Spin-Statistic Selection Rules for Multiphoton Transitions: Application to Helium Atom

0.17 MB·

by  T. Zalialiutdinov

Spin-Statistic SelectionRules forMultiphoton Transitions: Applicationto Helium Atom
T. Zalialiutdinov1, D. Solovyev1, L. Labzowsky1,2 and G. Plunien3
1 Department of Physics, St.Petersburg State University,
Ulianovskaya 1, Petrodvorets, St.Petersburg 198504, Russia
2 Petersburg Nuclear Physics Institute, 188300, Gatchina, St.Petersburg, Russia
3 Institutfu¨rTheoretischePhysik,TechnischeUniversita¨tDresden,Mommsenstrasse13,D-10162,Dresden,Germany
Atheoreticalinvestigationofthethree-photontransitionrates21P →21S ,11S and23P →21S ,11S
1 0 0 2 0 0
fortheheliumatomispresented.Photonenergydistributionsandprecisevaluesofthenonrelativistictransition
6
ratesareobtainedwithemployment of correlatedwavefunctionsof theHylleraastype. Thepossibleexperi-
1
mentsforthetestsoftheBose-Einsteinstatisticsformultiphotonsystemsarediscussed.
0
2

n I. INTRODUCTION
a
J
Thispaperpresentsatheoreticalinvestigationofthethree-photondecayratesfrom21P and23P statesintheheliumatom.
6 1 2
1 TheprimarymotivationforthesecalculationsistherecentseriesofspectroscopictestsofBose-Einsteinstatisticsforthesystem
oftwophotons[1]-[4].Theexperimentsuseaselectionrule[2,3]foratomictransitionsthatiscloselyrelatedtotheLandau-Yang

] theorem(LYT)[5,6]. Theselectionrulestatesthattwoequivalentsphotonscannotparticipateinanyprocessthatwouldrequire
h
themtobeinastatewithtotalangularmomentumequaltoone.Anevidentexampleinthehigh-energyphysicsistheprohibition
p
- ofthetwo-photondecayfortheneutralspin-oneZ0-boson.Thesameconcernstheannihilationdecayoforthopositronium(also
m spin-1 state). However both these decays are also forbidden by charge-parity conservation law. Positronium presents a real
o neutralsystem (it coincideswith itself after chargeconjugation),thereforeit possesses a definite chargeparity [9], connected
t withthetotalspinvalueS: parapositronium(S =0)ischarge-positiveandorthopositronium(S =1)ischargenegative. Since
a
. the chargeparity of a system of Nγ photonsequals ( 1)Nγ [10], parapositroniumcan notdecay into an odd numberof (not
s necessarilyequivalent)photonsandorthopositroniumc−annotdecayintoanevennumberofphotons.TheZ0-bosonasacharge-
c
i parity-negativeparticlecannotdecayintoanevennumberofphotons.RecentlyLYTwasemployedfortheproofthattheheavy
s
particlediscoveredatLHC[11]hasspinS =0,i.e. presentsHiggsboson.Asimilarsituationexistsforatomictransitions. The
y
h earlycalculationsofthetwo-photondecayofthesinglet21S0 (1s2s)1S0 andtriplet23S1 (1s2s)3S1 excitedstatesofHe
p andHe-likeionstotheground11S0 (1s)21S0 state reveale≡dthecrucialdifferenceinthe≡photonfrequencydistributionsin
≡
[ bothcases[12], [13]. Thedecayprobabilityforthe tripletcase turnstozero whenthefrequenciesof theemitted photonsare

equal. Latertheseconclusionswereconfirmedwithinthefullyrelativisticcalculations(see,forexample[14]). Aneutralatom
1
unlikepositroniumisnotarealneutralsystemanddoesnotpossessadefinitechargeparity.Thereforethedifferenciesinatomic
v
6 transitionsareexclusivelyduetotheBose-Einsteinstatistics.
5 In[15]theSpin-StatisticsSelectionsRules(SSSRs) formultiphotontransitionswith equalphotonsinatomicsystemswere
1 establishedwhichpresentanextensionofLYTtothe3-and4-photonsystems. TheSSSR-1soundslike:twoequivalentphotons
4 cannotparticipateinanyatomictransitionthatwouldrequirethemtohavetotal(common)oddangularmomentum.TheSSSR-2
0
is: threeequivalentdipolephotonscannotparticipateinanyatomictransitionthatwouldrequirethetohavetotalevenangular
.
