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Adiabatic Theory of Electron Detachment from Negative
Ions in Two-Color Laser Field
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Negative ion detachment in bichromatic laser field is considered within the adiabatic
theory. The latter represents a recent modification of the famous Keldysh model for
multiphoton ionization which makes it quantitatively reliable. We calculate angular
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aAdiabaticTheoryofElectronDetachmentfromNegativeIonsinTwo-ColorLaserFieldM.Yu.Ku
chievandV.N.OstrovskySchoolofPhysics,UniversityofNewSouthWales,Sydney2052,Austra
liaAbstractNegativeiondetachmentinbichromaticlaser?eldisconsidehttp://doc.xuehai
.net/b9d11adc64072bb66b8b04fc4.htmlredwithintheadiabatictheory.Thelatterrepresen
tsarecentmodi?cationofthefa-mousKeldyshmodelformultiphotonionization[L.V.Keldysh
,Zh.Eksp.Teor.Fiz.47,1945(1964)[Sov.Phys.-JETP20,1307(1965)]]whichmakesitquantit
ativelyreliable.Wecalculateangulardi?erentialdetachmentrates,partialratesforpart
icularATD(AboveThresholdDetachment)channelsandtotaldetachmentratesforH?ioninabic
hromatic?eldwith1:3frequencyratioandvariousphasedi?erences.Reliabilityoftheprese
nt,extremelysim-pleapproachistesti?edbycomparisonwithmuchmoreelaborateearliercal
culations.PACSnumbers:32.80.Rm,32.80.Fb
TypesetusingREVTEX
I.INTRODUCTION
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Interesttothephotoionizationofatomsinabichromaticlaser?eldbothintheory(see,forin
stance,Ref.[1]–[13])andinexperiment[1][14]–[16]seemstostem?rstofallfromthee?ec
tofthephasecontrol,i.e.dependenceoftheobservablesonthedi?erenceof?eldphases?.
Thecalculationshavebeencarriedoutpreviouslyforionizationofthehydrogenatomintwhtt
p://doc.xuehai.net/b9d11adc64072bb66b8b04fc4.htmlolaser?eldswithafrequencyratio1
:2[4],1:3[5]and2:3[7].PotvliegeandSmith
[6]presentedresultsforvariousfrequencyratiosandinitialstates.Di?erentschemeshave
beenemployed,butallofthemimplynumericallyintensivework.
Forthemultiphotonelectrondetachmentfromnegativeionssomeanalyticaltreatmentexists
[8][9]whichaimstoinvestigatequalitativefeaturesoftheprocess,mostlyinthecasewheno
neorboth?eldsareweak.Thepresenceoflargenumberofparametersintheproblemsometimesma
kesresultsofanalyticalstudiesnotdirectlytransparent.Quantitativereliabilityofthe
seapproacheshasneverbeenassessed.Thissituationlooksparticularlyunsatisfactorysin
cethemultiphotonelectrondetachmentfromnegativeionspresentsuniquesituationwhenqua
ntitativeresultscanbeobtainedbyanalyticalmethodinabroadrangeofparameterscharacte
ristictotheproblem.Indeed,ithasbeendemonstratedrecentlybyGribakinandKuchiev[17][
18]thatproperapplicationofthewell-knownKeldysh[19]modeltomultiphotondetachment[2
0]phttp://doc.xuehai.net/b9d11adc64072bb66b8b04fc4.htmlrovidesanextremelysimples
chemethatgivesveryreliableresultsforthetotalratesaswellasforATD(AboveThresholdDe
tachment)spectrumandATDangulardistributions.Thisadiabaticapproximationensuresana
ccuracywhichiscomparablewiththatofthemostelaboratenumericaldevelopmentsandworksu
nexpectedlywellevenoutsideitsformalapplicabilityrange,i.e.evenforsmallnumbernofp
hotonsabsorbed.TheevidencesofgoodperformanceoftheKeldyshmodelforthetotalrateswer
epresentedalsointheearlierpaper[25].
