wave packet interferometry for short-time electronic energy transfer: multidimensional optical...

Wave packet interferometry for short-time electronic energy transfer: Multidimensionaloptical spectroscopy in the time domainJeffrey A. Cina, Dmitri S. Kilin, and Travis S. Humble Citation: The Journal of Chemical Physics 118, 46 (2003); doi: 10.1063/1.1519259 View online: http://dx.doi.org/10.1063/1.1519259 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/118/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coherent control and time-dependent density functional theory: Towards creation of wave packets by ultrashortlaser pulses J. Chem. Phys. 136, 064104 (2012); 10.1063/1.3682980 Attosecond nonlinear Fourier transformation spectroscopy of CO 2 in extreme ultraviolet wavelength region J. Chem. Phys. 129, 161103 (2008); 10.1063/1.3006026 A photoelectron-photoion coincidence imaging apparatus for femtosecond time-resolved molecular dynamicswith electron time-of-flight resolution of σ = 18 ps and energy resolution Δ E ∕ E = 3.5 % Rev. Sci. Instrum. 79, 063108 (2008); 10.1063/1.2949142 Correlation of femtosecond wave packets and fluorescence interference in a conjugated polymer: Towards themeasurement of site homogeneous dephasing J. Chem. Phys. 120, 9870 (2004); 10.1063/1.1704635 Simulation of nonadiabatic wave packet interferometry using classical trajectories J. Chem. Phys. 112, 7345 (2000); 10.1063/1.481333

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Wave packet interferometry for short-time electronic energy transfer:Multidimensional optical spectroscopy in the time domain

Jeffrey A. Cina,a) Dmitri S. Kilin, and Travis S. HumbleDepartment of Chemistry and Oregon Center for Optics, University of Oregon, Eugene, Oregon 97403

~Received 18 July 2002; accepted 16 September 2002!

We develop a wave packet interferometry description of multidimensional ultrafast electronicspectroscopy for energy-transfer systems. After deriving a general perturbation-theory-basedexpression for the interference signal quadrilinear in the electric field amplitude of fourphase-locked pulses, we analyze its form in terms of the underlying energy-transfer wave packetdynamics in a simplified oriented model complex. We show that a combination of optical-phasecycling and polarization techniques will enable the experimental isolation of complex-valuedoverlaps between a ‘‘target’’ vibrational wave packet of first order in the energy-transfer couplingJ,characterizing the one-pass probabilityamplitudefor electronic energy transfer, and a collection ofvariable ‘‘reference’’ wave packets prepared independently of the energy-transfer process. With thehelp of quasiclassical phase-space arguments and analytic expressions for local signal variations, thelocation and form of peaks in the two-dimensional interferogram are interpreted in terms of thewave packet surface-crossing dynamics accompanying and giving rise to electronic energy transfer.© 2003 American Institute of Physics.@DOI: 10.1063/1.1519259#

I. INTRODUCTION

Advances in controlling1,2 or monitoring3 the interpulsedelays within sequences of femtosecond laser pulses withsuboptical-period accuracy have triggered the developmentof new electronic interference spectroscopies in which thesignal is quadrilinear in external fields bearing preciselyspecified optical phase relationships to one another.4–6 Thepower of these techniques for elucidating chemical dynamicslies in generating coherent signals that are directly propor-tional to the overlap between multipulse target and referencenuclear wave packets~in a pure-state description! or to amultipulse density matrix increment~in a mixed-state treat-ment!. This feature stands in contrast to more standardhomodyne-detected four-wave mixing spectroscopies, whichmeasure quantities proportional to the absolute-value-squared of the corresponding overlap or density matrix in-crement and do not require phase-controlled pulse se-quences. It also differs from ultrafast pump–probe7 and time-resolved fluorescence8 spectroscopies, which are linear in adensity matrix increment butbilinear in each of two externalpulses~pump and probe or gate! and which to first approxi-mation monitor time-dependent nuclear probability densities9

rather than overlaps between distinct amplitudes. The lattertwo examples additionally require a compromise betweentime resolution~pulse duration! and frequency resolution~spectral selectivity! in at least one of the pulses, whereasamplitude-sensitive interference spectroscopy approaches itsideal realization with pulses short enough to freeze nuclearmotion on the fastest time scales.

The experimental study of amplitude-sensitive nonlinearelectronic spectroscopy was initiated by Wiersma and co-workers with their measurement of phase-locked heterodyne-detected stimulated photon echoes~pl-HSPE! from dye mol-

ecules in solution.4,10 The current state of progress in thefield is exemplified by Jonas and co-workers’ recent two-dimensional Fourier-transform study of electronic transitionsin another solvated laser dye.6 In addition to incorporating anumber of technical improvements, those experiments6 mayreveal additional information about nonresonant effects be-cause they employ an external reference field that does notpass through the sample. There have been parallel advancesin heterodyne-detected multidimensional vibrational spec-troscopy of both solvated chromophores11 and neat liquids.12

Theory has guided all of these developments. The phase-locked heterodyned photon echo~and the closely relatedthree-pulse phase-locked pump–probe absorption! was sug-gested by Choet al.13 Tanimura and Mukamel14 analyzed theprospects for multidimensional vibrational spectroscopyprior to its successful implementation. Theoretical studies byMetiu and Engel15 prefigured the original experiments onlinear interference spectroscopy with phase-locked pulses.1,2

Despite the experimental progress and the contributionsfrom theory in elucidating the physical content of ultrafastelectronic interference measurements, much work remains tobe done toward interpreting these signals in terms of theunderlying molecular processes. In this context, it is desir-able to seek detailed pictures of many-body condensed-phasedynamics similar to those that have emerged from relativelysimpler pump–probe experiments16 and time-resolved coher-ent anti-Stokes–Raman scattering~tr-CARS! signals fromchromophores in low-temperature matrices.17 The most illu-minating interpretations in each case have been wave packetdescriptions akin to those originally devised for linearabsorption18 and resonance Raman19 data.9,20–23

In previous research that helped develop wave packetpictures for multidimensional interference experiments, weanalyzed the ability of nonlinear wave packet interferometrymeasurements equivalent to the pl-HSPE to prepare anda!Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 1 1 JANUARY 2003

460021-9606/2003/118(1)/46/16/$20.00 © 2003 American Institute of Physics


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measure superpositions between differently handed states inthe double-well potential of a chiral molecule.24 More re-cently, we have shown that nonlinear wave packet interfer-ometry has the capacity to reveal the complex-valued over-laps between a given short-pulse-generated target wavepacket on an excited potential energy surface of a polyatomicmolecule and an exhaustive collection of variable referencewave packets.25,26 Among other uses, this form of wavepacket interferometry could serve as a diagnostic tool forquantum control27 and molecule-based quantum informationprocessing.28

Here we make a detailed study of multidimensionaltime-domain electronic interference spectra for a model com-plex supporting electronic energy transfer, and interpret theirform in terms of the amplitude dynamics of the nuclear wavepackets on and between donor-excited and acceptor-excitedelectronic potential energy surfaces. The ultrafast dynamicsof electronic energy transfer in photosynthetic light-harvesting complexes,29 J-aggregates,30 and various modelcomplexes31–34has been the focus of intensive investigation.This prototypical process in chemical dynamics was de-scribed by Fo¨rster in an insightful heuristic treatment.35 Amore rigorous treatment, which generalizes the original de-scription by incorporating electronic coherence and inter- aswell as intramolecular motion was later put forward byRackovsky and Silbey36 and Soules and Duke.37,38While thefocus in prior studies has mostly been on donor- andacceptor-state population dynamics or nuclear probabilitydensities~proportional to the square and higher even powersof the transfer matrix elementJ!, we show here that wavepacket interferometry measurements are sensitive to thenuclear probability amplitude~linear in J! for electronic en-ergy transfer.

The desirability of obtaining amplitude-level dynamicalinformation for energy-transfer systems stems in part fromgeneral considerations similar to those that make such infor-mation valuable in the simpler context of light-inducednuclear motion within a single electronic state: There is nomore complete description of the time development of a sys-tem following interaction with an arbitrarily shaped resonantlaser pulse than that given by the excited-state nuclear wavefunction ~or density matrix increment!. For instance, while itis possible, in principle, to predict the future behavior of amolecule in a given electronic state~or states! from knowl-edge of the molecular Hamiltonian and nuclear wave func-tion~s! at some instant, the same is not true starting from theHamiltonian and an initial spatial probability density alone.It can also be argued that, as the most complete possibledescription of the state of nuclear motion, the time-dependent nuclear wave function should be uniquely sensi-tive to parameters defining what may be an only approxi-mately known molecular Hamiltonian. The predictedproportionality between the wave packet interferometry sig-nal and the energy-transfer matrix element is a vivid exampleof the latter feature of amplitude-sensitive measurements.

Within the specific context of energy-transfer systems,for which many conventional experimental methods havebeen crafted to monitor the change in population of donorand acceptor electronic states, the application of amplitude-

sensitive spectroscopies will represent a significant qualita-tive advance. It has been clear for some time that coherencesbetween electronic states, and hence the nuclear amplitude inthose states, can influence the course of electronic energytransfer.36 Moreover, there is every reason to believe thatrational attempts to control the energy transfer process exter-nally in real time will benefit from the development of ex-perimental methods for directly tracking donor-excited andacceptor-excited nuclear wave packets.

