Collective many-body resonances in condensed phase...

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 12 22 MARCH 2002

Collective many-body resonances in condensed phase nonlinearspectroscopy

Andreas Tortschanoff and Shaul MukamelDepartment of Chemistry, University of Rochester, Rochester, New York 14627-0216

~Received 31 July 2001; accepted 23 October 2001!

The optical response of assemblies of electronic and vibrational chromphores may show two typesof collective resonances induced by either direct short-range coupling~multiple quantum coherence!or by long-range macroscopic local field and cascading processes. Using a unified approach for bothtypes of resonances, we demonstrate how specific signatures in line shapes, phase profiles, anddensity dependence may be used to distinguish between the two. New high harmonic resonances atcombinations and multiples of optical frequencies of the single exciton transitions are predicted inthek11k22k3 four wave mixing signal for several model systems. ©2002 American Institute ofPhysics. @DOI: 10.1063/1.1427721#

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I. INTRODUCTION

The nonlinear optical response of systems with high dsity of chromophores depends on the fact that each cmophore is driven by an external local fieldEl which isdifferent from the average~Maxwell! field E. This gives riseto macroscopiclocal fieldandcascadingcontributions1,2 thatcan induce new resonances and other interesting collecmany-body effects that need to be accounted for in the inpretation of multidimensional spectroscopies.3–5 Chromo-phores with nonoverlapping charge distributions coupletwo ways:6 short-range microscopic interactions dependthe longitudinal electric field and may be described by rplacing the chromophore eigenstates by those of aggregLong-range coupling occurs via thetransverseelectric fieldwhich is generated by one group of chromophores and inacts with the others. These interactions can be describethe mean-~local-! field approach,1,7,8 where the effects of in-terparticle interactions are incorporated through an effeclocal field which is related to the external Maxwell field bthe Lorentz Formula.2 This implies that the coherent polaization generated within the sample adds to the electric fiand creates new interactions. The mean-field approximais justified for the long-range interactions where microscodetails are averaged out. Such details are included inshort-range direct interactions. Combining both contributioprovides a rigorous description of the optical response.

The connection between macroscopic susceptibiliand microscopic polarizabilities is crucial for comparincomputed polarizabilities with condensed phase measments and has drawn considerable attention since the edays of nonlinear optics. In the simplest approach, the Csius Mossotti expression for the dielectric function basedthe local-field formulation of the linear response9 has beenextended to the nonlinear response.2 This level of theory hasbeen primarily used to compare computed frequency-domoff resonant polarizabilities with bulk measurements, andcommonly used for the design of optical materials. Ttheory has been extended to the time domain using equa

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of motion for a single molecule driven by the local field8

The limitations of the local-field approximation for modelinresonant techniques were pointed out and a unified treatmof nonlinear signals that goes beyond the local-field appromation, and includes genuine many-body effects, was suquently obtained using the nonlinear exciton equationsmotion ~NEE!,10–12 which include additional dynamic variables involving few molecules. The local field approximatiis then recovered as the lowest order in a systematic hiechy when all dynamical variables are factorized into proucts of single molecule variables.

Coherent femtosecond measurements provide a dprobe for resonant transitions of coupled electronic avibrational chromophores.1,3,4,13–19Signatures of local fieldand cascading were found in femtosecond four wave mixsignals in GaAs quantum wells20–22 and in the gasphase,23–25 femtosecond fifth order Raman measurementsmolecular liquids26,27 and enhanced magnitudes of off resnant polarizabilities.10,11,28

In this paper we provide a unified treatment of the snatures ofboth types of coupling in the third order nonlinearesponse.29 We predict and analyze new high harmonic resnances originating from destroying the time ordering of tincident fields by the local field which were found iNMR30,31 and should be directly observed optically.

Third order time-resolved nonlinear spectroscopy offa variety of different techniques characterized by their pulsequence, wave-vector geometry and pulse frequencies.1,5 Ina four wave mixing experiment, three electric fields interawith the system and generate a polarization~and a signal! inthe directionsks56k16k26k3.12 To clarify the origin ofdifferent kinds of many-body resonances, we will concetrate on one technique, the reverse photon echo~RPE! withks5k11k22k3, where the first two pulses (k1 andk2) aretime coincident. However, similar effects will show upother four wave mixing techniques and can be treated usthe present approach. The RPE technique is realized ifphoton echo pulse sequence is reversed in time~hence its

7 © 2002 American Institute of Physics

AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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5008 J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 A. Tortschanoff and S. Mukamel

name!. We focus on the RPE since for independent two-lesystems at low density the RPE signal vanishes;7,21,22 thesignal thus results exclusively from many-body effects,ther short or long range, making this technique especisensitive and particularly suitable for the observation of clective resonances.

II. LOCAL FIELD AND CASCADING EFFECTS IN THENONLINEAR RESPONSE

The heterodyne signal in nonlinear spectroscopy is pportional to the induced polarization which in turn canexpanded perturbatively in the average electric field. Thenthorder nonlinear polarizationP(n) is:1

P(n)~r ,tc!5E2`

tcdtnE

2`

tndtn21 . . . E

2`

t2dt1

3R(n)~tn ;tn21 , . . . ,t1!

3E~r ,tn!E~r ,tn21! . . . E~r ,t1! ~1!

Here the Maxwell fieldE(r ,t) is the average transverselectric field which interacts with the system under invesgation at timest5t1 , . . . tn andtc is the observation timeThe nth order response functionR(n) is given by the sum ofall possible Liouville-space pathways. Due to its timordered~causal! structure,R(n) is nonzero only fort1,t2

, . . . ,tn .The local-field approximation~LFA! provides a simple

way to relate the microscopic polarizabilities of isolated mecules to the macroscopic susceptibilities.1,8,10,11,28At thislevel of theory the response of an ensemble of particlereduced to that of a single particle interacting with a lofield. In the long wavelength limit the local fieldEl is relatedto the external fieldE by the Lorentz formula2,9,29

El~ t !5E~ t !14p

3P~ t !, ~2!

whereP(t) is the polarization per unit volume. The polariztion of a single chromophore~i.e., a single molecule or anaggregate of coupled molecules! can be expanded in terms oits response functions~polarizabilities! a, b, g, . . . to vari-ous orders in the local field, and the total nonlinear polarition per unit volume of a macroscopic sample is

PNL~r ,tc!

5r0E2`

tcdt3E

2`

t3dt2E

2`

t2dt1s~tc2t3!

3b~t3 ,t2 ,t1!El~r ,t2!El~r ,t1!

1r0E2`

tcdt4E

2`

t4dt3E

2`

t3dt2E

2`

t2dt1

3s~tc2t4!g~t3 ,t2 ,t1!El~r ,t2!El~r ,t1!1 . . . , ~3!

whereP[P(1)1PNL and r0 is the molecular number density.

Both the total polarizationP and the local fieldEl can be


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expanded perturbatively in powers of the Maxwell fieldE,and we denote thejth order terms byP( j ) and El

( j ) , respec-tively. The expansion of the local field is obtained by substuting the expansion ofP in Eq. ~2!. The linear part of thelocal field El[El

(1) is connected to the Maxwell fieldE bythe Clausius Mossotti relation:

El~r ,t ![El(1)~r ,t !5E

2`

t

s~ t2t!E~r ,t!dt, ~4!

where,32

s~v!51

12 ~4p!/3r0a~v!. ~5!

The symbola is the linear response function~polariz-ability! of a single aggregate:

a~ t !5 iu~ t !(e9

me9g2 exp@~2 i«e9g2Geg!t#2c.c., ~6!

whereg denotes the ground state and the sum runs oveone-exciton statesue9& in the system. Plugging Eq.~4! in Eq.~2! we obtain

El~r ,t!5E2`

t

dt8s~t2t8!E~r ,t8!

14p

3 E2`

t

dt8s~t2t8!PNL~r ,t8!. ~7!

The first term in the r.h.s. of Eq.~7!, which correspondsto the first order termEl is responsible forlocal field effects,while the second term describes effects of the nonlinearlarization that generatecascadingcontributions. Substitutionof Eq. ~7! in Eq. ~3! yields an integral equation for the nonlinear polarizationPNL. An iterative solution of this equationresults in the expansion ofPNL in powers of the Maxwellfield E. Using these relations, we can thus express the poization order by order in the Maxwell field.

The linear polarization is given by

P(1)~r ,tc!5r0E2`

tcdt2E

2`

t2dt1

3a~tc ,t2!s~t22t1!E~r ,t1!. ~8!

The second order polarization has no cascading contributand the many-body corrections enter solely through theear first term ofEl

P(2)~r ,tc!5r0E2`

tcdt3E

2`

t3dt2E

2`

t2dt1s~tc2t3!

3b~t3 ,t2 ,t1!El~r ,t2!El~r ,t1!. ~9!