Recentlytheadiabaticapproachwasextendedbythepresentauthors[26]tothecaseofbichrom
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atic?eld.Thepracticalapplicationswerecarriedoutforthecaseoffrequency
ratio1:2wheninadditiontothephasee?ectsanotherunusualphenomenonexists,namelythepo
larasymmetryoftheangulardistributionofphotoelectrons.Unfortunatelynootherquantit
ativedataforphotodetachmentinthiscaseisavailablewhichmakescomparisonimpossible.
Themainobjectiveofthepresentstudyistoassessquantitativelyanaccuracyofthttp://doc
.xuehai.net/b9d11adc64072bb66b8b04fc4.htmlheadi-abaticschemebycomparisonwiththep
reviouscalculationscarriedoutbyTelnovetal[11]incaseof1:3frequencyratio.Forthisra
tiothepolarasymmetryisabsent,butthephasee?ectspersist.ThecalculationsbyTelnoveta
l[11]arebasedonsu?cientlysophisticatednumericalschemeprovidingausefulbenchmark.W
epresent(Sec.II)completecomparisonoftheresultsbyconsideringangulardi?erentialdet
achmentrates,heightsofATDpeaksandtotaldetachmentrates.Itshouldbeemphasizedthatth
eangulardi?erentialratesaremostsensitivetotheformulationofthemodelrepresentingan
ultimatetestforthetheory,asdiscussedinSec.III.Wedrawalsosomegeneralconclusionont
herelationbetweentheadiabaticapproachandthenumericalcalculationswithintheone-ele
ctronapproximation.
II.RESULTS
Theadiabatictheoryoftwo-colordetachmentwasoutlinedinourpreviouspaper[26]wherethe
readercan?ndallthedetailsofcalculation.Hereweonlywritedowntheexpres-sionfortheel
ectric?eldstrengthinthebichromaticlaser?eldwith1:3frequencyrahttp://doc.xuehai.n
et/b9d11adc64072bb66b8b04fc4.htmltioinordertospecifythede?nitionofthe?eldphasedi
?erence?
??(t)=F??1cosωt F??2cos(3ωt ?).F(1)
??1,F??2aretheamplitudevectorsforthefundamentalfrequencyωanditsthirdharmonicsF
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respectively.Belowweconsider,justasinRef.[11],thecasewhenbothfundamental?eldand
??1??F??2.Thenthedi?erentialphotoionizationitsthirdharmonicsarelinearpolarizedwi
thF
ratedependsonlyonthesingleangleθbetweenthephotoelectrontranslationalmomentum
??1??F??2.Atomicunitsareusedthroughoutthepaperunlessstatedp??andthevectorsF
otherwise.
OurcalculationsforH?detachmentarecarriedoutfortheparametersofH?asbefore[26](κ
=
0.2354,A=0.75).Wechoosetwosetsof?eldintensitiesI1andI2forthefundamentalfrequency
ω=0.0043(thatofCO2laser)anditsthirdharmonics,sameasinthepaperbyTelnovetal[11],n
amely(i)I1=1010W/cm2,I2=109W/cm2and(ii)I1=1010W/cm2,I2=108W/cm2.
Incaseofthefrequenciesratio1:3consideredherethhttp://doc.xuehai.net/b9d11adc6407
2bb66b8b04fc4.htmle?eld(1)doesnotpossesspolar
??1??F??2).Thereforeasymmetry(i.e.asymmetryunderinversionofthezaxisdirectedalong
F
thedi?erentialdetachmentratedoesnotchangeunderthetransformationθ?π?θ.Thisallo
wsustoshowplotsonlyfor1
2and
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?=π.Thetransformation??π??leavestheHamiltonianinvariantonlyiftisreplacedby?t.A
sstressedinRefs.[2],theproblemisinvariantunderthetimeinversionoperationprovidedt
he?nal-stateelectroninteractionwiththeatomiccoreisneglected.Thisisthecaseinthepr
esentmodel.Thereforeourdi?erentialionizationratesarethesamefor?and??.Thecalculat
ionsbyTelnovetal[11]dotakeintoaccountthe?nalstateelectron-coreinteraction.Theref
oretheyshowsomedi?erencebetweentheangulardi?erentialratesfor?and??.However,itpro
vestobequitesmallforlowATDchannelsasseenfromtheplots.Theimportanceoftheinteracti
onbetweentheemittedelectronandthecorehasbeen?rstpointedoutbyoneofthepresentautho
rs[27].Inthispaperseveralphenomenahasbeenpredictedforwhichhttp://doc.xuehai.net/
b9d11adc64072bb66b8b04fc4.htmlthisinteractionplayscrucialrole.Therelatedmechanis
mwasnamed“atomicantenna”.Intherecentliteraturethe?nalstateinteractionisusually
referredtoasrescattering.Inourproblemtherescatteringe?ectsareenhancedforhighATDc
hannelsasdiscussedbelow.