Our model energy-transfer complex is a simple one ame-nable to detailed analysis. It comprises a pair of interactingtwo-level chromophores whose electronic transition dipolemoments are fixed in space. Donor and acceptor chro-mophores each support a single intramolecular vibration. Wetouch on the issue of inhomogeneous broadening of the elec-tronic transition frequencies, but defer until later the incor-poration of important features such as multiple intra- andintermolecular vibrational modes, orientational disorder,electronic dephasing, and thermal congestion.39 By illustra-tion with this model complex, we show that nonlinear wavepacket interferometry, together with the tools of opticalphase control and polarization spectroscopy, is directly sen-sitive at the amplitude level to the dynamics of internuclearmotion accompanying and giving rise to energy-transfersurface-crossing transitions. It has the capacity to monitorthe basic process of electronic-nuclear state entanglementthat underlies energy transfer, and ameliorates a long-perceived shortcoming of conventional measurements ontwo-electronic-state systems.40

II. BASIC THEORY

We consider a dimer complex whose Hamiltonian,

H5u0&H0^0u1u1&H1^1u1u18&H18^18u1u2&H2^2u

1J$u18&^1u1u1&^18u%, ~1!

comprises four electronic levels:u0&5ugagb& with both mol-ecules unexcited,u1&5ueagb& with the ‘‘donor’’ excited,u18&5ugaeb& with the ‘‘acceptor’’ excited, andu2&5ueaeb&with both molecules excited. The corresponding nuclearHamiltonians,

H j5pa

2

2m1

pb2

2m1n j~qa ,qb!, ~2!

with potential energy surfaces,

n05mv2

2~qa

21qb2!, ~3!

n15e11mv2

2~~qa2d!21qb

2!, ~4!

n185e181mv2

2~qa

21~qb2d!2!, ~5!

n25e21mv2

2~~qa2d!21~qb2d!2!, ~6!

govern the motion of one intramolecular vibration in eachchromophore. The equilibrium position of a vibrational mode

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is displaced by a distanced when the host molecule is elec-tronically excited. The site energy of the two-exciton state istypically e2>e11e18 .41,42

Figure 1 shows a contour plot of the four site potentialenergy surfaces.43 A state change from 1 to 18—energytransfer from donor to acceptor—is expected to proceed ef-ficiently at positions wheren15n18 . This intersection occursalong a diagonal lineqb5qa2(e12e18)/mv2d whose loca-tion depends on the site-energy difference between donorand acceptor moieties.44 As energy transfer ensues, an en-tangled state develops through the processu1&uc1&→u1&uc1&1u18&uc18&. While the transferred population,^c18uc18&, is often measured, and some information aboutthe time evolution of the probability density,uc18(qa ,qb)u2,has already been obtained from ultrafast experiments, theredoes not yet appear to have been a direct determination ofthe entangled state itself, or in particular, of the transferredamplitudeuc18&.

We shall see that wave packet interferometry with opti-cally phase-locked ultrashort-pulse sequences can reveal thecomplex-valued overlap of a ‘‘target’’ wave packet describ-ing the energy-transfer amplitude with a collection of refer-ence wave packets of specified structure. The ultrashortpulses that generate the target and reference wave packetswill be part of a phase-controlled sequence, so we treat theevolution of the system under the time-dependent Hamil-tonianH(t)5H1VI(t), where

VI~ t !52m•EI~ t !, I 5A,B,C,D ~7!

describes the interaction with one of four pulses,

EI~ t !5eIAI~ t2t I !cos~V I~ t2t I !1F I !, ~8!

each of which has a well-defined polarization, envelopefunction, arrival time, carrier frequency, and phase. The in-tervals between pulses are referred to astp5tB2tA , tw5tC

2tB , and,td5tD2tC .45 The electronic dipole moment op-erator,

m5ma~ u1&^0u1u2&^18u!1mb~ u18&^0u1u2&^1u!1H.c.,~9!

allows transitions in which the exciton number changes byone. For our purposes, it is important that the molecular di-poles be nonparallel, so that pulses of different polarizationcan selectively address either the donor or the acceptor.

The experimental observable will be the portion of thepopulation of a specific excited electronic state that isquadrilinear in the field amplitudes~i.e., proportional toAAABACAD) immediately following the four-pulsesequence.46 To calculate that population, it is sufficient tohave the perturbative wave function,

uC~ t !&5@ t2tD#~11D !@ td#~11C!@ tw#

3~11B!@ tp#~11A!@ tA2t0#uC~ t0!&, ~10!

in which the molecular evolution operators47 are written as@ t#5exp(2iHt) and pulse overlap has been neglected.48 TheoperatorsI 5A, B, C, andD are pulse propagators,49

I 52 i E2`

`

dt@2t1t I #VI~t!@t2t I #, ~11!

whose forms are simplified by neglecting energy transferduring the pulses and adopting the rotating waveapproximation.50

In calculating the amplitude in each of the excited states,we take note of the fact that an odd number of laser–molecule interactions~one or three! are required to reacheither of the one-exciton states, while an even number~twoor four! are needed to reach the two-exciton state. For thenuclear wave function in the acceptor-excited state, we find

uc18~ t !&5^18u@ t2tD#$D@ td1tw1tp#1@ td#C@ tw1tp#

1@ td1tw#B@ tp#1@ td1tw1tp#A

1D@ td#C@ tw#B@ tp#1D@ td#C@ tw1tp#A

1D@ td1tw#B@ tp#A

1@ td#C@ tw#B@ tp#A%u0&un0&. ~12!

Similar expressions can readily be found for the nuclearprobability amplitude in the other electronic states.51

Since we are interested in the amplitude for 1→18 elec-tronic energy transfer of first order inJ, we need to examinethe contributions touc18& that are first and zeroth order inJ.The former are possible target wave packets while the latterare available as reference states. Keeping contributions to thefree evolution operator that are zeroth and first order inJ,52

@ t#5@ t#01@ t#1 , ~13!

we can rewrite Eq.~12! as

uc18~ t !&5uA18&1uB18&1uC18&1uD18&1u~JA!18&

1u~JB!18&1u~JC!18&1u~JD!18&

1u~DCB!18&1u~DCA!18&1u~DBA!18&

1u~CBA!18&1u~JDCB!18&1u~DCJB!18&

1u~JDCA!18&1u~DCJA!18&1u~JDBA!18&

1u~DBJA!18&1u~JCBA!18&1u~CBJA!18&,

~14!

FIG. 1. Schematic contour plots of potential energy surfaces for electronicground state, donor excited state, acceptor excited state, and two-excitonstate. The minimum of energy in each potential is at the corresponding siteenergy, as in Eqs.~3!–~6!. A sketch of the spatial path for a possible con-tribution to the target wave packet is also shown.

48 J. Chem. Phys., Vol. 118, No. 1, 1 January 2003 Cina, Kilin, and Humble


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which uses the short-hand notation

u~DCJB!18&5^18u@ t2tD#0D@ td#0C@ tw#1B@ tp#0u0&un0&,

~15!and so forth.

In order to calculate the interference populationP18 , wemust identify the quadrilinear contributions53 to ^c18uc18&: asum of four terms that are zeroth order inJ ~e.g., 2Re (DCB)18 u(A)18&) and twelve terms that are first order inJ~e.g., 2 Re(DCB)18u(JA)18&).

54 But the situation can simplifyconsiderably when we take account of laser polarization, aseach of the amplitudes in Eq.~14! depends on the relativeorientation of the field polarizations and molecular dipolemoment operators.55

III. CASE STUDY

We consider a simple example illustrating some basicfeatures of interference measurements of energy transfer: thedimer is assigned a well-defined internal and lab-framegeometry—as in a cryogenic matrix, macromolecular crystal,or layered structure56—with ma5m i and mb5m j along thespace-fixed xand y axes, respectively. Thenx-polarizedpulses can drive the transitions

~16!

while y-polarized pulses drive the transitions

~17!

Target amplitude on the acceptor state of first order inJcan result from energy transfer following excitation of thedonor state by anx-polarized laser pulse:

~18!

Preparation of a reference wave packet~zeroth order inJ!that can interfere with this 18 target requires one~or three!y-polarized pulses and two~or no! x-polarized pulses. Theformer case~with a total of threex-polarized pulses and oney-polarized pulse! is suitable for our purposes, as it affords awide variety of reference packets with both modes of vibra-tion set in motion. This feature is useful in generating over-lap with a target state whosea mode is set moving aftershort-pulse excitation of the donor and whoseb-mode mo-tion is initiated by energy transfer.

We consider four different polarization combinations:AyBxCxDx , AxByCxDx , AxBxCyDx , and AxBxCxDy . Un-

der the first of these, the acceptor-state amplitude reduces to

~19!

The resulting quadrilinear contribution toc18(t)uc18(t)&specifies the interference contribution to the acceptor-excitedstate population:

P18~AyBxCxDx!52 Re$^~Ay!18u~JDxCxBx!18&

1^~Ay!18u~DxCxJBx!18&

1^~DxCxAy!18u~JBx!18&

1^~DxBxAy!18u~JCx!18&

1^~CxBxAy!18u~JDx!18&%. ~20!

As expected, this signal contains no terms of zeroth orderin J.

