COLLISIONAL BROADENING OF SPECTRAL LINE SHAPES ...mukamel.ps.uci.edu/publications/pdfs/076.pdf4 S. Mukamel, Collisional broadening of spectral line shapes proaches, etc.) and is well

COLLISIONAL BROADENING OFSPECTRAL LINE SHAPES IN

TWO-PHOTON AND MULTIPHOTONPROCESSES

Shaul MUKAMEL

Universityof Rochester,Rochester,NY 14627, US.A.

INORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)93, No. 1(1982) 1—60. North-HollandPublishingCompany

COLLISIONAL BROADENING OF SPECTRAL LINE SHAPESINTWO-PHOTON AND MULTIPHOTON PROCESSES

ShaulMUKAMELUniversityof Rochester,Rochester,NY14627, U.S.A.

ReceivedSeptember1982

Contents:

A. GeneralFormalism 5.1. Thedressedclusterexpansion 351. Introduction 3 5.2. Theperturbativeexpansion 372. The model Hamiltonian 4 5.3. Practical methodsfor the calculation of two-photon3. The descriptionof multiphotonprocessesusing thetetradic line shapes— An overview 38

Ymatrix 6C. StrongField Line Shapes— Multiphoton Pmcesses6. Absorption line shapesin a strong radiation field 44B. WeakField Line Shapes7. ResonanceRamanspectrain a strong radiation field 49

4. Singlephotonprocesses— Absorption line shapes 134.1. Thedressedclusterexpansion 14 D. Summary4.2. The perturbativeexpansion 19 8. Concludingremarks 564.3. Practicalmethodsfor thecalculation of single photon

References 58line shapes—Anoverview 21

5. Two-photonprocesses 31 Note addedin proof 60

Abstract:The dressedatom picture and thetetradic .~ matrix areusedto developa unified theoreticalframework for the microscopicdescriptionof

collisional broadeningin multiphoton processes.We considerordinary absorption(singlephotonprocesses),two photonprocesses(absorptionandRamanspectra)andsaturationeffects in multiphoton processes(absorptionandRamanspectrain astrong radiationfield). Various approximationschemessuchas theclusterandtheperturbativeexpansion,semiclassicalandstochasticmethodsandthe staticandthe impactlimits arediscussed.Special attention is given to the factorizationapproximation which appearsin the presentfonnalism in a naturalway, and allows us to use theenormousprogressmadein thestudiesof singlephotonline shapesto constructusefulapproximationsfor multiphotonprocesses.The inclusionofnonimpactphenomenais thereforestraightforwardwithin the presentapproach.Some numericalcalculationsarepresentedto demonstratetheapplicability of themethodsdevelopedin this article.

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S. Mukamel,Collisional broadeningofspectral line shapes 3

1. Introduction

Studiesof nonlinearoptical phenomenasuch as two-photon absorption,Ramanand fluorescencespectra,multiphotonabsorptionand ionization of atomsand molecules,four wave mixing, etc. havedrawn considerableattention to recentyears, by virtue of the enormousadvancesin experimentaltechniquesand resolution [1—9].The common theoretical model which applies to theseproblemsconsistsof a few-levelsystem(the“absorber”)which interactswith an electromagneticfield andwith anexternal bath consisting of many degreesof freedom. The bath usually causesrelaxation and theabsorber—bathinteractioninfluencessignificantly the timeevolution of the absorber.The conventionaltheoretical tool in thesestudieswhich was applied with a remarkablesuccessfor numerousdifferentexperimentalsituationsis the multilevel Bloch equations[10—121.In theseequationsthe radiationfieldis treatedclassicallyandthe effectsof the bathare incorporatedby the addition of phenomenologicallevel relaxation(T1)and dephasing(T2) ratesinto the Liouville equationfor the isolatedabsorberplusfield. The classicaltreatmentof the radiation field is justified in many cases[13—151and is usuallyequivalentto the fully quantummechanical“dressedatom” approach[16—18]with the exceptionofphenomenasuch as spontaneousemission. The phenomenologicaltreatmentof relaxation pausesseveralseriousfundamentalproblems.A direct consequenceof this treatmentis that an ordinary lineshape is predictedto have a simple Lorentzian form (eq. (131b)). Moreover, all few-photon andmultiphotoncorsssectionsaregiven as sumsof productsof Lorentziancomplexamplitudes(eq. (131)).This simplicity which provides a very convenientway for back of the envelope interpretationofexperiments,is obviouslyat the expenseof rigor, andis oftenin disagreementwith experimentalfacts.A few examplesare:

(1) It is well establishedboth experimentallyand theoretically that no ordinary line shape isLorentzian in its far wings, which meansthat even if the simple Bloch equationshold for smalldetuningsthey will fail in the wings [19,20]. The detailedstudy and analysisof spectralline shapesinthe wings may thereforeyield microscopic infonnation regardingthe dynamical interactionsin thesystem,providedan adequatetheory is available.

(2) Precisemeasurementsof spectralline shapesnearthe line centershow asymmetrieswhich areincontradictionto the simple Lorentzian form, evenin caseswhere the latter is most likely to hold[21,22].

(3) Single photon line shapesareoften adequatelyevaluatedin the staticlimit wherethe deviationsfrom a Lorentzianbehaviorarevery significant [19,20].

(4) In multiphotonprocessestherearepressureinducedeffectssuch as the collisional redistributionin resonanceRaman [23,24] and two photonabsorption[25] or collisional inducedresonancesin fourwavemixing [26,27] (the so-calledPIER4 or its solid stateanalogueDICE signals).In thesecasesthemagnitudeof the effect itself (and not just its behavior in the wings)dependson the behaviorin thewings of someline shapes.As a result theseeffectsprovideaconvenientandsensitiveway to study lineshapesat largedetunings[28]wherethe Lorentzianform fails. For resonanceRamanin a threelevelsystemthereis a coherentandan incoherent(redistribution)componentof the scatteredradiation.Thetheoreticaltreatmentbasedon the Bloch equations[29]predictsthat the ratio of thesetwo componentsis independenton the detuningof the exciting radiation.This is in markedcontrastwith reality, sincethe redistribution componentactually vanishesmuch faster than the coherentcomponentas thedetuningis increased[24](seesection5).

The problemof ordinary (singlephoton)absorptionline shapeshasbeentreatedin the literaturebynumerousmethods (cluster expansion, perturbation theory, semiclassicalmethods, stochastic ap-

4 S. Mukamel,Collisionalbroadeningof spectralline shapes

proaches,etc.) and is well understood[30—43].By now, thereexists the unified theory of spectrallineshapeswhich enablesus to calculate microscopically collisionally broadenedline shapes,using thepotentialsof interaction.The purposeof the presentarticle is to developandapply asimpleandunifiedmicroscopicdescriptionof two photon and multiphoton processesin the presenceof a thermalbathwithout relying on theBloch’s equations.Thebasictool in thepresentformulationis thetetradicscatteringformalism[34,38, 39], whichtogetherwith thedressedatompictureenablesusto write veryconvenientlyclosed formal expressionsfor the experimentalobservables.This is a naturalstarting point for variousapproximationswhichfinally yield aconvenientschemefor themicroscopicinterpretationof experimentaldata. In particular,we developthefactorizationapproximationwhich enablesus to write the crosssectionfor anymultiphotonprocessasaproductof singlephotoncomplexline shapefunctions.We canthenmakeuse of the enormousprogressmadein the theoreticaltreatmentof the latterquantities.

This review is divided into four major parts. Part A (sections1—3) is devotedto introducing theHamiltonian andthe generaltetradic~Tmatrix andthe dressedatom formulation.In partB (sections4and 5) we apply the generalformulation to weak field line shapes(single photonabsorptionandtwophoton processes)and in part C (sections 6, 7) we apply it to strong field phenomena(multiphotonprocesses),i.e. absorptionandRamanspectrain the presenceof astrongradiation field. Finally part Dcontainssomeconcluding remarks.Although the presentreview focusseson collisional broadening,many of the formal results apply to more general types of baths (phonons[44—49],intramolecularbroadening[50—53],etc.). Thepresentreviewis not intendedto providea comprehensivesurveyof theliterature and is basically a summaryof recentprogressin the field. The referencesgiven here werechosenfor the convenienceof the readersandin manycaseswe cite recentreviewarticlesratherthanthe original contributions.

2. The modelHamiltonian

In this sectionwe shall introducethe basic model Hamiltonian which will be usedthroughoutthepresentarticle. We considera fluid with N + 1 atomscontainedin a volume £1. We assumethat oneatom, denoteds (the absorber),hasa few internal states i~)= Ia), Ib), ë)... and interacts with anexternalradiationfield whereastheotherN atomsarestructurelessbathperturberswhich interactwiths and with eachother.This model allows us to treatforeignbroadening.Whenthe perturbersandtheabsorberareidenticalparticleswehaveselfbroadening.Many of theresultspresentedhereapply to selfbroadeningas well, providedthe rate of resonantexchangeof excitationis sufficiently slow. The totalHamiltonian H of the absorberplus bathplus the radiationfield will be written as

H=HM+HR+V. (1)

Here HM is the matterHamiltonian, HR is the Hamiltonian of the free radiation field and V is the

radiation-matterinteraction,

~ ~ (la)

HR=~wka~ak, (ib)

S. Mukamel,Collisionalbroadeningof spectralline shapes 5

and

~ (ic)k v,”

i runs overthe internalstatesof the absorberande,, andy,, arethe energyandthe inverselifetime ofan isolatedabsorberin the i) state.HR is given by the sumover all modesk of the radiationfield withfrequencyt0k = kc. a~ and a,. arethe usualcreationandanihilation operatorsof the field. ~t is thetransitionmomentbetweenthe i and jY statesof the absorber[13]

e /2irh - -,(2)

e, m and P being the electroniccharge, mass and the momentumoperator,respectively.11 is thevolume of the system.

The HamiltonianH~will be takenin the form:

FL. = H0+ ~ U

5V’(OS — Q~)+~ N U”(Q~— Q,,~) (3)p=1 p’p=1

p,

6 5. Mukamel,Collisional broadeningof spectralline shapes

the radiationmodesanda is the collectionof bathquantumnumbers.Out of the infinite numberof theradiationfield modeswe shall normally focuson oneor two andthe restmaybe assumedto be in thevacuumstate.n will thereforebe allowed to assumeonly a few values.It is convenientto define acombinedstateof the absorberplus the field [16—18]:

Iv)~l~,n). (7)

In eachapplicationwe shall first of all haveto identify the relevantvalues of n (and Iu)). This will bemadeclear throughoutthe presentwork. We shouldreiteratethat i) areeigenstatesof H,, In) aretheeigenstatesof HR and I i’) are thereforeeigenstatesof H. + HR.

3. The description of multiphoton processesusing the tetradic~Tmatrix

In this sectionwe shall introducethe basictheoreticaltool which allows us to describeall radiativestationaryprocessesin a unified andsystematicway, namelythe tetradic(Liouville space)Y matrix. Letusfirst definetheLiouville space.In this spacewe definea ket vector A)) correspondingto an ordinaryoperatorA anda bra ((Al correspondingto its hermitianconjugateA~.The Liouville equation,

dA/dt = i[H, A] (8)

can be written in a tetradicmatrix form [57]:

dAnm!dt= ~ LnmklAkl (9)

where

L=[H, ]. (10)

Upon comparingequations(8) and (9) we have:

I. *‘.-~nmkl — nk ml ml nk

We furtherdefinea scalarproductin Liouville space

((AIB)) Tr(AtB) (12)

andatetradic“matrix element”

((AILIB)) Tr(AtLB). (13)

For eachpart of the Hamiltonian H1, H0, HM, V etc. we definea correspondingLiouvillian L1, L0, LM,

~/ etc. definedin an analogousform to equation(10), e.g.L0 = [H0, ], V = [V, ] etc. Supposewe startat time (t —~_ce) with matterplus the radiationfield which arenon interacting(V = 0).Thetotal density

S. Mukamel,Collisional broadeningofspectralline shapes 7

matrix will be

p(—o) p,~(—~)PR(°°). (14)

If V is adiabaticallyswitchedon, wehaveat steadystate:

p(t 0) = [1+ G(w = 0)V] p(_ce) (15)

where

G(w)=~~. (16)

is the tetradicGreen’sfunction. Equation(15) is the analogueof the Lippman—Schwingerequationofscatteringtheory [58].We shallbe interestedin detectingtherateof radiativechangeof an operatorA

IA)) = dIA))/dt = iVIA)) . (17)

Using equations(14)—(17) we get at steadystate:

((AIp(0))) = —i((AIY(w = 0)lp(_ce))) (18)

where

(19)

Equation(18) togetherwith (19) is the basicequationto beusedthroughoutthe presentwork. It allowsus to describethe crosssection for anymultiphotonprocessassociatedwith the Hamiltonian(1) as anappropriatematrix elementof the tetradic T matrix ~Y..9’ is the tetradicanalogueof the ordinary Tmatrix commonlyusedin scatteringtheory [58],

T(E)naV+VE+~+iV. (20)

We shall nowconsiderseveralformal propertiesof ~. Let us first of all takeA to bethe unit operator

A1=~Iv)(vI (21)

and

p(_cc)= a)(aJpa(O). (22)

Here

pa(Q) = exp(—HJk )/Tr exp(—HJkT), (23)

8 S. Mukamel,Collisional broadeningof spectralline shapes

T is thetemperature,andPais thedensitymatrixof theperturbers.In Liouville (tetradic)notationonemayrewrite eqs. (21) and (22) in the form:

(24a)

= p~Jaa)). (24b)

Upon substitutionof eqs.(23) and(24) in eq. (18) andsincethe unit operatorcommuteswith anyotheroperator,i.e.