TheresultsofourextremelysimpletheoryarecomparedinFigs.1-6withthepreviousnumerica
lcalculationsbyTelnovetal[11]whichareratherinvolving.Beingcarriedoutintheone-ele
ctronapproximation,theyemployanaccuratemodelforthee?ectiveone-electron
potentialinH?[28],complex-scalinggeneralizedpseudospectraltechnique[29]todiscret
izeandfacilitatethesolutionofthetime-independentnon-HermitianFloquetHamiltonianf
orcomplexquasienergiesandeigenfunctions,andcalculationoftheelectronenergyandangu
lardistributionsbythereversecomplex-scalingmethod[30].Asalucidillustrationofsimp
licityofthepresentapproachitisworthwhiletostressthatitdoesnotrelyonanyparticular
formofane?ectiveone-electronpotentialalbeitemploysonlytwoparameterhttp://doc.xue
hai.net/b9d11adc64072bb66b8b04fc4.htmlsκandAgoverningtheasymptoticbehaviorofthe
initialboundstatewavefunction.
FromFigs.1-6onecanseethattheadiabaticapproximationensuresgoodquantitativeagreeme
ntwithcalculationsbyTelnovetal[11].Inparticular,positionsofmaximaandminimainthea
ngularphotoelectrondistributionarewellreproduced.Thisdemonstratesthattheadiabati
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capproachcorrectlydescribesthenatureofthestructureasduetointer-ferencebetweenthe
electronwavesemittedatvarious(complex-valued)momentsoftime.Indeed,withintheadiab
atictheory[26]theionizationamplitudeisexpressedasasumofanumberofinterferingcontr
ibutions.Mathematicallytheycomefromdi?erentsaddlepointsintheapproximateevaluatio
noftheintegralovertimethatemergesintheKeldysh[19]model.Physicallytheycorrespondt
othecoherentemissionofphotoelectronatdi?erentmomentsoftime.Forourparticularfrequ
encyratio1:3thesumcontains6interferingcontributionsascomparedwith4termfor1:2freq
uencyratio[26]and2termsforone-colordetachment[17][18].Generallythishttp://doc.xu
ehai.net/b9d11adc64072bb66b8b04fc4.htmlsuggeststhatintheformercasemorecomplicate
dangularpatternsemerge.Probablyonecan?ndhereacorrelationwithanalternativein-terp
retationinthemultiphotonabsorptionframework.Thelatterargues[7][11]thattheangular
distributionstructurein1:3caseismorecomplicatedthanfor1:2ratiosinceallthepathway
sleadingtoacontinuumstatewiththesameenergyinterfereinthe1:3casewhereasaconsidera
blepatternofnon-interferingpathwaysexistsforthethe1:2caseduetoparityorenergyrest
rictions(eachpathwayischaracterizedbythenumberofphotonsofdi?erentcolorsabsorbeds
uccessively).
ThepartialdetachmentrateforeachATDchannelareshownintablesIandIIfortwosetsof?eldi
ntensities.TheagreementisgoodforlowATDchannels;notethattherescattering
e?ectswhichgeneratedependenceonthesignof?aremanifestedinthepartialratesevenlesst
hanintheangulardistributionsshowninFigs.1-6.ForhigherATDchannelswithlowratesthed
i?erencebetweenthepresentresultsandthoseofTelnovetal[11]becomesmorepronounced.Th
isbehttp://doc.xuehai.net/b9d11adc64072bb66b8b04fc4.htmlhaviorcouldbeinterpreted
asincreasingimportanceofrescatteringforhighATDpeaks.Themanifestationsofthise?ect
wereobservedrecentlyinexperiment[31]andarecurrentlyvividlydiscussedintheliteratu
re[27][31][32][33].
