Some Recent Advances in Bound-State Quantum Electrodynamics

0.16 MB·English

by  U. D. Jentschura

1
Some Recent Advances in Bound–State
Quantum Electrodynamics
U. D. Jentschura and J. Evers
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0
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2 Abstract: Wediscuss recent progress invarious problems relatedtobound-state quantum
electrodynamics: thebound-electron g factor, two-loop self-energycorrections and thelaser-
n
dressed Lamb shift.Theprogress relieson various advances inthebound-state formalism,
a
J including ideasinspired byeffective fieldtheoriessuch asNonrelativisticQuantum
Electrodynamics. Radiative correctionsindynamical processes represent apromising fieldfor
4
further investigations.
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Keywords:Calculationsandmathematicaltechniquesinatomicandmolecularphysics,
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quantumelectrodynamics–specificcalculations;
v
PACSnumbers:31.15.-p,12.20.Ds.
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2
2
1 1. Introduction
0
5 Thisisabriefsummaryofanumberofrecentadvancesinourunderstandingofbound-statequan-
0 tum electrodynamic (QED) effects. The topics are (i) two-loop corrections to the bound-electron g
/
h factor,(ii)higher-ordertwo-loopcorrectionstotheself-energyofaboundelectron,and(iii)thelaser-
p dressedLambshift.
-
Thefirsttwooftheseratherdiversetopicsarerelatedtotwo-loopeffects.Theinvestigationofthese
p
e is simplified considerably by the use of effective field-theory techniques inspired by Nonrelativistic
h QED (NRQED) [1–3]. The Wilson coefficients multiplying the effective operators in the NRQED
:
v Lagrangianarematchedagainstthoseofthefullrelativistictheory,providingasimplifiedframework
i forthecalculationofbound-stateeffects.Scale-separationparameterssuchasthephotonmassµare
X
cancelled at the end of the calculation. The analysis of higher-order corrections to the g factor of
r
a the bound electron is simplified further by a transformation to the length gauge, which results in a
lessernumberoftermstobeconsideredthanwouldbenecessaryinthevelocitygauge.Thisfacthas
inspired the developmentof Long-wavelengthQED (LWQED) [4], a theory which is obtained after
Power–Zienauand Foldy–Wouthuysentransformationsof the first-quantizedLagrangian;the second
quantizationiscarriedoutbyformulatingthepathintegral.Consequently,animprovedunderstanding
andatremendoussimplificationresultsforthecalculationofanumberofQEDcorrectionsforbound
states,suchasthegfactorandhigher-ordercorrectionstotheself-energy.
Afurtherfieldofrecentstudieshasbeenconcernedwiththeinteractionofa laser-dressedbound
electronwiththeradiationfield[5–8].Thisprocessentailscorrectionswhichcanonlybeunderstood
iftheanalysisiscarriedoutrightfromthestartintheframeworkofthelaser-dressedstates,whichare
theeigenstatesofthequantizedatom-laserHamiltonianintherotating-waveapproximation[9].
U.D.JentschuraandJ.Evers.Max–Planck–Institutfu¨rKernphysik,Saupfercheckweg1,69117Heidelberg,
Germany
CanadianJournalofPhysics1:1–12(2004) 2004NRCCanada
2 CanadianJournalofPhysicsVol.1,2004
2. Bound-electron g factor
Inthissection,webrieflysummarizetheresultsofarecentinvestigation[10]ofthebound-electron
g factor,whichisbasedonNRQED.Thecentralresultofthisinvestigationisthefollowingsemiana-
lyticexpansioninpowersofZαandln(Zα)forthebound-electrongfactor(nSstate)inthenon-recoil
limit,whichisthelimitofaninfinitenuclearmass:
2(Zα)2 (Zα)4 1 2
g(nS)=2 + + (Zα)6
− 3n2 n3 (cid:18)2n − 3(cid:19) O
Breit(1928),Diractheory
| {z }
α 1 (Zα)2 (Zα)4
+ 2 1+ + a ln[(Zα)−2]+a + (Zα)5
π (cid:26) × 2(cid:18) 6n2 (cid:19) n3 (cid:26) 41 40(cid:27) O (cid:27)
one-loopcorrection
| {z }
α 2 (Zα)2 (Zα)4
+ 0.656958 1+ + b ln[(Zα)−2]+b + (Zα)5
(cid:16)π(cid:17) (cid:26)− (cid:18) 6n2 (cid:19) n3 (cid:26) 41 40(cid:27) O (cid:27)
two-loopcorrection
| {z }
+ (α3). (1)
O
This expansion is valid through the order of two loops (terms of order α3 are neglected). The
notation is in part inspired by the usual conventions for Lamb-shift coefficients: the (lower case)
a terms denote the one-loop effects, with a denoting the coefficient of a term proportional to
kj
α(Zα)klnj[(Zα)−2].Thebtermsdenotethetwo-loopcorrections,withb multiplyingatermpro-
kj
portionaltoα2(Zα)klnj[(Zα)−2]. In[10],completeresultsarederivedforthecoefficientsa ,a
41 40
andb .