1 momentum.TheSSSR-3claimsthatfourequivalentdipolephotonscannothavetotaloddangularmomentum.
0 FortheexperimentaltestsofSSSRsitispossibletouselasersource. UnlikeoriginalLYTformulation(thiswasmentioned
6 both by Landau [5] and Yang [6]) the derivation [15] is valid also for the collinear i.e. for the laser photons. As particular
1
examplesthetwo-,three-andfour-photontransitionsinHe-likeUwereconsideredin[15]. Thischoicewasexplained,fromthe
:
v onesidebytherelativesimplicityofthecalculationsintwo-electronionswiththefullneglectoftheinterelectroninteraction.
i ThisneglectdoesnotchangetheSSSRsbutmakesthenumericalresultsinaccurate. Theinaccuracydropsdownwithincrease
X
ofthenuclearchargeandisminimal(about1%)forU.Fromtheothersidethespectrumintwo-electronatomsisreachenough
r
a toprovideallinterestingmultiphotontransitions. HoweversuchatomicsystemasHe-likeuraniumarenotsuitableforpossible
experimentaltestsofSSSRs. ForZ =92theenergydifferencebetweenstates21P ,23P andgroundstate11S isofhundreds
1 2 0
KeVandpreventstheuseofopticallasers. Anadvantageoftheuseofthelasersourceisthatallthephotonswillhavethesame
frequency. Ifwedividetheenergyintervalω betweenappropriateatomicstatesbyanintegernumberN andadjustthelaser
a γ
frequencyω tothisvalue,ω =ω /N ,thenumberofphotonsN intheabsorptionprocesswillbefixed.Thevalueofthetotal
l l a γ γ
angularmomentumJ forN -photonsystemcanbefixedbychoosingtheappropriatevaluesJ andJ fortheinitial(lower)
γ ei ef
andfinal(upper)atomiclevelsinthetransitionprocess.Forexample,ifwechooseJ =0,J =2andN =3theSSSR-2can
ei ef γ
betestedforN = 3,J =2whichprohibitsthreedipolephotonswiththesameparityandequalfrequencytobeinaquantum
γ
statewithtotalangularmomentumJ =2. Whatwecannotfixisthemultipolarityofphoton,i.e.atotalangularmomentumjof
everyseparatephoton.Alaserlightinthebeamcanbedecomposedinallpossiblemultipolarities.Thismeans,forexamplethat
togetherwithE1E1E1transition,alltransitionswiththesametotalparityconstructedwiththehighermultipoles,i.e. E1M1E2,
E1E1M2etc. willbealwaysabsorbed. Howeverthe processeswiththe photonsofhighermultipolaritiesareusuallystrongly
suppressed in atoms, therefore the E1E1E1 transition will be dominant. Measuring the absorption rate at the ω = ω /N
l a γ
2
frequencyonecanestablishthevalidityornon-validityoftheparticularSSSR:theatomicvapourshouldbetransparentforthe
laserlightatthefrequencyω = ω /N . Notealsothatunlikethe spontaneousemissionwhichisveryweakformultiphoton
l a γ
transitions,themultiphotonabsorptiondependsonthelaserintensityandcanbewellobservedintheexperiments[16].
Forthe testofSSSR-2 onecan choose,in particularthe transition21S 23P with transitionenergy0.35eV. Thistran-
0 2
→
sitionshouldobeytheSSSR-2, i.e. three-photonabsorptionofthelaserphotonswithequalfrequenciesshouldbeabsent. For
comparisonitisconvenienttoemploy21S 21P transitionwiththetransitionenergy0.60eV,whichshouldnotexhibitsuch
0 1
−
propertiessince in this case three equalphotonsshouldhave the totalangularmomentumJ = 1. It is importantthatin both
casestheinitial21S stateismetastablewhatenablestoperformlaserabsorptionexperiments.
0
ThemostadvantageousfortheexperimentaltestsofSSSRsisneutralheliumatomwheretransitionscorrespondtotheoptical
orinfraredregion.ForexampleforthetestofSSSR-2three-photontransitionsbetweenthe21S stateandexcited21P and23P
0 1 2
statescanbeused. UnlikeU90+ ionthecalculationsforneutralHeatomcanbeperformedwithhighaccuracy.Theapplication
ofSSSRstoneutralHeisthemainpurposeofthepresentpaper.Accuratevariationalcalculationsofthethree-photontransitions
aregiven,basedontheemploymentofthevariationalwavefunctions. UnlikethecaseofU90+ thesecalculationscanbedone
with nonrelativisticwave functions(overallaccuracyof about0.01%). A powerfulmethodfor obtainingthe variationalwave
functionsofHylleraastypedevelopedin[17]-[19]wasemployedforthesecalculations.