It is useful to consider specific values of the relativeoptical phases of the pulses. Under a generalization of thephase-locking scheme developed by Scherer and co-workersfor linear wave packet interferometry with phase-lockedpulse pairs,1,2 it is possible4 to make sequences of the form~8! comprisingpairs of pulse-pairsAB andCD for which

FB5FA1Vptp1fp , FD5FC1Vdtd1fd , ~21!

respectively. The phase shiftfp(fd) is termed the lockingangle of theAB (CD) pulse pair at the locking frequencyVp(Vd). The interpulse-pairphase shifts need not be con-trolled, so for our purposes angles such asFC2FB may beformally assumed to sample a full range of values from 0 to2p over many laser shots.57

Expressions~7!, ~8!, and~11! show that parts of a pulsepropagatorI which induce upward transitions~proportionalto u1&^0u, u18&^0u, u2&^1u, andu2&^18u) contain a phase factorexp(2iFI). Parts ofI that induce downward transitions~pro-portional to u0&^1u, u0&^18u, u1&^2u, and u18&^2u) contain a



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phase factor exp(iFI). As a result, we see that the last twoterms in Eq.~20! are not phase controlled and hence averageto zero over many repetitions. Using the optical phases~21!,we find that

P18~fp ,fd!52 Re$exp~2 ifp2 ifd!

3~^~Ay!18u~JDxCxBx!18&~0!

1^~DxCxAy!18u~JBx!18&~0!!1exp~2 ifp

1 ifd!^~Ay!18u~DxCxJBx!18&~0!%, ~22!

in which the superscript~0! designates the overlap with bothpulse pairs in-phase~i.e., fp5fd50). While three distinctoverlaps can contribute to the interference signal~22! for anarbitrary choice of the phase-locking angles, it is possible tocombine signals with different phase shifts in order to isolatethe combinations

^~Ay!18u~JDxCxBx!18&~0!1^~DxCxAy!18u~JBx!18&

~0!

51

4 H P18~0,0!1P18S p

2,2

p

2 D1 iP18S p

2,0D

1 iP18S 0,p

2 D J ~23!

and

^~Ay!18u~DxCxJBx!18&~0!5

1

4 H P18~0,0!2P18S p

2,2

p

2 D1 iP18S p

2,0D2 iP18S 0,

p

2 D J .

~24!

Combining signals with different values offp andfd by theprescriptions~23! and~24! is an example of ‘‘phase cycling’’in optical spectroscopy.58,59 Equation ~24! illustrates a keyprediction of our analysis, that nonlinear wave packet inter-ferometry with pairs of pulse pairs is capable of isolating thefull complex overlap60 ^a18uj18&5^(Ay)18u(DxCxJBx)18&

(0)

between a given energy-transfer target

uj18&5exp~ iFB!^18u@ tw#1Bxu0&un0&, ~25!

generated by first-order energy transfer duringtw , and themembers of a collection of variable reference wave packets

ua18&5$exp~ iFB!^18uCx†@2td#0Dx

†@ td1tw1tp#0Ay

3@2tp#0u0&un0&%ufp5fd50 . ~26!

The C and D pulse propagators are reassigned to the refer-ence wave packet in Eq.~26! in order to highlight the corre-spondence between Eqs.~25! and ~18!; some phase factorshave been introduced in both bra and ket to render the targetwave packet independent oftp andfp . Notice that the ref-erence state evolvesbackwards in the two-exciton stateduring td .

The remaining polarization combinations of immediateinterest are considered in Appendix A. A complete scan ofinterpulse delays would also include the two interleavedpulse sequencesACBD andACDB; these alternative order-

ings provide some interesting overlaps, but we do not pauseto analyze them here. The analysis of this section has fo-cused on isolating a particular contribution toP18 , thequadrilinear acceptor-excited state population. Section V willaddress some practical issues associated with measuringP18itself, by polarized fluorescence for example, in the presenceof possible quadrilinear populations in the other excitedstates 1 and 2.

IV. ENERGY-TRANSFER WAVE PACKET DYNAMICS

A quasiclassical analysis of phase-space trajectories canindicate when the overlapa18uj18&, experimentally isolableaccording to Eq.~24!, should be nonvanishing. A schematictrajectory for the dominant portions of the target wave packetuj18& is shown in Fig. 2. We specialize to energy-transferwaiting times equal to half a vibrational period,tw5tvib/25p/v, in order to allow time for only one crossing of theintersection between then1(qa ,qb) andn18(qa ,qb) surfaces,and show the average values ofa-mode andb-mode positionand momentum that should result when the energy-transfertransition from 1 to 18 occurs afteratw in state 1. After thesurface-crossing transition, thea-mode trajectory sweeps outan anglep(12a) about (vqa ,qa)5(0,0); this istwice theangular displacement about the same origin that would haveoccurred if energy transfer had not taken place.61,62 Mean-while, the b-mode undergoes an angular displacement byp(12a) about (vqb ,qb)5(vd,0) after energy transfer.

A schematic diagram of the phase-space motion for thereference wave packetua18& is shown in Fig. 3. In order forthe target and reference wave packets to overlap signifi-cantly, their phase-space coordinates must nearly coincide.Comparison with Fig. 2 indicates that sizable overlap shouldoccur for

tp5S m112a

2 D tvib . ~27!

and

td5S n1a

2 D tvib , ~28!

wherem andn are non-negative integers.It is possible to obtain a closed-form expression for the

isolable overlap~24! in the short-pulse limit~pulse durationmuch less than the inverse absorption bandwidth or inverseFranck–Condon energy!. Analytic forms are given in Appen-dix B for the target~B5! and reference~B7! wave packets;from the commutation properties of harmonic oscillator cre-ation and annihilation operators, we find their overlap

^a1uj18&5^~Ay!18u~DxCxJBx!18&~0!

5 iJS m

2 D 4

areaA areaB areaC areaD

3ei ~e182Vp!tp1 i ~e182e21Vp!td

3E0

tvib/2

dt exp$ i ~e182e1!t1d2eivt

3~eivtp1e2 ivtd21!1d2e2 ivt22d2%. ~29!



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The pulse areas, areaI , and dimensionless displacement,d,are as defined in Appendix B.

A. Calculated interferograms

For chosen parameter values, Eq.~29! can be evaluatedeasily by numerical integration. We takev58.331024 a.u.>2pc(182.17 cm21), m599 000 a.u.>53.917 u, d50.3 a.u.>0.158 75 Å (d250.369 7655EFC/v, whereEFC

5mv2d2/2 is the Franck–Condon energy!. Both phase-locking frequencies@see Eq.~21!# are chosen to match thevertical transition energy for donor excitation (Vp5Vd

5e11EFC). Figure 4 shows the calculated two-dimensional~2D! interferogram for the case of equal site energiese185e1 . The interferogram for a downhill case withe185e1

22EFC is shown in Fig. 5.63

The locations of maximal interference signal are inqualitative accord with the quasiclassical predictions of Eqs.~27! and ~28! for both equal-energy and downhill cases. For

FIG. 2. Phase-space trajectories for thea-mode~above! andb-mode~below!of a target wave packet prepared by anx-polarizedB pulse. 18←1 energytransfer is shown occurring afteratw of motion in the donor-excited state.

FIG. 3. Phase-space trajectories for thea-mode~above! andb-mode~below!of a reference wave packet prepared byAy , Cx , andDx pulses. The last twopulses operate in reverse order, with an intervaltd of backward evolutionbetween them.

FIG. 4. Calculated interferogram for the case of equal site energies,e1

5e18 , and energy-transfer waiting timetw5tvib/2. The sign and size ofboth the real part and the imaginary part are physically meaningful, givingthe complex-valued overlap~29! @divided by J(m/2)4areaA areaB areaCareaDtvib] as a function of the intrapulse-pair delaystp and td . Positive~negative! contours are given by solid~dashed! lines spaced by 0.008 373,with a maximal~minimal! value1~2!0.125 594.



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the equal-energy interferogram, maxima inuâ18uj18&u arefound when tp5(m1120.155/2)tvib and td5(n10.155/2)tvib . Maximal signals in the downhill interfero-gram occur farther ‘‘off diagonal,’’ when tp5(m1120.5/2)tvib and td5(n10.5/2)tvib . The deviation of theeffective energy transfer timeatvib/2 from zero in the case

of equal site energies can be attributed to the finite timerequired for the quantum mechanical target wave packet toleave the potential-crossing line at the Franck–Condon point.

The fringe structure in the 2D interferograms of Figs. 4and 5 reveals amplitude-level information about the formand dynamics of the corresponding target wave packets. Therates of change with tp and td of the phase G52 i lnâ18uj18& are given in Eq.~C7! @or ~C9!# and Eq.~C8!@or ~C10!#, respectively~see Appendix C!. The last term onthe right-hand side of both Eqs.~C7! and~C8!, involving theposition matrix elements a18uqb2qbuj18& and â18uqa

2qauj18&, respectively, would vanish if the reference wavepacket were the same as the target within a constant factor, aswould be expected at the signal maxima in the limit of qua-siclassical target-state dynamics (EFC@v).