41
Ingeneral,theexpressioncorrespondingto(1)forafreeelectronisobtainedbylettingtheparam-
eterZα 0ineverytermoftheloopexpansion(expansioninpowersofα).Inthislimit,theknown
→
free-electrontwo-loopresultisrecovered[11–17].
Uptotherelativeorder(Zα)2,thefree-electroncontributioninone-,two-,andhigher-looporder
ismultipliedbyarelativefactor
(Zα)2
1+(Zα)2a =1+(Zα)2b =1+ . (2)
20 20 6n2
This result consequently holds for the three-loop and the four-loop term not shown in Eq. (1). The
applicability of the relative factor (2) to the two-loop term, valid through (Zα)2, had been stressed
previouslyin[18].TheresultinEq.(2)hadbeenobtainedoriginallyin[19–23](forthe1Sstate).As
isevidentfromEq.(1),thecorrectionofrelativeorder(Zα)2 isdifferentonthelevelofthetree-level
diagramsandreadsg(nS) 2 [1 (Zα)2/(3n2)].
∼ × −
Explicitresultsforthecoefficientsin(1),restrictedtotheone-loopself-energy,read[10]
32
a (nS)= , (3a)
41
9
73 5 8 8
a (nS)= lnk (nS) lnk (nS). (3b)
40 0 3
54 − 24n − 9 − 3
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JentschuraandEvers 3
(cid:1) (cid:2) (cid:3)
(cid:4) (cid:5) (cid:6)
(cid:7) (cid:8) (cid:9)
Fig. 1. Feynman diagrams for the two-loop self-energy corrections to the
bound-electrong-factor.
Here, lnk (nS) is the Bethe logarithm for an nS state, and lnk (nS) is a generalization of the
0 3
Bethe logarithm to a perturbativepotentialof the form 1/r3. Vacuum polarization adds a further n-
independentcontributionof( 16/15)toa [24].TheBethelogarithmsfor1Sand2S[25]read
40
−
lnk (1S) = 2.984128555, (4a)
0
lnk (2S) = 2.811769893, (4b)
0
andthecorrespondingvaluesforlnk read[10]
3
lnk (1S) = 3.272806545, (5a)
3
lnk (2S) = 3.546018666. (5b)
3
Thequantitylnk (nS)isdefinedas,
3
ǫ 1 1 1 (Zα)3 µ 5
dkk2 φr r φ = 4 ln + lnk (6)
Z h | E H k r3 E H k | i − n3 (cid:20) (Zα)2 6 − 3(cid:21)
0 − S − − S −
where theultravioletcutoffǫ is to beunderstoodin the sense of[2],and thematchingofthe nonco-
variantcutoffǫtothecovariantphotonmassµisgivenas(see[26],pp.361–362)
2ǫ µ 5
ln ln + . (7)
(cid:18)m(cid:19)→ (cid:16)m(cid:17) 6
However, this replacement is not unique and the constant term depends on the actual form of the
integrand.Adifferentreplacementhastobeusedforsomeofthelow-energyphotoncorrectionstothe
gfactor[10].