For the third order polarization, which is the focusthis article we need to calculateEl

(2) in addition toEl bysubstituting Eq.~3! into Eq. ~7! and making use of Eq.~2!.We then get


5009J. Chem. Phys., Vol. 116, No. 12, 22 March 2002 Collective many-body resonances

P(3)~r ,tc!5r0E2`

tcdt4E

2`

t4dt3E

2`

t3dt2E

2`

t2dt1s~ t2t4!g~t4 ,t3 ,t2 ,t1!El~r ,t1!El~r ,t2!El~r ,t3!

1r0E2`

tcdt3E

2`

t3dt2E

2`

t2dt1s~ t2t3!b~t3 ,t2 ,t1!@El

(2)~r ,t2!El~r ,t1!1El~r ,t2!El(2)~r ,t1!#, ~10!

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El(2)~r ,t!

54pr0

3 E2`

t

dt8E2`

t8dt3E

2`

t3dt2E

2`

t2dt1

3s~t2t8!s~t82t3!b~t3 ,t2 ,t1!El~r ,t2!El~r ,t1!.

~11!

For completeness we also present the corresponfrequency domain expressions for the first, second,third order susceptibilities (x (1), x (2), andx (3), respectively!obtained by the Fourier transform of the above exprsions1

x (1)~2v;v!5r0a~v!s~v!, ~12!

x (2)~2vs ;v1 ,v2!5r0b~v1 ,v2!s~v1!s~v2!s~vs!,~13!

and

x (3)~2vs ;v1 ,v2 ,v3!

5r0g~v1 ,v2 ,v3!s~v1!s~v2!s~v3!s~vs!

12p

3r0

2s~v1!s~v2!s~v3!s~vs!

3 (perm

b~v1 ,v21v3!b~v2 ,v3!s~v21v3!, ~14!

where perm stands for the sum over all permutations offrequencies of the electric fieldsv1 , v2, and v3 . vs5v1

1v21v3 is the signal field frequency~different signals withall possible choices of sign invs56v16v26v3 are givenby simply changing the signs of various frequencies!.

III. THE REVERSE PHOTON ECHO IN AGGREGATESWITHOUT LOCAL-FIELD EFFECTS

In this paper we will focus on local-field effects and ontake into account the linear~free induction decay! contribu-tions to Eq.~7!. Higher-order~cascading! processes comingfrom the second term in Eq.~7!,1,8 which are interesting ontheir own and were observed experimentally,26,27,33–35willnot be considered here. By neglecting the second term in~7!, we obtain for the third order polarization,29


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q.

P(3)~r ,tc!5E2`

tcdt4E

2`

tcdt3E

2`

t3dt2E

2`

t2dt1s~tc2t4!

3R(3)~t4 ,t3 ,t2 ,t1!El~r ,t3!El~r ,t2!El~r ,t1!.

~15!

Note that in Eq.~15! we use the sample response functiR(3) and not the molecular polarizabilityg. This allows oneto include local interactions directly into the response funtion. However for a system of noninteracting molecules,R(3)

is related to the molecular polarizabilities simply byR(3)

5r0g.To establish a reference for discussing local-field effec

we shall summarize in this section what should be expecin their absence. We then setE(r ,tc) in Eq. ~1! to be theexternal electric field, which is directly controlled expermentally. The RPE experiment involves a sequence ofpulses a,b peaking at timesta , tb and described by thei~complex! field amplitudesEa , Eb , frequenciesva , vb andwave vectorska , kb . The pulses are assumed to be impsive, i.e., very short compared to all relevant time sca~except for the optical period! and are given by:

E~r ,t !5Ea~r ,t !1Eb~r ,t !1c.c., ~16!

where c.c. denotes the complex conjugate and

Ej~r ,t ![eikjrEj~ t !5Ejd~ t2t j !eikjre2 iv j (t2t j ), j 5a,b.

~17!

We assume that pulsea arrives first (ta,tb) and denote thedelay between the pulses bytba , and the time between thsecond pulse and the detection bytcb (tcb[tc2tb , tba

[tb2ta! ~see Fig. 1!. Note that the amplitudesEa andEb

are complex. The complex fields provide a convenient bokeeping of the phase. These pulses are defined in the ‘‘elope delayed form’’12,31,36as generated by an interferometpathlength difference. All electric field parameters~labeledwith the subscriptsa,b!, and in particular the time delaytba , tcb , can be controlled in the experiment. This is in cotrast to the microscopic interaction timest1 , t2 , t3 in Eq.~15! which need to be integrated out.

The level scheme displayed in the inset of Fig. 2 cosisting of a ground state and well separated one-, two-, thexciton manifolds, etc., is very general and can be useddescribe aggregates of coupled two or three level systemwell as coupled anharmonic vibrations. We consider resonexperiments and only allow transitions between adjacmanifolds which are in resonance with the driving electfield. Therefore, only the single and the two-exciton mafolds contribute to the third order nonlinear response fution. By invoking the rotating wave approximation~RWA!,1


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we find that there are eight Liouville space pathways contuting to the response function. The corresponding Feynmdiagrams are shown in Fig. 2.

Four wave mixing signals are generated in several wdefined directions, given by the different combinations ofelectric field wave vectors.5 We shall label the pulses in thorder they interact with the molecule producing the signak1 , k2, andk3. If local-field effects are ignored, the externfields directly act on the molecule and the time orderingthe interactions is controlled by the sequence of short noverlapping pulses. Thus the signal with a wave veckRPE52ka2kb results exclusively from the pathways prducing the polarization in the directionk11k22k3, i.e.,pathways 7 and 8 in Fig. 2.~Note that we assume impulsivpulses andta,tb .)

Our numerical simulations used the sum-over-statespression for the response function given in the followinHowever, for systems with many chromophores it mayadvantageous to calculate the response functions usingnonlinear exciton equations10,11,28,37–40which have a morefavorable scaling with system size.

We consider an aggregate with the general level schshown in the inset of Fig. 2.ug& denotes the ground stateue&, ue8& . . . the one-exciton states, andu f &, u f 8& . . . the two-exciton states.«nn8 denotes the energy difference betwetwo statesn andn8. The transition dipoles between adjacemanifolds are given bymeg andm f e . Gn8n are phenomenological dephasing rates associated with eachn8←n transi-tion. For this model the RPE part of the response functresponsible for the signal in theks5k11k22k3 directioncontains contributions of diagrams~7! and ~8! in Fig. 21

FIG. 1. Schematic representation of the individual terms contributing topolarization@Eq. ~27!#. The vertical bars indicate the pulse sequence,down ~up! arrows correspond to the interaction with the system on the~bra-! side. The decreasing line indicates a FID decay which was initiatethe pulse, where the line starts. See text for details.


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n

RIII(3)~tc ,t3 ,t2 ,t1!

5 i 3~R7~tc ,t3 ,t2 ,t1!1R8~tc ,t3 ,t2 ,t1!!

5 i 3Qr0 (e,e8, f

megm f em f e8~mge8

3exp@~2 i«e8g2Ge8g!~tc2t3!#2me8g

3exp@~2 i« f e82G f e8!~tc2t3!# !

3exp@~2 i« f g2G f g!~t32t2!#

3exp@~2 i«eg2Geg!~t22t1!#. ~18!

Q[u(tc2t3)u(t32t2)u(t22t1) stands for the product oHeavyside functions, that ensure proper time ordering witthe response function. Note that in Eq.~18! we have explic-itly included all three interaction times, even though in thsection we assumet15t2, because when local fields arincluded, this is no longer the case. In the sum the indie,e8 run over all one-exciton states andf runs over all two-exciton states.

The reverse photon-echo signal generated in the dition kRPE52ka2kb is proportional to the nonlinear polarization, which can be calculated from Eqs.~1! and~18!, assum-ing impulsive fields@Eq. ~17!#,

eet-y

FIG. 2. Double-sided Feynman diagrams representing the Liouville sppathways contributing to the third order response in the rotating waveproximation. Each column shows the diagrams contributing to a four wmixing signal in a distinct directionks , as indicated.t1 ,t2 ,t3 are the time-ordered interaction times with the fields. The inset shows the general lscheme.


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PkRPE

(3) ~ t,tb ,ta!5Ea2Eb* RIII

(3)~tc ,tb ,ta ,ta!

5 i 3Ea2Eb* r0 (

e,e8, f

megm f em f e8me8g

3~exp@~2 i«e8g2Ge8g!tcb#

2exp@~2 i« f e82G f e8!tcb# !

3exp@~2 i« f g2G f g!tba#. ~19!