The fringe structure of the peaks in Fig. 4 corresponds to]G/]tp52]G/]td>20.53v at the signal maxima, andthese values imply

â18uqa2qauj18&52â18uqb2qbuj18&

>0.53â18uj18&

mvdei2p~0.0775!

for the case of equal site energies. The phase derivatives atthe signal maxima in Fig. 5 are]G/]tp>27.53v and]G/]td>0.14v. Equations~C7! and ~C8! in our case ofdownhill energy transfer then yield

â18uqa2qauj18&52â18uqb2qbuj18&

>0.14iâ18uj18&

mvd,

a smaller~but still non-negligible! deviation from quasiclas-sical behavior than in the equal-energy case.

B. Target and reference wave packets

In order to clarify the relationship between the experi-mentally measurable overlapâ18uj18& and the spatial formof the target and reference wave packets, we have calculatedthe target wave packet using Eq.~B5! and the reference withwhich it interferes at intrapair delays producing maximaloverlap using Eq.~B7!. These are shown in Figs. 6 and 7 forthe case of equal site energies and Figs. 8 and 9 for downhillenergy transfer. The target wave packet plotted as a functionof (qa ,qb) is seen to bear a discernible resemblance to anysingle peak in the corresponding interference signal plottedwith respect to (td ,tp) ~i.e., transposed about the diagonalfrom our Figs. 4 and 5!, especially in the equal-energy case.While this vivid correspondence is a feature of the two-vibration model, it serves to illustrate that the 2D interfero-grams are sensitive records of the energy-transfer surfacecrossingamplitude.

Sources for detailed features in the wave packet interfer-ometry signals of Figs. 4 and 5 can be identified in the cor-responding target amplitudes of Figs. 6 and 8, respectively.For instance, the temporally varying signal oscillation fre-quency observed in Fig. 4 can be attributed in part to thespatially varying local de Broglie wavelengths seen in Fig. 6.The equal energy interferogram~Fig. 4! exhibits small satel-

FIG. 5. Calculated interferogram for the case of downhill energy transfer.Same parameters as in Fig. 4 excepte185e122EFC . Also shown is theabsolute signal intensity, in which the temporal locations of certain satellitepeaks are readily discernible.



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lite peaks near (tp ,td)5(m10.5,n10.5)tvib. Phase-spacediagrams make it clear that reference wave packets preparedwith these delays~not shown! will overlap the small trailingregion of target amplitude near (qa ,qb)5(2d,0) that is vis-ible in Fig. 6. This trailing region of target probability am-plitude arises from nonresonant electronic energy transferlate in the waiting period. The wave packet launched by theBx pulse reaches the outerqa turning point on the donor-excited surface after half a period, and amplitude transfer canoccur because the wave packet lingers there for a time;2p/EFC, which is only slightly longer than the local elec-tronic nutation period 2p/(n182n1);2p/4EFC.

There is also a series of satellite peaks near (tp ,td)5(m10.5,n10.5)tvib in the downhill interferogram of Fig.5. These derive only in part from electronic nutation at theouter a-mode turning point@with local period 2p/(n182n1);2p/2EFC, somewhat longer than in the equal-energycase#. In the downhill case, the edge of the donor-excitedwave packet prepared by theBx pulse still penetrates the

n15n18 crossing region in the vicinity of (qa ,qb)5(3d/2,d/2) as the packet reaches the outer turning point,and resonant energy transfer ensues. Both of these processescontribute to the trailing region of target probability ampli-tude near (qa ,qb)'(2d,0) in Fig. 8 and the correspondingsatellites in the interferogram. The downhill interferogramhas additional satellite peaks near (tp ,td)5(m,n)tvib . Thesecome once again from both nonresonant electronic nutation@near (qa ,qb)5(0,0)] and resonant transfer@for (qa ,qb)'(d/2,2d/2)] when theBx wave packet is in the Franck–Condon region. The resulting contributions to the targetwave packet evolve as a leading region of probability ampli-tude that is localized between (qa ,qb)5(0,2d) and(qa ,qb)5(2d/2,5d/2) by the end of the waiting period.

V. DISCUSSION

A. Measuring P18

In Sec. III and Appendix A, we determined the contribu-tions toP18(ABCD), that are isolable by phase cycling un-

FIG. 6. Real and imaginary parts of the target wave packet for the equalenergy case. Equation~B5! is plotted as ^qa ,qbuj18& divided byJ(m/2)areaBtvib /d. Contour lines of the same sign are spaced by 0.013 151and take maximal~minimal! values1~2!0.197 261. Locations of potentialminima for donor-excited~1! and acceptor-excited (18) states are alsoshown, as is the line of intersection betweenn1 andn18 .

FIG. 7. Real and imaginary parts of the reference wave packet for the equalenergy case at (tp ,td)max5(0.9225,1.0775)tvib Equation~B7! is plotted as^qa ,qbua18& divided by (m/2)3areaA areaC areaD /d. Contours of the samesign are spaced by 0.102 27 and take maximal~minimal! values1~2!1.534 11.



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der various polarization combinations. WhileP18 is indepen-dently observable in the sense of being the expectation valueof the Hermitian operatoru18&^18u ~or the quadrilinear con-tribution to that quantity!, we still need to consider how itcould be measured in practice.

One strategy would be to monitor the time- andfrequency-integratedy-polarized emission from 18→0. Butany quadrilinear contribution to the population of the two-exciton state 2 could give rise toy-polarized 2→1 emissionthat would obscure the sought-for signal. This complicationcan perhaps be overcome by the simple expedient of spec-trally filtering the y-polarized emission. While we have as-sumed thate25e11e18 in our calculations, this is an ines-sential choice not strictly obeyed in practice. In actuality, thepeak frequency of relaxed emission from the acceptor chro-mophore will depend slightly on whether the acceptor mol-ecule is or is not electronically excited. Thusy-polarizedemission fromP18 and P2 should be spectrally distinguish-able.

Spectral filtration of the emitted light may not be neces-sary in the case of downhill energy transfer, however. In thiscasex-polarizedemission can serve as an independent mea-sure of the relevant contribution toP2 under AyBxCxDx .With these polarizations, the amplitudes that overlap to pro-duce a quadrilinear contribution to the population of state 2result from the electronic transitions

~30!

~31!

~32!

The phase structure of each term is indicated; there is nocontribution toP2 of zeroth order inJ.

We showed in Sec. III that phase-cycling selection of thesignal proportional to exp(2ifp1ifd) could be used to ex-tract the single overlap(Ay)18u(DxCxJBx)18&

(0) from theother contributions toP18 . Expression~32! shows, though,that a portion ofP2 with the same optical phase could makean additional contribution toy-polarized emission. State 2can also emit withx polarization, and we can check the phasestructure of the quadrilinearx-emitting donor state popula-

tion, P1(AyBxCxDx), to see whether an exp(2ifp1ifd)term exists there as well. It happens that two contributions toP1(AyBxCxDx), those arising from overlaps between theamplitudes,

FIG. 8. Same as Fig. 6, but for the downhill case. The absolute value of thetarget wave packet is also plotted in order to emphasize the spatial locationsof local maxima in the probability amplitude due to nonresonant electronicnutation and resonant transfer from the edges of the nuclear wave packet.Both of these secondary energy-transfer mechanisms proceed efficiently atthe inner and outer turning points of motion in the donor-excited potentialwell.



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~33!

~34!

both carry an exp(2ifp2ifd) phase factor; so phase selec-tion of thex-polarized emission would not generally give anunobscured view of the exp(2ifp1ifd) contribution toP2 .But the overlap~33! and~34! involve backwardenergy trans-fer from the acceptor to the donor. If the acceptor site energyis sufficiently far below that of the donor~as in our downhillcase, where the acceptor-excited Franck–Condon point is it-self EFC below the intersection energy betweenn18 andn1),the backwards transition cannot occur for energetic reasons,and the corresponding contribution toP1 vanishes. Thus forthe downhill case, the exp(2ifp1ifd) component ofP2 canbe determined as the sole contribution tox-polarized emis-sion having this phase signature. Having been deter-mined independently, thisP2 contribution can be unam-biguously removed from they-polarized emission withoutspectral filtration, leaving the sought-for overlap^(Ay)18u(DxCxJBx)18&

(0) as the only remaining signal.

B. Prospects for state determination

The collection of reference wave packets~26! availableunderAyBxCxDx polarization is limited to botha-mode andb-mode Franck–Condon energy shells~see Fig. 3!. The in-terpair delaytw5tvib/2 places the target~25! on the sameenergy shell, allowing sizable overlap. Since referencepacket formation occurs first on then18 surface~where onlyqb is displaced! and then onn2 ~where bothqa and qb aredisplaced!, the prospects for determining overlaps with anexhaustive collection of reference packets—including manywith average energyoff the Franck–Condon shell—mightappear dim. But measuring overlaps between target wavepackets withtw slightly different from tvib/2 and on-shellreference packets can be nearly equivalent to measuringoverlaps between targets withtw5tvib/2 and off-shell refer-ence wave packets.

To exhibit this equivalence we note that

@ tw1dtw#15@dtw#0@ tw#11@dtw#1@ tw#0 ~35!

~see Ref. 52!. For slight increments in waiting time, the sec-ond term in Eq. ~35! can be neglected~unless e12e18>4EFC, a still greater energy difference than in our down-hill case!, and

^a18uj18&u tw1dtw5^n0u^0uAy

†@2tp2tw2dtw2td#0

3Dx@ td#0Cxu18&^18u@ tw1dtw#1

3Bx@ tp#0u0&un0&

>^n0u^0uAy†@2tp2tw2dtw2td#0

3Dx@ td#0Cx@dtw#0u18&

3^18u@ tw#1Bx@ tp#0u0&un0&

5^a18uj18&u tw. ~36!