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Theresultsforthetwo-loopcoefficientsread
56
b (nS)= , (8a)
41
9
b (1S)= 18.5(5.5). (8b)
40
−
Here, the result for b is an estimate based on an explicit calculation of a large contribution due
40
to low-energyvirtualphotons, and an estimate of the remaining,unknowncontributiondue to high-
energyvirtualphotons.Thedominantlogarithmictwo-looptermb iscausedexclusivelybythetwo-
41
loopself-energydiagramsinFig.1alone.Theothertwo-loopdiagrams,whichincludeclosedfermion
loops,canbefoundinFig.21of[27].Thelogarithmictermb is,however,exclusivelyrelatedtothe
41
gauge-invariantsubsetdisplayedinFig.1.
Thenewlycalculateda ,a andb arebound-statecorrectionstotheelectrong-factoroforder
41 40 41
(α/π)(Zα)4and(α/π)2(Zα)4,multipliedbylogarithmicterms.Thesecorrectionsare(atZ =1)for-
mallyoforderα5 andα6 andthereforeofthesameorderofmagnitudeasthetenth-andtwelfth-order
correctionstothe free-electronanomaly,whichbarelyareofexperimentalortheoreticalsignificance
atthecurrentlevelofaccuracy.Onemaythereforeaskwhythesebindingcorrectionsareofanyphe-
nomenologicalsignificance.ThereasonisthatatsomewhathigherZ,thesituationchangesdrastically,
due to Z4 scaling of the binding corrections. In addition, due to numerically large coefficients and
logarithmicfactors,the“hierarchy”ofthecorrectionschangesdrastically.Roughly,onemaysaythat
atZ = 1,the bound-electronanomalousmagneticmomentisapproximatelyindependentofbinding
correctionsoforder(α/π)(Zα)4 andhigher,whereasforhigherZ,thesituationisreversed,andthe
bindingcorrectionstotheone-andtwo-loopcontributionsarenumericallymuchmoresignificantthan
thehigher-loopfree-electroncorrections.This“transitionfromfreetobound-statequantumelectrody-
namics”asafunctionofZ isasomewhatpeculiarfeatureofthebound-electrong-factor.
For,example,weconsidertheratio
α
(Zα)4 ln[(Zα)−2]
π
r (Z)= (cid:16) (cid:17) , (9)
1 (α/π)4
whichgivesanorder-of-magnitudeestimateforthetheratiooftheone-loopself-energybindingcor-
rectiontotheeighth-orderanomalousmagneticmomentofthefreeelectron.Wehave
r (Z =1) 2, r (Z =10) 104 1. (10)
1 1
≈ ≈ ≫
Forthetwo-looplogarithmicbindingcorrection,wehave
α 2
(Zα)4 ln[(Zα)−2]
π
r (Z)= (cid:16) (cid:17) (11)
2 (α/π)4
andconsequently
r (Z =1) 0.2<1, r (Z =10) 2 103 1. (12)
2 2
≈ ≈ × ≫
As is evident from these considerations, the (Zα)4 one-loop and two-loop binding corrections are
roughlyof the same order of magnitudeas the highly problematicfour-loopcorrections[28–30] for
the free electron. However, the situation changes drastically even at very moderate nuclear charge
numbers, and the binding corrections to the one-loop and two-loop contributions become dominant
overthehigher-loopeffects.
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JentschuraandEvers 5
The NRQED one-loopcalculation[10] is dividedinto three parts, the first of which entails fully
relativistic form-factor corrections including lower-order terms, the second of which correspondsto
a spin-dependentscattering amplitude, and the third of which is a low-energy Bethe-logarithm type
correctionthatcontainslnk andlnk .Theone-loopcorrectioninEq.(1)canthereforebewrittenin
0 3
anaturalwayasδg(1) =g(1)+g(1)+g(1),where
1 2 3
α (Zα)2 (Zα)4 7 5 16
g(1) = 1+ + + lnµ , (13a)
1 π(cid:20) 6n2 − n3 (cid:18)6 24n 3 (cid:19)(cid:21)
α (Zα)4 16
g(1) = 4+ lnµ , (13b)
2 π n3 9
(cid:16) (cid:17)
α (Zα)4 32 µ 5 lnk 3
g(1) = ln 0 lnk . (13c)
3 π n3 9 (cid:20) (Zα)2 − 12 − 4 − 4 3(cid:21)
The new contribution of order (Zα)4 can be compared with the numerical results for the self-
energycorrection[31,32]completetoallordersinZα.Assumingcorrectnessofthelogarithmicterm
inEq.(13),afittonumericaldatayieldsa(1)(1S)= 10.2(1)anda(1)(2S)= 10.6(1.2)forthecon-
40 − 40 −
stantterm,inexcellentagreementwiththeanalyticresultswhichreada(1)(1S)= 10.236524318(1)
40 −
anda(1)(2S)= 10.707715607(1).