Here PkRPE

(3) is the time-resolved polarization component

the kRPE direction of the general third order polarizatio(P(3)(r ,t)5( ie

iki rPki(3)(t)). From Eq.~19! we note that dur-

ing tba we have a coherence between a two-exciton statethe ground state, whose phase-oscillation is given by theergy difference of the two-exciton state and the ground stwith dephasing rateG f g . The variation of the polarizationwith tba will contain the signatures of a superposition oftwo-exciton states, which is also retrieved by a Fourier traform ~see the following!. During tcb , we have two contribu-tions with a similar frequency and a different sign; in tcase of an harmonic oscillator with equally spaced energthese will interfere destructively causing the signal to vaniThis is expected since the harmonic oscillator is linear ahas no nonlinear response.1 Another important point is thaall RPE pathways must involve a two-exciton resonanThis is the reason for our earlier statement that no signaexpected for this pulse-configuration for uncoupled two-lesystems; within the RWA, a two-level system cannot hatwo consecutive interactions on the same~bra-, or ket-!side.7

Experiments performed on semiconductor quantwells21,22 and in the gas phase23,25 have found a RPE signaThat obviously cannot be explained by the simple pictureelectric fields interacting directly with these two band or twlevel systems. However it is possible to adequately descthese signals using local field effects,1,7,10,29as will be shownnext.

IV. LOCAL-FIELD EFFECTS IN THE REVERSEPHOTON ECHO

To simplify the discussion of local-field effects, we epand the local field@Eq. ~4!# to first order inr0 and theresponse is then calculated to second order inr0. To firstorder inr0, Eq. ~5! gives in the time domain

s~ t !5d~ t !14p

3r0a~ t !. ~20!

Using the pulse configuration Eq.~16! and Eq.~4!, weobtain for the local fieldEl

a associated with the externapulseEa:

Ela~r ,t !5Ea~r ,t !1

4p

3r0E a~ t2t!Ea~r ,t!dt1c.c.

5eikarS Ea~ t !1 iu~ t2ta!4p

3r0Ea(

e9me9g

2

3exp@~2 i«e9g2Geg!~ t2ta!# D 1c.c. ~21!


ndn-e,

s-

s,.d

.isle

f

e

The local field induced by an external short pulse issum of the original pulseEa and the free-induction deca~FID! Fa induced by this pulse.

Ela~r ,t ![Ea~r ,t !1Fa~r ,t !1c.c., ~22!

whereEa is given by Eq.~17!

Fa~r ,t ![eikarFa~ t ![4p

3r0E a~ t2t!Ea~r ,t!dt

5 iu~ t2ta!eikar4p

3r0Ea(

e9me9g

2

3exp@~2 i«e9g2Ge9g!~ t2ta!#. ~23!

Here the sum runs over all single exciton states. Note thatfree induction decay described bya(t) is initiatedon a dif-ferentmolecule. This is underlined throughout this articleusing a double-prime as a superscript whenever the secsystem is involved. Static inhomogeneous broadening caincluded by replacing this sum with an integration over tinhomogeneous distribution. Note that whileEa is a Deltafunctiond(t2ta), Fa has a step functionu(t2ta) ensuringcausality. WhileEa represents a short pulse peaking at timt5ta , Fa is a superposition of exponential decays startingt5ta and decaying with the dephasing ratesGe9g . Thismeans that for pulse delays shorter than 1/Ge9g the time or-dering of interactions of the system with the electric field cbe reversed compared to the external pulse sequence.same principle holds in the frequency domain whereEa hasa broad spectrum centered atva , while Fa has a Lorentzianline shape, corresponding to theg←e9 transition.

The total local fieldEl is the sum of the contributions oboth pulses:

El~r ,t !5Ela~r ,t !1El

b~r ,t !. ~24!

El enters Eq. ~15! in the form of the productEl(t1)El(t2)El(t3). Since we are only interested in thirdorder signals that contain two contributions from the fipulse and one contribution from the second, we only retproducts ofEl

aElaEl

b . We further need to sum over all pemutations of the time variablest1 ,t2 ,t3 since, unlike theMaxwell fields, the local fields are no longer impulsive atime ordering cannot be enforced.

El~t1!El~t2!El~t3!5 (perm

Ela~t1!El

a~t2!Elb~t3!. ~25!

The kRPE signal must result from terms containingEa2Eb* .

Furthermore, to second order inr0 we can neglect all termsthat contain more than one free-induction contribution (F).This leaves us with


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El~t1!El~t2!El~t3!

' (perm

Ea~t1!Ea~t2!E* b~t3!1Ea~t1!Ea~t2!F* b~t3!

12Ea~t1!Fa~t2!E* b~t3!52Ea~t1!Ea~t2!E* b~t3!

12Ea~t1!Ea~t2!F* b~t3!12Ea~t1!Fa~t2!E* b~t3!

12Ea~t1!E* b~t2!Fa~t3!, ~26!

where the second step shows explicitly the permutationinteraction times, taking into account that the impulsive cotributions must be time ordered and causality, i.e., theduced FID part can only interact after being generated. Inlast term on the r.h.s. of Eq.~26! the interaction time with thefield originating from pulsesa andb is reversed compared tthe pulse sequence. After interacting once with the fieEa the system subsequently interacts withE* b and finallywith Fa, originating from pulsea. Even though the signal isgenerated in the direction ofkRPE52ka2kb , in terms ofmicroscopic interactions it is a signal withks5k12k21k3,resulting from the Liouville space pathways~4!–~6! of Fig. 2

onain

ca

th

ID

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or-eio

ittr


f--e

,

which correspond to a transient grating. Consequentlyterms of the response function that need to be takenaccount under the RWA areRII 5R41R51R6, rather thanpathwaysR7 andR8. The contributions to the nonlinear polarization are schematically shown in Fig. 1.

To second order inr0 we only keep terms with, at mosone FID part. This implies that the time ordering of only ointeraction can be delayed and the first interaction still mbe from the ket-side. The photon-echo terms~1!–~3! of Fig.2 do not contribute in this approximation. Using Eq.~15!,Eq. ~26!, and Eq.~20! and retaining terms up to second ordin r0, we find that the nonlinear polarization generatedkRPE has two terms denoted ordinary and local-field induc

PRPE(3) ~tc ,tb ,ta!5POR

(3)~tc ,tb ,ta!1PLF(3)~tc ,tb ,ta!,

~27!

where

POR(3)~tc ,tb ,ta!5Ea

2Eb* RIII(3)~tc ,tb ,ta ,ta!, ~28!

PLF(3)~tc ,tb ,ta![PLFI

(3) ~tc ,tb ,ta!1PLFII

(3) ~tc ,tb ,ta!1PLFIII

(3) ~tc ,tb ,ta!1PLFIV

(3) ~tc ,tb ,ta!

5Ea2E

tb

tcdt3RIII

(3)~tc ,t3 ,ta ,ta!F* b~t3!12EaEb* Eta

tbdt2RIII

(3)~tc ,tb ,t2 ,ta!Fa~t2!

12EaEb* Etb

tcdt3RII

(3)~tc ,t3 ,tb ,ta!Fa~t3!1Ea2Eb*

4p

3r0E

tb

tcdt4a~tc2t4!RIII

(3)~t4 ,tb ,ta ,ta!. ~29!

th

sising

lts.

er

This is the final expression for the nonlinear polarizatigenerated in the direction of the two-pulse RPE signal cculated using the local-field approximation to first orderthe local field and to second order inr0. The five termsrepresent distinct physical processes, and are schematidepicted in Fig. 1, where the pulse sequence is indicatedthe vertical bars and the interactions with the system onbra- ~ket-! side by arrows pointing down~up!wards. The sys-tem interacts either directly with the pulse, or with the Fgenerated by a pulse ona differentchromophore. This FID isindicated by the decaying line in Fig. 1. The first term@Eq.~28!# corresponds to the ordinary RPE. The last term cosponds to an ordinary RPE signal, that interacts with anomolecule and initiates a FID which is detected. Note thaLFIII the sequence of interactions is reversed.

Using Eq.~18! for RIII(3) and an analogous expression f

RII(3) @Eq. ~A2!#, the time integrations are performed in Ap

pendix A. Numerical results are presented below. The expmental signal can be directly calculated from the polarizat@Eq. ~27!# for a variety of detection schemes.1 Mixing thesignal with an additional heterodyne pulse attc allows themeasurement of the time-resolved polarization, includingphase.41–43The heterodyne signal, which depends paramecally on the delay timestba and tcb , is given by12

l-

llybye

-ern

ri-n

si-

S~ tcb ,tba!5E2`

`

dtEh~t!P(3)~r ,t,tb ,ta!, ~30!

whereEh(t) is the heterodyne field.For the following discussion, suffice it to note that bo

the amplitude and phase of the polarization@Eq. ~15!# can bemeasured experimentally. It may be helpful for the analyto display the results in the frequency domain by performa double Fourier transform5,44,45 The resulting 2D-FTsignal12 is then given by

S~V2 ,V1!5E0

`

dtcbE0

`

dtbaS~ tcb ,tba!

3exp@ iV2tcb1 iV1tba#. ~31!