((1IV=0, (25)

we get

((1I..9(w)palaa)) = 0. (26)

Using eqs.(26) and(24), we finally have

~ ((ivI~(w)pa~aa))= 0 (27a)

i.e.

((aal~(w)paIaa))= —~ ((~vL9(w)paIaa)). (27b)

The relationbetweenthe matrix elementsof the tetradicandordinary (dyadic) T matrix is [38,39]:

((~‘v’I~(0)Iv~))= [T~,(E~) — T~,(E~,)]5~+ 21TiIT~’~(E~,)I25(E~,— E~,). (28)

Upon substitutionof eq. (28) in eq. (27b) weget

—Im(Taa(Ea))= IT (I T~,a(Ea)I2~ — Ea)) (29)

where(...) comesfor the averagingover the bath degreesof freedom.We havethusshownthateq.(27) isnothingbut theopticaltheoremof scatteringtheory [58]rewrittenin

tetradicnotation.We are now in a position to developthe perturbativeseriesfor Y. To that end we introducethe

following Green’sfunctions:

G,(w)’ ~—L+~= —if drexp(iwr)G,(r) (30)

S. Muka,nel,Collisionalbroadeningofspectralline shapes

where

G,(r) = exp(—iL,r). (31)

Similarly we have:

O,(oi)” .~ - (32)

w — L, + l~

G1(w) = — L + ~ (33)Making useof the Dysonequation[58]weimmediatelyhavefrom eqs. (19) and(30)

= ~ + n=2 (34)

~ [VG,(w)]”~V. (35)

~T~’°(co)is to nth order in V. Since in all our applicationsthe physicalobservablesareobtainedusingevenordersof V, we shall hereafterconsider~ In the time domainwemayrewrite eq. (35) in theform:

= —i Jdt~V G,(t5) V exp(iwt1) (36a)ti t2~—2

= ( i)~”~1Jdt

1 J dt2 . . . J dt~...1exp(iwt5)x VG,(t1—t2) VG,(t2— t3) . G,(t2~2—t2~_1)VG,(t2~_1)V. (36b)

SinceL1 andL, commuteandsinceL1p(—oo)= LiIvi’)) = 0, we maywrite the matrixelementsof ~9Tintheform:

0ap i((vpI5~2’°(0)palaa))

= (j)2fl Jdti ~ dt2~- 12 dt2~1((vvIV(t1)O,(t1— t2) V(t2)

X G,(t2 t3)~ . G,(t2~_2 t2~_1)‘V(t2~_1)G5(t2~_1)Y(0)palaa)) (37)


where

V(t)= exp(iLit) V exp(—iL1t). (38)

Alternatively we mayusethe ordinary T matrix (eq. (20)) togetherwith eq. (22) andget

((v~IY(0)palaa)) =2ITi TrB{T~a(Ea)Pa T~a(Ea)ô(Ea— H)} (39)

i.e.

2n

((pv~~2i~)(O)p~Iaa))= 2iri TrB ~ (p1 T°’~(Ea)la)p~(aITt~2””~(E,)ô(Ea H)Ii’). (40)p=O

Here ~ is analogousto y(P) andmaybeevaluatedusingeqs.(34)—(36)via thesubstitutionV—~V andG,(r) —~exp(—iH,’r).The sumin eq. (40) is over all the possibilitiesof applyingp perturbationsfrom theleft and 2n — p perturbationsfrom the right to get an overall expressionto order 2n. Eq. (40) is arearrangementof eq. (37) andprovidesan alternativebookkeepingdevice.Eq. (37) is fully time ordered(t

1 > t2~ > ~ andkeepstrackof the order in time in which the variousperturbationsareapplied.Using the relation V = [V. ], eq. (37) resultsin a sum of all sequencesin time in which we can apply 2nperturbations.In eq. (40), on the otherhand, we keeptrack of the total numberof left (p) andright(2n — p) perturbationsandwe disregardtherelativeorderin time of the left andright perturbations.Ofcourse,the final result is the samewhetherwe useeq. (37) or (40).

Eq. (37) is our fundamentalequation and this entire article is devotedto its evaluationunderdifferent conditions.The detailswill be presentedin the coming sections.We shall howeversketchtheessentialsteps inherent in that evaluationhere.Let us considera commonsituationin multiphotonexperimentsin which a strong, single mode, laser field is interactingwith a multilevel system andcouplessuccessivestates,i.e.

Ia)nela,n) (41a)

Ib)nalb,n_1) (41b)

Ic)nele,n_2) (41c)

Here a) is coupledby js to Ib), Ib) to a) and Ic) etc. Thestatesa), Ib), Ic).. . thusform a “linear chain”of coupledstates.The couplingschemeis shownin fig. 1. In an n photonprocess,wego from Iaa)) toI vii)) by absorbingn photons(ii = b, c,. . . for n = 1, 2, . . . respectively).Theappropriateoperatorwhichshouldbeconsideredis Y~

260sinceit is the lowestorderin which wecan go from Iaa)) to I vii)). (We need~-(2) for a single photonprocess,y(4) for a two photonprocess,etc.) We nextaskhow manypathways

with 2n linesdo existwhich lead from Iaa))to vii)). A simple inspectionof fig. 1 showsthat for ii = b, cand d (n = 1, 2, 3) thereare 2, 6 and20 pathwaysrespectively.It mayeasilybe verified that for an nphotonprocesswe have(2n)!I(n !)2 pathways.(Wehaveto operatewith V n timesfrom the left and ntimes from the right in all possiblesequencesand this resultsin the abovecombinatorialfactor.)

S. Mukamel,Collisional broadeningof spectralline shapes 11

/ /no ab OC

/// / I /

/ 1/____ / 1/

ba bb bc/ /‘1

/ // /

/ /Ca cb CC Cd

// /

/ //

do db dC dd/

/

Fig. 1. Thepathwaysin Liouville spacewhich contributeto themultiphotoncross section(eq. (37)). Eachbond denotesaradiativecoupling ~ Foran n photon processwe need consider the (2n)!/(n!)2 pathwayswhich start from and end at (~. (v= b, c, d,... for n = 1,2,3,respectively.)The pathwaysalwayscomein complexconjugatepairsobtainedby thetransformation —~ for all thepoints in thepathway.The stateslying alongthebroken lines are“even” and appearin the projectionoperatorP (eq. (43a)) which is usedto constructthefactorizationapproximation.

Usingthe Liouville conjugationsymmetry [38,39]

Gkl,nm(0) = G~c,mn(0) (42)

we see that for each pathway there is a complex conjugatepathwayobtainedby using the trans-formation ki ~ 1k for all pointsalongthe path.Thereforewe needactuallycalculateonly (2n)!I(2(n!)2)pathways(andtaketheimaginary part).The basicprocedurefor the calculationof an n photonprocessinvolves thereforethe following steps:

(i) Identify the n + 1 relevantmatterplus radiationstatesIa), Ib), jc),.. . ii).(ii) Using fig. 1 identify the (2n)!I(2(n !)2) independentpathwayswhich leadsfrom Iaa)) to I vii)).(iii) For eachpathwayidentify the phasefactorsG, and write the appropriatecorrelationfunction

(eq. (37)).(iv) Evaluate the correlation function by one of the available methods (cluster, perturbative,

stochastic,semiclassical,etc.).In concludingthis sectionwe shall introducethe factorization approximation which is very useful in

manycases.As we seefrom eq. (37), the honestevaluationof thecrosssectionfor an n photonprocessinvolvesthe calculationof a 2n-timecorrelationfunction, followed by 2n — 1 time integrations.Both ofthesestepsarevery complicated.In manycasesthe informationcontentof the experimentis muchlessdetailedthanthe completeinformation containedin the 2n time correlationfunction. A useful simpleapproximationmay be obtainedas follows: We define the tetradicMon projectionoperators[57,59]:

P = ~ A~p~))S~((A,~I (43a)

O=i—P. (43b)

12 S Mukamel,Collisional broadeningof spectralline shapes

The set IAN)) is a partialset of systemoperatorswhich shouldbe sufficient to representp(_cs~)aswell asour final observablevii)), i.e. P~p(_cc)))= Ip(—~°))),P~uv))=

Sr,.. is the “overlap” of A~andA,~p,.in Liouville space,i.e.

S~. Tr(A~A,~pM); (43c)

p~isthebathdensityoperator.Thesimplestchoiceisp~= Pa (eq.(23))althoughmoreelaboratechoiceswillbemadelater (seeeqs.(194), (213) and(236)).A convenientchoicefor A~is to takeall “even” operators(thoselying alongthe brokenlinesin fig. 1, whichmaybe reachedvia the applicationof evenpowersof Vto Iaa))). We then makeuseof the following formally exactexpression[58]:

P~(w)P= PR(o4P(1- PG,(0)PR(~)Py’ (44a)

where

R(w) = V + VO ~ OV. (44b)w—OLQ

Thesignificanceof eqs.(44) is as follows: Insteadof evaluatingY(w),we evaluateR(w) approximately.Upon substitutionof R(w) in (44a) we get an approximationfor ~

9’~ü). A low order approximationforR(w) results in an infinite order approximationfor Y(w). Eqs. (44), therefore,correspondto a partialresummationof the perturbativeseries(34) for Y. Upon expandingR to secondorderwehave:

PR~2~(w)P= PVP+ PVOG,(w) OVP. (45)

For the choiceof P discussedpreviously(seefig. 1), wehaveOVP= VP so that

PR(2)(w)~ = p~(2)(w) P = PVG,(w)VP. (46)

We thus have:

p~(2) P(i - PG,~~y(2) Py’. (47)

If we now expandthis to 2nthorderwe get

~ ,~j(2n)~ô ~,9(2) P(PO,~ y(2) ~)n_1 (48)

or alternatively

p&r(2n) P = (PVG,VPG,P)~’PVGSVP. (49)

Thelowestorderapproximationthusgivesusanexpressionfor multiphotonprocessesin termsof productsof singlephoton line shapespj9~(2)P andPG,P (eq. (48) or (49)).

If we further invokethe Markov (impact)approximation,wereplacethe PG.P matrix by


PGS(w)P(WES_LR+if)1 (49a)

whereI’ is arelaxationmatrix which containtsT1 and T2 contributions(seeeq. (130)). Eq. (49) togetherwith (49a) is the resultobtainedusing theBloch equations.This limit will bediscussedin detaillater. Atthis stagewejustwish to emphasizethat theBloch equationsareequivalentto applyingthefactorization(49) plus impact(49a) approximationsto the exactexpression(eq. (37)).

4. Single photon processes— Absorptionline shapes

In this section we shall consider the simplest radiativeprocess,i.e. a two level system 1 = a, binteractingwith a single modeof the radiation field occupiedwith flL photonswith frequencyCOL. Weassumethat Wba — ra ~ kT so that

PM(°) = IaXaI pa(O) (50)

wherepa(O)wasdefinedin eq. (23). The relevantsystem(matterplus radiation)stateswill be

Ia)~la,n1), (51a)

Ib)nalb,flL—1), (Sib)

sothat

p(_cc)= pM(—cc) PR(—°°)= Ia) (aJpa(Q). (52)

We further introducethe definitions

4 W~Wb~ (53)

/

5L I /.LbaI. (54)

Theabsorptionline shapefrom Ia) to Ib) is given by (seeeq. (37)):

Oab(Ll) = —i ~((bblg-(2)(o) pa~aa))= 2 ((aalg-(2)(o) pajaa))IT1.~L

= ~— [((abiG,(0)pa~ab))+ ((baIG,(0)palba))] (55)

wherethe 2irjs ~ factor was introducedfor the sakeof normalization,i.e.