40 −
Havingverifiedtheconsistencyoftheanalytic[10]andnumericalresults[31,32],aninterpolation
procedure[33]maynowbeusedtoextractamoreaccuratetheoreticalpredictionatlowandinterme-
diatenuclearchargenumbers,ifcombinedwithnumericalresultsathigherZ [31].Thus,theresultsin
Eqs.(3)and(8)maybeusedinordertoinferimprovedtheoreticalpredictionsforthebound-electron
g factor,notablyinthe experimentallyimportantspecialcases ofhydrogenlikecarbon[34]andoxy-
gen[35].Alternatively,theimprovedstatusofthetheorymaybeusedinordertoinferamoreaccurate
value of the electronmass. Specifically,the value fromthe carbonmeasurement[34],using the new
theory,reads
m(12C5+)=0.00054857990941(29)(3)u. (14)
The first error comes from the experiment [34], and the second error corresponds to the theoretical
uncertainty. The conclusion is that a further improvement of the experiment could lead to a much
better determination of the electron mass; the new theory provides room for at least an improved
determinationbyoneorderofmagnitude.
Forthecalculationofyethigher-orderbindingcorrectionstotheone-loopandtwo-loopcontribu-
tions, a detailed understandingof the two-loop form-factors,including their slopes, is required. The
mostrecentcalculationsoftheseeffects,inbothdimensionalandphoton-massregularizations,canbe
foundin[36–38].
3. Two-loop Bethe logarithms
Asiswellknown[39–46],thetwo-loopLambshift∆E(2),inthelimitofaninfinitenuclearmass,
maybewrittenas
α 2 (Zα)4m c2
∆E(2) = e H(Zα). (15)
π n3
(cid:16) (cid:17)
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(2LSE) (SVPE)
(SEVP)
Fig. 2. Thetwo-loop corrections totheLambshift inhydrogenlike systemsfall naturallyintothree gauge-
invariantsubsets,whicharethepuretwo-loopself-energyterms(2LSE),thevacuum-polarizationcorrectionto
thevirtual-photonlineintheself-energy(SVPE),andtheself-energyvacuum-polarizationandpuretwo-loop
vacuum-polarizationcorrections(SEVP).Thetwo-loopBethelogarithmisanumericallylargecorrectiontothe
nonlogarithmictermoforderα2(Zα)6anditisexclusivelyrelatedtothe2LSEsubset;however,foracomplete
resultinthisorder,contributionsfromtheothergauge-invariantsubsetswillhavetobeconsideredaswell.
ForSstates,thedimensionlessfunctionH(Zα),hasasemianalyticexpansionoftheform
H(Zα)=B +(Zα)B
40 50
+(Zα)2 B ln3[(Zα)−2]+B ln2[(Zα)−2]+B ln[(Zα)−2]+B , (16)
63 62 61 60
(cid:2) (cid:3)
where we ignore higher-order terms, and upper case is used for the B coefficients that multiply
ij
termsoforderα2(Zα)i lnj[(Zα)−2]m c2.Thecoefficients,restrictedtothetwo-photonself-energy
e
diagrams(Fig.2),readasfollows
8
B(2LSE)(nS) = = 0.296296, (17a)
63 −27 −
16 16
B(2LSE)(1S) = ln(2)= 0.639669, (17b)
62 27 − 9 −
16 3 1 1
B(2LSE)(nS) = B(2LSE)(1S)+ + ln(n)+Ψ(n)+C . (17c)
62 62 9 (cid:18)4 4n2 − n − (cid:19)
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JentschuraandEvers 7
Then-dependenceofB hasbeenclarifiedin[46,47]
61
4
B(2LSE)(nS) = B(2LSE)(1S)+ [N(nS) N(1S)]
61 61 3 −
80 32 3 1 1
+ ln(2) + ln(n)+Ψ(n)+C , (18)
(cid:18)27 − 9 (cid:19)(cid:18)4 4n2 − n − (cid:19)
whereC =0.577216... isEuler’sconstant,Ψ(n)isthelogarithmicderivativeoftheGammafunction,
andN(nS)isrelatedtoacorrectiontotheBethelogarithminducedbyaDirac-deltapotential.Explicit
valuesforN(nS)canbefoundin[47](n=1,...,8).