III.CONCLUSION
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Asasummary,theadiabaticapproachprovidesquantitativelyreliabletoolforinvesti-gati
ngtwo-colorphotodetachmentofnegativeions.Inparticular,theinterferencestructurein
thephotoelectronangulardistributionsaswellasthephasee?ectsarecorrectlydescribed.
Sincegenerallytheinterferencephenomenaareknowntobemostsensitivetothedetailsofthe
oreticaldescription,onecanconcludethatthepresenttheoryhadsuccessfullypassedthest
ringenttest.
TheKeldyshscheme[19]isknowntobegauge-noninvariant.Importantly,thecalcula-tionswi
thintheadiabaticapproach[17][18][26]employthedipole-lengthgaugeforthelaser?eldth
usstressingcontributionofthelong-rangeasymptoteoftheinitialboundstatewavefunctio
n.Theuseofthelengthgaugetogetherwiththeadiabaticahttp://doc.xuehai.net/b9d11adc6
4072bb66b8b04fc4.htmlpproach(i.e.integrationovertimebythesaddlepointmethod,seeRe
fs.[17][18][26]anddiscussioninSec.II)ren-derself-consistentcharactertothetheoret
icalscheme.Indeed,theexactevaluationoftheintegralsdoesnotaddtotheaccuracyofthere
sultascomparedwiththeuseofthesaddlepointmethod.Thisisbecauseintheformercasethein
tegralabsorbsthecontributionsfromthewavefunctionoutsideitsasymptoticdomain,where
infactitisknownwithmuchloweraccuracy(being,inparticular,in?uencedbythee?ectsbeyo
ndthesingleactiveelectronapproximation).
Themethodisstraightforwardlyapplicabletothenegativeionswiththeouterelectronhavin
gnon-zeroorbitalmomentum,suchashalogenions,whichcouldbeeasieraccessible
fortheexperimentalstudies(fortheone-colordetachmentsuchapplicationscouldbefoundi
nRef.[17]).Technicallythecalculationswithintheadiabaticapproachareextremelysimpl
ereducingto?ndingtherootsofpolynomialandsubstitutingthemintoananalyticalexpressi
on[26](therelatedMathematica[34]prograhttp://doc.xuehai.net/b9d11adc64072bb66b8b
04fc4.htmlmtakesonlyfewlines).Itshouldberecognizedthatthesingleactiveelectronapp
roximationitselfintroducessomeintrinsicerror.Itseemsthatoftenthiserrorcouldbecom
parablewiththedi?erencebetweentheresultofnumericalone-electroncalculationsandthe
seoftheadiabaticapproximation.Uncertaintyoftheone-electronapproachinprinciplecou
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ldberemovedwithinthetwo-electronapproachwhichhoweverconsumesmuchmoree?orts.Thetw
o-electroncalculationswhichhasbeencarriedoutrecentlyshowthattheone-electronappro
ximationisgenerallysu?cientunlessoneisparticularlyinterestedinthesubtleresonance
e?ects[25][35][36](thecalculationsbeyondone-electronapproximationarecurrentlypos
sibleonlyforsmallnumberofabsorbedphotons).ForhighATDchannelswithlowintensitiesth
eadiabaticapproximationbecomeslessreliableduetoincreasingroleofrescatteringe?ect
sneglectedinthepresentformoftheapproximation.Itseemshoweverthatrelativelysimplem
odi?cationsoftheadiabaticapproximationcouldbecarriedtoincluderescatteringe?ectht
tp://doc.xuehai.net/b9d11adc64072bb66b8b04fc4.htmls.
Reliabilityoftheresultsobtainedaboveforthesimpleone-electronproblemwithrescat-te
ringneglectedishighlyimportantinperspective,sincetheyaretobeincludedasacon-stitu
entpartinthetreatmentofmuchmoresophisticatedone-electronandmany-electronproblems
governedbytheantennamechanism[27][37][38].