The effective reference wave packetua18& of Eq. ~36! fol-lows a phase-space trajectory that lies off the Franck–Condon shell for the internal vibration of the donor, increas-ing the dimensionality of the accessiblea-mode phase spacefrom one to two. This is a step toward providing an exhaus-tive set of reference wave packets to interfere with the targetpacket. To gain a second dimension in theb-mode space mayrequire additional optical transitions to access state-0 orstate-1 surfaces during reference state preparation.

C. Echolike versus nonecholike signals

When experiments of the kind considered here are car-ried out on chromophores in low-temperature solids, it willbe necessary to consider the effects of inhomogeneousbroadening. As a result of differences in local environment,the site energiese1 , e18 , ande2 in Eqs.~3! through~6! mayvary with location in the sample. This spatial inhomogeneityin the site energies could affect the nonlinear wave packetinterferometry signal from a bulk sample,64 but in energy-transfer systems, the effects of inhomogeneity would dependon the degree of correlation between donor and acceptor en-ergy shifts.

The simplest situation would entail perfect correlationbetween donor and acceptor site energies, so thate15 e1

1de, e185 e181de, and e25 e212de. In this case, thesingle overlap~24! that is isolable underAyBxCxDx polar-ization depends on the site-energy shift as

^~Ay!18u~DxCxJBx!18&~0!}exp$ ide~ tp2td!%, ~37!

and the overlap that is isolable underAxByCxDx polarization@see Eq.~A3!# goes as

^~By!18u~DxCxJAx!18&~0!}exp$2 ide~ tp1td!%. ~38!

In this limiting situation, we would naturally identify theoverlaps~37! and ~38! as arising from echolike and non-echolike signals, respectively; the former overlap suppressesthe effects of~correlated! inhomogeneous broadening alongthe td'tp diagonal and the latter does not.65

Dynamical considerations come into play as well, how-ever. The semiclassical criteria~27! and~28! suggest that theoverlap ^(Ay)18u(DxCxJBx)18&

(0) can be nonzero when thedelay difference in Eq.~37! takes on values

tp2td5~m2n112a!tvib , ~39!

some or all of which may appear off the diagonal. On theother hand, Eq.~A7! indicates that (By)18u(DxCxJAx)18&

(0)

can be nonzero when



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tp1td5~n1a!tvib . ~40!

This time can be sufficiently short~e.g., forn50) that thenonecholike signal arising from the overlap in Eq.~38!would not be suppressed by inhomogeneous dephasing.

D. Related theoretical work

Szocs et al.66 very recently made a theoretical study of2D photon echo spectroscopy on an excitonic two-site modelsystem somewhat related to the present investigation. As astep toward a full analysis of vibrational and electronic co-herence effects in photon echo signals from conjugated poly-mers, they considered the frequency resolved time-dependentthird-order polarization from a purely electronic equal-energy two-site~four-level! system. The formal treatmentand specializing conditions differ in several respects fromthose adopted here. Szo¨cs et al. found that the positions ofoff-diagonal peaks in the frequency domain interferogramcarry information on the energy-transfer coupling strength,the relative heights of four characteristic peaks reflect theangle between the site-localized transition dipole moments,and the peak shapes depend on the ratio of homogeneous toinhomogeneous dephasing. Consistent with the conclusionsof the present study, they concluded that the~heterodynedetected! frequency domain signal carried more structuraland dynamical information than its~homodyne detected!time-domain counterpart.

In an earlier and more general study, Zhanget al.39 alsoanalyzed two-dimensional electronic spectroscopy fromsmall aggregates coupled to a vibrational bath. Like those ofSzocset al.,66 the specializing conditions of Zhanget al. arerather different from ours, and the possibility of coherentvibrational motion is suppressed. Emphasizing the structuralinformation content of these methods, they showed how vari-ous 2D techniques could provide information on intermo-lecular coupling strengths and patterns. Their study did notdwell explicitly on the possibility of separately addressingdonor and acceptor moieties with differently polarizedpulses.67 Interestingly, Appendix F of Ref. 39 contains ex-pressions for the contributions to various 2D four-wave mix-ing signal of first order in the electronic coupling constantJ.

E. Relationship to time-resolved CARS

Nonlinear wave packet interferometry measurements ofthe kind suggested here and elsewhere24,25 share some ex-perimental and theoretical features with ultrafast time-resolved coherent anti-Stokes–Raman scattering measure-ments recently made by Karavistaset al.17 The tr-CARSmeasurements did not involve electronic energy transfer,but—like wave packet interferometry—are sensitive to co-herently excited electronic transitions and nuclear dynamicsin a low-temperature medium (I2 in an cryogenic argon ma-trix!, where quantum mechanical wave packet motion is ob-served to play a significant role. Similar samples, along withmolecular beams, could naturally be studied by wave packetintereferometry measurements as well.

Electronic dephasing~or decoherence! effects of the kindobserved by time-resolved CARS will undoubtedly comeinto play in nonlinear wave packet interferometry measure-ments as well. Because of the nearly harmonic nature of lowtemperature host lattices, both time-resolved CARS andwave packet interferometry experiments on chromophores insolid matrices should be valuable testing grounds for quan-titative models of electronic decoherence.68–70

FIG. 9. Real part, imaginary part, and absolute value of the reference wavepacket for the downhill case at (tp ,td)max5(0.75,1.25)tvib . Note the quasi-classical coherent-state structure of the reference wave packet. Plotting pa-rameters are the same as in Fig. 7.



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F. Concluding remarks

Our analysis of nonlinear wave packet interferometry foran energy-transfer complex illustrates the potential power ofthis form of multidimensional electronic spectroscopy forobserving coupled electronic and nuclear dynamics of many-body condensed molecular systems at the amplitude level.Our calculations for a simple model complex show that thisform of ultrafast multidimensional electronic spectroscopy—along with the tools of polarization spectroscopy and opticalphase control—has the capacity to measure not just the evo-lution of electronic populations and nuclear probability den-sities, but the time development of nuclear wave functionsaccompanying energy-transfer surface-crossing transitions.

Further research along the lines initiated here can ad-dress the important issues of multiple intra- and intermolecu-lar modes, electronic dephasing, thermal effects, orienta-tional disorder, and for gas phase samples, rotationaldynamics and congestion. The neglect of vibrational dissipa-tion specifically, a prerequisite for the onset of incoherentenergy transfer as described by Fo¨rster theory, is in keepingwith our specialization to very short-time dynamics~i.e., be-fore the initial nonequilibrium vibrational distribution in thedonor-excited state has decayed to quasithermal equilib-rium!. The amplitude-level dynamics studied here may nolonger be readily discernible once the regime of Fo¨rstertransfer ensues, but wave packet interferometry could be asensitive means for observing theonsetof incoherent hop-ping and the decay of donor–acceptor electronic coherencewith increasing transfer intervals. In this connection it isworth mentioning that while the treatment given in this paperis based on a pure-state description of the energy-transfercomplex, the approach is also directly applicable to isolatedor condensed-phase systems with population distributed overthermally occupied levels. This generalization is accom-plished formally by the elementary step of summing withBoltzmann weight over the initially populated energy statesof the complex plus bath.21

In the interest of conceptual and computational simplic-ity, we here assumed weak energy-transfer coupling~J lessthanv!. Weak coupling should not be a requirement for in-terwell coherence to be observed by wave packet interferom-etry, however, and it might be anticipated that stronger cou-pling would lead to more intense~if less simplyinterpretable! interference spectroscopy signals. Further nu-merical studies of wave packet interferometry in the strongand intermediate coupling regimes will be a natural topic forfuture research. Since the numerical calculations reportedhere ~but not the basic theoretical expressions! use laserpulses that are arbitrarily abrupt on the vibrational timescale, it will be necessary to further investigate the practicalconsequences of nonzero pulse duration and finite spectralbandwidth. We have specialized to pulse sequences with afew specific polarization combinations that are sensitive tothe nuclear wave function arising from energy transfer to theacceptor-excited potential energy surface. Future studies caninclude pulse sequences of arbitrary polarization.71

APPENDIX A: ALTERNATIVE POLARIZATIONS

Here we summarize the interference signal under alter-native polarization conditions for the oriented model systemconsidered in Sec. III. Under polarizationsAxByCxDx , theamplitude in the acceptor-excited state through first order inJ becomes

uc18~ t !&5u~By!18&1u~JAx!18&1u~JCx!18&1u~JDx!18&

1u~DxCxBy!18&1u~DxByAx!18&

1u~CxByAx!18&1u~JDxCxAx!18&

1u~DxCxJAx!18& ~A1!