We shall use this representation to display our resuNote thatV1 is the frequency conjugate totba and V2 isconjugate totcb . V1 thus reflects the dynamics taking placbetween the two pulses, whileV2 shows the evolution aftethe second pulse.


nguconimrop

ne

tio

r

s

ngbe

-tos

ofibu-r-

heheeEq.


V. RPE SIGNALS FOR TWO AND THREE LEVELMODEL SYSTEMS

In this section we present the RPE signal@Eq. ~31!# forvarious model systems with the goal of clearly highlightithe many-body resonances. Although local fields can indnew peaks, they do not carry new microscopic informatiunlike the peaks caused by microscopic couplings. Theportant differences between the two will be discussed fofew examples, using typical parameters for IR spectroscof anharmonic vibrations.

In all the numerical calculations we assumed equal oexciton transition dipolesmeg and dephasing constantsGeg .From Eq.~A6! and Eqs.~A7!–~A10! @or similar Eq.~A11!and Eqs.~A12!–~A15!# we can see that in this case the raof the first and second order terms inr0 ~the ordinary RPEand the local field-induced signal! is determined by a factoof (4p)/3r0umegu2, which has the same dimensions~of en-ergy! as Geg . This factor was assumed to be 0.1Geg in allcalculations for illustration purposes. Based on crude emates, we expect this low order truncation inr0 to hold evenfor neat liquids.

qfeouinle

nt treciumn

teb

r-

le


e,-ay

-

ti-

In the case of very high chromophore density and strooscillator strengths, an expansion to higher order wouldstraightforward. An expansion of the local field inr0 couldbe totally avoided, by calculatings(t) via a numerical Fou-rier transform directly from Eq.~5!. However, the perturbative expansion is extremely valuable, since it allows usdistinguish the different effects and clarify their origin, ashown in the following.

A. Uncoupled two level systems

As discussed in the previous section, for a collectionuncoupled two level systems the response-function contrtions forkRPE vanish (RIII

(3)50) and no RPE signal is geneated in the direction 2ka2kb . However, Eq.~27! also con-tains a purely local-field contribution, which depends onRIII

(3)

andRII(3) @Eq. ~29!#. The expression forRII

(3) becomes particu-larly simple in the present model, since it is the sum of tindividual response functions. Only the third term on tr.h.s. of Eq. ~29! is nonzero. Neglecting evolution in thexcited- or ground-state population periods, we get from~A9!

Pks(3)2LS~ tcb ,tba!5 i 3

4p

3r0

2(e9,e

ume9gu2umegu4Ea2Eb* exp@~2 i ~«e9g1«eg!2~Ge9g1Geg!!tba#

3exp@~2 i«e9g2Ge9g!tcb#2exp@~2 i«eg2Geg!tcb#

«eg2«e9g1 i ~Ge9g2Geg!, ~32!

n-

tone

af

alel

where we wrote the signal which isexclusively induced bythe local fieldas a function of the time delays (tcb , tba! andthe sum runs over all different pairs of molecules. From E~32! we clearly deduce the expected effects and their difence from two-exciton resonances resulting from direct cpling. The most interesting temporal evolution occurs durtba where the signal shows oscillations at sums of singexciton energies. During this period, two molecules are icoherence and the many-body density matrix oscillates asum of their frequencies.29 The observed frequencies atherefore simple sums of one-exciton resonance frequenno multiple quantum coherence between separated molecis involved, and the new resonances carry no additionalcroscopic information. This is in contrast to multiple quatum resonances resulting from intermolecular coupling~videinfra!, where oscillations at frequencies of« f g are observedduring the periodtba . Also the relaxation during this timeinterval is given by the sum of one-exciton damping ra(Geg1Ge9g). This doubling of the dephasing rate was oserved experimentally in semiconductor quantum wells.21,22

During the time periodtcb , the system oscillates at a supeposition of single-exciton frequencies.

In the special case of a sample of identical uncouptwo level systems, Eq.~32! reduces to

.r--

g-ahe

es,lesi-

-

s-

d

Pks(3)TLS~ tcb ,tba!5 i 4

4p

3r0

2umegu6Ea2Eb* tcb

3exp@~2 i«eg2Geg!tcb#

3exp@~2 i2«eg22Geg!tba#, ~33!

where we used the limit

lime2→ e1

S exp@2 i e1t#2exp@2 i e2t#

e22 e1D 5 i t exp@2 i e1t#.

~34!

Even though the denominator of Eq.~32! vanishes, thelimit in Eq. ~34! is well-defined and leads to a secular cotribution initially increasing linearly withtcb . P(3) does notdiverge, because of the exponential damping terme2Gegtcb.When inhomogeneous broadening is included, we needintegrate Eq.~32! over distributions of energies which cafurther cure this divergence. The simple linear rise with timresults from the neglect of population decay (Gee5Ggg50)and static broadening, so that the FID isexactlyon resonancewith the e←g transition. This problem does not occur ifmore realistic relaxation model is included.~See page 167 oRef. 1.!

The physical meaning of the initial rise of the signduring tcb can be understood from Fig. 1. For a two lev


t tID

on

itInn

eyte

th

l.nayth

totut

in

alitw

was

ly

er

stemofs,

by-ohe-

o-

ln-

ocalys-ingters

sas-

l

in-of

ms

enandfoureylay

the

di-Fig.

a


system, the entire signal results from theLFIII term, wherethe time ordering of interactions is reversed with respecthe pulse-delay times and the last interaction is with the Finitiated by the first pulse and takes place after the secpulse. The larger thetcb , the more time there is for thisinteraction to occur. Note also that after the interaction wthe second pulse attb , the system is in a population state.the absence of population relaxation, the only contributioto the decay of the signal is the decay of the FID that intacts with the system, and after the interaction, the decathe generated signal. Since for our simple two level systhese two contributions have the same relaxation rate~be-cause the FID, as well as the final signal results fromsame transition!, we get the simpletcbe

2Gtcb behavior in Eq.~33!. This initial rise as a function oftcb is only seen in thelocal-field contributions but not in the ordinary RPE signa

From Eq.~33!, we expect the polarization as a functioof tba to oscillate with twice the optical frequency and decwith 2Geg . Homodyne detected experiments measuretime integrated signalSA

H(tba)5*0`dtcbuPA(tcb ,tba)u2 and in

this case we expect the signal to be proportional toe24Gegtba.This effect of fast dephasing with twice the rate of a phoecho experiment was observed in semiconductor quanwells.21 Using heterodyned detection it should be possibledirectly observe the high harmonic frequency correspondto twice the optical transition energy.

The time-resolved ordinary RPE signal and the locfield contributions for a three level system are comparedFig. 3, where we used a three level system because for alevel system no ordinary RPE contribution exists. The tcolumns show the squared amplitude of the polarizationfunction of tba and tcb ~with the other time variable heldfixed!, respectively. While the ordinary RPE signal onshows exponential decay, the local field contribution~whichfor a three level system consists of a superposition of sev

FIG. 3. The squares of the amplitudes of~a! the ordinary RPE contribu-tions and~b! the local field contributions to the time-resolved RPE signof a three level system as a function oftba ~with tcb511 ps) andtcb ~withtba50) in the first and second column, respectively.


o

d

h

sr-ofm

e

e

nmog

-no

oa

al

contributions! initially increases as a function oftcb . Theparameters used are the same as for the three level sydescribed following. The oscillation results from beatingthe f←e and e←g transitions and has a period of 1.7 pcorresponding to the anharmonicity (D5«eg2« f e

520 cm21).As discussed previously, for two level systems thekRPE

signal can be generated either by local-field effects orintermolecular coupling.29 Let us compare the local-field effects to the ordinary RPE signal for a simple model of twstrongly coupled two level systems. The level scheme scmatically depicted in the inset of Fig. 4~b! consists of aground stateug&, two single excited statesue1&, ue2& withan energy spacing depending on the couplingJ, and onedoubly excited stateu f &. These states are obtained by diagnalizing the 434 Hamiltonian of this system.

The polarization responsible for the ordinary RPE 2ka

2kb signal for two coupled two level systems~Fig. 4! is

POR(3)cTLS~ tcb ,tba!

5 i 34p

3r0

2Ea2Eb* ~me2,gm f ,e21me1,gm f ,e1!

3 (e5e2,e1

megm f ee(2 i« f g2G f g)tba

3@e(2 i«eg2Geg)tcb2e(2 i« f e2G f e)tcb#. ~35!