II. THEORY
Wehavecalculated23P 11S ,23P 21S ,21P 11S and21P 21S three-photontransitionrates. Transitions
2 0 2 0 1 0 1 0
→ → → →
from23P levelexhibitthebehaviourgovernedbySSSR-2andtransitionsfrom23P levelwerecalculatedforcomparison.
2 1
Thethree-photondecayof21P stateisqualitativelydifferentfrom23P three-photondecaysincetheSSSR-2playssignif-
1 2
icantrolein23P 11S +3γ(E1)transition. Thisdifferenceaffectsboththeangularandtheenergydistributionsofthree
2 0
photons.In[15]itw→asshownthatinHe-likeuranium23P 11S +3γ(E1)transitionrateturnstozerowhenthefrequencies
2 0
→
ofthreephotonsareequalwhile21P 11S +3γ(E1)transitionratehasthemaximuminthesamesituation. Thisisalso
1 0
→
truefortheneutralheliumbecausetheangularpartremainsthesame.
Fullyrelativisticformofthedifferentialtransitionrateinconjunctionwiththeintegrationoverphotondirections~ν =~k/~k
| |
andsummationoverthephotonpolarizations~eofallthephotonswiththeaccountforallpermutationsofphotonsis[15]
dW (ω ,ω ) ω ω ω
i→f 1 2 = 3 2 1 (1)
dω dω (2π)5
1 2 λ3Xλ2λ1jγ3Xjγ2jγ1mγ3mXγ2mγ1
Q(λ3) Q(λ2) Q(λ1) Q(λ3) Q(λ1) Q(λ2)
jγ3mγ3ω3 fn′ jγ2mγ2ω2 n′n jγ1mγ1ω1 ni jγ3mγ3ω3 fn′ jγ1mγ1ω1 n′n jγ2mγ2ω2 ni
+ +
(cid:12) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:12)(cid:12)Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω2) Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω1)
(cid:12)
(cid:12)
(cid:12) Q(λ1) Q(λ3) Q(λ2) Q(λ1) Q(λ2) Q(λ3)
(cid:12) jγ1mγ1ω1 fn′ jγ3mγ3ω3 n′n jγ2mγ2ω2 ni jγ1mγ1ω1 fn′ jγ2mγ2ω2 n′n jγ3mγ3ω3 ni
+ +
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
n′n (En′ −Ef −ω1)(En−Ef −ω1−ω3) n′n (En′ −Ef −ω1)(En−Ef −ω1−ω2)
X X
2
Q(λ2) Q(λ3) Q(λ1) Q(λ2) Q(λ1) Q(λ3)
jγ2mγ2ω2 fn′ jγ3mγ3ω3 n′n jγ1mγ1ω1 ni jγ2mγ2ω2 fn′ jγ1mγ1ω1 n′n jγ3mγ3ω3 ni
+ ,
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:12)
Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω3) Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω1) (cid:12)(cid:12)
(cid:12)
(cid:12)
(cid:12)
whereωi (i=1,2,3)arethephotonfrequencies,jγ,mγ arethephotonangularmomentaandtheirprojections,λi =0(cid:12), 1, 1,
the indices i, n, f representthe initial, intermediate and final atomic states respectively. Summation over n, n′ in Eq.−(1)
extendsoverentireatomicspectrum.Thefollowingnotationsareused[20]
Q(0) =α~A~(0) (~r), (2)
jγmγω jγmγ
Q(1) =α~A~(1) (~r)+G A(−1)(~r), (3)
jγmγω jγmγ γ γmγ
whereA~(0) (~r),A~(1) (~r)aremagneticandelectricvectorpotentials,A~(−1) (~r)isthescalarpotential
jγmγ jγmγ jγmγ
A~(j0γ)mγ(~r)=ijγgjγ(ωr)Y~j∗γjγmγ(~n), (4)
3
j j +1
A~(1) (~r)=ijγ+1 γ g (ωr)Y~∗ (~n) γ g (ωr)Y~∗ (~n) (5)
jγmγ (s2jγ +1 jγ+1 jγjγ+1mγ −s2jγ +1 jγ−1 jγjγ−1mγ
j +1 j
+G γ g (ωr)Y~∗ (~n)+ γ g (ωr)Y~∗ (~n) ,
jγ s2jγ +1 jγ+1 jγjγ+1mγ s2jγ +1 jγ−1 jγjγ−1mγ !)