ω
2 6
mixed relativistic
integration integration
region(bM) region(bF+bH)
ǫ
2
nonrelativistic mixed
integration integration
region(bL) region(bM)
-
ǫ ω
1 1
Fig. 3. The integration regions for the two virtual photons
in the two-loop self-energy problem comprise a low-energy
regime with two low-energy photons, which gives rise to bL.
Themiddle-energyregionswithonelow-energyandaonehigh-
energy photon give rise to bM. The high-energy contribution
bF +bH isasyetunknown.
TheB coefficientsarethesumofseveralcontributions
60
B(2LSE)(nS) = b +b + b +b +b , (19a)
60 L M { F H VP}
b = b (nS) two-loopBethelogarithm,twosoftphotons, (19b)
L L
∼
10
b = b (nS) = N(nS) onesoft,onehardphoton, (19c)
M M
9 ∼
b softelectronmomenta,twohardphotons, (19d)
F
∼
b hardelectronmomenta,twohardphotons, (19e)
H
∼
b vacuum-polarizationcorrections. (19f)
VP
∼
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8 CanadianJournalofPhysicsVol.1,2004
Only the terms b + b are currently known [48,49] (see also Fig. 2). The contributions in curly
L M
bracketsinEq.(19a)remaintobeevaluated.However,anestimateforthetotalvalueofB maybe
60
obtained,
B (nS) = b +b + b +b +b ,
60 L M F H VP
{ }
B (nS) b +b 15%. (20)
60 L M
≈ ±
Thisestimate[48,49]is basedoncorrespondingone-loopcalculations,wherethe low-energyvirtual
photonsgive the by far dominantcontributionto the constant term [2]. The results for the two-loop
BethelogarithmsofSstatesread[48,49]
b (1S) = 81.4(3), (21a)
L
−
b (2S) = 66.6(3), (21b)
L
−
b (3S) = 63.5(6), (21c)
L
−
b (4S) = 61.8(8), (21d)
L
−
b (5S) = 60.6(8), (21e)
L
−
b (6S) = 59.8(8). (21f)
L
−
A few clarifying remark might be in order. The B coefficient multiplies a correction of order
60
α2(Zα)6, which is effectively an order-α8 contribution to the energy levels of hydrogen (Z = 1).
Inordertocompletethecalculationatthisorderofmagnitude,itwouldalsobenecessarytoconsider
the four-loopDirac form-factorslope of the electron,as well as the three-loopbindingcorrectionof
orderα3(Zα)5m c2 [50].Thethree-loopslopehasrecentlybeenevaluatedin [51],completingthe
e
theoryofenergylevelsinhydrogenuptotheorderofα7.
4. Laser-dressed Lamb shift
In the recentpast, seminaladvanceshavebeen obtainedboth in the techniquesof high-precision
spectroscopy(e.g.,[52]),andinthecoherentpreparationandmanipulationofmediabyexternalelec-
tromagneticfields[53,54].Thusitisdesirabletostudytheboundelectronsinteractingsimultaneously
bothwiththequantizedradiationfieldandwith anexternaldrivingfield.An accuratetheoryofsuch
systems, includingall dynamic effects, mighteventuallyopen a possibility for a whole new class of
high-precisionexperiments,providedthat technical problemsrelated to the required highly accurate
intensity stabilization of the laser (and others) can be solved. Traditionally, radiative and relativistic
correctionsare treatedwith methodsof QED, whereasstudiesrelated to the dynamicalnatureof the
interaction of matter with driving laser fields are the domain of Quantum Optics (QO). Obviously,
a treatment of bound electrons in the presence of both the radiation field and external driving fields
requires a combinationof ideas from both subject areas: While, a priori, the essential-state approx-
imation of QO [53] is not sufficient to obtain the accuracy of QED, a perturbative treatment of the
interactionoftheboundelectronwithastrongexternal(laser)fieldasinQEDishopelessbecauseof
thelargecouplingparameter.