@compare Eq. ~19!#. The four-pulse contribution to^c18(t)uc18(t)& gives the interference signal

P18~AxByCxDx!52 Re$^~By!18u~JDxCxAx!18&

1^~By!18u~DxCxJAx!18&

1^~DxCxBy!18u~JAx!18&

1^~DxByAx!18u~JCx!18&

1^~CxByAx!18u~JDx!18&%, ~A2!

which can be compared to Eq.~20!. As in Sec. III, we canidentify the optical phase-structure of the terms in Eq.~A2!in order to determine which portions of the signal are isol-able by phase cycling. The last two overlaps in curly bracesare not phase locked because the pulses within each pairinduce transitions in the same direction in one of the over-lapped states or in opposite directions in each of the over-lapped [email protected]., in the fourth overlap of Eq.~A2!, Ax andBy both induce upward transitions in the three-pulse wavepacket, whileCx drives an upward transition in the one-pulsewave packet andDx drives a downward transition in thethree-pulse wave packet#. The last two overlaps in Eq.~A2!are therefore expected to average to zero over many repeti-tions, and the signal becomes

P18~fp ,fd!52 Re$exp~ ifp2 ifd!

3~^~By!18u~JDxCxAx!18&~0!

1^~DxCxBy!18u~JAx!18&~0!!

1exp~ ifp1 ifd!

3^~By!18u~DxCxJAx!18&~0!%, ~A3!

underAxByCxDx polarization@compare Eq.~22!#. Again itproves possible to isolate a single overlap,^(By)18u(DxCxJAx)18&

(0), by combining signals with differ-ent intrapulse-pair phase shifts. In the single overlap isolablefrom Eq. ~A3!, electronic energy transfer takes place duringthe interval tp1tw ; this differs slightly from the situationwith Eq. ~24!. More significantly, as is discussed in Sec. V,the overlap isolable from Eq.~A3! may exhibit nonecholikebehavior under some regimes of inhomogeneous broadening,an issue that could be the subject of an in-depth investiga-tion.



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For the two remaining polarization combinations, nosingle overlap is strictly isolable by phase cycling. With po-larizations AxBxCyDx , the quadrilinear portion of theenergy-transfer signal becomes


3~^~Cy!18u~JDxBxAx!18&~0!

1^~DxCyBx!18u~JAx!18&~0!!

3exp~2 ifp2 ifd!

3~^~DxCyAx!18u~JBx!18&

1^~CyBxAx!18u~JDx!18&~0!!%. ~A4!

And finally, under polarizationAxBxCxDy , we find


3~^~Dy!18u~JCxBxAx!18&~0!

1^~Dy!18u~CxBxJAx!18&~0!!

3exp~2 ifp1 ifd!

3~^~DyCxAx!18u~JBx!18&~0!

1^~DyBxAx!18u~JCx!18&~0!!%. ~A5!

When ~for which interpulse delays, if any! a sum of twooverlaps in Eq.~22!, ~A3!, ~A4!, or ~A5! that is isolable byphase cycling reduces to a single overlap can only be deter-mined by a detailed analysis of the nuclear dynamics.72

Let us consider the wave packet dynamics associatedwith the single overlap, (By)18u(DxCxJAx)18&

(0), that isisolable by combining signals~A3! with different phase lock-ing angles. This overlap is equivalent to that between refer-ence and target wave packets resulting from the sequences ofelectronic transitions

~A6!

respectively. Phase-space diagrams for the reference packetcan be drawn which are analogous to those in Figs. 2 and 3.If we specialize to the case with energy-transfer intervaltp

1tw5tvib/2 just large enough to ensure one crossing of thedonor–acceptor intersection, then coincidence between tar-get and reference phase-points requires

tp5atvib

2, tw5~12a!

tvib

2, td5S n1

a

2 D tvib .

~A7!

APPENDIX B: TARGET AND REFERENCEWAVE PACKETS

Here we derive analytic expressions for the target~ref-erence! wave packet defined by Eq.~25! @Eq. ~26!#. Using

the short-pulse limit of theBx pulse propagator~11! and thefree-evolution operator of first-order inJ given in Ref. 52,we find

uj18&5mJ

2areaBE

0

twdte2 iH 18~ tw2t!e2 iH 1tu0,0&. ~B1!

The target wave packet~B1! is assumed to originate from thevibrational ground state of the electronically unexcited com-plex; areaI5*dt AI(t) is the integrated pulse envelope.

We can use harmonic-oscillator creation and annihilationoperators to go further with Eq.~B1!. Adopting the usualdefinitions qa5(2mv)21/2(a†1a), pa5 i (mv/2)1/2(a†

2a), qb5(2mv)21/2(b†1b), pb5 i (mv/2)1/2(b†2b), andintroducing the corresponding phase-space translation opera-tors Ta(a)5exp(aa†2a*a) and Tb(b)5exp(bb†2b*b),73

we have

H15Ta~d!~H01e1!Ta†~d!, ~B2!

H185Tb~d!~H01e18!Tb†~d!, ~B3!

H25Ta~d!Tb~d!~H01e2!Tb†~d!Ta

†~d!. ~B4!

From relations~B1!–~B4! and some operator algebra73 it fol-lows that

uj18&52mJ

2areaBe22d22 i e1tvib/2

3E0

tvib/2

dt exp$ i ~e182e1!t22id2 sinvt

1d~12eivt!a†

1d~11eivt!b†%u0,0&. ~B5!

The dimensionless displacement isd5(mv/2)1/2d, and thewaiting time has been set totw5tvib/2.

In the short-pulse limit, the reference wave packet~26!prepared from the vibronic ground state remains a quasiclas-sical coherent state of the form

ua18&52 i S m

2 D 3

areaA areaC areaD eiVptp2 iVdtdeiH 2td

3e2 iH 18~ tp1tw1td!eiH 0tpu0,0&. ~B6!

Relations~B3! and ~B4! plus some operator manipulations~and the choicetw5tvib/2) lead to the more explicit expres-sion

ua18&5 i S m

2 D 3

areaA areaC areaD expH 2 ie18tvib

2

1 i ~Vp2e18!tp2 i ~Vd1e182e2!td1d2eivtd

2d2e2 ivtp22d21d~12eivtd!a†

1d~11e2 ivtp!b†J u0,0&. ~B7!

The inner product of Eqs.~B5! and~B7! leads to the experi-mentally isolable complex overlap, Eq.~29!, that is plotted inSec. IV.



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APPENDIX C: INTERFEROGRAM FRINGESTRUCTURE

In order to better understand the local fringe structure ofthe peaks in the 2D interferograms presented in Sec. IV, it isuseful to consider the first derivatives with respect totp andtd of the complex overlap between the target~25! and refer-ence ~26! wave packets. In the short-pulse limit adoptedthere, the reference wave packet~though not the target! pre-pared from the vibrational ground state is a minimum-uncertainty Gaussian wave packet,74

^qa ,qbua18&5expH 2mv

2~qa2qa!22

mv

2~qb2qb!2

1 i pa~qa2qa!1 i pb~qb2qb!1 igJ ,

~C1!

whose time evolution is governed by the quasiclassical mo-tion of the average values of the position and momentumexpectation values

qa[â18uqaua18&

â18ua18&5d~12cosvtd!, ~C2!

pa[â18upaua18&

â18ua18&52mvd sinvtd , ~C3!

qb[â18uqbua18&

â18ua1&5d~12cosv~ tw1tp!!, ~C4!

pb[â18upbua18&

â18ua18&5mvd sinv~ tw1tp!, ~C5!

and the~complex! phase

g5g02p

22 i lnH S m

2 D 3

areaA areaC areaDJ 1Vptp

2Vdtd1fp2fd2vtw2e18~ td1tw1tp!1e2td

1mvd2

4~sin 2vtd2sin 2v~ tw1tp!!. ~C6!

Contributions from the short-pulse limit of the pulse propa-gators~11! have been included in Eq.~C6!; g0 is the arbi-trary initial phase attached toqa ,qbun0&. The wave function~C1! is the position-space representation of Eq.~B7!.

Writing the target-reference overlap in terms of a com-plex phase, a18uj18&5exp(iG), leads to the sought-for timederivatives

]G

]tp52Vp1e181

mv2d2

2

2mv2deiv~ tw1tp!â18uqb2qbuj18&

â18uj18&, ~C7!

]G

]td5Vd1e182e22

mv2d2

2

1mv2 de2 ivtdâ18uqa2qauj18&

â18uj18&. ~C8!

As could have been anticipated from Fig. 3, the derivativewith respect totd(tp) is sensitive to only the donor~acceptor!vibration. The interferogram fringe structure yields informa-tion on the spatial form of the target energy-transfer ampli-tude through the last term on the right-hand side of Eqs.~C7!and ~C8!.

Our closed-form expression forâ18uj18& gives equiva-lent alternative forms for the delay derivatives ofG. By dif-ferentiation of Eq.~29! we obtain

]G

]tp52 i

]â18uj18&/]tp

â18uj18&52Vp1e181vd2eivtpR

~C9!

and

]G

]td52 i

]â18uj18&/]td

â18uj18&5Vd1e182e22vd2e2 ivtdR

~C10!

where, in both cases,

R5*0

tvib/2dt exp$ i ~e182e11v!t1d2eivt~eivtp1e2 ivtd21!1d2e2 ivt22d2%

*0tvib/2dt exp$ i ~e182e1!t1d2eivt~eivtp1e2 ivtd21!1d2e2 ivt22d2%

. ~C11!