This is markedly different from the local-field signa@Eq. ~32!#, where all combinations of one exciton frequecies ~including 2«eg) contribute to the signal intba . This isshown in Fig. 4~a!, which displays the 2D signal for twoindependent two level systems. Figure 4 compares the lfield induced RPE signal of two independent two level stems with the ordinary RPE signal caused by the couplbetween two interacting two level systems. The parameused in the simulations were:«e1,g52014 cm21, «e2,g

52085 cm21, the transition dipole momentsm as well as thedampingG51cm21 of the independent two level systemwere assumed to be equal. For the coupled system wesumed a coupling constantJ5210 cm21. The dephasingrates for the coupled system were set toGeg5G f e51 cm21

~for e5e2,e1), G f g52Geg .As can be seen in Fig. 4~b! the ordinary RPE signa

@Eq. ~35!# only shows peaks whenV1 is resonant withthe two-exciton state. It thus provides new microscopicformation about the two-exciton manifold. The structureEq. ~35! coincides with the case of two three-level systewith identical « f g and an amplitude ratio A1 /A2

5(me1gm f e1)/(me2gm f e2) ~as discussed in the next section!.In principle it should also be possible to distinguish betweresonances induced by coupling between different sitesintramolecular resonances of a three-level system. In awave mixing experiment with three different pulses thshould have different phases as a function of the time debetween the first two interactions which is set to zero inpresent calculations.

The phase profiles of the local field-induced and ornary RPE resonances displayed in the second column of

l


epl

on

o-. 5,

ing

in

l-

ecel

he

m

spioingsim

eqing

firign

t


4 show pronounced differences. Equation~35! represents thepure RPE without local-field contributions. In general wshould see a superposition of resonances caused by couand by local field, and this case is shown in Fig. 4~c!. Newpeaks can be found atV152«e1,g and V152«e2,g . The

FIG. 4. ~Color! ~a! The total local field-induced RPE signal of a systecomposed of two independent two level systems~b! the ordinary RPE and~c! the total RPE signal~including local-field effects! for two coupled twolevel systems. The first two columns show the amplitude and phase, retively of the 2D-FT signal. The dotted lines indicate the relevant transitenergies~see text!. The insets in the last column show the correspondlevel schemes. The parameters for the two level systems used in thelations were:«1eg52014 cm21, «2eg52085 cm21, the transition dipolemoments of the independent two level systems were assumed to beand the dampingG51cm21. For the coupled system we assumed couplconstantJ5210 cm21 between the two systems.

FIG. 5. ~Color! ~a! total RPE signal and~b! the ordinary RPE contributionsfor a three level model with the parameters as indicated in the text. Thecolumn shows the amplitude and the second column the phase of the sThe dotted lines indicate the relevant transition energies at«eg , « f e and2«eg , « f g . ~c! The level scheme and~d! a slice through the 2D-phase ploat V152«eg showing the total signal~solid!, ordinary RPE signal~dashed!and local-field contributions~dotted!.


ing

local-field contributions will have a different dependencemolecular density and a different profile of the phase.

B. Identical three level systems

A system of identical uncoupled three level chrmophores whose level scheme is shown in the inset of Figis the simplest model where all terms of Eq.~27! contribute.We assume a ground stateug&, a first excited stateue& and asecond excited stateu f & with an anharmonicityD5«eg

2« f e . We further neglect population relaxation (Gee50)but keep all other dephasing rates general. The followparameters were used in the simulations:«eg52085 cm21,« f g54150 cm21, D520 cm21, m f e51.3meg , G51 cm21.

For this model the polarization@Eq. ~27!# can be calcu-lated analytically and we next discuss the various effectsmore detail.

1. Ordinary RPE signal of a three level system

The time domain RPE signal resulting from the impusive fields alone is determined by the polarization@Eq. ~A6!#

POR(3)3LS~ tcb ,tba!5 i 3r0Ea

2Eb* umegu2um f eu2e(2 i« f g2G f g)tba

3@e(2 i«eg2Geg)tcb2e(2 i« f e2G f e)tcb#. ~36!

The two terms oscillating at frequencies (« f g ,«eg) and(« f g ,« f e) as a function of (tba ,tcb) can be clearly seen. Notthat due to the minus sign, the two terms will exactly canfor a system with vanishing anharmonictyD50 and equaldampingGeg5G f e . The two resonances are recovered in t

ec-n

u-

ual

stal.

FIG. 6. ~Color! Local-field effects for a three level system:~a! amplitudeand phase of the total local field contributions.~b!–~e! The individual con-tributions. The diagrams forLFI , LFII , andLFIV are multiplied by a factorof 5 with respect toLFIII .


igg-n

ceaah

n

y

-

eaion

de

epo

ieaeicID

si-al

gd

fn

ul

he

al

llyroes.

sys-n-

hee.

ces

ashisn.


2D-FT signal, shown in Fig. 5 which compares the total snal including local-field effects, with the ordinary RPE sinal. The Fourier transform can be performed analytically afrom Eq. ~A11! we get

SOR3LS~V2 ,V1!5 i 3r0Ea

2Eb* meg2 m f e

2

3F 1

~V22«eg1 iGeg!~V12« f g1 iG f g!

21

~V22« f e1 iG f e!~V12« f g1 iG f g!G . ~37!

While the ordinary RPE signal only shows resonanV15« f g , inclusion of local fields leads to new peaksV152«eg . Also the phase profile is different and the locfield influences and modifies the original resonances. Tcan be clearly seen in Fig. 5~d!, where the phase variatioalong a slice withV152«eg is shown for the total RPEsignal, the ordinary RPE and the LF contribution.

When examining the variation with different delatimes, we note that as a function oftba the polarization is adamped oscillation with frequency« f g and a phase determined by the second delay periodtcb , whereas as a functionof tcb , the signal shows an amplitude modulation with a bfrequency corresponding to the anharmonicity. As a functof tba we expect a single damped oscillation. The homodysignal of our simple model should show an exponentialcay proportional toe22G f gtba ~cf. Fig. 3!.

2. Local field effects

The local field induced terms@Eq. ~29!# are evaluated inAppendix A @Eqs.~A7!–~A9!# and expressions for the threlevel system can be easily obtained. These terms corresto the different LF contributions as schematically sketchedFig. 1. Figure 6 shows the absolute value and the phasthe total local-field contribution and of each of the individucontributions to the 2D-FT RPE signal. The first term dscribes the time ordering, where the first pulse interacts twwith the system and the final interaction occurs with the Fgenerated on another chromophore by the last pulse.

PLFI

(3)3LS~ tcb ,tba!5 i 34p

3r0

2Ea2Eb* umegu4um f eu2e(2 i« f g2G f g)tba

3Fe(2 i« f e2(G f g1Geg))tcb2e(2 i«eg2Geg)tcb

~«eg2« f e!1 iG f g

1e(2 i« f e2G f e)tcb2e(2 i« f e2(G f g1Geg))tcb

i ~G f g1Geg2G f e!G .~38!

In Fig. 6~b! we see two peaks at (« f g ,«eg) and (« f g ,« f e) forLFI and Eq.~38! shows that they result from the superpotion of different contributions. As in the pure RPE case,resonances occur atV15« f g and as a function oftba , Eq.~38! is similar to Eq.~36!, as far as frequency and dampinare concerned, but has a different phase because of theferent variation withtcb . The denominators in the r.h.s. oEq. ~38! add complex amplitudes to the terms, which depeon the damping and the anharmonicity. This term res


-

d

stlis

tne-

ndnofl-e

l

if-

dts

from the interaction of the FID with the system. Since tFID is caused by ag←e transition of another~identical!molecule, it can only be resonant with«eg and is detuned byD from the f←e transition. For the last two terms the repart of the denominator vanishes~since for our model, whichneglects inhomogeneous broadening, the FID isexactlyonresonance!. However, the dephasing rates do not generacancel and will give a purely imaginary prefactor. For zeanharmonicity and equal damping the entire term vanish

2D-Fourier transform of Eq.~38! gives

SLFI

3LS~V2 ,V1!

5 i 34p

3r0

2Ea2Eb* umegu4um f eu2

31

V12« f g1 iG f gH 1

~«eg2« f e!1 iG f g

3F 1

V22« f e1 i ~G f g1Geg!2

1

V22«eg1 iGegG

21

i ~G f g1Geg2G f e!

3F 1

V22« f e1 i ~G f g1Geg!2

1

V22« f e1 iG f eG J . ~39!

The next term corresponds to the process where thetem interacts once with pulse one, then with the FID of aother molecule induced by pulse one and finally with tsecond pulse attb . As in the above-mentioned case the timordering is not reversed with respect to the ordinary RPE

PLFII

(3)3LS~ tcb ,tba!52i 34p

3r0


31

« f g22«eg1 i ~2Geg2G f g!

3$e(2 i2«eg22Geg)tba@e(2 i«eg2Geg)tcb

2e(2 i« f e2G f e)tcb#2e(2 i« f g2G f g)tba

3@e(2 i«eg2Geg)tcb2e(2 i« f e2G f e)tcb#% ~40!