Aj(−γm1)γ(~r)=ijγgjγ(ωr)Yj∗γmγ(~n). (6)
g =4πj (ωr) (7)
jγ jγ
Here G is the gauge parameter defining gauge for the electromagnetic potentials, j (kr) is a spherical Bessel function,
jγ jγ
~n= ~r .
|r|
Actuallyforourpurposes(calculationsforHeatom)thefullynonrelativisticcalculationoftransitionenergiesaswellasfully
nonrelativisticexpressionforthephotonemissionoperatorsarequitesufficient.Wehaveonlytotakeintoaccountthespin-orbit
interactionfordescribingtheintercombinationtransitions.
Inthenonrelativisticlimit(kr 1)withG = jγ+1 andj =1forthedipolephotonsEq. (3)reducestotheform
≪ jγ jγ γ
q
4√2
Q(1) =i πωrY∗ (~n), (8)
1mγω 3 1mγ
Usingdefinitionforthesphericalcomponentsofvector~r
4π
rm = ~r Y∗ (~n) (9)
1 3 | | 1m
r
Eq. (8)canbepresentedintheform
8π
Q(1) =i ω rm . (10)
1mγω 3 1
r
SubstitutionofEq. (10)totheexpressionforthedifferentialthree-photontransitionrateEq. (1)yields
dW (ω ,ω ) 16
i→f 1 2 = (ω ω ω )3 (11)
dω dω 27π2 3 2 1
1 2 mγ3mXγ2mγ1
(rmγ3)fn′(rmγ2)n′n(rmγ1)ni (rmγ3)fn′(rmγ1)n′n(rmγ2)ni
+ +
(cid:12)(cid:12)Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω2) Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω1)
(cid:12)(cid:12) (rmγ1)fn′(rmγ3)n′n(rmγ2)ni (rmγ1)fn′(rmγ2)n′n(rmγ3)ni
(cid:12) + +
n′n (En′ −Ef −ω1)(En−Ef −ω1−ω3) n′n (En′ −Ef −ω1)(En−Ef −ω1−ω2)
X X
(rmγ2)fn′(rmγ3)n′n(rmγ1)ni (rmγ2)fn′(rmγ1)n′n(rmγ3)ni 2
+ .
Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω3) Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω1)(cid:12)(cid:12)
(cid:12)
IntheNe-electronatomrmγ = Ne rimγ,whererimγ arethesphericalcomponentsofradius-vectorfori-th(cid:12)(cid:12)electron. Thenthe
i=1
totaltransitionratecanbedefinedPas
1 1 dW (ω ,ω )
W = i→f 1 2 dω dω . (12)
i→f 1 2
3!2j +1 dω dω
ei meXi,mef ZZ 1 2
Summationoverallprojectionsappearingintheexpression(11)canbeperformednumericallyforeachvalueofcorresponding
angularmomenta.
Toperformthenumericalcalculationsofthree-photontransitionsintwo-electronatomicsystemsweusethevariationalwave
functionsofHylleraastype[19]. ThewavefunctionforastatewithatotalelectronorbitalangularmomentumL ,itsprojection
e
Meandtotalspatialparityπe =( 1)Le isexpandedasfollows
−
Ψ (~r ~r )= YLeMe(~n ,~n )GLeπe (r ,r ) (1 2) , (13)
LeMe 1 2 le1le2 1 2 le1le2 1 2 ± ↔
le1+Xle2=Leh i
4
whereGLeπe istheradialpart,correspondingtoacertainbipolarharmonicsYLeMe [21]. Thesefunctionareobtainedbythe
le1le2 le1le2
variational method developed in [17]-[19]. The method consists of expansion of GLeπe in the exponential basis set with a
le1le2
complexparametersα ,β andγ generatedinaquasirandommanner[17]withthesizeofbasisdefinedbyintegernumberN
i i i
N
GLeπe (r ,r )= U Re[exp( α r β r γ r )]+W Im[exp( α r β r γ r )] , (14)
le1le2 1 2 { i − i 1− i 2− i 12 i − i 1− i 2− i 12 }
i=1
X
wherer = r~ r~ ,U andW arelinearparameters.