Since the same ratioR appears in both Eqs.~C9! and ~C10!, and the integrand in the numerator of Eq.~C11! simplycontains an extra factor exp(ivt), we have the remarkable result

eiv~ tp1td!

vd2 S Vd1e182e21 i]

]tdD â18uj18&e1e18 ,e2

51

vd2 S Vp1e182 i]

]tpD â18uj18&e1 ,e18e2

5â18uj18&e1 ,e181v,e21v . ~C12!

The rate of change of the interferogram along either time axis specifies the complex-valued interference signal from adifferentenergy-transfer complexwith the acceptor and two-exciton site energies increased by one vibrational quantum. We have usedthe first equality in Eq.~C12! as a consistency check on thetp andtd fringe structure of the calculated interferograms presentedin Sec. IV.



128.138.73.68 On: Fri, 19 Dec 2014 20:30:22

ACKNOWLEDGMENTS

One of the authors~J.A.C.! thanks Rick Dahlquist, Uni-versity of Oregon, for encouraging him to consider opticalphase cycling in the context of wave packet interferometry;Howard Carmichael, Oregon, and Sid Nagel, The Universityof Chicago, for helping him understand harmonic oscillators;and David Jonas, University of Colorado, for asking aboutthe observability of the phase change associated with energytransfer mentioned in Ref. 52. Chris Bardeen, University ofIllinois, kindly brought the work of Ref. 46 to our attention.We benefited from many helpful conversations with Profes-sor Tom Dyke and Professor Andy Marcus, UO, who arepresently attempting experiments along the lines discussedhere. This work was supported by a grant from the US Na-tional Science Foundation. Additional support from the UODepartment of Chemistry and Oregon Center for Optics isgratefully acknowledged.

1N. F. Scherer, R. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys.95,1487 ~1991!.

2N. F. Scherer, A. Matro, R. J. Carlson, M. Du, L. D. Ziegler, J. A. Cina,and G. R. Fleming, J. Chem. Phys.96, 4180~1992!.

3A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, J.Chem. Phys.111, 10934~1999!; 115, 5691~2001!.

4W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, Chem. Phys.233, 287 ~1998!.

5L. Lepetit and M. Joffre, Opt. Lett.21, 564 ~1996!.6J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, J. Chem. Phys.115, 6606~2001!.

7A. H. Zewail, Science242, 1645~1988!.8A. A. Mokhtari and J. Chesnoy, IEEE J. Quantum Electron.25, 2528~1989!; P. F. Barbara and W. Jarzeba, Adv. Photochem.15, 1 ~1990!; R.Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Nature~Lon-don! 369, 471 ~1994!; M. Chattoraj, B. A. King, G. U. Bublitz, and S. G.Boxer, Proc. Natl. Acad. Sci. U.S.A.93, 8362~1996!.

9L. W. Ungar and J. A. Cina, Adv. Chem. Phys.100, 171 ~1997!.10See also S. Saikan, T. Nakabayashi, Y. Kanematsu, and N. Tato, Phys.

Rev. B38, 7777~1988!; W. H. Hesselink and D. A. Wiersma, Phys. Rev.Lett. 43, 1991~1979!.

11M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, J. Am. Chem. Soc.122, 11541~2000!; O. Golonzka, M. Khalil, N. Demirdo¨ven, and A. Tok-makoff, J. Chem. Phys.115, 10814 ~2001!; M. Zanni, N.-H. Ge, Y. S.Kim, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. U.S.A.98, 11265~2001!.

12L. J. Kaufman, J. Heo, L. D. Ziegler, and G. R. Fleming, Phys. Rev. Lett.88, 207402~2002!; K. J. Kubarych, C. J. Milne, S. Lin, V. Astinov, andR. J. D. Miller, J. Chem. Phys.116, 2016~2002!.

13M. Cho, N. F. Scherer, G. R. Fleming, and S. Mukamel, J. Chem. Phys.96, 5618~1992!; S. Mukamel,Principles of Nonlinear Optical Spectros-copy ~Oxford University Press, New York, 1995!.

14Y. Tanimura and S. Mukamel, J. Chem. Phys.99, 9496~1993!.15H. Metiu and V. Engel, J. Opt. Soc. Am. B7, 1709~1990!. See also W. S.

Warren and A. H. Zewail, J. Chem. Phys.78, 2279 ~1983!; 78, 2298~1983!.

16W. Domcke and G. Stock, Adv. Chem. Phys.100, 1 ~1997!; W. T. Pollard,S. L. Dexheimer, Q. Wang, L. A. Peteanu, C. V. Shank, and R. A. Mathies,J. Phys. Chem.96, 6147 ~1992!; D. M. Jonas, S. E. Bradforth, S. A.Passino, and G. R. Fleming,ibid. 99, 2594 ~1995!; T. J. Smith, L. W.Ungar, and J. A. Cina, J. Lumin.58, 66 ~1994!; Z. Li, J. Fang, and C. C.Martens, J. Chem. Phys.104, 6919 ~1996!; C. J. Bardeen, J. Che, K. R.Wilson, V. V. Yakovlev, V. A. Apkarian, C. C. Martens, R. Zadoyan, B.Kohler, and M. Messina,ibid. 106, 8486 ~1997!; Y.-S. Shen and J. A.Cina, ibid. 110, 9793~1999!.

17M. Karavitas, R. Zadoyan, and V. A. Apkarian, J. Chem. Phys.114, 4131~2001!; see also I. Pinkas, G. Knopp, and Y. Prior,ibid. 115, 236 ~2001!.

18E. J. Heller, J. Chem. Phys.68, 2066~1978!.19E. J. Heller, R. L. Sundberg, and D. Tannor, J. Phys. Chem.86, 1822

~1982!; J. L. McHale, Acc. Chem. Res.34, 265 ~2001!.

20S. Mukamel, J. Phys. Chem.88, 3185~1984!; Y. J. Yan and S. Mukamel,J. Chem. Phys.88, 5735~1988!.

21R. A. Harris, R. A. Mathies, and W. T. Pollard, J. Chem. Phys.85, 3744~1986!.

22A more general approach based on a moment expansion of physical ob-servables, but independent of a Gaussian ansatz on the wave function ordensity matrix, has been recently put forward by O. V. Prezhdo and Y. V.Pereverzev, J. Chem. Phys.113, 6557~2000!; 116, 4450~2002!.

23See also D. J. Tannor,Introduction to Quantum Mechanics: A Time De-pendent Perspective~University Science Books, New York, 2003!.

24J. A. Cina and R. A. Harris, J. Chem. Phys.100, 2531 ~1994!; Science267, 832 ~1995!. See also R. P. Duarte-Zamorano and V. Romero-Rochı´n,J. Chem. Phys.114, 9276 ~2001!; C. S. Maierle and R. A. Harris,ibid.109, 3713~1998!.

25J. A. Cina, J. Chem. Phys.113, 9488~2000!.26T. S. Humble and J. A. Cina, Bull. Korean Chem. Soc.~to be published!.27D. J. Tannor and S. A. Rice, J. Chem. Phys.83, 5013 ~1985!; J. Cao, J.

Lumin. 87–89, 30 ~2000!; R. J. Levis, G. M. Menkir, and H. Rabitz,Science292, 709 ~2001!.

28R. Zadoyan, D. Kohen, D. A. Lidar, and V. A. Apkarian, Chem. Phys.266,323 ~2001!.

29R. van Grondelle, J. P. Dekker, T. Gillbro, and V. Sundstrom, Biochim.Biophys. Acta1187, 1 ~1994!; X. Hu and K. Shulten, Phys. Today50, 28~1997!.

30J-Aggregates, edited by T. Kobayashi~World Scientific, Singapore, 1996!.31H. S. Cho, D. H. Jung, M.-C. Yoonet al., J. Phys. Chem. A105, 4200

~2001!; C.-K. Min, T. Joo, M.-C. Yoon, C. M. Kim, Y. N. Hwang, D. Kim,N. Aratani, N. Yoshida, and A. Osuka, J. Chem. Phys.114, 6750~2001!.

32F. Zhu, C. Galli, and R. M. Hochstrasser, J. Chem. Phys.98, 1042~1993!;erratum,98, 9222~1993!.

33A. Albrecht Ferro and D. M. Jonas, J. Chem. Phys.115, 6281~2001!.34Y. Sakata, T. Toyoda, T. Yamazaki, and I. Yamazaki, Tetrahedron Lett.33,

5077~1992!; J. A. Pescatore and I. Yamazaki, J. Phys. Chem.100, 13333~1996!.

35Th. Forster, inModern Quantum Chemistry, Part III, edited by O. Sinano-glu ~Academic, New York, 1965!, p. 93.

36S. Rackovsky and R. Silbey, Mol. Phys.25, 61 ~1973!.37T. F. Soules and C. B. Duke, Phys. Rev. A3, 262 ~1971!.38For a recent relaxation-theory study of vibrational effects in electronic

energy transfer see M. Yang and G. R. Fleming, Chem. Phys.275, 355~2002!.

39W. M. Zhang, V. Chernyak, and S. Mukamel, J. Chem. Phys.110, 5011~1999!; see also V. Chernyak, W. M. Zhang, and S. Mukamel,ibid. 109,9587 ~1998!.

40A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, andW. Zwerger, Rev. Mod. Phys.59, 1 ~1987!; our reference is to a statementon p. 15 to the effect that for practical purposes, only the populationdifference between the two@electronic# states was then accessible in real-istic experiments.