In[5–7],anatomwithtworelevantenergylevelsdrivenbyastrongnear-resonantmonochromatic
laser field is studied as the easiest model system for the above problem. The incoherent part of the
resonance fluorescence spectrum emitted by this system in QO is known as the Mollow spectrum,
wherethecouplingstrengthischaracterizedbythetheRabifrequencyΩdefinedas(~=ǫ =c=1)
0
Ω= q ex ǫ g , (22)
L L
− h | · | iE
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JentschuraandEvers 9
foradrivinglaserfieldE (t) = ǫ cos(ω t)withfrequencyω ,macroscopicclassicalamplitude
L L L L L
E
andpolarizationǫ .Here,q = q istheelementarycharge.Thecorrespondingcouplingconstant
L L
E −| |
g fortheinteractionofaquantizeddrivinglaserfieldwiththemainatomictransitionisdefinedby
L
g = q g ǫ xe (γ), (23)
L − h | L· | iEL
where (γ) = ω /2V is the electric laser field perphotonandV is the quantizationvolume.The
EL L
matchingofthepelectricfieldperphotonwiththecorrespondingclassicalmacroscopicelectricfield
L
E
isthengivenby
2√n+1 (γ) . (24)
EL ←→ EL
IfΩislargerthanthenaturaldecaywidthΓofthetransition,thentheMollowspectrumapproxi-
matelyconsistsofonecentralandtwosidebandpeaksofLorentzianshape,whicharelocatedsymmet-
ricallyaroundthedrivinglaserfieldfrequency.Thesidebandpeaksareshiftedfromthedrivingfield
frequencybythegeneralizedRabifrequencyΩ =√Ω2+∆2,where∆=ω ω isthedetuningof
R L R
−
thelaserfieldfrequencyω withregardtotheatomictransitionfrequencyω .TheshapeoftheMollow
L R
spectrummayeasilybeexplainedintermsofthedressedstates, whicharedefinedastheeigenstates
ofthequantumopticalinteractionpictureHamiltoniandescribingthematter-lightinteraction.Intrans-
ferring to the dressed state picture, the interaction with the driving laser field is accountedfor to all
orders.
Thus,whenevaluatingradiativeandrelativisticcorrectionstotheMollowspectrum,itisnaturalto
start the analysisfrom the dressed-state basis as opposedto the unperturbedatomic bare-state basis.
In [5,6], it was shown that this distinction in fact has to be made. It is not sufficient to modify the
energies(whichenterintheformulaforthedressedstates)accordingtotheusualbare-stateLambshift
inordertoobtainthecorrectresultforthecorrectionstotheMollowspectrum.Instead,atnonvanishing
detuningandnonvanishingRabifrequency,atreatmentstartingfromthedressed-statebasisleadstoan
additionalnontrivialcorrectionterm.ThistermgivesrisetoashiftoftheMollowsidebandsrelativeto
thecentralpeakgivenby
Ω2
δω(C) = , (25)
±
∓C √Ω2+∆2
where
α p2 + p2
= ln[(Zα)−2] g e (26)
C π (cid:10) (cid:11) m2(cid:10) (cid:11)
is a dimensionlessconstant. Here, the notation . and . denotesthe expectationvalue evaluated
g e
hi hi
withthegroundorexcitedatomicstate,respectively.
Inspired by the interpretation of the bare Lamb shift correction in terms of a “summed” shift as
in[5,6],thisadditionalcorrectioncanbeinterpretedasamodificationtotheRabifrequency:
δω(C) = Ω2(1 )2+∆2 Ω2+∆2 , (27)
±
±(cid:16)p −C −p (cid:17)
withδω(C) δω(C)becauseofthesmallnessofthecorrection.Itshouldbenotedthatthisinterpreta-
± ±
≈
tionintermsofasummationisnottrivialandwasshowntobevaliduptofirstorderinthecorrection.