J0ab(4)d4~ri. (56)


SFig. 2. The two pathwaysin Liouville spacewhich contributeto theordinary (singlephoton)absorptionline shape(eq. (55)).

As is clearly seenfrom fig. 2 we needevaluateonly onepathway(plus its complexconjugatepath)inthis case.We thereforedefinethecontributionof the pathwaythat passesvia lab)) (the first term in thebracketsof eq. (55)) as:

‘ab(4) ((abl G,(0)palab)) _iJ d’r exp(—i4r— ~yr)Iab(T) 1ab(/~1)— lIab(/i) (57)

where

P~L Iab(T) Tr(Vab(T) Vba(O)Pa) (Vab(T) Vba(0)). (58)

Using eqs. (ic) and (58) we get

Iab(T) Tr[exp(iHaT) exp(—iHbT)Pa] (59)

so that

Uab(4)Iab(4). (60)

The function Iab(4) is the singlephoton line shapefunction. Its real and imaginarypartsarerelatedbythe Kramers—Kronigrelation[58]

Iab(4) = PPJd4’ ~ (61)Thereforetheabsorptionline shapeI’~bis sufficientto determineIab(Ll). Note alsothatIab(I-l) =

We shall now considervariousmethodsfor the evaluationof Iab(4).

4.1. The dressedclusterexpansion

The purposeof the clusterexpansionis to expressthe line shapein termsof the dynamicsof smallclustersof atoms (one perturber, two perturbers,etc.). We shall develophere a “dressedcluster

S. Mukamel,Collisionalbroadeningofspectralline shapes 15

expansion”DCE which retainsthe exactstatic information (i.e., n-particlestaticcorrelationfunctions)of the materialsystem[42,43]. Theusefulnessof the DCE arisessincethe line shapeis easilycalculablein termsof well knownquantities(staticcorrelationsplus few body dynamics).

We start by rearrangingIab(T) eq. (59)) in the form:

Iab(T) = (1 + fab(T)), (62)

where

fab(T) exp(iHar)exp(—iHbr) — 1. (63)

The ~ ~)operationwas definedin eq. (58). fab(T) hasthe desirablepropertythat it vanisheswhenall

particlesarefar apart.We nextpartition the Hamiltonianeq. (3) as follows:HanaHo+Ua, (Ma)

HbnaHo+Ub, (64b)

where

H0na T5+~7’,,,

(65a)

Ua~U”+ ~ U””’,p p>p’

and

Ub ~ Ui” + ~ U~’- (65b)

Let usintroducethe positiveandnegativetime orderedexponentials,as follows:

exp(—iHbr)= exp(—iHor)exp±[_iJ dr1 Ub(r1)] (66a)exp(iHa’r) = exp_[i Jdr1 Ua(Ti)] exp(iHor) (66b)

where

A(r) exp(iH~’r)A exp(—iH11r), A = (Ia, Ub (67)

exp+[_i Jdr1A(ri)] 1— i I dr1 A(ri) + (—i)2 I dr1 J dr2 A(r1) A(r2) ~ (68a)


exp [~J d~1A(Tl)] i + i J d~1A(ri) + i2 Jdr1 J dT2 A(r2)A(ri) +“~ (68b)To simplify oursubsequentmanipulationswe shall definethe timeorderingoperatorsT~and T as

follows: T~( T) which operateson a productof Ub(Ua) at different times,rearrangesthem in orderofdecreasing(increasing)timesfrom left to right, i.e.

T~Ub(r~)Ub(T~) - - Ub(’r’k) = Ub(Tl) Ub(T2) - (J~,(r,.) (69a)

T Ua(T’i) Ua(T~i) Ua(T~)= Ua(Tk) Ua(T~_i) Ua(Ti) (69b)

Tt>T2 .

Ti Tk is a permutationof r~ ‘r’~.We thushave:

1+ fab(T) I + T~T exp[i I dT1 Ua(Ti)] exp[_i Jd~1Ub(Tt)]. (70)To proceedfurtherwe shall now introducethe functions:

f~2(r)enexp[iJdrtU~’(ri)]exp[—iJdri U~’(Tl)]_ 1 (71)

where j and k run over the ab~orberas well as the perturbersj, k = s, 1,. . - N. In addition let usintroduce a third ordering operator T*. When T* operateson a product of Ua and Ub terms itrearrangesthem suchthat all Ua areto the left andall Ub areto the right. We can nowdefine:

‘t ~p—’-r~p*ab — 1 a ~ b ~

sothatN±1

i+fab(T) Tab fl [i+f~,(T)]. (73)j,k= I

Eq. (73) is an importantstartingpoint for the clusterexpansion.We now define I~(’r)as the dressedp perturberspectrum.It is calculatedfor a systemconsistingof

the absorberatom+ p perturbersin a volume (1. The “dressing” is reflectedin the fact that for thedistributionfunction we usethe reducedp + 1 particledistributionfunction of the system[60]

q5~±1(X~,X1, X2, - . . X,,) = Tr pa(Xs,X1,. - . XN) (74)p±1,

where

X,ne(Q.,,P,,). (75)

S. Mukamel.Collisional broadeningofspectralline shapes 17

Eq. (74) is valid both classically where 0~~,P,. are classicalcoordinates,andquantummechanicallywhere 0~,P,, are operators.~ shouldbe distinguishedfrom the barecluster distribution function~ definedfor a systemwith oneabsorberandp perturberswhich is commonly usedin the ordinaryclusterexpansions(OCE) of line broadening[41]:

~±1(X~,X1,. . - X,,) = exp[—$H~f~(X~,X1, - . . X,,)]/Z°’~ (76a)

where

~ X1, - . . X,,)]. (76b)

Herethe superscript(p + 1) denotesthat the systemcontainsonly p + 1 particles.We thusdefine:

I~(T)neTr{exp(iH~”~r)exp(—iH~°T)4~+’}. (77)

Usingthesedefinitions we mayrewrite eq. (77) in the form:

ç p±i

I~(r)=Trj Tab fl [1+f’~(T)]pa~. (78)j,k=I

Eq. (78) isour startingpoint for theclusterexpansion.To proceedfurtherwe shallconsiderthe seriesofp perturberspectraI~:

Jo = I (79a)

‘1= 1+(f~2(X~,Xi))nel+~~i (79b)

‘2 = 1 + [(f’~J~.çb2(X~,X1)) +(f~42(X,, X2))] + [(f~f~,çb3(X~,X1, X2))

+ (f~f~43(X~,X1, X2)) + (f’~f~43(X~,X1,X2))

+ ~ ~3(X~,X5, X2))] 1 + + ~X2. (79c)

Here 11 is the volume, 4.~+1(X~,X1,. . - X~)is the reducedp+ 1 particle distributionfunction (eq. (74))and

(f~bfab. . - çbp÷i) Tab Tr (f~bfab.. . ~p±~). (80)s,1...p

A pictorial representationof the two perturberspectrum‘2 (eq. (79c)) is displayedin fig. 3. The sixtermsin (79c) arerepresentedby diagramsshowingwhich atomsareinteracting.In eq. (79) eachtermcontainingkdifferentatomsrequiresthemto bemicroscopicallycloseandis thus0(1/fl”). It iseasyto see

18 £ Mukamel,Collisional broadeningof spectralline shapes

s~. s~

(a) (b)

ss~1so~saz:1(c) (d) (e) (f)

Fig. 3. Pictorial representationof theclusterfunction 12 (eq. (79c)) correspondingto the absorbers plus two perturbers1 and2. Diagrams (a)—(f)correspondtothesix termsof eq.(79c)respectively.Eachf” is denotedbyaline connectingatomsi andj. Diagrams(a)and(b)give2XI whereasdiagrams(c)—(f) give X2.

that the generalterm I,, will begiven by:

(81)

the kth contributionis of the (~)clustersof k perturbers.Eqs. (81) maybe invertedto yield:

X2fl(122h1+1), (82)

— oPEl — jP\ 1 .L (P\l — .— ~ t1p ku1 p—t ~~tp_~

In the thermodynamiclimit weconsiderN perturberswhereN -~ ~, 12 —+ andN/fl is finite. We thenhave:

/N\ 1 n”i I—--~—, (83)~qjfl” q!

where

n=N/fl. (84)

Upon substitutionof eq. (83) in eq. (81) we get in the thermodynamiclimit:

1ab(T)~1+~~fXq(T). (85)

The final stepin the DCE is madeby introducingthe ansatz


1ab(T)exP[~ ~Jq(r)]. (86)q=i q.

Jq maybe obtainedby equatingthe expansions(85) and(86) term by term, i.e.

(87)

Using eq. (87), Jq(T) will be expressedin termsof Xq’(T) whereq’ � q. i.e.

J1(T) = Xi, (88a)

J2(T) = X2—XL (88b)

J3(T) = X3 —3X2X1 + 2x~etc. (88c)

In concludingthis sectionwe notethe following:1. Eq. (86) with (88) is our final result for the DCE expansion.J,, are given in terms of the

p-perturbersspectrumL?. The pth order DCE is obtainedby calculatingthe spectrumof our absorberplusp-perturberseq. (77) usingthe staticdistributionfunctions(74). We thengetXi, . . - x,- J

1,. . - .1,,, areobtainedfrom Xi,.. . x,~.usingeqs. (88).

2. An alternativeclusterexpansionis theordinaryclusterexpansion(OCE). The OCEis obtainedbysimply replacingeach4,, in eq. (79) by ~, the “bare” p-particlestatic distributionfunction. Both theDCE and the OCE areformally exact,howeverthe DCE is a resummedversion of the OCE andalow-order DCE containsmany terms (i.e. static contributionsoften called “initial correlations”) [55]which appearin the OCE in higher orders.

3. The exactlysolvableAnderson—Talman[31]model is definedby taking U”” = 0in eq. (3), i.e. thebath atomsdo not interactwith eachotherbut only interactwith the absorber.For this model it is easyto showthat Jq(’r) with q � 2 vanishidentically andwe havethe exactresult:

Jab(T)~

T~= exp[n .11(T)]. (89)

4.2. Theperturbativeexpansion

We shall now develop an alternative expansion for Iab(T) this time in termsof dynamiccorrelationfunctionsof the fluid [42].The lattermay thenbe evaluatedusingkineticor hydrodynamictechniques[60,61]. Let usdefinethe perturbationresponsiblefor the line broadening:

AUnaHbHa~Us1~, (90)

where

Us~=U~”—U7’. (91)

20 S. Mukapnel,Collisional broadeningof spectralline shapes

We may thenwrite:

exp(—iHbT)= exp(—iHar)exp+[_iA JdT~U(ri)], (92)where

U(r) exp(iHar)U exp(—iHar), (93)

andwhereexp±was definedin eq. (68).Upon substitutingeq. (92) in eq. (59) we get:

Jab(T)= (exp÷[_iAJdT~ U(Ti)] Pa). (94)For the subsequentmanipulationslet usintroducethe following moments:

m~(Ti,. . - r~)na(U(r1)U(r2)~U(r~))

Tr[ U(Ti) U(r2).~U(TP)Pa. (95)

Alternatively if we introducethe Fourierrepresentation:

U(Q)= ~ exp(ikQ) Uk (96)

and

Nk =~exp(ik - Q~), (97a)

Nk = exp(ik - Q~), (97b)

wehave

m~(-ri,T2, - . - T~) ~ Uk1Uk2 - - Uk~(J~~k1(ri)N_k1(Tu)I’~k2(T2)N_k2(r2). . . P~Jk5(rP)PJ_k~(TP))klk2,.. .k~

(98)

HereN,. is the particlenumberdensityandi’Q. is the densityof a taggedparticle.We note that m1= (U) is time independent.Without loss of generality,we may include(U) ~fl CUba

i.e. usethe transformation:

Wba

4 CUba + ((J) (99a)


U—*U—(U), (99b)

so that:

m1=0. (99c)

Using eqs.(94), (95) and(99) weget:

Iab(T) = 1 + (—iA)2 Jdr~J dT

2 m2(r1, r2) + (—iA)3 Jdr~J dr

2 Jd~m3(T1,T2, r3) +--- (100)The final stepin the perturbativeexpansionis to introducethe ansatz:

Iab(T) exp[~ (_iAY1Kq(T)], (101)

Kq(T) areobtainedby expandingeq. (101)in A andcomparingtermby term with eq. (100) resultingin:

i.e. ~ (iA)” Kq(’r) log{1 + (exp+[iA JdT~U(Ti)] — i)} (102)K1=0, (103a)

K2 = I dT1 J dT2 m2(ri, r2), (103b)K3 = I dT1 J dT2 J dT3 m3(r1, r2, r3), (103c)K4 = JdTi J dr2 J dr3J dr4[m4(r5, T2, T3, T4) — m2(r1, r2) m2(r3, r4)

— m2(r1, r3) m2(r2, 74)— m2(71,74) m2(r2, 73)] - (103d)

Eq. (101) togetherwith (103) is our final resultfor the perturbativeexpansion[42].