ACKNOWLEDGMENTS
WeappreciatefruitfuldiscussionswithG.F.Gribakin.Wearegratefultotherefereeofthepr
esentpaperforattractingourattentiontoRef.[25].ThesupportfromtheAustralianResearc
hCouncilisthankfullyacknowledged.V.N.O.acknowledgesahospitalityofthestu?oftheSch
oolofPhysicsofUNSWwherethisworkhasbeencarriedout.
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TABLES
TABLEI.PartialratesfortheH?detachmentbythelaserwavewiththefrequencyω=0.0043andi
tsthirdharmonicswiththeintensitiesrespectivelyI1=http://doc.xuehai.net/b9d11adc6
4072bb66b8b04fc4.html1010W/cm2andI2=109W/cm2.Thenumberofabsorbedphotonsnrefersto
thefundamentalfrequency.Ineachblocktheupper?guregivespresentresultandthelowerone
theresultobtainedbyTelnovetal[11].Thenumberinsquarebracketsindicatethepowerof10.
One-colour
nfundamentalTwo-colour——————————————————–
?=0?=π?=1
2πOne-colourharmonic
130.21[–12]
0.33[–12]0.38[–10]0.53[–10]0.36[–9]0.27[–9]0.20[–9]0.20[–9]0.20[–9]0.85[
–10]
140.64[–13]
0.14[–12]0.88[–11]0.14[–10]0.11[–9]0.97[–10]0.58[–10]0.69[–10]0.58[–10]0
.32[–10]
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150.16[–13]
0.47[–13]0.20[–11]0.32[–11]0.32[–10]0.33[–10]0.16[–10]0.23[–10]0.16[–10]
0.17[–10]0.16[–15]0.31[–15]
160.36[–14]
0.14[–13]0.52[–12]0.72[–12]0.82[–11]0.12[–10]0.44[–11]0.78[–11]0.44[–11]
0.10[–10]
170.68[–15]
0.35[–14]0.15[–12]0.22[–12]0.21[–11]0.43[–11]0.12[–11]0.27[–11]0.12[–12]
0.60[–11]
Total0.92[–9]
0.96http://doc.xuehai.net/b9d11adc64072bb66b8b04fc4.html[–9]0.62[–7]0.56[–7]0
.15[–7]0.14[–7]0.36[–7]0.33[–7]0.36[–7]0.33[–7]0.46[–7]0.30[–7]
TABLEII.Sameasintable1,butfortheintensitiesI1=1010W/cm2andI2=108W/cm2.
One-colour
nfundamentalTwo-colour———————————————————
?=0?=π?=1
2πOne-colourharmonic
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150.16[–13]
0.47[–13]0.66[–13]0.11[–11]0.96[–12]0.17[–11]0.46[–12]0.70[–12]0.46[–12]
0.25[–11]0.16[–21]
160.36[–14]
0.14[–13]0.11[–13]0.52[–12]0.21[–12]0.60[–12]0.99[–13]0.21[–12]0.99[–13]
0.12[–11]
170.68[–15]
0.35[–14]0.17[–14]0.22[–12]0.42[–13]0.22[
–
12]0.21[–13]0.64[–13]0.21[–13]0.50[–12]
Total0.92[–9]
0.96[–9]0.66[–8]0.66[–8]0.79[–9]0.74[–9]0.36[–8]0.35[–8]0.36[–8]0.35[–8
]0.46[–13]0.30[–13]
FIGURES
FIG.1.DetachmentofH?ioninbichromatic?eldwiththefrequenciesω=0.0043and3ωandinte
nsitiesI1=1010W/cm2andI2=109W/cm2respectively.Di?erentialdetachmentrate(inunits1
0?8a.u.)asahttp://doc.xuehai.net/b9d11adc64072bb66b8b04fc4.htmlfunctionoftheelec
tronemissionangleθisshownforthe?rstATDpeak(correspondingtoabsorptionofn=8photon
soffrequencyω)andvariousvaluesofthe?eldphasedi?erence?asindicatedintheplots.Ope
nsymbolsshowresultsofcalculationsbyTelnovetal
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[11](inthe?=±1
?=?1
and?=?12πandopentrianglesthesefor2π
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