In[5,7],theleadingrelativisticandradiativecorrectionsuptorelativeorders(Zα)2 andα(Zα)2,
respectively, have been evaluated, as well as all other relevant correction terms up to the specified
orderofapproximation.Itturnsoutthatallcorrectionsmaybeinterpretedaseithercorrectionstothe
2004NRCCanada
10 CanadianJournalofPhysicsVol.1,2004
Rabi frequency Ω or as corrections to the detuning ∆, such that one can define the fully corrected
generalizedRabifrequencyΩ(j) by
C
2 2
Ω(j) = Ω2 1+Ωˆ(j) + ∆ ∆(j) . (28)
C r ·(cid:16) rad(cid:17) (cid:16) − rad(cid:17)
Here,Ωˆ(j) containsallcorrectionstotheRabifrequency,namelytherelativisticandradiativecorrec-
rad
tionstothetransitiondipolemoment,field-configurationdependentcorrections,higher-ordercorrec-
tionstotheself-energy,andcorrectionstothesecularapproximation.∆(j) consistsofallcorrections
rad
to the detuning,i.e. the bare Lamb shift, Bloch-Siegertshifts, and off-resonantradiativecorrections.
The superscript (j) indicates the dependence of the result on the total angular momentum quantum
number.
Equation(28)summarizesthemainresultofthisstudy:Inthepresenceofdrivinglaserfields,the
usualbarestateLambshiftoftheatomicstatesisaugmentedbyadditionalcorrectionterms.Thesein
partdependonthelaserfieldparametersΩand∆,whichspanatwo-dimensionalparametermanifold
determiningtheactualvalueofthedynamicalLambshift.
A promising candidate for the experiment are the hydrogen 1S 2P and 1S 2P
1/2 ↔ 1/2 1/2 ↔ 3/2
transitions.Weconsiderhereasaspecificexamplethe1S 2P transitionwithΩ = 1000 Γ
1/2 ↔ 1/2 · 1/2
and ∆ = 50 Γ as the laser field parameters. The Rabi frequency is shifted with respect to the
· 1/2
leading-orderexpressionΩ =√Ω2+∆2byrelativisticandradiativecorrectionsasfollows,
R
Ω(1/2) Ω = 738.282(60) 106Hz. (29)
C R
±(cid:16) − (cid:17) ± ·
Thisdrivinglaserfieldparametersetisexpectedtobewithinreachofimprovementsofthecurrently
availableLyman-αlasersources[55]inthenextfewyears.Thecorrespondingresultforthe1S
1/2 ↔
2P transitionwithΩ=1000 Γ ,∆=50 Γ is
3/2 · 3/2 · 3/2
Ω(3/2) Ω = 734.871(60) 106Hz. (30)
C R
±(cid:16) − (cid:17) ± ·
Allgivenuncertaintiesareduetounknownhigher-orderterms[7].
By a comparison to experimental data, one may verify the presence of dynamical leading-
logarithmic correction to the dressed-state radiative shift in Eq. (25), which cannot be explained in
termsofthebareLambshiftalone.Thisallowstoaddressquestionsrelatedtothephysicalrealityof
the dressed states. On the other hand, the comparison with experimental results could also be used
to interpret the nature of the evaluated radiative corrections in the sense of the summation formulas
whichleadtotheinterpretationoftheshiftsasarisingfromrelativisticandradiativecorrectionstothe
detuningandtheRabifrequency.
5. Conclusions
One of the obvious conclusions to be drawn from the recent advances in bound-state quantum
electrodynamicsisasfollows.A widespreadopinionhasbeeninvalidatedwhichsuggestedthattwo-
loopcorrectionsto the bound-electrong factor,and two-loopcorrectionstothe Lambshiftin higher
ordermightbeasevere,if notinsurmountable,obstacleagainstfurthertheoreticalprogress.Quiteto
the contrary, the recent advances have shown that an understanding of these effects is feasible to a
goodaccuracy.Whilesomeimportantcontributionsremaintobeevaluated,thefurtherprogram(e.g.,
theevaluationoftheremaininghigh-energyparts)isclearlydefined,anddecisivefirststepstowarda
muchimprovedunderstandinghavebeenaccomplished.
Further advances are possible both on the experimental as well as on the theoretical side: con-
cerningtheg factor,therecenttheoreticalprogressallowsforanorder-of-magnitudeimprovementof
2004NRCCanada