4.3. Practicalmethodsfor the calculationofsinglephotonline shapes— An overview

We shall now evaluatethe lowest order approximationfor Iab(Ii) (eq. (57)) using the expansionsdeveloped in the previous sections, and summarizethe approximations.Both the cluster and the

22 £ Mukamel,Collisionalbroadeningof spectralline shapes

perturbativeexpansions,to lowestordermaybe written in the form:

Iab(4) —i JdT exp[i4r + i(U)T — ~(Ya + Yb)T— J dTt (T — r~)~(Ti)] (104)where

~(T) d2g(T)Id’r2. (105)

(i) TheclusterexpansionTo lowestorder (eq. (86))we have:

g(T) = —nflTr[exp(iH~r)exp(—iH~T)4~2— 1]. (106)

Here~ and~ includethe systematom andoneperturber.Denotingthe eigenstatesof Ha andHbby a) and 13) respectively,we have:

~(r) = nfl ~ w~(aIf3) (I3Ia’) (4~2)~’~exp(iw~,,,T)- (107)

To lowestorder in density(OCE)we set

(4~2)a~’~(4~°2)ac~’ P(~)8aa~ (108)

andwe get:

= nfl ~ w~0P(a)I(aI /3)12 exp(iw~T). (109)

aIS

(ii) SemiclassicalclusterespansionIf we wish to evaluate(106) usingclassicalmechanicswe first rewrite it as:

g(T) = —nfl {Tr exp+[_i I dr1 U(ri)] ~ i}. (110)Classicallywe set

U(T) = U(Q(T)IQO,P0), (lii)

which is obtainedby solving Hamilton’s equationsusingthe Ha Hamiltonianandwhere Q0 and Pt) are

the initial values of the relativecoordinatesandmomentaof the perturberandabsorber.Also,

Tr—~JdQodPo. (112)


We thenhave:

g(’r) = —nflJ dQodP0 {exp[_i Jdi-1 Usi(ri)] — i} g2(Q0)F(P0). (113)g2(Qo) is the pair distributionfunction (eq. (143))andF(P0) is the Maxwell’s distributionof momenta:

F(P0) = 4~_1/2 (2mkT)~3”2exp(—P~/2mkT). (114)

(iii) Theweakcouplinglimit

Taking eq. (101) to lowest orderwe have:= (U(r) U(0)) = ~ UkUk’(Nk(T) N_k(T)Nk’ N_k’). (115)

Eq. (115) showsthat (U(r) U(0)) includes two and threeparticle correlationfunctions. There is onelimit in which (U(r) U(0)) may furtherbe simplified. This is the Brownian particle limit in which s ismuchheavierthan p sothat we mayignore the timeevolution of Q~in (115) andset O~(T)= Q~(0).Inthis casewehave:

~ U~U,~(N_,,(r)N_k’). (116)

Using the definition of the dynamicstructurefactor [61]:

~— J dr exp(—iCUT)(Nk(T) ~ = S(k,w) ~kk’ (117)we have[42]:

~(T) = J dw ~ I U,.~2S(k,w) exp(iw’r). (118)Eq. (118) when substitutedin eqs. (104) resultsin our lowest orderperturbativeapproximationfor theline shape.The relevantinformation which enterseq. (118) is the interactionpotential U,., andS(k,w)which is well knownfrom otherexperiments(neutronscatting,light scattering,etc.).

Eqs. (115) and (118) provide a simple connection between light scatteringand line broadeningexperiments [61]. In the former, one directly probes S(k,w) since the density fluctuations areresponsiblefor the light scattering.Eq. (118) showsthat in the limit of weak line broadening(secondorder perturbation)then the line broadeningof a heavy two-level impurity in a fluid is basicallymonitoring the same function S(k,w). We haverecentlyusedeq. (115) to studythe line broadeninginfluids neartheir critical points[62].A universalline shapefunction which broadensdueto the increasein densityfluctuationsnearcriticality was obtained.


(iv) Thestatic limitThe static limit holdswhen the motionsof the perturbers(typical time scaleof ~(T), A’ duration

of a collision) aremuchslower thanthe observedline width F, i.e.

F/A>>1. (119)

Formally we get this limit by neglectingthe kinetic energytermsin the HamiltoniansHa andHb ineq. (59). We then have:

Iab(T) = (exp(iUT) Pa) = (iii [1+ Js~(T)]Pa) (120)where

f5~(T)naexp(iU~PT)_1. (120a)

In orderto expresseq. (120)in termsof well knownpropertiesof the fluid let us introducethefollowingstaticcorrelationfunctions[60]:

n2(r1, r2) N (5(r1 — Q~)ö(r2—(121)

n~+~(r1,.. - r~+~)(NN~),(~(ri— Qu) ~(r~— 02)- - - 3(~±~— Q5)).

Eq. (120) may thenbe expandedaccordingto the numberof perturbers:

Jab(T) = 1 + ~ ~((fs1fs2~ - fsp)n~+~(r~,r2,. . - r~,r5)). (122a)

Upon introducingthe ansatz:

Iab(T) = exp[~ ~ J~~(T)]~ (122b)

andcomparingterm by term with eq. (122a), weget:

J~(r)= Jdr1~. dr,,(123)

- c~ IJst ~ - . - r~,r5

where~,, arethe Ursell distributionfunctions[60]

(124a)

= n2(r1, r2) — ni(ri) n1(r2) , (124b)

S. Mukamel,Collisionalbroadeningofspectralline shapes 25

= n~— ~ (k ~ 1) n~(r1,- . - r~)~,.-,(r±i, - . - r,.). (124c)

To lowestorder, the line shapeis given by eq. (104)where:

JdT~(r — Ti) gsiai(Ti) = —12 J dO{exp[i UT] — 1} n2(Q,0). (125)No trajectoriesneedto be calculatedin this limit andthe line broadeningis purely inhomogeneous.

(v) TheMarkovian (impact) limitTheMarkovian limit is the importantcasewhen all the relevantcorrelationtimes of the perturbers

areshort comparedwith the observedbroadening.In this limit the line shapeassumesa simplealmostLorentzianform. Mathematicallythe impact limit is realizedwheneverthe typical time scaleof ~(r),A is muchshorterthan the observedline width F, ie.

F/A ~ 1. (126)

(This is the oppositeof the static limit.) Condition (126) implies that the short time behaviorof g(r) isirrelevantfor our line shape.For timesAr ~ 1 wethen have

Jdr1 (7— Ti) ~(Ti) 7 J dTi ~(Ti) — JdT r~g(r1) = (—i4’ + t)r + i~. (127)Here

—izi’+tanJdrg(T) (128a)

and

i~_=nF+iqllnaJ dTg(T), (128b)

whereg(r) is given by any of theexpressionsdevelopedpreviouslyin thissection,i.e. (107), (109), (113),(115), (118) or (125). We then have:

I F 41 — 4 -Iab(4YeXp(fl ) [.(4_4P)2+F2cosfl ~(4 4P)2+F2Slfl i~ j. (129)

where

Fab~(Ya+ Yb)+ tab. (130)

26 S. Mukaniel, Collisional broadeningof spectralline shapes

Typically ~is quitesmall [21,22,41] so that the line shape(129) is basically a Lorentzianwith a slightasymmetrycausedby the dispersiveterm (the secondterm in the brackets).In the extremecasewhen

I nHOwe have:

Iab(4) 4’4+~Fab (131)

so that

Iab(/i) = (4’ ~)2+ F~b (131a)

and

Iab(4) (4 41)2+ F~h (131b)

To summarize, in the Markovian limit the line shape is given by the parameters F, 4’ and ii” which aregiven in terms of ~(T).

If we useeqs. (118) we maywrite:

t 1rlim~IUkI2S(k,w), (132a)a,0 k

4’=PPJ dw~JUkI2S~~, (132b)

?i’PPJ ~ (132c)

and

~“ ~ (132d)

For the sakeof illustrationwehavecalculatedg(r) for acollinearcollision modelwhere:

U?(Q) = Aaexp(—OIL) (133a)

and

U~”(Q)= Ab exp(—Q/L). (133b)

The model is characterizedby two parameters,the rangeof the potential L and a dimensionless

£ Mukamel,Collisionalbroadeningofspectralline shapes 27

couplingparameterc:

c (Ab — Aa)IAa. (134)

Wehave calculated the dimensionless single particle line broadening function [63]:

J(w)=~ J d~(r)exp(—iwr) (135)wheref is the frequency of binary collinear collisions, using the quantum expression (eq. (109)) and thesemiclassicalexpression(eq. (113)). We thencalculatedthe line shapefunction (eq. (104)). In figs. 4 and5 we presentthe resultsof our calculations[63] for a two-level absorberin liquid nitrogen (T = 77°K),andin liquid helium (T = 4.2°K). For each casewevariedthe magnitudeof the interactionresponsible

I ~ 9.6,dci~o 200 400 OC: ~Q() - 0 0 200 400 600 800

8.7:0. . ~ 2~:~ i j’\C~2.- 0 200 400 600 800 - 0 200 400 600 800

I - ~4.2~:~-~’\ I I I~~~•I0 200 400 600 800 0 200 400 600 800

5.6~IO-~ I I I

0 200 400 600 ~ 0 20 30 2

:‘

0 200 400 600 800 -0.2 0 0.2 4.0 4.2w (cm

1) A (cm’)

Fig. 4. Quantum(eq. (109)) andsemiclassical(eq. (113)) calculationsfor a two level absorberin N2, for severalvaluesof the interactionstrength

parameterc (eq. (134)) [63].The temperatureT = 7’7°K, the absorber-perturberreducedmass~s= 14 amu, the rangeof the interaction potentIalL = 0.2A andthefrequencyof collinearcollisionsf= 0.8 X 1013Hz. (a)The dimensionless singleperturberspectralfunctionJ(w) (eq.(135)). (b) Theabsorptionline shape(eqs.(60) and (104)) o~b(

4)’ (l/1r)I~’b(4).


~ [~IaaI 2.5dc~2

35~2 ~ ~ ~ I I II

4b I I

~ ~ I I II 1.4 ~ C~0.1 10 200 400 600 800 I I 4 I I

8.8~i:~~‘~III C ~O.65~ 0~2 2 .9~IO~

o 200 400 600 800 o I I I ~ I Iw (cm~) -0.002 0 0.002 0.478 0.480 0.482

A (cm~)

Fig. 5. Quantum (eq. (109)) and semiclassical(eq. (113)) calculationsfor a two level absorberin He, for severalvalues of interaction strengthparametersc (eq. (134)) [63].ThetemperatureT = 4.2°K,theabsorber-perturbersreducedmass~s= 3.5amu,therangeof the interactionpotentialL= 0.2A andthefrequencyof collinearcollisionsf = 0.8 x 1013Hz. (a)The dimensionlesssingle perturberspectralfunction J(w) (eq. (135)). (b) Theabsorptionline shape(eqs.(60) and(104))o’~b(zl)= (lI1r)Iab(~).

for the line broadening by varying c (eq. (134)). The variousparameterswere chosenas foll~pws[64]:The two-level absorberwasassumedto havethe massof N2 (M1 = 28), L was takento be0.2A andthefrequency f of binary collinear collisionswas given the typicalvalueof 0.8x l0’~Hi. Figure4ashowsthequantummechanicaland classical J(w) for the two-level absorberin N2 whereasfig. 4b gives thecorrespondingline shapes crab(

4)= (1/7r)I’~b(4).Figure 5 contains the samequantitiesfor the othersolvent(He).

The following pointsshouldnow bemade:1. Thedurationof collisionA~is given by the inversespectralwidth of J(w)which is of the orderof

0.01 psecin all cases.By decreasingthe line broadeninginteractionc we aredecreasingthe observedline width and,when c is smallenough,the resultingline is narrowerthan the durationof collision A’(Markovian limit) (eq. (126)). Our calculationsfor c = 0.65 x 10_2 are in the Markovian limit andas cincreaseswe tend towards the static limit, althoughthe condition Fr,, ~- 1 is never realized for ourparameterssothat we do not reachthe static limit. In thestrong couplingcase(c = 4) the line is broadand asymmetricanddoesnot tendtowardsa Gaussianas implied by simplestochasticmodels[20].