12 1 2 i i
| − |
SinceweareinterestedinthetransitionsbetweenstateswithacertainvaluesoftotalmomentumJ~ =L~ +S~ ,theL S J M
e e e e e e e
couplingschemeneedstobeused,whereS~ isthetotalspinvalue[9,21]. Then
e
(rmγ)n′L′eSe′Je′Me′nLeSeJeMe =δSe′Se(−1)Je′−Me′ (cid:18)−JMe′e′ m1γ MJee(cid:19) (15)
× (2Je′ +1)(2Je+1)(−1)L′e+Se′+Je+1 LJe′e LJee′ S1e′ hn′L′e||r||nLei
(cid:26) (cid:27)
p
EvaluationofreducedmatrixelementinEq. (15)onthebasisofHylleraas-typewavefunctionsisdescribedin[22].
As an illustration of SSSR-2 for three equal photons we consider 23P 11S +3γ(E1) transition in the helium atom
2 0
→
and demonstrate that the value J = 2 of the total angular momentum for the three equivalent dipole photons is prohibited.
In principle, this decay can proceed via several channels. First, 23P n3S + γ(E1) n′3P n′1P + 2γ(E1)
2 1 1 1
→ → →
11S + 3γ(E1); this channel is prohibited since the transition n3S 11S + 2γ(E1) is prohibited by SSSR-1. Second,
23P0 n3D +γ(E1) n′3P n′1P +2γ(E1) 11S +3γ1(E→1);this0channelisalsoprohibite(cid:2)dbyS(cid:3)SSR-1sincethe
2 1 1 1 0
→ → →
transitionn3D 11S +2γ(E1)isprohibitedbythisrule. Thecontributionofthethirdchannel23P n3D n1D +
1 → 0 (cid:2) (cid:3) 2 → 2 2
γ(E1) n′3P n′1P +2γ(E1) 11S +3γ(E1)doesnotturntozerosoevidently. Thestatesadmixedbythespin-orbit
interact→ionarep1lacedin1 thesquarebr→ackets,0n, n′aresequentialnumbersforthestateswiththesamesymmetryintw(cid:2)o-elect(cid:3)ron
(cid:2) (cid:3)
atoms.
The23P 11S +3γ(E1)transitionunlikethe21P 11S +3γ(E1)transitionproceedsonlythroughspin-orbitmixing
2 0 1 0
→ →
oftheintermediatestates. Following[13,23]wewritethetrueP andDwavefunctionsintheform
∞
n′ 3P = n′ 3P + ǫ(P) n′′ 1P , (16)
1itrue 1i n′n′′ 1i
n′′=2
(cid:12) (cid:12) X (cid:12)
(cid:12) (cid:12) (cid:12)
∞
n′ 1P = n′ 1P ǫ(P) n′′ 3P , (17)
1itrue 1i− n′′n′ 1i
n′′=2
(cid:12) (cid:12) X (cid:12)
(cid:12) (cid:12) (cid:12)
∞
n′ 3D = n′ 3D + ǫ(D) n′′ 1D , (18)
2itrue 2i n′n′′ 2i
n′′=3
(cid:12) (cid:12) X (cid:12)
(cid:12) (cid:12) (cid:12)
∞
n′ 1D = n′ 1D ǫ(D) n′′ 3D , (19)
2itrue 2i− n′′n′ 2i
n′′=3
(cid:12) (cid:12) X (cid:12)
(cid:12) (cid:12) (cid:12)
where
n′ 3P H n′′ 1P
ǫ(P) = h 1| 3| 1i , (20)
n′n′′ E(n′ 3P) E(n′′ 1P)
−
n′ 3D H n′′ 1D
ǫ(D) = h 2| 3| 2i (21)
n′n′′ E(n′ 3D) E(n′′ 1D)
−
and H is the spin-orbit interaction operator [24]. In the absence of external electric and magnetic fields, the lowest-order
3
spin-dependentrelativisticcorrectionsH consistofthespin-orbitterm(inatomicunits)
3
Z ~r p~
H = i× i s (22)
so 2c2 r3 · i
i (cid:20) i (cid:21)
X
b
5
andthespin-other-orbitterm
1 ~r p~
H = ij × i (s +2s ) . (23)
soo 2c2 i6=j" ri3j #· i j
X
b b
MatrixelementsofEqs. (22)and(23)canbereducedto[25,26]
hLeSe′JeMe|Hso|LeSeJeMei= c22(−1)Le+Se′+Je(cid:26)J1e LSee′ LSee(cid:27)(cid:28)Le(cid:12)(cid:12)~r1r×13p~1(cid:12)(cid:12)Le(cid:29)hSe′||s1||Sei , (24)
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) b