41In our numerical calculations,e2 is taken toequal e11e18 unless other-wise stated.

42John Jean investigated the short-pulse-driven dynamics of a similar sys-tem in contact with a thermal reservoir in a ground-breaking study of thecompetition between coherence and relaxation in wave packet surfacecrossing: J. M. Jean, J. Phys. Chem. A102, 7549 ~1998!. See also A.Matro and J. A. Cina, J. Phys. Chem.99, 2568~1995!; C. Kalyanaramanand D. G. Evans, J. Chem. Phys.115, 7076~2001!.

43For plots of the adiabatic potential energy surfaces of this model understrong and weak coupling, see Fig. 2 of Fo¨rster’s article~Ref. 35!.

44In the case of equal site energies,e15e18 , the intersection between donorand acceptor surfaces includesqb5qa50, so energy transfer proceedsefficiently at the Franck–Condon point. Whene15e181mv2d25e1812EFC , the acceptor surface passes through the donor minimum; forcontinuous spectra this would be the case of maximal ‘‘Fo¨rster overlap’’between donor emission and acceptor absorption. A further redshift in theacceptor absorption spectrum would lead to an inverted regime of decreas-ing energy-transfer efficiency.

45The lower-case subscripts designate ‘‘preparation,’’ ‘‘waiting,’’ and ‘‘de-lay.’’

46Experiments on fluorescence detection of femtosecond accumulated pho-ton echoes have measured a similar quantity that is quadrilinear in thefield amplitudes of phase-related continuous wave beams. The time reso-lution in those experiments was set by the field correlation time of acontinuous-wave laser rather than the pulse duration. See K. Uchikawa, H.



128.138.73.68 On: Fri, 19 Dec 2014 20:30:22

Ohsawa, T. Suga, and S. Saikan, Opt. Lett.16, 13 ~1991!; S. Saikan, K.Uchikawa, and H. Ohsawa,ibid. 16, 10 ~1991!.

47We set\51 throughout this paper.48We specify the arbitrary initial phase so that,@ tA2t0#uC(t0)&5u0&un0&,

whereun0& is a vibrational eigenket ofH0 with energyEn0.

49As defined here, the pulse propagators are anti-Hermitian:I †52I .50Under these approximations, the integrand of Eq.~11! becomes

@2t1t I #VI(t)@t2t I #>$@2(ma•eI)/2#(u1&^0ue2 iH 1(t I2t)eiH 0(t I2t)1u2&3^18ue2 iH 2(t I2t)eiH 18(t I2t))2@(mb•eI)/2#(u18&^0ue2 iH 18(t I2t)eiH 0(t I2t)

1u2&^1ue2 iH 2(t I2t)eiH 1(t I2t))%AI(t2t I)e2 iV I (t2t I )2 iF I1H.c.

51The expression foruc1& is the same as Eq.~12!, but with ^1u instead of^18u on the right-hand side. The nuclear wave function in the bi-excitonicstate is given by uc2&5^2u@ t2tD#$D@ td#C@ tw1tp#1D@ td1tw#B@ tp#

1@ td#C@ tw#B@ tp#1D@ td1tw1tp#A1@ td#C@ tw1tp#A1@ td1tw#B@ tp#A1D@ td#C@ tw#B@ tp#A%u0&un0&.

52Where @ t#05exp(2iHt)uJ50 and @ t#152 iJ*0t dt$u18&exp(2iH18(t

2t))exp(2iH1t)^1u1u1&exp(2iH1t)exp(2iH18(t2t))^18u&%. Notice that thefirst-order evolution operator carries a phase factor2 i 5e2 ip/2 associatedwith the energy-transfer process, which our amplitude-sensitive measure-ments will be able to detect.

53An experimental strategy for isolating the interference contribution to thepopulation of a given electronic state with beam choppers and lock-inamplifiers was demonstrated in Refs. 1 and 2.

54We use notation in which a bra retains the name of the ket to which it isdual: ^BAu is the bra dual to the ketuBA&.

55Through the pulse propagators~11!, which involve the dot productsma•eI andmb•eI .

56P. Andrews and W. L. Barnes, Science290, 785 ~2000!.57In the phase-locking scheme of Scherer and co-workers~Refs. 1 and 2!,

the pulses in a phase-locked pair differ by a pure delay, and are otherwiseexact copies of each other. For example, the phases-at-arrivalFA andFB

differ at most by an integer multiple of 2p. In that method, the intrapulse-pair delays (tp) are actively stabilized and selectively undersampled sothat FB2FA52pN5Vptp1fp for a chosen value of the locking anglefp . The same situation applies for theCD pulse pair. With respect to theinterpulse-pairphase shifts, timing jitter on the unstabilized delaytw al-lows us to regard the interpulse-pair phase shifts as random variables. Fora detailed discussion of the relationship between phase shifts and timedelays, see Ref. 3.

58D. Keusters, H.-S. Tan, and W. S. Warren, J. Phys. Chem. A103, 10369~1999!; D. Keusters, P. Tian, and W. S. Warren, in TheThirteenth Inter-national Conference on Ultrafast Phenomena, OSA Trends in Optics andPhotonics, Vol. 72, OSA Technical Digest, Postconference Edition~Opti-cal Society of America, Washington, D.C., 2002!, pp. 480–481.

59A rudimentary example of phase-cycling in linear ultrafast spectroscopycan be found in Ref. 2. There it was shown that in-phase and in-quadratureinterferograms could be combined to yield the resonant part of the time-dependent linear dipole susceptibility.

60It may be noted that while the overlaps in Eq.~23! depend explicitly onthe observation timet, the isolable single overlap in Eq.~24! ist-independent. In both contributions to Eq.~23!, one of the overlappedwave packets evolves aftertD with a time evolution operator zeroth orderin J and the other with an operator linear inJ. In the overlap~24! on theother hand, both wave packets experience evolution aftertD under thesame unitary operator@ t2tD#0 @see Eq.~13!#; so their overlap does not

change. It happens similarly, that the single overlap isolable withAxByCxDx polarization ~see Appendix A! is observation-time indepen-dent. For both polarization combinations, however, it may be desirable incases of relatively strong energy-transfer coupling to time gate the mea-surement of acceptor-state population—on about a picosecond timescale—in order to ensure that the portion linear inJ dominates theacceptor-state population.

61This result has the special implication that after (12a)tw of motion in the18 state, thea-mode target trajectory reaches a phase-point lying on theFranck–Condon energy shell; the phase-space location of the target-statea-mode will hence be accessible to a reference wave packet after sometd

interval of ~backwards! motion in state 2. This interesting elementary fea-ture of the transferred amplitude’s motion in an energy-transfer systemdoes not seem to have been noted previously.

62The basic physics of this process is the same as follows the instantaneousupward displacement of the hanging point of a Hooke’s law spring hold-ing a stationary weight: If the hanging point is returned to its originalposition afteratvib/2, then (12a)tvib/2 later, the mass passes downwardthrough the same vertical position it had at timeatvib/2.

63Equation~29! was integrated with a time steptvib/2400, yielding numeri-cal errors of about a part in 106. The interference signals presented heresupersede the preliminary calculations of D. Kilin and J. A. Cina,Ul-trafast Phenomena~Springer, Berlin, in press!, Vol. 13, conference pro-ceedings. The latter calculations were performed less accurately, by nu-merical diagonalization of the one-exciton subspace in the presence of asmall specified value forJ.

64There does not appear to be any fundamental reason why fluorescence-detected wave packet interferometry signals could not be measured onsingle molecules; in which case the issue of inhomogeneous broadeningwould be moot if the site energies were stable for the duration of dataacquisition.

65See Chap. 10 of Shaul Mukamel’s book, cited in Ref. 13.66V. Szocs, T. Pa´lszegi, A. Torschanoff, and H. F. Kaufmann, J. Chem. Phys.

116, 8218~2002!.67This issue is addressed, however, by O. Golonzka and A. Tokmakoff, J.

Chem. Phys.115, 297 ~2001!.68J. A. Cina and R. A. Harris,Ultrafast Phenomena IX, edited by W. Knox

and P. Barbara~Springer, Berlin, 1994!, p. 486.69R. Silbey and R. A. Harris, J. Phys. Chem.93, 7062~1989!.70S. Jang, J. Cao, and R. J. Silbey, J. Chem. Phys.116, 2705~2002!.71For instance, the uniformly polarized sequenceAxBxCxDx should reveal

the loss of amplitude in a donor-state wave packet of orderJ2 due toelectronic energy transfer. The corresponding quadrilinear signal shouldbe resistant to degradation by orientational disorder, and even provideuseful amplitude-level information from a randomly oriented sample.

72The Appendix of Ref. 25 analyzed the dynamics underlying theC andDterms~in the notation of that paper! in the nonlinear interferogram for apolyatomic molecule thatdoes notundergo energy transfer. Conditionswere found under which these two terms do not coincide, and argumentsmade that under these same conditions, theA and B term signals wouldvanish except at very shorttw . It is worth noting that theC andD termscan more generally be separated from each other by phase cycling, whichisolatesA1D andB1C. See Ref. 26.

73C. Cohen-Tannoudji, B. Diu, and F. Lalo¨e, Complement GV , QuantumMechanics Vol. I~Wiley, New York, 1977!.

74E. J. Heller, J. Chem. Phys.62, 1544~1975!.



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