S. Mukamel.Collisional broadeningofspectralline shapes 29

2. The line shapes all exhibit a blue shift. This arises as we are using a purely repulsive potential andtook Ab > A,. so that U is always positive. By including an attractive part in the potential we may get ared shift, as is usually the case for solvent shifts.

3. The quantum effects on J(w) and tTab(Ll) increase dramatically as we go from N2 to He since themass of the perturber becomes smaller and the temperature decreases. N2 is almost classical and thedifference between the quantum and classical results is small. For He the classical calculations of J(w)are very different from the quantum ones. Thus when studying line broadening in cryogenic liquids, it isnecessary to perform quantum Franck—Condon type calculations of the line broadening.

4. The microscopic information content of Uab(Ll) is very small in the Markovian limit (small c), sincethen0ab(4) depends essentially on J(0) (eq. (128a))and is insensitive to the details of J(w). Only for thelarger values of c, Uab(Ll) contains a significant microscopic information.

(vi) A stochasticmodel— TheGaussianrandommodulationThis simple two-parametermodeldevelopedby Kubo [20],was applied with remarkable success to

many systems.The model assumesthat the frequency CUba undergoesa stationaryGaussianprocessresultingin:

= ~2 exp(—Ar) (136a)

so that

Jdr~(r — T~)~(Ti) = [exp(—Ar)—1+ Ar].Here8 is ameasureof the couplingstrengthresponsiblefor the line broadening(the amplitudeof thefrequencyfluctuations)andA_i is a correlationtime which measuresthe typical timescaleof ,~(r)(i.e.the duration of a collision). The nature of the line shape function I’,.’b(zl) is dominated by thedimensionlessparameterK,

K = A/o. (136c)

When K ~ 1 wearein theMarkovian (impact) limit,

exp[_J dr1 (r — Ti) ~(Ti)] exp(— ~ r), (K ~- 1) (137)

and

Iab(Ii) = TabI(42+ F~b), (137a)

wherey = Ya+ Yb and

Fab = ~Y+ 82/A (13Th)

30 5. Mukamel,Collisionalbroadeningof spectralline shapes

(see eqs. (131)). Whereaswhen K 4 1 we arein the static limit,

exp[_J dr1 (r — r1) ~(ri)] = exp(—~82r2), (K 4 1) (138)

and the line shape assumes the Voigt profile

Iab(Ii) = - 6Jd4’ (4 — 4P)2 + ~72~~(~2) (139)which when v ~ 8, becomes a Gaussian:

Iab(4) J~2exP(—~-2). (140)

In fig. 6 we displayafew calculationsof the line shapefunction (eq. (104) togetherwith eq. (136)). Thisline shapewill be extensivelyusedlater in our numericalcalculations.

In concludingthis sectionwe shall makea few commentsregardingthe momentsof the line shapefunction (104). The spectralmomentsof the line shapefunction 1b(

4) providea convenientmeansforevaluatingthe accuracyof variousapproximationsandarewidely usedin comparingexperimentalandtheoreticalresults.The kth momentis definedas:

Mk=~!~J d4I’~b(4)4”~, k=0,1 (141)We shall nowwrite explicit expressionsfor the variousmoments.Using eqs. (101) and(103) andwell

~

Fig. 6. The stochasticGaussianmodulation line shapefunction of Kubo (eq. (104) togetherwith (136)) 120], for y = 0.


knownpropertiesof Fourier transforms:

(142a)

M1=0, (142b)

and

Mk = (U’~), k � 2, (142c)

we now introduce the definition of another type of static correlationfunction of the fluid [60]

g~÷1(r1,r2, - - - r) 12 n~±i(r1,r2,. - - ~ 0) (143)

notethat g~+1aredimensionless.Making use of the definition (143) we have:

M2 = n Jdr U2(r) g2(r) + n2Jdr1 dr2 U(r1) U(r2) g3(r1, r2) - (144)In generalM~will dependon all the q-body staticcorrelationfunctionsgq with q � p + 1 (p perturbersplus the absorberatom).

5. Two-photonprocesses

The simplestmultiphotonprocessis a two-photonprocessin a three-levelsystem[65—74].We shallconsiderheretwo modesof theradiationfield (with frequenciesCUL andw5), andtreatin aunified way aRamanprocess(fig. 7a) anda two-photonabsorption(TPA) (fig. Th). In a Ramanprocessaphotonisabsorbedfrom the

10L modeandan w5 photonis beingscattered.The relevanteigenvectorsof H. + HR

within the rotatingwaveapproximation(the “system” states)aretherefore:

Ia) Ia, flL, 0) (145a)

Ib)anlb, flL~1,0) (145b)

flL~1,1) (145c)

with the eigenvalues:

E,.’~e,.+wL, (146a)

Ebeb, (146b)


— Ic>

I I ~>

(A) (B)Fig. 7. Two-photonprocessesin athreelevel system(Ia), Ib) andIc)) interactingwith two modesof theradiationfield with frequenciesWL and cu,.The correspondingfrequencydetuningsaredenotedby ‘iL and ~ (a)A resonanceRamanprocess.(b) Two-photonabsorption.

and

E,,=r,,+w~. (146c)

In a TPA experimentonephoton is being absorbedfrom eachof the modes.The relevant“system”states are therefore:

Ia) a, nL, n5), (147a)

Ib) an Ib, nL— 1, n5), (14Th)

Ic) an le, nL—1, fls 1), (147c)

with eigenvalues:

EaEa+WL+CUs, (148a)

Eb = ~b + (Us, (148b)

and

E,,=s,,. (148c)

Weshall define the detuning parameters (fig. 7)

4LEaEb’10L(Oba (149a)

4S=EC_Eb={:sb_~~ (Ran) (149b)


The different definition of 4,. (eq. (149b)) is the only differencebetweenthe theoriesof Raman andTPA. Fromnowon we shall considerRamanspectrabut all the resultsof this sectionapply to TPA aswell.

For a two-photon processwe will evaluate ~ (eqs. (36) and (37)) to fourth order and thecorrespondingline shapefunction is givenby:

I(4L, 4,.) an—i((ccIY~4~(w= 0)palaa))= (—i)4 J dt~f dt2 J dt3x ((cci CV(ti) G,.(t~— t2) 1I(t2) G~(t2— t3) V(t3) G,.(t3)


The subscript s signifies that the tracein eachmatrix elementis taken over the system only and theentireproductshould then be averagedover the bath via the operationTrB(. - - Pa)- At this stagewewish to evaluatethe matrix elementsof eqs. (152). To that endwe recall that eachtime the left (right)index changesin the matrix element((i.~~~ilIIVIp~t))it meansthat the operatorV actedfrom left (right).If we focus on pathway~ note that V(0) and V(t1) operatefrom right whereasV(t2) and 7/(t3)operatefrom the left. We thushave:

(j’1= exp[- - -] TrB [V,,b(t2)Vba(t3)Pa(~°)Vab(0)Vb~(tI)]= exp[~ - ] (Vab(0) Vb,,(tl) VCb(t2) Vba(t3)) -

(153a)

In the secondequalit in (153a wehaveusedthecyclic permutationinvarianceof thetrace.Similarlywe have for pathways II and III

= exp[- - .] TrB[ VCb(tl) Vba(t3) Pa(°°)Vab(0)VbC(r2)]

= exp[- - ] (Vab(O) Vb,,(t2) V,,b(tl) Vba(t3)) (153b)

~fii~= exp[- - -] TrB[ VCb(tI) Vba(t2)pa(_0~) Vab(0)VbC(ts)]= exp[- - ] (Vab(0) Vb,,(t3) VCb(tl) Vba(t2)) (153c)

wherethe exp[~] factors are the sameas in eqs. (152). Eqs. (153) show that the various pathwaysresult in essentiallythe samefour point correlation function with a simple permutationof the timearguments.Using eqs. (150)—(153)we thus get:

1(44,.)= ~ J dt~J dt2 J dt3[~(t~,t2, t3) F(11, t2, t3)+ çb(t2, t1, t3) F(t2, t1, t3) + ~(t3, t~, t~) F(t3, ti, ta)] + CC. , (154)

where

/.LLI/2abI, /.L,.—I/2b,,I,

~(t1, t2, 13)= exp[—i4Lt3 — ii,.(ti — t2) — ~Yat3— ~Yb(ti + t2 — t3) b’,,Iti — t2I] , (155)

1a2~a,.2F(t

1,t2, tl)an(Vab(0)Vb,,(tI) V,,b(t2) Vba(t3))

TrB[Vab(0) Vbc(tI) V,,b(t2) Vba(t3)Pa] , (156)

A(r) anexp(iHtr)A exp(—iHir). (157)

The significanceof eq. (154) is clear.As is obviousfrom fig. 8, the variouspathwaysareformedby thesuccessiveapplicationof two perturbationsfrom left and two from the right. The various pathwayssimply differ in the order in time in which thesefour perturbationsareapplied. Eq. (154) is nothingbuta sum of all the possiblesequencesin time which leadfrom Iaa)) to cc)). Figure8 is thereforea verycompactbookkeepingdeviceand is equivalentto the Feynmandiagramsusedin othermethods[9].


In concludingthis subsectionwe note that eqs. (154)—(156)expressthe two-photoncrosssection intermsof the four time correlation function F(t~,t2, t3). Upon substitutionof eq. (ic) in eq. (156) wefinally get:

F(t1, 12, 13) = (exp(iHbtl)exp(—iH,,t1)exp(iH,,t2)exp(—iHbt2)exp(iHbt3)exp(—iHat3)). (158)

Therest of this sectionwill be devotedto varioustechniquesfor the evaluationof (158).

5.1. Thedressedclusterexpansion

The dressedclusterexpansionfor the four-point correlationfunction F(t1, t2, t3) proceedsalongthesamelines developedin section4 for the two-pointcorrelationfunction.The basicgoal is, as before,touseexactstatic correlationsplus few body dynamicstowardsthe evaluationof F. In analogywith eq.(63) we define:

fi,,,(ti) anexp(iHbtl)exp(—iH,,ti)— 1, (159a)

fcb(t2) anexp(iH,,t2)exp(—iHbt2)— 1, (159b)

and

fba(t3) anexp(iHbt3)exp(—iHat3)— 1, (159c)

sothat

F(t1, t2, t3) = ([1 +f1,,,(t1)] [1+fcb(t2)] [1+fba(t3)]) - (160)

The operatorsf havethe desirablepropertythat theyvanishwhenall atomsarefar apart.In completeanalogywith eq. (73) wemaynowwrite:

1 +fbC(r) an tb,, [I [1+f~(r)] (161a)j.k

1+fCb(T)an i’,,bfl[1+f~(r)] (161b)tm

1+fba(T)an Tbafl[1+fb:(T)] (161c)

wherej, k, 1, m, u, v run overall particles(absorberplus perturbers).We thushave:

F(t1, t2~t3) = TbCTCbTb,.flfl {I([1 +fi!c(ti)] [1+f~(t2)][1+f~(t3)]). (162)jk im

Eq. (162) (analogousto eq. (73)) is a key stepin the clusterexpansionandit allows usto rearrangetheproductsin the r.h.s. at will without havingto worry aboutcommutatorssincethe T operatorsin front


will finally take careof the appropriatetime ordering. The dressedclusterexpansionnow proceedsasin section4.1. We first considera systemconsistingof the absorberplus p “dressed”perturbersin avolume 0. The correspondingF will be denoted F~(t1,12, 13). For p = 1 (absorber plus a singleperturber)we have,usingeq. (162):

F1(t1, 12, 13)= 1 + xi(ti, t2, 13) (163)

where

Xi(ti, t2, t3) an(1[(f,,(t1) ~2(XS,X1))+ (f~,(t2)4i2(X~,X1))+ (f~(t3)4)2(XS,X1))

+ (f~(ti)f~~,(t2)42(X,.,Xi)) + (f~(ti)f~,(t3)~2(X,.,Xi))

+ (f~(t2)f~(t3)çb2(X,., Xi)) + (f c(ti)f~c~(t2)f~j’a(t3)4~2(X,.,X1))] - (164)

As in section4.1, the “dressing” arisessincewe areusingthe reducedtwo-particlestatic distributionfunction, 42 (eq. (74)) ratherthan4~(eq. (76)) which correspondto a clusterof two particles.Since allfactors f vanish unless (Q. — Q1) is microscopicallysmall, theneach(- - - -) factoris 0(1/fl) so that Xi is0(1). (This is the reason for the 12 factor in eq. (164).) Turning now to the two-perturberspectrum(p = 2) we havein a si; ~rmanner:

F2(t1, t2, t3) 1+~~i(ti,t2, t3)+~X2(t1, (2, t3) - (165)

X2 containsall the termswherethe absorberandthe two perturbersarecloseandthe staticdistributionfunction which enters is 43(X,., X1, X2). For a generalnumberof perturberswe have,as a naturalgeneralizationof eqs. (79) and(81):

Fp(ti, ~2, 13) = I + ~ ~ (J~)Xq(ti, t2, (3) (166)

where Xq depends on the reducedq + 1 particlesstaticdistribution function ~~±i(x,., Xi xq).In completeanalogywith eqs.(82) we nowhave:

Xi(ti, t2, 13) = fl[F1(t1, 12, t3) 1] (167a)

X2(ti, (2, t3) = fl2 [F2(11, t2, t~) — 2F1(t~,(2, (3)+ 1] (167b)

etc.