(cid:12)(cid:12) (cid:12)(cid:12)
hLeSe′JeMe|Hsoo|LeSeJeMei= c12(−1)Le+Se′+Je(cid:26)J1e LSee′ LSee(cid:27)(cid:28)Le(cid:12)(cid:12)~r12r×132p~1(cid:12)(cid:12)Le(cid:29)hSe′||s1+2s2 ||Sei . (25)
(cid:12)(cid:12) (cid:12)(cid:12)
EvaluationofreducedmatrixelementsinEqs. (24)-(25)ontheHylleraastypew(cid:12)(cid:12)avefunctio(cid:12)(cid:12)nswaspresbentedibn[13].Inthisway
(cid:12)(cid:12) (cid:12)(cid:12)
forthe23P 11S +3γ(E1)transitionprobabilityweobtain
2 0
→
2
dW (ω ,ω ) 16
i→f 1 2 = (ω ω ω )3 (U γ1,γ2,γ3 +Dγ1,γ2,γ3)+5permutations , (26)
dω1dω2 27π2 3 2 1 mγ3mXγ2mγ1(cid:12)(cid:12)n′X′n′n{ n′′n′n n′′n′n }(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Unγ′1′,nγ′2n,γ3 =(rmγ3)11S0n′′ǫnS′′Tn′(rmγ2)n′n(rmγ1)n23P2 × (27)
1 1
,
(ES E ω )(ET E ω ω ) − (ET E ω )(ET E ω ω )
(cid:26) n′′ − f − 3 n − f − 3− 2 n′ − f − 3 n − f − 3− 2 (cid:27)
Dnγ1′′,nγ′2n,γ3 =(rmγ3)11S0n′′(rmγ2)n′′n′ǫnS′Tn(rmγ1)n23P2 × (28)
1 1
.
(ES E ω )(ES E ω ω ) − (ES E ω )(ET E ω ω )
(cid:26) n′′ − f − 3 n′ − f − 3− 2 n′′ − f − 3 n − f − 3− 2 (cid:27)
Here ES and ET are the energies of the singlet or triplet n-states respectively. Permutations in Eq. (26) are understood as
n n
permutationsoftheindices1,2,3. FromEq. (26)itisclearthattheprobabilityof23P decayissuppressedbythesmallness
2
ofthespin-orbitinteraction. Thesamewasobservedalsoforthetwo-photondecayof23S state[13]and,inprinciple,should
1
holdforanyintercombinationtransitions.
Forthe23P 11S +3γ(E1)transitioninthechannel23P n3S +γ(E1) n′3P n′1P +2γ(E1) 11S +
2 0 2 1 1 1 0
→ → → →
3γ(E1)thereistheresonance(thesituationwhentheenergydenominatorturnstozero).Thepresenceofthecascade-producing
state 23S in the sum overall the intermediatestates in the transition amplitudeEq. (26) leads(cid:2)to the(cid:3)arrivalof the high, but
1
narrow ”ridge” in the frequency distribution dW(ω1,ω2). Analogous situation arises for 21P 11S +3γ(E1) transition.
dω1dω2 1 → 0
Howevertheexistenceofthis”ridge”doesnotinfluencethevalidityoftheSSSR-2,since”ridge”correspondstothenonequal
frequenciesfortheall3-photon.
III. TECHNICALDETAILSANDRESULTS
TransitionprobabilitiesobtainedfromtheexpressionEq.(12)werecheckedfortheconvergencewithadifferentlengthofthe
basissetN.ResultsofcalculationswiththeN =30, 50, 100,150forthe23P 21S +3γ(E1)and21P 21S +3γ(E1)
2 0 1 0
→ →
transitionsarepresentedinTableI.Correspondingplotforthe23P 21S +3γ(E1)transitionispresentedinFig.1.However
2 0
→
itismoreconvenienttopresent2-dimensionalsectionalcutofFig. 1,withthefixedfrequencyofthethirdphotonatthepoint
ω = ∆/3, where ∆ = E(23P ) E(21S ). This two-dimensionalcut is presented in Fig. 2. Two-dimensionalplots for
3 2 0
the 23P 11S +3γ(E1) and 2−1P 11S +3γ(E1) transition are presented in Fig. 3 and Fig. 4 respectively. The
2 0 1 0
→ →
maindifferencebetweenFig. 3andFig. 4,arisesatthepointwithcoordinatesω = ω = ∆/3inthefrequencydistribution.