This term is interesting because it shows four resonanwith all combinations ofV15« f g or 2«eg and V25«eg or« f e as can be seen in Fig. 6~c!. They all have the sameamplitude which depends on detuning. Note that as longDÞ0, the real part of the denominator does not vanish. Tis why this term generally makes only a minor contributioAgain, for vanishing anharmonicity~and equal damping! thisterm vanishes.

In the frequency domain this signal is


hathitIDsn

ulv

hi

.se

ak

orfo

.scra

efter

sesen-el

elhichcon-


SLFII

3LS~V2 ,V1!52i 34p

3r0


31

« f g22«eg1 i ~2Geg2G f g!

3H 1

V122«eg1 i2GegF 1

V22«eg1 iGeg

21

V22« f e1 iG f eD G2

1

V12« f g1 iG f g

3F 1

V22«eg1 iGeg2

1

V22« f e1 iG f eG . ~41!

The next term is very special since it is the only one tinvolves a reversal in the order of interactions. First,system interacts with the first pulse from the bra-, then wthe second pulse from the ket- side and finally with the Fof another molecule, initiated by the first pulse. In this cathe system is never prepared in a coherent superpositiou f & andug& and shows no resonances atV15« f g in Fig. 6~c!.But after the first pulse, two separated uncoupled molecare excited coherently and interact after the second pulsethe FID.

As shown above, for uncoupled two level systems tterm is the only one contributing in the direction 2kRPE. Forthe three level case we get from Eq.~A9!

PLFIII

(3)3LS~ tcb ,tba!52i 34p

3r0

2Ea2Eb* umegu4e(2 i2«eg22Geg)tba

3F2ume8gu2i t cbe(2 i«eg2Geg)tcb

2um f eu2e(2 i«eg2Geg)tcb2e(2 i« f e2G f e)tcb

« f e2«eg1 i ~Geg2G f e!G .

~42!

We find that all resonances occur atV152«eg . Themain contribution arises from the first term on the r.hwhereby the FID is exactly on resonance with the inductransitions. This term peaks at (V1 ,V2)5(2«eg ,«eg); nodouble-exciton state is involved. The other two terms, peing also at (2«eg ,«eg), as well as at (« f g ,«eg), contain acomplex prefactor with the detuning in the denominatmaking them small compared to the first term. Note thatspecial values of damping~i.e., when Geg2G f e5G f g

22Geg) they cancel exactly with the first term on the r.hof Eq. ~40!. Equation~42! only vanishes for the harmonicase, i.e., when the transition energy and the dephasingare equal for thee←g and f←e transitionsand the dipolemoments correspond to the harmonic case (m f e5A2meg).

46

In the frequency domain this signal is


teh

eof

esia

s

.d

-

,r

.

tes

SLFIII

3LS ~V2 ,V1!52i 34p

3r0

2Ea2Eb* umegu4

31

V122«eg1 i2GegH 2umegu2

~V22«eg1 iGeg!2

2um f eu2

« f e2«eg1 i ~Geg2G f e!F 1

V22«eg1 iGeg

21

V22« f e1 iG f eG J . ~43!

Finally, to take all local field effects into account, whave to consider the contributions of processes whereby acreating the RPE signal by three interactions with the pulthe generated signal interacts with another molecule to gerate a FID which is finally detected. For the three levsystem this term gives

PLFIV

(3)3LS~ tcb ,tba!5 i 34p

3r0

2Ea2Eb* umegu4um f eu2e(2 i« f g2G f g)tba

3H i t cbe(2 i«eg2Geg)tcb

21

«eg2« f e1 i ~G f e2Geg!

3@e(2 i« f e2G f e)tcb2e(2 i«eg2Geg)tcb#J . ~44!

Not surprisingly, we find only resonances withV15« f g andthe major contribution at (« f g ,«eg). In the frequency domainwe obtain

TABLE I. Contributions to the four peaks in the signal of a three levsystem. For each resonance peak, all contributions are listed, showing wterm contributes the prefactor and the dephasing characteristics of eachtribution. l[ i 3Ea

2Eb* umegu2um f eu2, l8[ i 3 4p/3Ea2Eb* umegu4um f eu2, and l9

[ i 3 4p/3Ea2Eb* umegu6.

Resonance Contribution Prefactor Damping

(« f g ,«eg) OR lr0 (G f g ,Geg)(« f g ,«eg) LFI 2l8r0

2/(D1 iG f g) (G f g ,Geg)

(« f g ,«eg) LFII 22l8r02/(2D1 i (2Geg2G f g)) (G f g ,Geg)

(« f g ,«eg) LFIV il8r02tcb (G f g ,Geg)

(« f g ,«eg) LFIV l8r02/(D1 i (G f e2Geg)) (G f g ,Geg)

(« f g ,« f e) OR lr0 (G f g ,G f e)(« f g ,« f e) LFI l8r0

2/(D1 iG f g) (G f g ,G f g1Geg)

(« f g ,« f e) LFI l8r02/( i (G f g1Geg2G f e)) (G f g ,G f e)

(« f g ,« f e) LFI 2l8r02/( i (G f g1Geg2G f e)) (G f g ,G f g1Geg)

(« f g ,« f e) LFII 2l8r02/(2D1 i (2Geg2G f g)) (G f g ,Geg)

(« f g ,« f e) LFIV 2l8r02/(D1 i (G f e2Geg)) (G f g ,Geg)

(2«eg ,«eg) LFII 2l8r02/(2D1 i (2Geg2G f g)) (2Geg ,Geg)

(2«eg ,«eg) LFIII 2il9r02tcb (2Geg ,Geg)

(2«eg ,«eg) LFIII 22l8r02/(2D1 i (Geg2G f e)) (2Geg ,Geg)

(2«eg ,« f e) LFII 22l8r02/(2D1 i (2Geg2G f g)) (2Geg ,G f e)

(2«eg ,« f e) LFIII 2l8r02/(2D1 i (Geg2G f e)) (2Geg ,G f e)


tenc

ion

hes a

m-rre-le.thetal

yne

of

ni-

edton

the

re

vee


SLFIV

3LS ~V2 ,V1!5 i 34p

3r0


1

V12« f g1 iG f g

3H 1

~V22«eg1 iGeg!2

21

«eg2« f e1 i ~G f e2Geg!F 1

V22« f e1 iG f e

21

V22«eg1 iGegG J . ~45!

The RPE signals from an anharmonic three level sysare summarized in Table I, which gives for each resona

FIG. 7. The amplitude of the 2D-FT signal for two coupled three lesystems as shown in the inset.~I! The total signal, which consists of thordinary RPE~II ! and the local-field contributions~III !. Smaller peaks arezoomed out by the factors indicated. The two indicated regions of~I! areshown in a 3D plot on the left side.


me

peak in the total RPE signal@Fig. 5~a!# all terms that con-tribute with their amplitude and dephasing rate~which in thefrequency domain determines the line width!.

Finally, when comparing the variation withtba , we seethat while the ordinary RPE signal is a damped oscillatwith frequency« f g and dephasing rateG f g , the local-fieldcontribution is a superposition of two oscillations with« f g

and 2«eg . In the homodyne-detected signal, where tsquare of the amplitude is measured, the former givesingle damped exponential with a dephasing rate 2G f g ,while the local-field contributions should show a more coplex pattern, where the beat frequency with a period cosponding to the anharmonicity, are clearly distinguishabSuch a behavior can be seen in Fig. 3, which showssquare of the amplitude of the two contributions to the toRPE polarization vs.tba for a fixed value oftcb . ~Note thatthis corresponds to a time-gated detection; in a homodexperiment one would have to integrate overtcb .! We seethat the anharmonicity ofD520 cm21 shows up in the os-cillation period of T51.7 ps and the dephasing ofG f g

52Geg52 cm21 corresponds to a dephasing time scaleT2/258 ps with tba . As a function oftcb the ordinary echosignal has a slowerGeg51 cm21 damping, while the local-field contributions show a complicated behavior, with an itial rise time.

C. Two coupled three level systems

To simulate the complete RPE signal of two couplanharmonic vibrations, we used the values found in phoecho experiments47,48 for the symmetric and asymmetric COstretches in rhodium~I! dicarbonylacetylacetonate~RDC!.46

The level scheme is shown in the inset of Fig. 7 andtransition energies are«e1,g52085 cm21 and «e1,g

52014 cm21. The values of the anharmonic splittings aD1510.6 cm21, D2512 cm21, and D3525 cm21 and thedipole moments are assumed to beme1,g5me2,g , m f 1,e1

5m f 2,e251.3me1,g , m f 3,e15m f 3,e250.9me1,g , and m f 1,e2

l

n
FIG. 8. ~Color! The phase of the 2D-FT signals showin Fig. 7. Lower row shows a slice for constantV2
5«e2,g , see text for details.


tos

teon

Ina

en

eser

s-ea

2D

adffefo

oausonop

iynu

teoodeo

ecvemsilio

eb

frbu

calul-in-

nu-

d inson-y-ex-tion,tical

s insed,alnalenicef.the

riede

tshar-

thn-

cial-

talal

ct

to


mf1,e25m f 2,e1520.04me1,g . The dephasing rate was setGe1,g5Ge2,g51cm21 and the double excited states were asumed to dephase with twice this rateG f i ,g52Gei,g , whilepopulation relaxation was neglectedGei,e j5Gg,g50.