In the thermodynamiclimit, in analogywith eq. (85) we get:

F(t1, 12, t3) = 1 + :~.::.~~Xq(ti, (2, t~). (168)

The final stepin our expansionwill now be made.The expansion(168)will be resummedby introducing


the ansatz:

F(t1, t2, t3)anexp[~.~!._J(tit2, t3)] - (169)

Jq maybe obtainedby comparingthe expansions(168) and (169)term by term, i.e.

~~_1Jq(ti,t2, t3)’ log[1+~~-~xq(ti,Z’2, 13)] - (170)

Using eq. (170) Jq will beexpressedin termsof x~’whereq’ � q; i.e.

J1(t1, (2, (3) = xi(t~,t2, t3) (171a)

J2(t1, t2, t~) = X2(ti, (2, 13)— x~(ti,t2, t3) (171b)

etc.Eqs.(169) and(171) areanalogousto eqs. (86) and(88).In concludingthis sectionwe notethe following:1. Eq. (169) togetherwith (171) is our final resultfor the clusterexpansionof two photonprocesses.

We may actuallyusetwo different clusterexpansions:the DCE andthe OCE which use~ (eq.(74))or ~ (eq. (76)) respectivelyas the staticdistributionfunctionsin the calculationsof the xv’s. A loworderDCE includesmany-bodystaticcorrelationsof the fluid which appearin the OCE only in higherorders.

2. J~is given in termsof the p-perturbersfunctionsF,, (eq. (166)). The pth order is obtainedbycalculating the spectrumof our absorberplus p-perturberseq. (166) using the static distributionfunctions(74) or (76). We thenget Xi,. . . i~- L,. . - J,,, areobtainedfrom Xi,. . - X~using eqs.(171).

3. The exactly solvableAnderson—Talmanmodel is defined by taking U” = 0 in eq. (3), i.e. weassumethat the bathatomsdo not interactwith eachotherbut only interactwith the absorber.For thismodel it is easyto show that Jq(T) with q � 2 vanish identically andwe havethe exact result:

F~T)(t1, t2, (3) = exp[n J1(t1, t2, t~)]- (172)

Eq. (172) is analogousto eq. (89).

5.2. Theperturbativeexpansion

The clusterexpansionexpressesthe two-photoncrosssectionin termsof staticdistributionfunctionsof the fluid ~ or 4~÷~)and the dynamicsof a few particles(singleperturber,two perturbers,etc.).An alternativetypeof expansionis perturbativein the interactionresponsiblefor the line broadening,which enablesus to usestandarddynamicalcorrelationfunctions as ourinput informationfor the linebroadening.The perturbativeexpansionis not uniqueand thereare vaious waysfor carrying it outdependingon the choiceof a zeroorder Hamiltonian (Ha, Hb or H,,) [75,44]. In this sectionwe chooseHb as our zeroorderHamiltonianandwrite:


exp(iHbtl)exp(—iH,,11)= exp±(—iJ dr UCb(r)) (173a)

exp(iH,,t2)exp(—iHbt2)= exp~(iJ dr uCb(r)) (173b)

exp(iHbt3)exp(—iHat3)= exp±(—iJ dT Uab(T)) (173c)where

U,,b(T)= exp(iHbT) (U,, — Ub) exp(—iHbr) (174a)

and

Uab(T)= exp(iHbr)(U,.— Ub) exp(—iHbr). (174b)

Upon substitutingeqs. (173) in eq. (158)we get:

F(t~, t2~ t3) = (exp±(—iAJdr uCb(r)) exp~(iAJ dr UCb(r)) exp+(_iA J dr Uba(T))). (175)The final stepin the perturbativeexpansionis achievedby introducing the ansatz (see eq. (101)):

F(t1, 12, 13) = exp[± ~ Kq(ti, t2, (3)] - (176)

The varioustermsKq maybe obtainedby expandingequations(175) and(176) in a powerseriesin Aandcomparingtermby term.We thus get:

K1(t1, ~2, (3) = J dr (U,,b(r)) — J dr (UCb(r))+ J dr (Uba(T)), etc. (177)More explicit expressionswill bepresentedin the nextsectionwherewe shallcarry out thisexpansionina slightly different way.

53. Practical methodsfor the calculationof two-photonline shapes— An overview

We shall now summarizeour resultsfor two-photonprocessesandconsidervariouspracticalwaysforthe evaluationof I(4~,4,.). As is evident from eq. (154), the problem boils down to the evaluationofF(t1, (2, 13) (eq. (158)).


(i) First order clusterexpansionUsing eq. (169) to lowestorder,we have:

F(t1, t2, 13) = exp[n Xi(ti, t2, (3)]. (178)

In order to evaluateXi(ti, ~2,t3) (eq. (164))we needconsideroneabsorberplus asingle perturber.Uponeliminating the centerof mass motion, Ha, Hb and H,, becomesingle particle Hamiltonianswitheigenstatesaa), Ib/3) andIcy) andeigenvaluesEaa, Eb,s andE,,,, respectively, i.e.

Halaa) = Eaalaa) (179a)

HbIbf3)= Eb~Ib/3) (179b)

H,,Icy) = E,,~Icy). (179c)

Eqs. (159) togetherwith (179) result in:

fbc(ti) = ~ bf3) (bf~Icy)[exp(iw~7t1)— 1] (cyl , (180a)137

fcb(t2) = ~ Ic-r) (cyIbf3) [exp(iw,,,,t2)— 1] (bf~t, (180b)ISi’

fba(t3) = ~ b13)(b$Iaa)[exp(iwsat3)— 1] (aal , (180c)a~3

~=~Iaa)P(a!)(aaI. (181)

By inspecting eq. (164) we notethat x~consists of three terms containing a single f factor, threetermswith two f factorsand one term with threef factors.We shall denotetheseterms W1, W2 and W3respectively. Using eqs. (164), (180) and (181) we finally get:

X1(ti, t2, t3) W~+W2+ W3 (182)

where

W1 = 11 ~ P(a)(ao~Ib$)(b/3Icy) (cylaa) [exp(iw137t1)— 1]“$7

+ 11 ~ P(a)(aa~Icy) (cyjb$) (b$Iaa) [exp(iw713t2)— 1]“$7

+ 12 ~ P(a) (aaIbf3)I2 [exp(iw~t

3)—1] (182a)

40 S. Mukamel.Collisionalbroadeningof spectralline shapes

W2= (1 ~ P(a)(aa1b13) (bf3Icy)(cy~b/3’)(b/3’Iaa)“1313,7

x {[exp(iw137ti) — 1] [exp(iw713.12)— 1] + [exp(iw07t~)— 1] [exp(iw13,,t3)— 1]}

+ 12 ~ P(a)(aalcy)(cyIb/~)(bf3Iaa)[exp(iw713t2)— 1] [exp(iw01,13)— 1] (182b)laTh’

W3 I? ~ P(a)(aaIbI3)(bI3Icy)(cyIbf3’)(b/3’Iaa)“$713,

x{[exp(iw1371)— 1] [exp(iw713.2)—1] [exp(iw13713)—1]}. (182c)

Eqs. (182) clearlydefinethe preciseinformation contentof two-photonspectrain the binary collisionapproximation.Togetherwith eqs. (178) and (154) theyyield a closedexpressionfor the spectrum.Wenotethat a two-photoncrosssectiondependson variousmultitime quantitieswhich do not appearinthe theory of singlephotonprocesses.Thus the threeline shapefunctionsI’~b(zl),I~,,(4)and I’~,,(z.l)donot containall the informationthat enterseqs. (182).Eq. (178)togetherwith (182) is a generalizationofarecentstudyof the samemodelsystem[74]in which severalperturbativeassumptionswere made,andis the exactexpressionto lowest orderin densityn.

(ii) ThesemiclassicalclusterexpansionEq. (175) can alsobe usedto developasemiclassicalnon-perturbativeapproximationfor F. It should

be notedthat F(t1, ~2,(3) is not an analyticfunctionof h so that its semiclassicallimit is not unique[75].In eq. (175), we choseHb asour zero-orderHamiltonian andthereforeif we useit as the startingpointfor our expansionwe get an approximationthat involves the calculationof classicaltrajectoriesusingthe Ub potential.Alternatively we could haveusedH,. or H,, to be ourzero-orderHamiltonian andthefinal semiclassicalresultwill be different. A detailedanalysisof thisnon-uniquenesswas recentlymade[75]. Comingbackto eq. (175), the semiclassicallimit is achievedusingthe following substitutions(seesection4.3):

Up,a(T)~tUp,a(Q(T)1P0,Qo), tiji = cb, ba (183a)

exp±—*exp, (183b)

and

TrB —~ JdP0dO0 (183c)where U~,,.,(Q(r)IPo,Q~)is the value of U,,cJQ) at 0 = 0(r). 0(T) is obtainedby solving a classicaltrajectoryon theHb Hamiltonianwith theinitial conditions0 = 00 andP = P0. Uponsubstitutionof eqs.(183) in eq. (175) we finally get:

Xi((t, (2, (3) = n{J JdP0dO0exp[_i J dr U,,b(Q(r)lPct, 00)+ i JdTU,,b(Q(r)IP~, Q~)- J dTUba(Q(T)IPO,Q~)]~2(P0,Q~)-1 }. (1~)


(iii) Thestatic limitThe static limit is obtainedby neglectingthe kinetic energytermsin the HamiltoniansHa, Hb, H,, in

eq. (175). We thenget an expansionanalogousto eqs. (122)—(124).To lowestorderwe have:

xi(ti, t2~ t3) = ~i{JJ dPodO0exp[i UCb(Qo)(12— t1) — i Uba(Qo) t3] 4s2(P0,Q~)— i} - (185)

The evaluationof Xi involvesthusmerelyastaticaverageover configurationsandno trajectoriesneedto be calculatedin this case.

(iv) Theweakcoupling limitThe weakcoupling limit may be obtainedby expandingK(t1, t2, (3) (eq. (176)) to secondorder in A.

However,as we havealreadymentionedin section5.2, this limit dependson ourchoiceof azero-orderHamiltonian.For the sakeof simplicity, weshall focushereon the specialcasewhereHa = H,, whichimpliesthat thestatesa) andIc) haveidenticalinteractionswith the bath.This is a commonsituationinmolecularRaman spectrawhere a) and Ic) are usually different vibrational statesof the groundelectronicstatewhereasIb) belongsto an electronicallyexcitedstate.This choicealsoeliminatessomeof theambiguityin thechoiceof thezero-orderHamiltonianandsimplifiesthefinal result.We thusdefine

AUHaHbHcHb; (186)

with thischoiceit is moreconvenientto chooseHa asourzero-orderHamiltonian.We thereforerewriteeq. (158) in the form:

F(t1, t2, t3) = (exp(iA Jdr U(r)) exP+(_iAJdr U(r)) exp~(iAJ dr U(T))), (187)where

U(r) an exp(iHar)U exp(—iHar). (188)

Without lossof generalitywe mayalso set

(U)=Tr~(Uçb2)=0, (189)

this can alwaysbe doneby including (U) in CUl,a andusingthe transformation

wba—* cob,.+(U) (190a)

U—~U—(U). (190b)

Theweakcouplinglimit is now obtainedby introducingthe ansatz(eq. (176)) expandingeqs. (187)and(176) in A andcomparingorderby order.Eq. (189) implies

K1=0 (191)

42 5. Mukamel.Collisionalbroadeningof spectralline shapes

so that thelowest order thatcontributesto eq. (176)is the secondorder.After a little bit of algebra,wefinally get [48]

F(t1, t2, 13) = exp{—A

2[g(t1)+ g(t3) + g(t2 — t~) + g(t3 — t2) — g(t2) — g(t3 — t~)]} (192)

where

g(r) = Jdr~(r - T~)(U(Ti) U(0)) = Jdr~J dr2 (U(r1) U(r2)). (193)Wenotethat in this case(unlike the moregeneralexpansioneqs.(182)), the informationcontentof thetwo-photonexperimentis identical to that containedin the single photon line shapeeq. (104), i.e. theline broadeningfunctiong(r). Severalwaysfor theevaluationof g(T) werealreadydescribedin section4 (quantum,semiclassical,hydrodynamic,etc.). A stochasticmodel was also developedrecently byTakagaharaHanamuraandKubo [71].Their final result was a specialcaseof eq. (192) with g(T) givenby eq. (136b).