1 2
Forthetransition23P 11S +3γ(E1)differentialtransitionratebecomeszeroduetoSSSR-2. Thesetwotransitionsare
2 0
→
notsuitableforexperimentsduetothetransitionfrequencieswhichisoutoftheopticalrange,butarecalculatedforadditional
demonstrationofSSSR-2.
Inordertointegrateoverphotonfrequency,the32pointsofGauss-Legendrequadraturemethodwasemployed.Thenumerical
valuesforthespin-orbit(Eq.(24))andspin-other-orbit(Eq.(25))matrixelementsforsomelowestP andDstatesarepresented
6
inTableIIandareinagoodagreementwiththepreviouscalculations[26],[25]. Nonrelativisticvariationalenergiesofbound
statesobtainedandusedinthisworkarepresentedinTableIII.Theyareinaverygoodagreement(13significantdigits)with
previouscalculations.Weshouldstressthatforourpurposesitissufficienttoemploynonrelativisticenergylevels:theyprovide
anoverallaccuracyinourcalculationsatthelevel0.01%.
In[15]wesuggestedamethodforapossibleexperimentaltestoftheSSSRsbymeansoflaserlightabsorptionandpresented
numericalexampleswith the highlychargedHe-like ions. The photonfrequenciesin this case are in the X-ray regionwhich
makesitdifficulttoperformthetests. Inthepresentpaperwesuggesttheheliumatomasthemostadequatesystemforthetest
of SSSRs with the opticallasers. The helium atom has long served as a testing groundfor both theoreticaland experimental
studiesandseveralreviewshavechronicledtheprogressin bothareas(see, forexample,[30]). Thenumericalcalculationsof
three-photon23P 21S +3γ(E1)and21P 21S +3γ(E1)transitionsinheliumarelistedinTableI.Thesetransitions
2 0 1 0
→ →
aremostsuitableforthelasertestingofSSSR-2duetothevaluesofthecorrespondingtransitionfrequencies(andhencetolaser
onesafterdividingbyN ),belongingtotheopticalorinfraredregion.
γ
Correspondingone-photondecayrates,arealsopresentedandareingoodagreementwithpreviouscalculations[27,28](see
TableI).Ourresultscoincidewiththenumbersgivenin[28]with4significantfigureswhichmeanstheaccuracyof0.01%.The
sameaccuracyshouldhavethe3-photontransitionprobabilities.
Fromthe energydifferenceslisted in Table I onecan see thatproposedtransitionsaresuitable forthe test ofSSSR-2 since
each emitted (absorbed)photon is in the optical (infrared) range. As it was mentioned in the introductionthe corresponding
one-photondecayissuppressedatthefrequencyω . Note,thattheenergydifferencebetweentripletandsingletP statesinHe
l
atomareresolvableforthelasersource.
Acknowledgments
TheauthorsareindebtedtoV.I.KorobovforthepermissiontousethecomputercodesfortheconstructionoftheHevariational
wavefunctionsandforvaluableconsultations. ThisworkwassupportedbyRFBR (grantNo. 14-02-00188). T.Z., D. S.and
L.L.acknowledgethesupportbySt.-PetersburgStateUniversitywitharesearchgrant11.38.227.2014. TheworkofT.Z.was
also supportedby grantof German-RussianInterdisciplinaryScience Center (G-RISC) P-2015b-10and non-profitfoundation
”Dynasty”(Moscow).
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7
FIG. 1: (Color Online) 3-dimensional plot for frequencies distributionof the transitionrate 23P → 21S +3γ(E1) in Helium. On the
2 0
verticalaxisthetransitionrate dW ins−1isplotted;onthehorizontalaxesthephotonfrequenciesareplottedinunitsω /∆,ω /∆where
dω1dω2 1 2
∆ denotestheenergy difference E(23P)−E(21S). Thelowest (zero) point isthepoint withcoordinates ω /∆ = ω /∆ = 1/3 at the
1 2
bottomofthe”pit”inthefrequencydistributionforthetransitionrate.This”pit”arisesduetoSSSR-2.
