombcorrectedamplitudetothestandardSFA
1+3Ip/c2. one)coincideswiththatderivedwithinthePPTtheory. The
ThestandardSFAyieldslargerionizationprobabilitiesthan differentialandtotalionizationratesarecalculatedanalytically.
themodifiedSFAwhichcanbeexplainedbyasimpletunneling Thecalculatedtotalionizationrateisslightlylargerthanthe
picture. Inthefirstcasethemagneticfieldactsontheelectron PPT rate at large ionization potentials. To improve the pre-
onlywhenitentersthebarrier.Inthiscaseduringthetunneling dictionsoftheCoulomb-correctedrelativisticSFA,wehave
theelectronkineticenergywillchangeontheamountofthe proposedamodifiedSFAapproachwhichisbasedonanother
interactionenergywiththemagneticfieldµB0 =−sE0/2c(s= partitionofthetotalHamiltonian. InthisapproachtheSFA
±1forspinalongthemagneticfieldoropposite),givingriseto matrixelementcontainstheeigenstateoftheenergyoperator
akineticenergysplittingfordifferentspinorientationsofthe inthelaserfieldasthewavefunctionoftheinitialboundstate.
electronand,consequently,todifferenttunnelingprobabilities. ThemodifiedSFAtakesintoaccountthedynamicalZeeman
Whereas,inthesecondcasethemagneticfieldisalsoacting splittingoftheboundstateenergyduetothespininteraction
beforetunnelingandnokineticenergysplittingoccurs. The withthelasermagneticfield(whentheelectroninitialpolar-
ratiosoftheKeldysh-exponentforthesetwoscenarioscanthen izationisalongthelasermagneticfield)andtheprecessionof
beexpandedinthespinenergy: theelectronspinintheboundstate(whentheelectroninitial
(cid:20) (cid:21) (cid:20) (cid:21) polarizationisalongthelaserpropagationdirection).
ΓΓmstaonddifiaerdd = exp −2(2(Ip+3EE020/2ecx))p3/2(cid:20)−+2(2e(xIpp))3−/2(cid:21)2(2(Ip−3EE00/2c))3/2 iicoanOlizcuaartlircoeunsluarltatistoensshoionfwqthutehaanrtetilttahatetiivvSiesFltAyicctreoecrghrienmcitqeudweiffhaeilcrlehonwttaiskatlehsaeniandntotaolatyactl--
3E0
counttheimpactoftheCoulombfieldoftheatomiccoreas
∼ 1+I /c2 (98)
p well as the electron spin dynamics in the bound state. This
methodcanbeviewedasanalternativetothePPT.Inthefol-
To decide which SFA partition is best suited to model the
lowingpaperofthesequeltheCoulomb-correctedrelativistic
ionizationprocess,acomparisonwithnumericalsimulations
SFAwillbeusedtoinvestigatespin-resolvedionizationprob-
orexperimentaldataisnecessary.
(cid:82) abilities. Whileforthetotalionizationratethepredictionof
Further,wenotethattheCoulomb-correctionterm dηα·
thestandardSFAisclosetothatofthemodifiedSFA,forspin
∇V(c)/cthatisneglectedinbothstrongfiel√dapproximations effectstheirpredictionsarequitedifferent. Thenextpaperof
givesacorrectionfactorofonly1+(I /c2) E /E ,whichis
p 0 a thesequelwillbedevotedtothisissue.
oftheorderofafewpercentsforthemostextremeparameters
(E /E =1/16,Z =90).
0 a
IV. CONCLUSION Acknowledgments
We have generalized the Coulomb-corrected SFA for the WeappreciatevaluablediscussionswithC.H.KeitelandC.
ionization of hydrogen-like systems in a strong laser field Mu¨ller.
[1] C.I.Moore,A.Ting,S.J.McNaught,J.Qiu,H.R.Burris,and [2] E.A.Chowdhury,C.P.J.Barty,andB.C.Walker,Phys.Rev.A
P.Sprangle,Phys.Rev.Lett.82,1688(1999).