(v) ThefactorizationapproximationThe factorization approximation (eq. (48)) developed in section 3 simplifies considerably the

evaluationof the Raman spectrum.We note that, in general,the evaluationof F(t1, t2, 13) is quitecomplicated.Moreover, even if we haveF, we still needto performthe triple integral eq. (154). Thefactorizationapproximationenablesus to expressthespectrumusingthesinglephotonline shapesI’~b,I~andI’~,,,.To that endlet us introducethe projectionoperator:

P = palaa))((aal+ pblbb))((bbl + ~(pa+ p~)[lac))((ad + ca)) ((cal] (194)

wherewedefine

p,. = exp(—H,,IkT)ITrexp(—HJkT), = a,b, c. (195)

Using eq. (194) andeq. (48), togetherwith figure 8, weget:

((ccIPYPIaa))= ((ccIP~2)PIbb))((bblPG,.PIbb))((bblP~2)PIaa))

+ ((ccIP5~2)PIac))((acIPG~PIac))((adIP~2)PIaa))

+ ((cdIPY(2)PIca))((calPG,.PIca))((caIPY(2)PIaa)). (196)

Eq. (196) togetherwith (150) immediatelyresultsin:

1(tiL, 4,.) = 2~s~a Iab(4L) I~(4,.)+ Im[I~(4L— LI,.) 1Cb(4,.) Iab(LIL)] }. (197)(vi) The impact(Markovian) limit

Theimpactlimit for the single photon line shapeswas discussedin detailin section4.3. Basicallyit

£ Mukamel.Collisional broadeningofspectralline shapes 43

results in

~ p=ab,cb,ac (198)

where f~laare given by eq. (130). Eq. (197) togetherwith (198) is the commonresult that is obtainedusingthe Bloch equations[29].

Upon the substitutionof eqs. (198) in eq. (197) we finally get:

I(LIL, LI,.) = ~ f.7r8(LIs — LIL) + ~1’bc~} (199)

wherefor the sakeof simplicity wehavealsotaken

YaYc°- (200)

We recallthat in thiscase

Tab = ~Yb+ tab (201a)

Tb,, ‘~Yb+tac. (201b)

We haveperformedsomenumericalcalculationsof eq. (197) whereI~,.1(zl)are given by eq. (104) with

g(r) denotedas g,,,,.(r) [76,77]. We have further assumedthat levels a) and Ic) have identicalinteractionswith the bath. This applies for instanceto molecularspectrawhen a) and Ic) aredifferentvibrationallevelsof thegroundelectronicstateandIb) belongsto an excitedelectronicstate.As aresultU,. = U,, in eq. (3) andweget:

g(r) an gab(T) = g,,b(r) (202a)

and

gac(r) = 0. (202b)

g(r) was taken from the stochasticline shapetheory of Kubo (eq. (136)), and we havecalculatedI(LIL, A~)usingK = 10 (impact)and w = 0.1 (static) line profiles for Iab(ii).

In fig. 9 we show the line profiles for both cases.At4L = 0 we see two components.A narrow

(Raman)componentat A,. = 4L and a broad (redistribution) componentat A,. = 0. The 6 and Aparametersfor the K = 10 and K = 0.1 lines are chosenso that Iab(LI) hasthe samefull width at halfmaximum. As 4L is detuned, the ratio of integratedintensities of the redistribution and Ramancomponentsdoesnot changein theimpact limit K = 10 (right column)andis equalto 2T~b/yb.This mayeasily be seen using eq. (199). In the otherextremeK = 0.1 (left column)we seehow theredistributionterm graduallydisappears.Thismaybe interpretedby sayingthat tab is actuallyfrequencydependentTab(LIL), andit vanishesatlargedetunings,wherethe impactlimit fails [24].In the presentformulationwe do not need to invoke this argumentand the vanishingof the redistribution componentarisesnaturally from the frequencydependenceof the actualline shapefunction I’~b(A).


6.1.1:2 :~ ~O 40 80

K’O.I K’IO

3.0.I~2 - I.2102 - I -

I I I _________~ 0 -40 0 40 80 -40 0 40 80

K’O.I K’IO

&-20 ~L20

8.4lO~ - 4.4lO~-

0 ~o 8~

K’O.I K’lO

~L.40

l.4l0~- I.2lO~- -

I -~ __ I __-40 0 40 80 -40 0 40 80

Fig. 9. TheresonanceRamanspectrumI(&, 4,.)(eq. (197))(divided by 2~s~jc~)[76,77j.We haveusedthestochasticline shapeof Kubo (seefig. 6)asa model for thesingle photonline shapes.g,b(r) = gcs(r)= g(r), g,,(r) = 0, y,= 0, y~= 1, y, = 0.5. The K = 10 lineswerecalculatedusing 8 = 100andA = 1000whereastheK = 0.1 lineswerecalculatedusing 8 = 8.49 andA = 0.849, so that the I~,(4) hasthesamewidth in both cases.Notethattheratio of intensitiesof the Ramanandthe redistribution termsdoesnot changewith thedetuningin the impact (K = 10) limit whereasin theK = 0.1 casetheredistributioncomponentvanishesat largedetunings.

6. Absorptionline shapesin astrongradiation field

In this section we makethe first application of the tetradic scatteringformulation towards thecalculationof a multiphotonprocess.We shall considerthesimplestexample,namelya two-levelsysteminteractingwith a single mode of a strong radiation field [78—81].Unlike the weak field spectra

S. Mukame4Collisionalbroadeningofspectralline shapes 45

consideredin sections4 and5, the T1 relaxationmechanismis playingnow an activerole in the process.In fact, the inclusionof a T1 relaxationis necessaryin order to definealine shapein a strongfield. Inthe absenceof a T1 relaxationmechanismthe populationsof the two levelswill be equalizedat steadystateandtherewill be no absorptionof radiation.The adequatedescriptionof the T1 relaxationwillnow be done as follows. We startwith the systemstateswhich are identical to thoseof section4, i.e.

Ia>=Ia,nL) (203a)

Ib) = Ib, nL— 1). (203b)

TheLiouville equationfor theentire system(absorber,plus bathplus field) is:

dpldt= —iLp (204)

whereL is theLiouville operator.

Lan[H, ]+f. (205)

Here H is given by eq. (1) (without the va and y~,terms which will be included in L). £ is the T1

relaxationmatrix given byLaa.aa = Lbb,aa = ~ (206a)

Lbb.bb = Laa.bb = iyb, (206b)

L,.b.,.b = Lba,ba = ~iy, (206c)

where

y=ya+yb. (206d)

L is assumedto arisefrom a coupling to someotherbathwith a shortcorrelationtime (e.g. spontaneousemission, inelastic collisions) and is responsible for the relaxation of our two-level system (in theabsenceof driving, ~ = 0) to thermalequilibrium, i.e.

Pb= y,./y, (207a)

Pa = ybly. (207b)

Usingeq. (27) andfig. 2, the strongfield absorptionline shapeis given by

I,.b(A) = i((aalY(0)Ip(—oo))). (208)

The initial densitymatrix p(—co) is

= Pap,.Ia)(al + Pbpblb) (bI; (209)


Pa andp~were introducedin eq. (195). In Liouville tetradicnotation,eq. (209) reads

Ip(~°)))= 1apaIa~))+ Pbpblbb)). (210)

Upon substitutingeq. (210) in (208)and recallingthat

‘VIaa)) = ~LL(Iba))— Jab)))= —l’ibb)), (211)

we get

Iab(LI) = i(~,.— ~b) ((aal.~i(0)paIaa)). (212)

At this stagewe introducethe tetradicprojectionoperator:

P = palaa))((aal+ pblbb))((bbl (213)

andthe complementaryprojection

O=i—P, (214)

in termsof which wemay write

I,.b(LI) (Pa — Pb) ((aaIPY(0) PIaa)). (215)

P~Pwill now be evaluatedusingeqs. (44). P convertsY, R and Gto 2x 2 matricesin the aa)), Ibb))space.Using eqs.(44b) and(211), it is clear that

((aaIPRPIaa))= ((bbIPRPIbb))= —((bbIPRPlaa))= —((aaIPRPIbb)) (216)

wethus have:

PR(w)P = ((aaIPR(w)PIaa))(~ (217)

similarly PG~(w)P is given by

PG,.(~)P=(°~~’ ~‘D ~= 1 - (‘°t”~ ~ ‘~hl. (218)\ —l)’b cu+lybJ w(w+iy)\ 1)’,. w+lyaJ

Multiplying eqs.(217) and(218) andletting w —~0 we get:

PG(o)PR(o)p = ((aaIPR(0)PIaa))(~ ~). (219)

Upon substitutionof eqs. (44), (217)and (219) in eq. (215)we finally get [81]:

S. Muka,nel,Collisionalbroadeningof spectralline shapes 47

1,.b(A) = (Pa — Pb) 1 + (~~(LI) (220)

where

X(A) = i((aaIPR(0)PIaa)). (221)

~(A) maybe expandedin powersof /5L resultingin:

x(’-l)= is2X12~ is4X’4~+,L6X’61+ -. (222)

sothat

I LI — p — is ~ x~2~(A)— is t X~(~)+ 223ab( ) — ( a b) 1+(2/y) [is~ X~2~(A)— is~ ~(A)+- -

To lowest orderin isL (eq. (46))

PR(2)J5 = j~y(2) 1~. (224)

Eqs. (221) togetherwith (224) yield

x~2kLI)= 2I’~b(A) (225)

if wetruncatethe expansion(222) andretainonly ,~(2)weget an expressionfor the line shapein astrongfield, expressedin termsof theordinary,weak-field, line shape,I’,~bi.e.

Iab(A) (Pa— Pb) 1+ (4/y)~~I~1,(LI) (226)

wherewe recall the normalization:

J Iab(1i)d1.1 = IT. (227)Eq. (226) with (227)is an importantresult of this sectionandenablesus to expressthe strong field

line shapeIab(/-l) in termsof the ordinaryweak-fieldprofile Iab(A). In general,however,the line shapedependsalso on ~ X~

6~etc. which contain higher order dipole correlationfunctions[81]. Thus, thenon-linearline shapecontainsmoredetailedinformation regardingthecollision thanthe ordinaryweakfield line.

We shallnow discusseq. (226)andconsidersomelimiting cases.We first notethat in weakradiationfields (to lowestorder in isL) eq. (226) reducesto:

1(A) = 2(P,.— Pb)is~.Iab(A). (228)


In the impact limit we have(eq. (131)with LI’ = 0):

Iab(/i) = LI (229)

Upon substitutionof eq. (229) in eq. (226) we finally get

IA—P ~ 2is~.Tabab( )— ( a — b) A2 + T~b+ 4is~Tai,/y (230)

This is the well known formula describinga Lorentzianline in a weak field (isL—*0) which is powerbroadenedat strong fields when isL is comparableto \/Taby. Eq. (230) may beobtaineddirectly fromthe Bloch equationsand was first derived in the context of pressurebroadeningby Karplus andSchwinger[79].

Wehavecalculatedtheline shapefunction eq. (226),where I’~b(A)wasevaluatedusingeqs.(104) and(136) [81].The resultsaregiven in figs. 10 and 11 for various valuesof theRabi frequencyisL- Figure10 showsthe impact line shape(K = 10) andfig. 11 showsa non-impactcase(K = 1).

As is clearly seen from fig. 10 I,.b(zI) remainsa Lorentzian in this caseevenfor strong saturatingfields isL- In fig. 11, on the otherhand,the line shapechangesits form significantly as isL increases.Itmay be shownthat in the impactlimit ~ = X~6~= - - = 0 andeq. (226) becomesexact [81].

.0.1: 1II~1a0221~2~

2.4 .I:3 JIL’lO

-100 0 00 -50 0 50

Fig. 10. Thesaturationbehavior of the absorptionline shape[811(eq.(226)togetherwith (104) and(136))in theMarkovian limit K = 10. Thefrequencyscaleis setby taking v = 1. In theseunits we have B = 100, A = 1000.


I 61:’ 2410~ I

2.4I:~~~~1I0102 JFig. 11. Sameasfig. 10 but in a non.impactcaseK = 1 (8 = 10, A = 10).