FIG.2: (Color Online) 2-dimensional sectional cut for frequencies distributionof thetransitionrate23P → 21S +3γ(E1) inHelium.
2 0
On the vertical axis the transition rate dW in s−1 is plotted; on the horizontal axis the photon frequency is plotted in units ω /∆ where
dω1 1
∆ = E(23P)−E(21S). Second frequency ω isfixedatthepoint ω /∆ = 1/3. Thelowest (zero) point isthepoint withcoordinates
2 2
ω /∆=ω /∆=1/3atthebottomofthe”pit”inthefrequencydistributionforthetransitionrate.This”pit”arisesduetoSSSR-2.
1 2
FIG.3: (Color Online) 2-dimensional sectional cut for frequencies distributionof thetransitionrate23P → 11S +3γ(E1) inHelium.
2 0
On the vertical axis the transition rate dW in s−1 is plotted; on the horizontal axis the photon frequency is plotted in units ω /∆ where
dω1 1
∆ = E(23P)−E(11S). Second frequency ω isfixedatthepoint ω /∆ = 1/3. Thelowest (zero) point isthepoint withcoordinates
2 2
ω /∆=ω /∆=1/3atthebottomofthe”pit”inthefrequencydistributionforthetransitionrate.This”pit”arisesduetoSSSR-2.
1 2
8
TABLEI:Probabilitiesforthethree-photon23P →21S +3γ(E1),21P →21S +3γ(E1)andone-photon23P →21S +1γ(M2),
2 0 1 0 2 0
21P →21S +1γ(E1)transitionsins−1.Thenumberinparenthesesindicatesthepoweroften.Transitionenergies∆EineVarelistedin
1 0
twolastcolumns.
N W233γP(E2−1)21S0 W231γP(E1−1)21S0 W213γP(M2−22)1S0 W211γP(E1−1)21S0 ∆E23P−21S ∆E21P−21S
30 3.727(−27) 4.430(−15)
50 3.763(−27) 6.651(−15)
100 3.763(−27) 6.575(−15) 0.35 0.60
150 3.764(−27) 6.573(−15)
500 − − 1.1953(−8) 1.9746(6)
1.1952(−8)[27], [28]
TABLEII:Valuesofsinglet-triplet(ST)spin-orbitandspin-other-orbitmatrixelementscomputedwithN =500(ina.u.)
State hH i hH i
so ST soo ST
2P −0.0000019049679 −0.000000688044
−0.000001907[26] −0.000000689[26]
−0.000001904968[25] −0.000000688043[25]
3P −0.000000558930 −0.000000191302
−0.0000005604[26] −0.0000001926[26]
−0.000000558929[25] −0.000000191301[25]
4P −0.00000023446 −0.000000078773
−0.0000002346[26] −0.0000000793[26]
−0.000000234440[25] −0.000000078772[25]
3D −0.00000016289 −0.000000080
−0.00000016270[26] −0.00000007996[26]
−0.000000162750[25] −0.000000079992[25]
TABLEIII:Nonrelativisticenergiesforboundstatesofheliumina.u.
State/N 30 50 100 150 500 Drake[29]
11S −2.903701 −2.903722 −2.903724357 −2.90372437162 −2.903724377034044 −2.9037243770341195
23S −2.175178 −2.175225 −2.175229365 −2.17522937745 −2.175229378236739 −2.17522937823679130
21S −2.144864 −2.145681 −2.14597379 −2.14597398 −2.145974046042 −2.145974046054419
23P −2.13272 −2.13314 −2.133164 −2.133164171 −2.133164190756 −2.133164190779273
21P −2.123819 −2.123839 −2.123843017 −2.123843067 −2.123843086358 −2.123843086498093
33D −2.055521 −2.055634 −2.055636 −2.055636027 −2.055636047 −2.055636309453261
31D −2.055553 −2.055606 −2.05561892 −2.05561895 −2.05561897 −2.055620732852246
9
FIG.4: (Color Online) 2-dimensional sectional cut for frequencies distributionof thetransitionrate21P → 11S +3γ(E1) inHelium.
1 0
On the vertical axis the transition rate dW in s−1 is plotted; on the horizontal axis the photon frequency is plotted in units ω /∆ where
dω1 1
∆=E(21P)−E(11S).Secondfrequencyω isfixedatthepointω /∆=1/3.
2 2