Figure 7 shows the resulting 2D-FT RPE signal. Nothat due to the anharmonicity, otherwise forbidden transiticontribute as well and give weak signals atV25« f 1,e2 andV25« f 2,e1, which are shown magnified by a factor of 20.addition to the ordinary RPE resonances, we find new peat twice, and at the sum of the two single-exciton frequcies. Thus, instead of resonances only atV1

5« f 2,g ,« f 3,g ,« f 1,g @as expected for the ordinary RPE~case Iin Fig. 7!#, we find six ‘‘columns’’ with peaks because thlocal-field resonances can also occur at all combinationsingle exciton energies. Local-field effects thus show vdistinct features in the RPE signal.

Note that the local field contributions@shown in Fig. 7~III !# have their most pronounced peaks at resonanceV152«e1,g,2«e2,g ,(«e1,g1«e2,g), but also make a contribution to the resonances of the ordinary RPE signal. Local-fieffects also show a very different variation of phaseshown in Fig. 8, which displays the phase of the entirespectrum, as well as a slice along theV1 axis for a fixedvalue ofV25«e2,g .

VI. CONCLUSIONS

Comparison of the resonances of coupled systemsthe local field-induced resonances found in a system of inpendent chromophores shows several fundamental diences. While interaction induced resonances contain inmation about the local microscopic environment,5 themacroscopic local field-induced resonances simply shcombinations of one-exciton resonances and do not yieldditional microscopic information. When inhomogeneobroadening is included, we might get different informatiabout the macroscopic sample than from linear spectroscsince the average of a product of correlation functionsdifferent from the product of averages. For the heteroddetected four wave mixing signal of a general aggregate bfrom interacting chromophores, we expect a complex patresulting from both short- and long-range effects, as demstrated here for the reverse photon echo in several msystems. In addition to the two-exciton bands, we expect nlocal-field resonances, with a different dependence on ccentration.

Local fields can also be viewed as a retardation effwhich leads to breakdown of time ordering in an impulsiexperiment. The third order response function in the tidomain, which describes ideal impulsive experiments, haterms. The corresponding frequency domain susceptibincludes 48 terms corresponding to all six permutationsthe three fields.1 The former has absolute control of timorders. The latter has no time control at all; see e.g., douresonant vibrationally enhanced IR-FWM.49,50As the pulsesbecome longer, impulsive experiments start to assume aquency domain character and realistic experiments willintermediate and may contain more terms than in the imp


-

s

ks-

ofy

of

lds

nde-r-r-

wd-

y,seiltrnn-elwn-

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e8

tyf

ly

e-el-

sive case. Also in an ideal impulsive experiment the lofields that interact with the molecules are no longer impsive and macroscopic many-body effects can have majorfluences on the signal.

Other types of collective resonances related to contious manifolds of levels were predicted6,51 Intermolecularresonances were observed in fifth order Raman.15 Local-fieldeffects in fifth order Raman ofCS2

26,52 were simulatedrecently.53 High harmonic resonances have been observeNMR.30,31 In optical k11k22k3 signals these resonancehave not yet been directly resolved. In studies on semicductor quantum wells21,22a RPE signal was observed, decaing with twice the dephasing rate in a transient gratingperiment. Since those experiments used homodyne detecthey could not observe the resonances at twice the opresonance frequency. Recent experiments23,24 on iodine va-por also revealed the RPE signal for negative time delaya photon-echo setup. Again homodyne detection was ubut due to the special level structure in iodine, vibrationquantum beats are observed. A careful analysis of this sigas a function oftba should be able to reveal differences in thwave packet structure, which however, for a nearly harmosystem are very small and were not resolved. Further in R23 the focus was on a different three pulse scheme, wheretime delay between the first and second pulse was va~this was assumed to be zero here! and the time between thsecond and the third pulse~corresponding to ourtba) wasfixed. Heterodyne detected four wave mixing experimen43

are necessary in order to observe the predicted highmonic resonances in optical signals.

ACKNOWLEDGMENTS

The support of the National Institutes of Heal~GM59230-01A2! and the National Science Foundatio~CHE-9814061! is gratefully acknowledged. The stay of author A.T. at Rochester was funded by the Austrian SpeResearch Program F016 ‘‘ADLIS’’~Austrian Science Foundation Vienna/Austria!.

APPENDIX A: THE FOUR WAVE MIXING K RPE SIGNAL

For the model aggregate described in Sec. III, the tosignal @Eq. ~27!# can be evaluated analytically and the finexpressions for the individual terms in Eqs.~28! and~29! aregiven in the following. In this appendix we use the companotation

e i j [« i j 2 iG i j , ~A1!

i.e., we include the dephasing in the site energies.~In themain text we showed the dephasing rates explicitly.!

Due to the breakdown of time ordering, we also needtake into accountRII ~see Fig. 2! for one of the terms in Eq.~15!. Three Liouville space pathways contribute toRII :

RII 5 i 3~R41R51R6!, ~A2!


re-In


R4~ t,t3 ,t2 ,t1!5Q (e,e8,g

megme8gme8gmeg

3exp@2 i eeg~ t2t3!#

3exp@2 i eee8~t32t2!#

3exp@2 i eeg~t22t1!# ~A3!

R5~ t,t3 ,t2 ,t1!5Q(e,e8

megmgeme8gmge8

3exp@2 i ee8g~ t2t3!#

3exp@2 i egg~t32t2!#

exp@2 i eeg~t22t1!# ~A4!


R6~ t,t3 ,t2 ,t1!5Q(e,e8

megme8gm f eme8 f

3exp@2 i e f e8~ t2t3!#

3exp@2 i eee8~t32t2!#

3exp@2 i eeg~t22t1!# ~A5!

Using Eqs.~18!, ~A2!, ~17!, and~23!, the integrations inthe expressions of Eqs.~28! and ~29! can be performed andwe easily find the expressions for the polarization corsponding to each of the contributions depicted in Fig. 1.terms of the time intervalstcb and tba they are given by

POR(3)~ tcb ,tba!5 i 3r0Ea

2Eb* (e,e8, f

megm f eme8 f@mge8exp@2 i ee8gtcb#2me8gexp@2 i e f e8tcb##exp@2 i e f gtba# ~A6!

PLFI

(3) ~ tcb ,tba!5 i 34p

3r0

2Ea2Eb* (

e,e8,e9, f

ume9gu2megm f eme8 fexp@2 i e f gtba#Fmge8

exp@2 i ~ e f g2 ee9g* !tcb#2exp@2 i ee8gtcb#

ee9g* 1 ee8g2 e f g

2me8g

exp@2 i ~ e f g2 ee9g* !tcb#2exp@2 i e f e8tcb#

ee9g* 1 e f e82 e f g

G ~A7!

PLFII

(3) ~ tcb ,tba!52i 34p

3r0

2Ea2Eb* (

e,e8,e9, f

ume9gu2megm f eme8 f@exp@2 i ~ eeg1 ee9g!tba#

2exp@2 i e f gtba##mge8exp@2 i ee8gtcb#2me8gexp@2 i e f e8tcb#

e f g2 eeg2 ee9g

~A8!

PLFIII

(3) ~ tcb ,tba!52i 34p

3r0

2Ea2Eb* (

g,e,e8,e9, f

ume9gu2megexp@2 i ~ eeg1 ee9g!tba#

3Fme8gmge8mge

exp@2 i ~ eee81 ee9g!tcb#2exp@2 i eegtcb#

eeg2 eee82 ee9g

1mgeme8gmge8

exp@2 i ~ ee9g1 egg!tcb#2exp@2 i ee8gtcb#

ee8g2 egg2 ee9g

2me8gm f eme8 f

exp@~2 i ~ eee81 ee9g!tcb#2exp@2 i e f e8tcb#

e f e82 eee82 ee9gG ~A9!

PLFIV

(3) ~ tcb ,tba!5 i 34p

3r0

2Ea2Eb* (

e,e8,e9, f

ume9gu2megm f eme8 fexp@2 i e f gtba#Fmge8

exp@2 i ee8gtcb#2exp@2 i ee9gtcb#

ee9g2 ee8g

2me8g

exp@2 i e f e8tcb#2exp@2 i ee9gtcb#

ee9g2 e f e8G . ~A10!

The corresponding 2D-frequency domain expressions can be evaluated easily using Eq.~31!, resulting in

SOR~V2 ,V1!5 i 3r0Ea2Eb* (

e,e8, f

megm f eme8 f

1

V12 e f gF mge8

V22 ee8g

2me8g

V22 e f e8G ~A11!