7. ResonanceRamanspectrain a strongradiationfield

We shall now consideramorecomplicatedexampleof a multiphoton process:A threelevel systeminteractingwith a strong radiationfield [76,77,82—88]. The modelsystemis essentiallyidenticalto thatof section5 andwill againapply to Ramanspectraandto two-photonabsorptionas well (seefig. 7 andfig. 8). However there are two differences:We are looking for a solution of ((cc19(0)Iaa))which isnon-perturbative(infinite order) in isL but still to secondorder in ~ Secondly,we shall introduceamore elaborateT1 relaxationmatrix in which the three-level systemis closedunder relaxationandrelaxesto level a) in the absenceof driving (isL = 0). As wasalreadymentionedin the previoussection,the T1 mechanismis playingan importantrole in a strongfield andour resultdependson thedetails ofthe T1 relaxationmechanism.The time evolution of the entire (absorberplus bath plus laserandscatteredfield modes)densitymatrix p is given by the Liouville equation:

dp/dt—iLp, (231)

where

Lan[H, ]+!anL0+L (232)

is the Liouville operator.Here H is the Hamiltonian (eq. (1), with y,. = y~= = 0) and£ is the T1relaxationmatrix. We assumethat thethree-levelsystemis closedunderrelaxationwhich proceedsonlydownwardson the energyscale,i.e. L is a 3 X 3 matrix in the spaceof level populations,

50 S. Mukamel,Collisionalbroadeningofspectralline shapes

£ = -i (i~: -~:~) (233)wherethe rowsand columnsarelabelledin decreasingorderon the energyscale(bb, cc andaa).

The level relaxationrates(T1) for the levels Ib) and c) are given by y~,and Ye respectively,andweassumeYa = 0 (Ia) being the groundstateof the system).The b—* c andb—*a crossrelaxationratesycband vab, satisfy

ycb±yab= yb. (234)

We assumethat initially (at t—* —ce) the bath andthesystemareuncorrelatedandp(—co) is given by eq.

(52). Using eq. (37) andfig. 8, the Ramanspectrumfor arbitraryfield strengthis given byI(A,., AL) = _i((ccI~(0)Ip(—x))). (235)

We now introducethe following tetradicprojectionoperator:

P = palaa))((aal+ pblbb))((bbl + p,,Icc)) ((ccl + ~(pa + Pc) [Ica))((cal+ lac)) ((ad] (236a)

andits complementaryprojectionoperator

O=i+P. (236b)

The spectrumin eq. (235) is given by matrix elementsof P.~T(o)P, i.e.

I(AL, A,.) = —i((ccIPY(0) PIaa)). (237)

The P projectionoperator(eq. (236a))converts.9 into a 5 x5 matrix in thespace(Iaa)),Ibb), icc)), ac))andIca))). This matrix will be evaluatedusingeqs. (44). We havethusto find the 5 x5 PRP matrix.

We recallthat sincewe areusinga weakprobe(iss is small) we needto evaluateP~P(andPRP) tosecondorder in iss andto infinite orderin isL- Thereforesomeof the matrix elementsof R areto zeroorder in iss, othersare first order in iss andthe rest aresecondorder in iss. We shall denotethesebyR~°

1,R~’~andR~2~respectively.After somestraightforwardalgebraicmanipulationswe may rewrite eq. (237) in the form [76,77]:

I(AL, A,.)= — (2Iy~)xi(AL) ~ LI,.) — ~-xi(AL) [x6(AL, A,.)+ X7(AL, A,.)]

+ Im[ x3(AL, A,.) [x4AL~ A,.) — -~—xi(AL)[x4(AL, A,.) + X5(&, A~)]]]}[Iac( L s)] X2(AL,AS)~ Yb

(238)where

= yb(1 + ~j’. (238a)


Here

xi(AL) = —i((aa!PR~°~(0)Plaa)) (239a)

X2(~L, LI,.) = ((acIPR~°’(0)PIac)) (239b)

X3(AL, A,.) = ((cdIPR~°(0)Plac)) (239c)

X4(LIL, A,.) = ((acIPR”~(0)PIaa)) (239d)

X5(’iL, A,.) = ((acIPR”~(0)PIbb)) (239e)

Xo(AL, A,.)= _i((cciPR(2)(0)PIaa)) (239f)

L, A,.)= —i((ccJPR(2)(0) Plbb)). (239g)

The functionsX~(i = 1, - . - 7) contain the interactionwith the external field to an arbitraryorder (allordersin isL). The functionsXi andX2 areto zerothorder in ~ X3, X4 andX5 areto first orderin n,.,andX6 andX7 areto secondorder in ~ They can be expandedin powersof isL as follows:

x~= ~ (i = 1,2), (240a)

= issisLX~ + /L,./2 ~ +- (j = 3~4, 5), (240b)

and

Xk = is~X~+ is~is~x~±-~(k = 6,7). (240c)

In general,the ~ (i = 1, 2,.. - 7; n = 2, 4, - - .) termsinvolve n-timecorrelationfunctionsandthereforeknowledgeof successivelyhigher-ordercorrelation functionsis neededfor the calculationof the lineshape when the field strength is increased(already in the weak-field limit, four time correlationfunctionshad to be introduced).Thus, in principle, the information contentof the PRSexperimentsincreaseswith the applied field strength. However, although formal expressionsfor the Xa’~wererecentlygiven [76,77], their evaluationin termsof n-timecorrelationfunctions(n >2) is quite tediousandwill not be discussedhere.

In the following we consideronly the contributionof the two-time correlationfunctionsX~2~to theRamanline shape.Termsinvolving X~4tandhigher ordersin the expansion(240) containintegralsovern-time correlation functions (with n � 4) resulting in increasingpowers of isiIA whereA1 is thecorrelationtime usuallyassociatedwith the durationof a single collision. Hence,if the externalfield isnot too strong,i.e. if thecondition 4 A is satisfied,wecan consideronly the ~ (i = 1, - . - 7) termsin the expansion(24.0).

In this case,the operatorPRP (eq. (44b)) can be written as

PR(w)P=P’VP+PVO .~ .~ - OvP, (241)w-QL

0Q-QLQ


and within the spacespannedby the statesIbc)), Icb)), lab)) and Iba)), ~ can be replacedby the unitoperator;wecan write

PR(~)PzfrVG,.(00)VP, (242)

usingeqs. (239) and(240) we obtain

x~2~’=—2I~b(AL)= Ib,,(A,.)

= Ib,,(A,.)

(243)= Iab(AL)

[Iat,(Ai) + Ib,,(A,.)]

and

= 2I~,,(—A,.).

Upon substitutionof eqs. (243)in eq. (238) we finally get [76,77]:

I’A A~— 2~~js~ 1-~—I”’A ~“‘A~. L, ~‘ — 1 + (4is~Iy’~)I’~b(AL) ~Yb ab~ LP cb~~s

+Im[ * —t 2 [I:b(AL)+~1~-~I’a’b(AL)ICb(A,.)]]} (244)[I,.,,(AL—A,.)I,,b(A,.)] + isL Yb

wherewerecall thatthesinglephotonline shapefunction I,.b(A) isgivenby eq. (104)and‘cb andIac arethecorrespondingline shapesfor thecb andac transitionsrespectively.

In the impactlimit (eq. (131)) wehave:

I~(A)=A+~T, ~is=ab,cb,ac. (245)

Upon substitutingeqs. (245) in eq. (244) we finally get [88b1:

IA A — 2isLis~ I f~,.A Tac)(2TabIy~ 1)+A,.+i(F,.,,+T,.b) 246( L, ~ ~ m (A,.+~Tcb)(A,.AL+~Tac)is~ ( )

In conclusion,we havederiveda formal expression(eq. (238)) for the collision broadenedRamanlineshape,that is valid for arbitrary field strengths(isL) and detunings(AL, A,.). The line shapewas thenexpandedin termsof the ~~‘° (i = 1, 2,. . - 7; n = 2, , 6,. . .) functions(eqs. (239)). The ~ functionscan

S.Mukamel,Collisional broadeningof spectralline shapes 53

be expressedin termsof Laplacetransformsof two-time correlation functionsof the transition-dipoleoperators(in the interactionpicturewith respectto Ha+ Hb+ H,,). Similarly, the ~ (n>2) functionsare related to n-time correlation functions [80,81]. We should point out that since higher-ordercorrelation functionsappearwith higher powers of the field strength (isL) and the time integrationscontributepowersof thecorrelationtime A ‘ (usuallyassociatedwith the durationof asinglecollision),the expansion(240) is actuallyan expansionin the dimensionlessparameterisL/A. We haveconsideredthe mediumfield strengthcase,definedby the condition

isLIA ~ 1 (247)

i.e., althoughthe field can be strongenoughto saturatethe ab transition(if isL � WybTab), the period

(is~1)of the associated(on-resonance)Rabioscillationis much longer thanthe collisiontime (A1) andhence, only single-photon absorption can occur during the collision with the perturber. In thisapproximationonly the lowestorder terms(x~2~)in the expansionof the xi’s contributeto the Ramanspectrumwhich is thus completelydeterminedby the two time correlationfunctions (g,.,,, gcb, gac). Asimilar result (valid only in the mediumfield limit) was recentlyobtainedfor the resonancefluorescencespectrumin atwo-levelsystem[89].The advantageof the presentapproachis that the evaluationof thecorrections(due to higher-ordercorrelation functions) to the line shape is straightforwardthoughsomewhattedious(usingeqs.(238), (239) and(240) andexpandingthem in the radiativeinteraction/LL).A furtheradvantageof the presentmethodis that sincethe interactionswith the laserandthescatteredmodes are treatedon the same footing, it does not necessarilyinvoke the fluctuation regressionapproximation[90] which is perfectly valid only in the Markovian limit. Figures 12 and 13 show thevariation of the RRSspectra(eq. (244))with thepump intensityfor resonancepumping (AL = 0) andforoff-resonancepumping (AL = 25) [76,77]. As in fig. 9, we haveusedthe stochasticline shapefunction(eqs. (104) with (136))for the singlephotonline shapeI’~b(A).The right (K = 10)column is in theimpactlimit and the left (K = 0.1) column is in the static limit. The strong field is obtainedwheneverthegeneralizedRabifrequency

0= (A~+4,i.s~”~, (248)

is the largestparameterin the problem(exceptfor the inversedurationof a collision). The spectrumseparatesin this caseinto two (non-Lorentzian)peakscenteredat

A~=~(AL±0). (249)

In the weak field limit, the A~ peakreducesto the coherentRamancomponent(aroundLI,. = AL) andtheA ~ gives the incoherentredistributioncomponent(aroundA,. = 0). Thisgradualchangeis showninfigs. 12 and 13 as we move from bottom (strong field) to the top and is commonto the impact andnon-impact lines (right and left panels). The difference between the impact and non-impact lines is inthe shapeof the two peaks.We furthernotethat the two equalcomponentswhen isL is large,aremuchnarrower in the static (K = 0.1) case. This may again be interpretedby invoking the frequencydependenceof F,,b(A) which existsat largedetuningsin K = 0.1casebut is absent,by assumption,in theimpact (K = 10) case[24].Figure 14 showsthe variation of the Ramanspectrawith the detuningof thepump field for a finite intensity isL = 20. We notethat as AL increasesweneeda strongerisL in ordertoachievethe asymptoticbehaviormentionedabove.Thus a isL = 20 field is “strong” for AL = 0 (upper


~I 00 I .~l ~ I ~I ~ I 8

______2o~ ~

_____ 8

~ (S~~l~,7)i

I I I I I I I

.o o~

i~____~1__-~//_L~—~

~~ Q ~ 0 “‘~

i2____....c~~(ø~

0 [7~~i;(0 (SV~1V)I

V LL.r~I


- KS0.I - K~I0 -

2.3.: TIJ~:~- K~0.l - K~I0 -

L~I0

~ -~ 0 80K~0.l A KalO/ \ t~20

-4 -~4.4-10 - 5.710

0 ~0 ~ -~ ~ J4~ ~- K~0.I K~I0

~L~40

5.3l~~- 5.7l~~ / \

Fig. 14. Variation of theresonanceRamanspectrumI(4L, 4,) (dividedby 2je~j~)(eq. (244))with detuning4L for a finite Rabi frequency15L = 20.

Otherparametersare thesameas in fig. 12 [76.77].

panelsin fig. 14) andthe spectraaresplit into two equalcomponents.The samefield for AL = 40 (lowerpanel)is weakand the two componentsarenot equal,neitherin their intensitynor in their width. Wealso see that the lines are narrower in the non-impact (K = 0.1) case.Finally we note that the twocomponentsare roughly equal in magnitudein the impact but a dramatic changeoccurs for thenon-impactlines wherethe redistributioncomponentdisappearswith the detuning.


8. Concluding remarks

Documents

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