SLFI~V2 ,V1!5 i 3

4p

3r0

2Ea2Eb* (

e,e8, f

ume9gu2megm f eme8 f

1

V12 e f gH mge8

ee9g* 1 ee8g2 e f g

F 1

V22~ e f g2 ee9g* !

21

V22 ee8gG

2me8g

ee9g* 1 e f e82 e f g

F 1

V22~ e f g2 ee9g* !

21

V22 e f e8G J ~A12!

SLFII~V2 ,V1!52i 3

4p

3r0

2Ea2Eb* (

e,e8, f

ume9gu2megm f eme8 fF 1

V12~ eeg1 ee9g!2

1

V12 e f gG 1

e f g2 eeg2 ee9g

3F mge8

V22 ee8g

2me8g

V22 e f e8G ~A13!

SLFIII~V2 ,V1!52i 3

4p

3r0

2Ea2Eb* (

e,e8, f

ume9gu2meg

1

V12~ eeg1 ee9g!H me8gmge8mge

eeg2 eee82 ee9gF 1

V22~ eee81 ee9g!2

1

V22 eegG

1mgeme8gmge8

ee8g2 egg2 ee9gF 1

V22~ egg1 ee9g!2

1

V22 ee8gG2

me8gm f eme8 f

e f e82 eee82 ee9gF 1

V22~ eee81 ee9g!2

1

V22 e f e8G J

~A14!

SLFIV~V2 ,V1!5 i 3

4p

3r0

2Ea2Eb* (

e,e8, f

ume9gu2megm f eme8 f

1

V12 e f gH mge8

ee9g2 ee8gF 1

V22 ee8g

21

V22 ee9gG

2me8g

ee9g2 e f e8F 1

V22 e f e8

21

V22 ee9gG J . ~A15!

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1S. Mukamel,Principles of Nonlinear Optical Spectroscopy~Oxford, NewYork, 1995!.

2D. Bedeaux and N. Bloembergen, Physica~Amsterdam! 69, 67 ~1973!.3Ultrafast Phenomena XII, edited by T. Elsasser, S. Mukamel, M. M. Munane, and N. F. Scherer~Springer-Verlag, Berlin, 2000!.

4S. Mukamel and R. M. Hochstrasser, Special Issue on MultidimensioSpectroscopies, Chem. Phys.266, 2 ~2001!.

5S. Mukamel, Annu. Rev. Phys. Chem.51, 691 ~2000!.6V. Chernyak, N. Wang, and S. Mukamel, Phys. Rep.263, 213 ~1995!.7J. A. Leegwater and S. Mukamel, J. Chem. Phys.101, 7388~1994!.8S. Mukamel, Z. Deng, and J. Grad, J. Opt. Soc. Am. B5, 804 ~1988!.9H. A. Lorentz,The Theory of Electrons~Dover, New York, 1952!.

10S. Mukamel, inMolecular Nonlinear Optics, edited by J. Zyss~Academic,New York, 1994!.

11J. Knoester and S. Mukamel, Phys. Rev. A39, 1899~1989!; ibid. 41, 3812~1990!.

12W. M. Zhang, V. Chernyak, and S. Mukamel, J. Chem. Phys.110, 5011~1999!.

13S. Woutersen, Y. Mu, G. Stock, and P. Hamm, Chem. Phys.266, 137~2001!.

14P. Hamm, M. Lim, W. F. DeGrado, and R. M. Hochstrasser, J. ChPhys.112, 1907~2000!.

15A. Tokmakoff, M. J. Lang, D. S. Larsen, G. R. Fleming, V. Chernyak, aS. Mukamel, Phys. Rev. Lett.79, 2702~1997!.

16Ultrafast Infrared and Raman Spectroscopy, edited by M. Fayer~DekkerInc., New York, 1999!.

17D. D. Dlott, Chem. Phys.266, 149 ~2001!.18J. Stenger, D. Madsen, J. Dreyer, E. T. J. Nibbering, P. Hamm, an

Elsaesser, J. Phys. Chem. A105, 2929~2001!.19S. Mukamel and D. S. Chemla, Special Issue on Confined Excitation

Molecular and Semiconductor Nanostructures, Chem. Phys.210, 1 ~1996!.20S. Wu, J. Appl. Phys.73, 8035~2001!.21K. Leo, M. Wegener, J. Shah, D. S. Chemla, E. O. Go¨bel, T. C. Damen, S.

Schmitt-Rink, and W. Scha¨fer, Phys. Rev. Lett.65, 1340~1990!.22S. Schmitt-Rink, S. Mukamel, K. Leo, J. Shah, and D. S. Chemla, P

Rev. A44, 2124~1991!.


al

.

T.

in

s.

23V. Lozovoy, I. Pastirk, M. G. Comstock, and M. Dantus, Chem. Phys.266,205 ~2001!.

24B. I. Grimberg, V. V. Lozovoy, M. Dantus, and S. Mukamel, J. PhyChem. A~in press!, ~2002!.

25S. T. Cundiff, J. M. Shacklette, E. A. Gibson, inUltrafast Phenomena XII,edited by T. Elsasser, S. Mukamel, M. M. Murnane, and N. F. Sche~Springer, Berlin, 2000!, pp. 344–346; S. T. Cundiff, J. M. Shacklette, anV. O. Lorenz, ‘‘Interaction Effects in Optically Dense Materials,’’ SPI2001, preprint.

26D. A. Blank, L. J. Kaufman, and G. R. Fleming, J. Chem. Phys.111, 3105~1999!.

27L. J. Kaufman, J. Heo, G. R. Fleming, J. Sung, and M. Cho, Chem. P266, 251 ~2001!.

28F. C. Spano and S. Mukamel, Phys. Rev. Lett.66, 1197~1991!.29S. Mukamel and A. Tortschanoff, Chem. Phys. Lett.~submitted!.30J. Jeener, A. Vlassenbroek, and P. Broekaert, J. Chem. Phys.103, 1309

~1995!.31W. Richter and W. S. Warren, Concepts Magn. Reson.12, 396 ~2000!.32Here we assume a single molecular species. This assumption simp

the notation, without changing the relevant physics. For a mixture we hto replacer0a(t) by ( ir0

i a i(t).33J. C. Kirkwood and A. C. Albrecht, J. Raman Spectrosc.31, 107 ~2000!.34T. Efthimiopoulos, M. E. Movsessian, M. Katharkis, and N. Merlemis,

Appl. Phys.80, 639 ~1996!.35I. Biaggio, Phys. Rev. Lett.82, 193 ~1999!.36A. W. Albrecht, A. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas,

Chem. Phys.111, 10934~1999!.37C. Scheurer, A. Piryatinski, and S. Mukamel, J. Am. Chem. Soc.123,

3114 ~2001!.38M. Dahlbom, T. Minami, V. Chernyak, T. Pullerits, V. Sundstro¨m, and S.

Mukamel, J. Phys. Chem. B104, 3976~2000!.39J. C. Kirkwood, C. Scheurer, V. Chernyak, and S. Mukamel, J. Ch

Phys.114, 2419~2001!.40V. Chernyak, W. M. Zhang, and S. Mukamel, J. Chem. Phys.109, 9587

~1998!.


hy

hy

ys

.

ho,

.


41M. Cho, N. F. Scherer, G. R. Fleming, and S. Mukamel, J. Chem. P96, 5618~1992!.

42N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, J. Chem. P93, 856 ~1990!.

43W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, Chem. PhLett. 238, 1 ~1995!.

44A. Tokmakoff, J. Phys. Chem. A104, 4247~2000!.45S. M. Gallagher Faeder and D. M. Jonas, J. Phys. Chem. A103, 10489

~1999!.46M. Khalil and A. Tokmakoff, Chem. Phys.206, 213 ~2001!.47O. Golonzka, M. Khalil, N. Demirdo¨ven, and A. Tokmakoff, Phys. Rev

Lett. 86, 2154~2001!.


s.

s.

.

48K. A. Merchant, D. E. Thompson, and M. D. Fayer, Phys. Rev. Lett.86,3899 ~2001!.

49M. Bonn, C. Hess, J. H. Miners, T. F. Heinz, H. J. Bakker, and M. CPhys. Rev. Lett.86, 1566~2001!.

50W. Zhao and J. C. Wright, Phys. Rev. Lett.84, 1411~2000!.51T. V. Shahbazyan, I. E. Perakis, and M. E. Raikh, Phys. Rev. Lett.84,

5896 ~2000!.52V. Astinov, K. J. Kubarych, C. J. Milne, and R. J. Dwayne Miller, Chem

Phys. Lett.327, 334 ~2000!.53T. I. C. Jansen, J. G. Snijders, and K. Duppen, J. Chem. Phys.113, 307

~2000!; ibid. 114, 10910~2001!.


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